fractional brownian motion trading strategies

Abstract

In the adenoidal-frequency limit, conditionally expected increments of fractional Pedesis converge to a white noise, peeling their dependance on the path story and the prediction horizon and making dynamic optimisation problems manipulable. We feel an explicit formula for locally mean–variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-attuned profits are linear in the trading visible horizon and boost asymmetrically A the Hurst exponent departs from Brownian motion, leftover finite arsenic the index reaches zero while diverging American Samoa it approaches one. Trading costs penalise numerous portfolio updates from temporary signals, leading to a finite trading frequency, which fundament embody chosen so that the event of trading costs is arbitrarily small, conditional the required speed of convergence to the altitudinous-frequency limit.

Introduction

First proposed as a mold of price dynamics by Mandelbrot [14], fractional Brownian motion (fBm) has since puzzled researchers and stirred controversy for its evasive properties, which have confounded both empirical and a priori work. Long-drift addiction in plus prices, the attribute that originally motivated the use of fBm to depict price dynamics, corpse undecided; view Greene and Fielitz [7], Fama and daniel Chester French [6], Poterba and Summers [17], Lo [13], Jacobsen [12], Teverovsky etdannbsp;al. [22], Willinger etdannbsp;al. [23], Baillie [1]. Arbitrage, which has plagued the espousal of fBm in models of optimal investment (Rogers [19], Salopek [20], Dasgupta and Kallianpurdannbsp;[5], Cheridito [2]) disappears with frictions (Guasoni [8], Guasoni etdannbsp;aluminium.dannbsp;[10], Czichowsky and Schachermayer [4], Czichowsky etdannbsp;al. [3]), leading to finite hoped-for profits; see Guasoni etdannbsp;al. [9].

This paper finds locally mean–variance optimal trading strategies in half Brownian move and characterises their overlap and performance in the high-frequency limit. Our analysis starts from a fixed trading oftenness, for which optimal strategies are directly proportional to the (conditionally) expected increment and inversely proportional to its variance. The central lineament of fractional Brownian apparent motion is that unlike dissemination models, the conditionally likely increment is not proportional to the length of the trading period of time, but to a power thereofdannbsp;– the Hurst power. Because the increment's standard deviation scales with the same power, the medium Sharpe ratio is unreactive to the length of the trading period.

The key insight (Theoremdannbsp;2.3) is that the high-frequency limit of such a estimate (the "latent tramp" of third Brownian motion) is a white-noise process with a variableness depending on the Hurst exponent, but invariant to any grading of the process (which would equally shell both expected increments and their standard deviation). This result in turn off leads to a cascade of implications for best continuous trading of fractional Brownian motion.

First, the optimal mean–variance public presentation from trading fractional Brownian motion is proportional to the length of the whole trading horizondannbsp;– as for Brownian motion with driftdannbsp;– in bitchiness of the different grading of mean and variance on individual periods. The grounds is that the cumulative performance of high-frequency trading fractional Brownian movement on a bounded horizon is essentially equivalent to the average public presentation of a separate-meter model with infinitely many periods and independent, identically distributed Sharpe ratios. Both performances are deterministic because randomness disappears through ergodicity.

Arcsecond, the resulting execution is asymmetric in the Hurst exponent (Fig.dannbsp;1 and Theoremdannbsp;2.2), remaining bounded as the process approaches a white racket (just about \(H= 0\)), but diverging as it approaches a near-straight line with random drift (near \(H= 1\)). This ensue is significant because it does non theme from the autocovariance properties of the strategies' prospective returns; indeed, for any Hurst power, the instantaneous forecast is a white noise. As an alternative, the result reflects the magnitude of the variant of the Caucasian-randomness forecast that is extracted from the paths of fBm for different values of \(H\): for \(H\) near zero, the weights of the white-hot-noise omen are small and extremely saved up along Holocene epoch increments, which results in a moderate variance. By contrast, for \(H\) near one, the figure's weights are large and get hold of far into the path's medieval story, guiding to a diverging disagreement.

Fig.dannbsp;1

Profits per unit of risk and time (vertical), i.e., the expression in (2.1) divided past \(T/\Gamma \), against Hurst exponent (horizontal)

Fraught size image

Third, and contrary to the intuition from preceding results, including our personal, we find that such performance is immune to itsy-bitsy frictionsdannbsp;– so much as proportionate transaction costs or immediate nonlinear Leontyne Price impact (Theoremdannbsp;3.1 and Corollarydannbsp;3.2). Specifically, while frictions detract from performance, their essence vanishes arbitrarily quickly by lento increasing the trading relative frequency as trading costs decrease, so that their asymptotic affect vanishes at any required rate. Similarly, holding trading costs continual, their effect also vanishes by increasing the skyline while appropriately calibrating the trading frequency.

Fourth, we observe that approximations of the potential trend of waist-length Brownian question meet weak, but not in norm. This observation highlights a soft difference between the familiar drifts of diffusions and their partial analogies for fractional processes. Non only are diffusive drifts of the range of infinitesimal time intervals (informally, \(dt\)) while fractional drifts are a exponent thereof (conversationally, \((dt)^{H}\)); in gain, diffusive drifts can be taken as close approximations of conditionally expected increments over any sufficiently small interval because such approximations converge (in norm) As random variables. By contrast, fractional drifts are critically depending on the specific interval: as the interval length declines to zero, the conditionally expected returns converge in law, but non as random variables in any tenable sentiency.

At last, IT is worthwhile comparing the findings in this paper to the recent results in Guasoni etdannbsp;aluminium. [9], as both articles study best trading strategies for fractional Brownian motion, though in very different settings. The main difference lies in the objective functions reasoneddannbsp;– here a local mean–discrepancy criterion on a finite time interval, while in [9] a risk-neutral aim with a long horizon. Particularly, the presence of a nonlinear friction is material to micturate the problem in [9] well posed, as it would otherwise lead to unbounded expected lucre. In contrast, the in attendance local anaesthetic mean–discrepancy standard is cured posed eve without frictions As the fast Sharpe ratio stiff bounded for any \(H\in (0,1)\), although arbitrage is executable on any time interval because arbitrage win remain dispersed. Both [9] and the present paper lead to finite maximal Sharpe ratios that are asymmetric in \(H\), but their skews are reversed and arise for different reasons: while the asymptotically optimal strategies in [9] have higher Sharpe ratios near zero than near one, they are not necessarily optimal as the strategies maximize a gamble-neutral objective, non the Sharpe ratio. By contrast, the Sharpe ratios obtained here are indeed optimum as they maximise the local mean–variance criterion over some exhaustible interval away ergodicity.

The rest of the composition is organised every bit follows. Sectiondannbsp;2 describes the model and the main result without frictions, discussing their significance and implications. Sectiondannbsp;3 considers frictions and shows how their touch can atomic number 4 mitigated aside a judicious choice of the trading frequency. Sectiondannbsp;4 concludes, and all proofs are in the Vermiform process.

Main results

An investor trades a safe and a risky plus. The dependable rate is assumed set to simplify notation, while the cost of the risky plus is a multiple of fractional Brownian motion.

Definition 2.1

Fractional Brownian movement (fBm) with Hurst index \(H\in (0,1)\) is a Mathematician operation \(B^{H}=(B^{H}_{t})_{t\geq 0}\) defined along a probability place \((\Omega ,\mathcal{F},P)\), with continuous trajectories such that \(E[B^{H}_{t}B^{H}_{s}]=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H})\), \(t,s \geq 0 \), and \(E [B^{H}_{t}]=0\), \(t\geq 0\).

The case \(H=1/2\) corresponds to accustomed Brownian motion, henceforward excluded. Thus \(H\in (0,1)\setminus \{1/2\}\) unless explicit otherwise. Consider a trading horizon \(Tdangt;0\) and a frequency \(n\geq 1\), which represents the number of trading periods in the interval \([0,T]\). The set \(\Sigma _{n}\) of strategies consists of sequences \(\private detective _{s}\), \(s\in \{T k/n:\ 0\le k\lupus erythematosus n-1\}\), of unselected variables such that \(\sherloc _{s}\) is \(\mathcal{F}_{s}\)-measurable for all such \(s\), where \((\mathcal{F}_{s})_{s\ge 0}\) is the augmented natural filtration of \(B^{H}\).

An investor who holds at the outset of for each one time interval \([T k/n, T (k+1) /n]\) adannbsp;number of shares equal to \(\pi _{T k/n}\) attains the mean–variance performance

$$\begin{aligned} \operatorname{MV}(\PI ,k,n,S) danamp;:= E_{T k/n}[\pi _{T k/n}(S_{T (k+1)/n}-S_{T k/n})] \\ danamp;\phantom{=::}-\frac{\gamma }{2} \operatorname{Var}_{T k/n}[\pi _{T k/n} (S_{T (k+1)/n}-S_{T k/n})], \last{aligned}$$

where \(E_{t}[X]\) and \(\operatorname{Volt-ampere}_{t} [X]:=E_{t}[X^{2}]- (E_{t}[X])^{2}\) respectively refer the conditional expectation and tentative variant of a haphazard varying \(X\) with respect to \(\mathcal{F}_{t}\), while \(S_{t} = \sigma B^{H}_{t}\) denotes the risky asset price at time \(t\). The parameter \(\gamma dangt;0\) represents the investor's aversion to danger (as measured by variance).

Assuming clock-completing preferences, the overall performance in the interval \([0,T]\) of a trading strategy is defined atomic number 3 the aggregate of the per-period performances, weighing each period by its duration \(T/n\), i.e.,

$$ R_{\Vasco da Gamma }(\pi ,n,S):= \frac{T}{n}\sum _{k=0}^{n-1} \operatorname{Mv}(\private investigator ,k,n,S) =\int _{0}^{T} \operatorname{MV}(\PI ,\lfloor t n\rfloor ,n,S)dt . $$

Thus the high-frequency objective for aliquot Brownian movement is

$$ V(H,\gamma ):=\limsup _{n\to \infty }\sup _{\pi \in \Sigma _{n}}E[R_{ \gamma }(\sherloc ,n,\sigma B^{H})] $$

and represents the supreme performance of a round-the-clock-time scheme that updates the portfolio at arbitrary frequency on the time interval \([0,T]\).

With this notation, the chief result of this paper is

Theorem 2.2

For each \(H\in (0,1)\),

$$ V(H,\Vasco da Gamma )=\frac{T}{\Vasco da Gamma }\bigg( \frac{\Gamma (H+1/2)\Gamma (2-2H)}{2\Gamma (3/2-H)}-\frac{1}{2}\bigg), $$

(2.1)

and the limit superior in the definition of \(V(H,\gamma )\) is in fact a limit.

Before discussing the details of this result, information technology is useful to compare information technology to the common benchmark of Brownian motion with drift, i.e.,

$$ S_{t}=\mu t+\sigma W_{t}, $$

for few Brownian move \(W\) and \(\mu \in {\mathbb{R}}\), \(\sigma dangt;0\). A simple calculation then showsdannbsp;that

$$ \swallow _{\pi \in \Sigma _{n}}E[R_{\gamma }(\pi ,n,S)] =E\bigg[R_{\gamma }\bigg(\frac{\mu }{\gamma \sigma ^{2}},n,S\bigg)\bigg] = \frac{\mu ^{2}}{2\gamma \sigma ^{2}} T . $$

(2.2)

In other speech, some the optimal scheme and its performance make out non count on \(n\), are inversely proportional to the squared volatility \(\sigma ^{2}\) and risk aversion \(\gamma \), and are respectively linear and quadratic equation in the freewheel. To boot, performance is linear in the investment apparent horizon. The linear dependence on the impetus and the inverse dependence on the unpredictability is at the substance of the put on the line–return tradeoff that arises in unselected-walk models: as returns are serially independent, their stochasticity is purely a seed of hazard, and its reducing is uniquely beneficial.

The fractional high-absolute frequency performance in (2.1) contains surprising features some in its departures and in its analogies with the usual mean–division performance (2.2). In contrast to (2.2), the performance in (2.1) is independent of volatility. (In fact, the resultant is also nonsymbiotic of an additional drift, as observed in Mentiondannbsp;A.11. Intuitively, the reason is that for a short time separation, the conditionally expected increment of fBm is of order \((dt)^{H}\), which makes an ordinary drift of enjoin \(dt\) worthless in the mean–variance optimal strategy and its performance.) As shown below, the optimal scheme inversely depends happening variableness, but this dependence is lost in functioning because the expected return directly depends on variance, thereby offsetting its effect.

In analogy to (2.2), the performance in (2.1) is linear in the investment horizon. Upon reflection, also such an analogy is surprising because the linearity in the horizon of the habitual mean–variance performance in (2.2) stems from the independence of increments of Brownian motion and the constant purport. As an alternative, the dependence in increments of fractional Brownian gesticulate is substantial and indeed crucial to engender positive returns.

The dependence on the Hurst exponent \(H\), displayed in Fig.dannbsp;1, is similarly confusing in view of its asymmetry. At matchless extreme, as \(H\) approaches nada and increments increasingly resemble white noise (Mishura [15, Lemma 4.1]), performance converges to a impermanent limit, i.e.,

$$ V(H,\gamma ) = \frac{T}{\Vasco da Gamma }\bigg( \frac{\Gamma (1/2)\Gamma (2)}{2\Gamma (3/2)}-\frac{1}{2}\bigg) + O(H) = \frac{T}{2\Gamma } + O(H) . $$

(2.3)

(The last equivalence follows from the identity \(\Gamma (\frac{3}{2})=\frac{1}{2}\Gamma (\frac{1}{2})\).) As \(H\) approaches \(1/2\), performance flattens around zero as the march mimics an routine Brownian movement. This expansion exploits identities involving the derivatives of the Gamma function (cf. Sun and Qin [21]), namely

$$ V(H, \gamma )=\frac{\pi ^{2}}{6} (H-1/2 )^{2}\frac{T}{\gamma } +O\big( (H-1/2 )^{3}\extensive) . $$

In fussy, this identity confirms the intuition from FIG.dannbsp;1 that functioning reaches its singular minimum of range in the dolphin striker showcase of \(H=1/2\), while slowly increasing in each direction. At the other immoderate, as \(H\) approaches one and the process resembles a straight line with random pitch, performance diverges, i.e.,

$$ V(H,\gamma ) = \bigg(\frac{1}{8 (1-H)}+\frac{-3 + \log 4 }{4}\bigg) \frac{T}{\gamma }+O(1-H ) . $$

(2.4)

To hold the terminal figure \(1/(8 (1-H))\), recall that \(\Gamma (x)\sim 1/x\) for \(x\) near zilch. The term \((-3 + \log 4)/4\) follows from Sir Thomas More compound higher-order asymptotics. Key to understanding these features is the prediction mechanics at the heart and soul of the problem. As our hateful–variance objective is meter-additive, the optimal trading strategies maximise public presentation in the next period. Because for a square-integrable random variable \(X\), the functional \(\varphi \mapsto E[\varphi X]- \frac{\gamma }{2} \operatorname{Var}[\varphi X]\) attains its maximum \(E^{2}[X]/(2\da Gamma \operatorname{Var}[X])\) at \(\phi ^{*}=E[X]/(\gamma \operatorname{Var}[X])\), the best strategy \(\operative (n)\in \Sigma _{n}\) is

$$ \shamus _{T k/n}(n):= \frac{E_{T k/n}[B^{H}_{T (k+1)/n}-B^{H}_{T k/n}]}{\gamma \operatorname{Var}_{T k/n}[B^{H}_{T (k+1)/n}-B^{H}_{T k/n}]}, \qquad 0\leq k\leq n-1. $$

(2.5)

To investigate the high-frequency limit, it is convenient to extend these strategies by starboard-continuity to the entire interval \([0,T]\), i.e., setting

$$ \operative _{t}(n):=\pi _{T k/n}(n),\qquad t\in [T k/n,T(k+1)/n), 0\leq kdanlt; n. $$

With this notation, the next theorem identifies the limit of such strategies, which is interpreted as the asymptotically optimal strategy in the high-absolute frequency regime.

Theorem 2.3

The sequence \((((T/n)^{-H}\pi _{t}(n))_{t\in [0,T]})_{n\in \mathbb{N}}\) consists of Gaussian processes that, atomic number 3 \(n\) increases, converge in finite-magnitude distributions to a Gaussian process \((B_{t})_{t\in [0,T]}\) so much that

1) \(B_{0}=0\) a.s.

2) \(E[B_{t}] = 0\), \(t\in (0,T]\).

3) \(E[B_{t}^{2}]=1-\frac{\Da Gamma (3/2-H)}{\Vasco da Gamma (H+1/2)\Gamma (2-2H)}\), \(t\in (0,T]\).

4) \(E[B_{t}B_{s}]=0\) for \(t\neq s\), \(s, t\in [0,T]\).

Trial impression

Follows from Propositiondannbsp;A.5, Lemmadannbsp;A.7 and Theoremdannbsp;A.8 below. □

This result has a striking message: Capable a scaling factor, the optimal strategydannbsp;– hence the anticipated return complete the adjacent perioddannbsp;– is essentially a white noise (the exception is \(t=0\), for which the outgrowth is conventionally pinned at zero). In other words, regardless of the Hurst exponent \(H\) and regardless of the autocorrelation of increments in fragmental Brownian motion, the forecasts of short-term increments (i.e., the trading signals) are near unrelated from unrivalled instant to the next. Figuredannbsp;2 illustrates the convergence answer in the theorem by plotting at increasing frequencies the autocorrelation of \(\pi _{T k/n}(n)\), which converges to the autocorrelation of a white noise.

Ficus carica.dannbsp;2

Autocorrelation (vertical axis) of the strategies \(\pi _{T k/n}(n)\) (equivalently, of the expected increments) against the clock lag (horizontal) as the oftenness \(n\) increases from 100 (top) to 200, 500 and 1000 (bottom). The autocorrelation converges to the white-dissonance limit of one at lag nought and zero elsewhere. Each planned curve is the average of 1000 sample autocorrelograms with \(H = 0.6\)

Full size image

The Hurst exponent controls the musical scale of the strategy: Denoting aside \(\Delta \) the length of each trading period, toll increments own conditional expectation of order \(\Delta ^{H}\) and probationary variableness of order \(\Delta ^{2H}\), which implies trading positions of order \(\Delta ^{-H}\) (cf. Proposaldannbsp;A.5 and Theoremdannbsp;A.8). This feature is in contrast to the Brownian benchmark, in which both the expected return \(\mu \Delta \) and its variance \(\sigma ^{2} \Delta \) are of the same order. Instead, the variance in the fractional setting has a smaller Holy Order, which means that bets become Thomas More favourable as the trading frequency increases, and thus their optimal size up increases.

Note, however, that the inexplicit performance in each historic period is proportionate to the dependant on expectation \(\Delta ^{H}\) times the position size \(\Delta ^{-H}\), therefore of order 1. As each trading period leads to the same performance (in view of the white-noise property established in Theoremdannbsp;2.3), trading over an interval of length \(T\) generates a performance proportional to \(T\). In particular, the results beneath show that the optimal trading position is asymptotically

$$\varphi _{t}(\Delta ) := \frac{\Delta ^{-H}}{\Gamma } \frac{\Gamma (H+1/2)\Gamma (2 - 2H)}{\Gamma (3/2-H)}B_{t}, $$

and that on the subsequent interval, the expected increase has the straight line (conditional) have in mind and division

$$ m_{t}(\Delta ) := B_{t} \Delta ^{H} ,\qquad v(\Delta ) := \frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2 - 2H)} \Delta ^{2 H} . $$

(2.6)

The performance formula in Theoremdannbsp;2.2 follows from

$$\begin{straight} Ti\bigg[\varphi _{t}(\Delta ) m_{t}(\Delta ) -\frac{\gamma }{2} \varphi _{t}(\Delta )^{2} v(\Delta ) \bigg] danamp;= \frac{T}{2\gamma } \frac{\Gamma (H+1/2)\Gamma (2 - 2H)}{\Gamma (3/2-H)} E[B_{t}^{2}] \\ danAMP;= \frac{T}{\gamma }\bigg( \frac{\Gamma (H+1/2)\Gamma (2-2H)}{2\Da Gamma (3/2-H)}-\frac{1}{2}\bigg) . \close{straight}$$

This analysis also offers an intuitive explanation for the asymmetric behaviour of the carrying into action in (2.3) and (2.4). For \(H\) approximately zero, the asset cost \(S\) itself is akin to a white noise, for which the entail and variance in (2.6) are of the same order. Accordingly, the performance converges to a finite limit. By contrast, for \(H\) close to one, the process degenerates to a heterosexual line with random slope, as noise vanishes from its increments. Thus the trading strategy generates return with virtually no risk, and the functioning diverges.

Note that the mean–variance optimal strategies \((\pi _{t}(n))\) are not arbitrage opportunities as the support of their payoffs is \((-\infty ,+\infty )\). Although continuous trading with fBm leads to arbitrage opportunities (see e.g. Rogers [19], Salopek [20]), it is exonerate that on any finite deterministic grid, fBm does not admit arbitrage because an equivalent martingale measure can live constructed through a backward recursion that aligns all not absolutely unsurprising increments to zero. (In fact, arbitrage disappears yet when a minimal time has to pass between two subsequent transactions; see Cheridito [2].)

A deeper question is whether the sequence of strategies \((\pi (n))_{n\ge 1}\) yields an arbitrage in some limit sense, and the answer is affirmative. The sequence of distinct-time mean–variance optimal policies offers a statistical arbitrage in this

$$ \lim _{n\rightarrow \infty } E[W(n)]=\infty , $$

where \(W(n)\) is the final wealth of the scheme \(\pi (n)\) starting from goose egg initial capital, i.e.,

$$ W(n)=\summation _{k=0}^{n-1}\pi _{T k/n}(n)(S_{T (k+1)/n}-S_{T k/n}). $$

This fact is pronto verified by observant that in contemptible–variance optimisation, the anticipation of the optimal strategy is ever doubly As astronomic as its variance, whence

$$ \lim _{n\rightarrow \infty } E[W(n)]= \lim _{n\rightarrow \infty } \frac{2n}{T}E\big[R_{\gamma }\big(\private detective (n),n,S\big)\big] = \infty , $$

because \(E[R_{\gamma }(\pi (n),n,S)]\) tends to a finite nonzero limit (for \(H\Nebraska 1/2\)) by Theoremdannbsp;2.2.

As Theoremdannbsp;2.3 establishes that the rescaled strategies essentially converge to a white noise in delimited-dimensional distributions, a natural question is whether such a convergence holds in a stronger sense, such As in square norm, so that its limit can be interpreted as a rescaled asymptotically optimum strategy in persisting prison term.

The close result provides a disadvantageous resolution to this question by exhibit that even focusing on a sequence of deuce partitions, the square average between each discretisation and the close corpse delimited away from zero.

Theorem 2.4

Let \(\Delta _{k} = T/{k}\). For all \(H\in (0,1)\setminus \{1/2\}\) and \(t\in (0,T)\),

$$ \lim _{n\to \infty } E\big[\full-size(\Delta _{2^{n}}^{-H}\pi _{t}(2^{n}) - \Delta _{2^{n+1}}^{-H}\pi _{t}(2^{n+1})\big)^{2}\big] dangt; 0 . $$

The significance of this result is that the optimal scheme is extremely sensitive to the trading frequency used, and that optimal strategies at acceleratory frequencies are non approximations of some underlying dogging-time strategy, which does non exist. In fact, even if such a strategy existed, IT would be of no purpose because the paths of a white-noise procedure are not even measurable (fibrocystic disease of the pancreas. Revuz and Yor [18, p. 37]).

At a many concrete even out, the to a higher place results show that as the frequency increases, the comparable trading strategies get on increasingly variable; thus in drill, their ostensible a priori performance May constitute more offset by the trading costs that such strategies entail. The adjacent department investigates this issue by identifying how the optimal trading frequency depends on the size of trading costs.

Trading costs

The optimal strategies identified in (2.5) imply that asset positions are both large and highly variable, thereby calling into dubiousness their robustness to trading costs. To investigate this issue, recall the sequence of strategies \(\operative (n)\), \(n\geq 1\), defined in (2.5) preceding.

Assuming that a portfolio change from \(\theta _{1}\) to \(\theta _{2}\) shares incurs the toll \(\lambda |\theta _{1}-\theta _{2}|^{\explorative }\) for some \(\alpha ,\lambda dangt;0\), the local mean–variance analysis for a trader applying the strategy \(\pi (n)\) leads to the functionals (setting \(\pi _{t}(n):=0\) for \(tdanlt;0\))

$$ \tilde{R}(n) := R(n) - \sum _{k=0}^{n-1}\lambda |\pi _{T k/n}(n)-\principal investigator _{T (k-1)/n}(n)|^{\alpha }, $$

where \(R(n)\) denotes resistance performance, i.e.,

$$\begin{aligned} {R}(n) danamp;:= \frac{T}{n}\sum up _{k=0}^{n-1} E_{T k/n} [\pi _{T k/n}(n)(S_{T (k+1)/n}-S_{T k/n}) ] \\ danamp;\phantom{=::}- \frac{\gamma }{2} \frac{T}{n}\sum _{k=0}^{n-1} \operatorname{Var}_{T k/n}[\pi _{T k/n}(n) (S_{T (k+1)/n}-S_{T k/n})] , \end{aligned}$$

while the second term in \(\tilde{R}(n)\) represents the effect of trading costs. The next result shows that expectable trading costs \(E[R(n) - \tilde{R}(n)]\) grow over with a superlinear power of the trading frequency \(n\) that increases with both the Hurst and the friction exponents. As a resultant role, for nonmoving transaction costs, the objective function arbitrarily deteriorates as the frequency increases, and the optimal trading frequency must be finite.

Theorem 3.1

\(E[R(n) - \tilde{R}(n)] = O(n^{1+\important H}) \) and hence \(\lim _{n\to \infty } E[\tilde{R}(n)] = -\infty \).

The future logical ill-trea is to understand the force of small trading costs along the overall objective. Hither the above result leads to an unexpected implication: with a judicious choice of the trading frequency, the effect of frictions is negligible at whatsoever order.

Corollary 3.2

Lashkar-e-Tayyiba \(n_{\lambda }= \lfloor \lambda ^{\frac{\beta -1}{1+\important H}} \rfloor \) for \(\exploratory \in (0,1)\). Then

$$ E[R(n_{\lambda }) - \tilde{R}(n_{\lambda })] = O(\lambda ^{\of import }) \qquad \textit{for } \lambda \to 0 , $$

that is, trading costs are of order \(\lambda ^{\beta }\).

Upon thoughtfulness, this result is a channelize consequence of Theoremdannbsp;3.1. However, its closing is unreasonable when compared to the results for frictions in usual diffusion models (cf. Guasoni and Weber [11, Theorem 4.1]) where the welfare personnel casualty is of the order of \(\lambda ^{\frac{2}{2+\alpha }}\); for exemplar, proportional dealings costs correspond to \(\exploratory =1\), preeminent to a upbeat loss of Holy Order \(2/3\).

Intuitively, the main difference is that in familiar diffusion models, the main determinant of optimal portfolios is the asset price's local stray which is typically smooth. So as the trading frequency increases, small and small adjustments are required, which means that holding trading costs constant quantity, the altitudinous-absolute frequency limit of the portfolio performance is finite.

In contrast, in the fractional setting reasoned here, the "latent drift" of the process is extremely irregulardannbsp;– in the set, it is a white noisedannbsp;–; hence it entails trading costs that raise with the trading frequency as implicit by Theoremdannbsp;3.1. However, this irregularity canful be controlled to make trading costs negligible in the in flood-frequency limit, aside choosing the trading frequency \(n_{\lambda }\) to grow slowly as \(\lambda \) decreases so that overall costs disappear in the limit. Naturally, lease \(n\) grow more slowly has the downside that the convergence of the strategy's performance to the optimum in (2.1) is too going to be slower.

Corollarydannbsp;3.2 besides identifies the maximal speed identifies the maximal speed at which the frequency may grow so that the strategy converges to the optimum. In particular, \(n_{\lambda }\) English hawthorn mature at a plac arbitrarily close to \(\lambda ^{-\frac{\-1}{1+\alpha H}}\), but non at this take range: at this scathing regimen, costs would not vanish but converge to a affirmative finite limit, which would be suboptimal.

American Samoa trading costs are fixed in applications, the signification of this result is as follows: In practice, the trading cost \(\lambda \) implies that the optimal trading interval should follow \(1/n_{\lambda }\approx \lambda ^{\frac{1-\beta }{1+\alpha H}}\), where \(\beta \) is close to indefinite. However, the nigher the \(\beta \) to one, the bigger the trading separation, which agency that the intersection of the strategy to the frictionless limit for a firm horizon \(T\) is slower, and its risk higher. Thus if the apparent horizon is not long enough to guarantee that the wages has a sufficiently miserable risk, one may choose to lessen the value of \(\important \) to reduce risk further, at the price of an increased trading monetary value.

Conclusion

This paper finds locally mean–variation best trading strategies for an asset price that follows fractional Brownian motion, and finds that the average Sharpe ratio is finite, noninterchangeable in the Hurst exponent, bounded near zero, and unbounded approximate one. The central upshot is that conditionally expected increments are asymptotically a Mathematician white disturbance, irrespective of the Hurst power, but with a disagreement that depends on that exponent.

The optimal performance is insensitive to small trading frictions, therein their impact can be slaked arbitrarily well past calibrating the trading frequency appropriately. This phenomenon is in sharp contrast to diffusion models for which the affect of lowly frictions has a regressive order of order of magnitude.

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Acknowledgements

We are grateful to the Co-Editor Masaaki Fukasawa, the Comrade Editor program and two unnamed referees for their perceptive and insightful comments, which helped improve the paper significantly.

Funding

Undecided get at funding provided by ELKH Alfréd Rényi Institute of Mathematics.

Author information

Affiliations

1. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

   Paolo Guasoni

2. Taras Schevchenko Home University of Kyiv, 64 Volodymyrska, 01033, Kyiv, Ukraine

   Yuliya Mishura

3. Alfréd Rényi Plant of Mathematics, Capital of Hungary, Hungary

   Miklós Rásonyi

Corresponding author

Mapping to Miklós Rásonyi.

Additional information

Publisher's Note

Springer spaniel Nature remains colorless with regard to territorial claims in published maps and institutional affiliations.

Guasoni is partially supported by SFI (16/IA/4443, 16/SPP/3347). Mishura is supported by the ToppForsk jut out Nr. 274410 of the Explore Council of Norway with title Force: Stochastics for Time-Space Risk Models. Rásonyi is supported by the NKFIH (National Enquiry, Development and Innovation Office, Hungary) grant KH 126505 and by the "Lendület" Cary Grant LP 2022-6 of the Hungarian Academy of Sciences.

Appendix: Proofs

Appendix: Proofs

Henceforward, the expectation \(E[X]\) of a random variable \(X\) is defined as \(-\infty \) when both \(E[X^{+}]\), \(E[X^{-}]\) are infinite. Recall also the whim of straight line equivalence, where \(f(x) \sim g(x)\) near \(x = x_{0}\) substance that \(\lim _{x\to x_{0}} f(x)/g(x) = 1\).

A.1 Auxiliary results on fractional Brownian motility

For \(s,t\geq 0\), introduce the kernel

$$\begin{aligned} Z_{H}(t,s) danamp;:= c_{H}\bigg(t^{H-\frac{1}{2}}s^{\frac{1}{2}-H}(t-s)^{H- \frac{1}{2}} \\ danamp;\phantom{=:}\qquad - \Big(H-\frac{1}{2}\Big)s^{\frac{1}{2}-H} \int _{s}^{t} u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}\, du\bigg)1_{\{0danlt; sdanlt; t\}}, \remainder{aligned}$$

(A.1)

where

$$ c_{H}=\bigg(\frac{2H\Gamma (3/2-H)}{\Vasco da Gamma (H+1/2)\Gamma (2-2H)} \bigg)^{1/2}. $$

Then, taking an (ordinary) Brownian motion \(W=(W_{t})_{t\geq 0}\), the chemical formula

$$ B^{H}_{t}:=\int _{0}^{t} Z_{H}(t,s)\, dW_{s},\qquad t\geq 0, $$

(A.2)

defines an fBm with parameter \(H\) which generates the same filtration as \(W\) (cf. [16, Theorem 3.2]). Moreover, any fractional Brownian question allows the delegacy (A.2) with about Wiener procedure \(W\), and some processes give the Saame filtration. Fixing such a representation, denote \(\mathcal{F}_{t}:=\sigma (W_{s},\ 0\leq s\leq t)\).

The kernel mental representation in (A.2) implies the following properties of fractional increments.

Proposition A.1

For whatever \(0\leq z\leq u\),

$$\begin{aligned} B^{H}_{u}-B^{H}_{z} =danamp; \int _{0}^{z} \big(Z_{H}(u,s)-Z_{H}(z,s)\expectant) \, dW_{s}+\int _{z}^{u} Z_{H}(u,s)\, dW_{s} , \end{aligned}$$

(A.3)

$$\Menachem Begin{aligned} E_{z}[B^{H}_{u}-B^{H}_{z}] =danamp; \int _{0}^{z} \big(Z_{H}(u,s)-Z_{H}(z,s) \big)\, dW_{s} , \final stage{allied}$$

(A.4)

$$\begin{aligned} E_{z}[(B^{H}_{u}-B^{H}_{z})^{2}] =danampere; \bigg(\int _{0}^{z} \big(Z_{H}(u,s)-Z_{H}(z,s) \queen-sized)\, dW_{s}\bigg)^{2}+\int _{z}^{u} Z_{H}^{2}(u,s) \, ds .\qquad\ \ \end{aligned}$$

(A.5)

Proof

The relations immediately follow from (A.2). □

In particular, note that for \(z=0\),

$$ E_{z}[B^{H}_{u}-B^{H}_{z}]=E[B^{H}_{u}-B^{H}_{z}]=0 . $$

As all the relations (A.3)–(A.5) contain the increment \(Z_{H}(u,s)-Z_{H}(z,s)\), it is useful to rewrite this expression in a more than convenient form.

Lemma A.2

Let \(0\leq sdanlt; zdanlt; u\). Then

$$ Z_{H}(u,s)-Z_{H}(z,s)=(H-1/2)c_{H} s^{1/2-H}\int _{z}^{u} v^{H-1/2}(v-s)^{H-3/2} \, dv. $$

(A.6)

Trial impression

For \(Hdangt;1/2\), (A.6) follows by integrating (A.1) by parts. For \(Hdanlt;1/2\), (A.2) implies that

$$\begin{straight} danamp;Z_{H}(u,s)-Z_{H}(z,s) \\ danamp;= c_{H} s^{1/2-H}\bigg(u^{H-1/2}(u-s)^{H-1/2} - (H-1/2)\int _{z}^{u} v^{H-3/2}(v-s)^{H-1/2} \, dv \\ danamp;\qquad \qquad \quad \ \ \,\, -z^{H-1/2}(z-s)^{H-1/2} \bigg). \last{aligned}$$

(A.7)

Integrating \(\int _{z}^{u} v^{H-3/2}(v-s)^{H-1/2}\, du\) by parts gives

$$\begin{straight} danA;\int _{z}^{u} v^{H-3/2}(v-s)^{H-1/2}\, dv \\ danamp;= \frac{u^{H-1/2}(u-s)^{H-1/2}}{H-1/2}- \frac{z^{H-1/2}(z-s)^{H-1/2}}{H-1/2} -\int _{z}^{u} v^{H-1/2}(v-s)^{H-3/2} \, dv. \end{straight}$$

Note that this construction can comprise integrated by parts, while the kernel \(Z_{H}(u,s)\) itself cannot, because the integral \(\int _{z}^{u} v^{H-1/2} (v-s)^{H-3/2}\, dv\) exists for \(zdangt;s\), but diverges for \(z=s\). Thence, for \(0\leq sdanlt; z\) and \(0danlt; Hdanlt;1/2\), (A.7) implies that

$$Z_{H}(u,s)-Z_{H}(z,s)=c_{H}(H-1/2) s^{1/2-H}\int _{z}^{u} v^{H-1/2} (v-s)^{H-3/2} \, dv, $$

which coincides with (A.6). □

Let \(\tilde{c}_{H}:=c_{H} (H-1/2)\). In view of (A.6), equations (A.3)–(A.5) take the following alternative representation.

Proposition A.3

For \(zdanlt; u\), denote

$$ \xi _{u,z}:=\int _{0}^{z} s^{1/2-H}\bigg(\int _{z}^{u} v^{H-1/2}(v-s)^{H-3/2} \, dv\bigg) \, dW_{s}. $$

For any \(H\in (0,1)\setminus \{1/2\}\), we have

$$\begin{aligned} B^{H}_{u}-B^{H}_{z} danamp;= \tilde{c}_{H} \bigg( \eleven _{u,z} + \int _{z}^{u} Z_{H}(u,s)\, dW_{s} \bigg), \\ E_{z}[B^{H}_{u}-B^{H}_{z}] danamp;= \tilde{c}_{H} \xi _{u,z}, \\ E_{z}[(B^{H}_{u}-B^{H}_{z})^{2}] danamp; = (\tilde{c}_{H} \xi _{u,z} )^{2} + \int _{z}^{u} Z_{H}^{2}(u,s)\, ds , \\ \operatorname{Volt-ampere}_{z}[B^{H}_{u}-B^{H}_{z}] danA;= \int _{z}^{u} Z^{2}_{H}(u,s)\, ds . \end{aligned}$$

Proof

Follows from Lemmadannbsp;A.2 and Propositiondannbsp;A.1. □

A.2 Variance bounds for conditionally expected increments

Henceforth, assume that \(zdangt;0\) . Think that \(\xi _{u,z}\) is a centered Gaussian random variable with variance

$$ E[\xi _{u,z}^{2}]=\int _{0}^{z} s^{1-2H}\bigg(\int _{z}^{u} v^{H-1/2} (v-s)^{H-3/2}\, dv\bigg)^{2}\, ScD. $$

The goal is to find lower and upper bounds for the right-hand side. Thus set

$$\begin{aligned} J_{u,z} danamp;:= (H-1/2)^{-2} \int _{0}^{\frac{z}{2(u-z)}} \bigg(1- \frac{u-z}{z}r\bigg)^{1-2H}\big((1+r)^{H-1/2}-r^{H-1/2}\big)^{2}\, dr , \\ I^{\flat }_{u,z} danamp;:= z^{2H-2} ( {u-z} )^{2} \int _{0}^{1/2} s^{1-2H} \bigg(\frac{u}{z}-s\bigg)^{2H-3}\, ds + ({u-z} )^{2H} J_{u,z} , \\ \widetilde{I}^{\flat }_{u,z} danamp;:= \frac{u^{2H-1}}{z} ( {u-z} )^{2} \int _{0}^{1/2} s^{1-2H}\bigg(\frac{u}{z}-s\bigg)^{2H-3}\, darmstadtium + ({u-z} )^{2H} J_{u,z} , \\ I^{\distinct }_{u,z} danamp;:= \frac{u^{2H-1}}{z} (u-z)^{2} \int _{0}^{1/2} s^{1-2H} (1-s )^{2H-3}\, ds + (u/z)^{2H-1} ({u-z} )^{2H} J_{u,z}. \end{aligned}$$

Lemma A.4

For \(Hdangt;1/2\),

$$ \tilde{c}_{H}^{2} I^{\flat }_{u,z}\leq \tilde{c}^{2}_{H} E[\cardinal ^{2}_{u,z}]= E\big[E_{z}^{2}[B^{H}_{u}-B^{H}_{z}]\big]\leq \tilde{c}_{H}^{2} {I}_{u,z}^{ \sharp }, $$

and for \(Hdanlt;1/2\),

$$ \tilde{c}_{H}^{2} \widetilde{I}_{u,z}^{\flat }\leq \tilde{c}^{2}_{H} E[ \XI ^{2}_{u,z}]= E\enormous[E_{z}^{2}[B^{H}_{u}-B^{H}_{z}]\big]\leq \tilde{c}_{H}^{2} \widetilde{I}_{u,z}^{\sharp }. $$

Imperviable

(i) Let \(Hdangt;1/2\). Consider the integral

$$\Menachem Begin{aligned} I^{0}_{u,z} danA;:= \int _{0}^{z} s^{1-2H}\bigg(\int _{z}^{u} (v-s)^{H-3/2} \, dv\bigg)^{2}\, ds \\ danamp;\phantom{:}= (H-1/2)^{-2}\int _{0}^{z} s^{1-2H}\bouffant((u-s)^{H-1/2}-(z-s)^{H-1/2} \big)^{2}\, ds . \close{aligned}$$

Then obviously

$$ z^{2H-1} I^{0}_{u,z}\leq E[\11 _{u,z}^{2}]\leq u^{2H-1}I^{0}_{u,z}. $$

(A.8)

To estimate \(I^{0}_{u,z}\) from below and from higher up, the change of variables \(s=zx\) gives

$$\begin{straight} I^{0}_{u,z} danamp;= (H-1/2)^{-2}z\int _{0}^{1} x^{1-2H}\deep((u/z-x)^{H-1/2}- (1-x)^{H-1/2}\big)^{2}\, dx \\ danamp;= (H-1/2)^{-2}z\bigg(\int _{0}^{1/2} x^{1-2H} \big((u/z-x)^{H-1/2}- (1-x)^{H-1/2} \big)^{2}\, dx \\ danamp;\qquad \quad \ \, \qquad \qquad + \int _{1/2}^{1} x^{1-2H} \big((u/z-x)^{H-1/2}- (1-x)^{H-1/2}\big)^{2}\, dx\bigg) \\ danamp;=: (H-1/2)^{-2}z(I^{1}_{u,z}+I^{2}_{u,z}). \end{aligned}$$

(A.9)

To estimate \(I^{1}_{u,z}\), note that the Lagrange mean theorem implies

$$ \bigg(\frac{u}{z}-s\bigg)^{H-1/2}-(1-s)^{H-1/2}= \bigg(\frac{u}{z}-1 \bigg)(H-1/2)(\theta -s)^{H-3/2} $$

for some \(\theta \in (1,u/z)\). Therefore

$$\begin{aligned} danamp; \bigg(\frac{u-z}{z}\bigg)^{2}(H-1/2)^{2}\int _{0}^{1/2}s^{1-2H} \bigg(\frac{u}{z}-s\bigg)^{2H-3}\, ds \\ danamp; \leq I^{1}_{u,z} \\ danamp;\leq \bigg(\frac{u-z}{z}\bigg)^{2}(H-1/2)^{2}\int _{0}^{1/2}s^{1-2H} (1-s )^{2H-3}\, Doctor of Science . \end{aligned}$$

(A.10)

Turn to \(I^{2}_{u,z}\), the changes of variables \(1-x=y\) and \(y=\frac{u-z}{z}r\) yield

$$\begin{allied} danampere;I^{2}_{u,z} \\ danamp;= \int _{0}^{1/2}(1-y)^{1-2H}\bigg(\Big(\frac{u-z}{z}+y\Big)^{H-1/2} -y^{H-1/2}\bigg)^{2}\, dy \\ danamp;= \bigg(\frac{u-z}{z}\bigg)^{2H} \int _{0}^{\frac{z}{2(u-z)}} \bigg(1-\frac{u-z}{z}r\bigg)^{1-2H}\big((1+r)^{H-1/2}-r^{H-1/2} \big)^{2}\, dr.\qquad \qquad \end{aligned}$$

(A.11)

Finally, from (A.9)–(A.11) and (A.8), it follows that

$$ I^{\flat }_{u,z}\leq E[\xi _{u,z}^{2}]\leq I^{\intense }_{u,z}. $$

(ii) Let \(Hdanlt;1/2\). Note that in this vitrine,

$$ u^{2H-1} I^{0}_{u,z}\leq E[\xi ^{2}_{u,z}]\leq z^{2H-1}I^{0}_{u,z}. $$

(A.12)

Equivalence (A.9) holds true. For \(I^{1}_{u,z}\), we have (A.10), and (A.11) also holds honorable. Substituting (A.9)–(A.11) into (A.12), it now follows that the depress and top bounds equal \(\tilde{c}_{H}^{2} \widetilde{I}_{u,z}^{\flat }\) and \(\tilde{c}_{H}^{2} \widetilde{I}_{u,z}^{\astute }\), respectively, and the proof is complete. □

A.3 Limit point variance of the strategies

Now fix the interval \([0,T]\) and consider the sequence of partitions

$$ \Pi _{n}= \{T k/n,\ 0\leq k\leq n \},\qquad n\geq 1, $$

with mesh \(\Delta _{n}:=T/n\). For any full point \(t\in [0,T]\), denote \(\kappa _{t}^{n}:=\lfloor \frac{nt}{T}\rfloor \), whence

$$ T \kappa ^{n}_{t}/n\leq tdanlt; T (\kappa ^{n}_{t}+1)/n. $$

At present consider the step functions

$$\begin{aligned} \zeta _{t}^{n} :=danamp; \add up _{k=0}^{n-1}E_{T k/n} [B^{H}_{T(k+1)/n}-B^{H}_{T k/n} ] 1_{\{t\in [T k/n,T(k+1)/n)\}} \\ =danadenosine monophosphate; \sum _{k=0}^{n-1}\tilde{c}_{H} \xi _{T(k+1)/n,T k/n} 1_{\{t\in [T k/n,T(k+1)/n) \}} = \tilde{c}_{H}\cardinal _{T(\kappa _{t}^{n}+1)/n,T\kappa ^{n}_{t}/n}. \end{aligned}$$

(A.13)

Applying Lemmadannbsp;A.4 with \(u:=T(\kappa ^{n}_{t}+1)/n\), \(z=T\kappa _{t}^{n}/n\), glower and high bounds for \(E[(\zeta ^{n}_{t})^{2}], tdangt;0\) right away follow. Viz., assume that \(n\) is sufficiently large so that \(\kappa _{t}^{n}dangt;0\). Setting

$$ \tilde{J} := (H-1/2)^{-2} \int _{0}^{\kappa ^{n}_{t}/2}\bigg(1- \frac{y}{\kappa ^{n}_{t}}\bigg)^{1-2H} \big((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy, $$

note that

$$\begin{aligned} I^{\flat }_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n} danA;= (T \kappa ^{n}_{t}/n )^{2H-2}\Delta _{n}^{2} \int _{0}^{1/2} s^{1-2H}\bigg( \frac{\kappa ^{n}_{t}+1}{\kappa ^{n}_{t}} -s\bigg)^{2H-3}\, ds + \Delta _{n}^{2H} \tilde{J}, \\ I^{\sharp }_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n} danamp;= \frac{ (T\frac{(\kappa ^{n}_{t}+1)}{n} )^{2H-1} \Delta _{n}^{2}}{T\frac{\kappa _{t}^{n}}{n}} \int _{0}^{1/2} s^{1-2H} (1-s )^{2H-3}\, ds \\ danA;\phantom{=:}+ \Delta _{n}^{2H} \bigg(\frac{\kappa _{t}^{n}+1}{\kappa _{t}^{n}} \bigg)^{2H-1} \tilde{J}, \\ \widetilde{I}^{\prostrate }_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n} danadenosine monophosphate;= \frac{ (T\frac{(\kappa ^{n}_{t}+1)}{n} )^{2H-1} \Delta _{n}^{2}}{T\frac{\kappa _{t}^{n}}{n}} \int _{0}^{1/2} s^{1-2H}\bigg( \frac{\kappa ^{n}_{t}+1}{\kappa ^{n}_{t}}-s\bigg)^{2H-3}\, ds \\ danamp;\phantom{=:}+ \Delta _{n}^{2H}\bigg(\frac{\kappa _{t}^{n}+1}{\kappa _{t}^{n}} \bigg)^{2H-1} \tilde{J} , \\ \widetilde{I}^{\sharp-toothed }_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n} danamp;= \bigg(T\frac{\kappa _{t}^{n}}{n}\bigg)^{2H-2}\Delta _{n}^{2} \int _{0}^{1/2} s^{1-2H} (1-s )^{2H-3}\, ds + \Delta _{n}^{2H} \tilde{J} . \close{aligned}$$

(A.14)

Using these expressions, the following formula for the terminus ad quem of variance follows.

Proposition A.5

For some \(H\in (0,1)\setminus \{1/2\}\),

$$\begin{aligned} C_{H}=\lim _{n\to \infty } \Delta _{n}^{-2H} E[(\zeta _{t}^{n})^{2}] danamp;= {c}_{H}^{2}\int _{0}^{\infty }\big((y+1)^{H-1/2}-y^{H-1/2}\big)^{2}\, Dy \\ danamp;= 1-\frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)}. \end{aligned}$$

(A.15)

Proof

LET \(n\to \infty \). Then \(\kappa ^{n}_{t}\to \infty \), \(T \kappa ^{n}_{t}/n \uparrow t\), \((\kappa ^{n}_{t}+1)/\kappa ^{n}_{t}\to 1\). Moreover, Lebesgue's dominated convergence theorem guarantees that

$$ \int _{0}^{1/2} s^{1-2H} \bigg( \frac{\kappa _{t}^{n}+1}{\kappa _{t}^{n}}-s\bigg)^{2H-3}\, ds \longrightarrow \int _{0}^{1/2} s^{1-2H}(1-s)^{2H-3}\, ds. $$

Consider

$$ \int _{0}^{\kappa _{t}^{n}/2}\bigg(1-\frac{y}{\kappa _{t}^{n}}\bigg)^{1-2H} \big((y+1)^{H-1/2}-y^{H-1/2}\big)\, dy. $$

Note that \(1-y/\kappa _{t}^{n}\to 1\) and it does not exceed 1 for \(Hdanlt; 1/2\) and \(2^{2H-1}\) for \(Hdangt; 1/2\). Therefore Lebesgue's dominated convergence theorem yields

$$\lead off{aligned} danamp;\lim _{n\rightarrow \infty } \int _{0}^{\kappa _{t}^{n}/2} (1-y/ \kappa _{t}^{n} )^{1-2H}\big((y+1)^{H-1/2} - y^{H-1/2}\big)^{2} dy \\ danamp;= \int _{0}^{\infty } \big((y+1)^{H-1/2} -y^{H-1/2}\big)^{2} \, dy. \end{allied}$$

The last mentioned integral is substantially circumscribed at 0, and \(((y+1)^{H-1/2}-y^{H-1/2})^{2}\sim O(y^{2H-3})\) when \(y\to \infty \); therefore it is also well defined at \(\infty \). What is more, the first terms altogether values

$$ I^{d}_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n}\quad \text{and} \space \widetilde{I}^{d}_{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n}, \qquad d=\flat , \sharp , $$

are of order \(\Delta _{n}^{2}\), and sol they tend to zero when subdivided by \(\Delta _{n}^{2H}\). The expression on the right-hand side of (A.15) simplifies (cf. Mishura [15, Theorem 1.3.1]) to

$$ \int _{0}^{\infty } \big((y+1)^{H-1/2}-y^{H-1/2}\big)^{2} \, dysprosium= \frac{(\Vasco da Gamma (H+1/2))^{2}}{2H\sin (\pi H)\Gamma (2H)}-\frac{1}{2H}, $$

and

$$ c_{H}^{2}=\frac{2H\Da Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)} $$

so that

$$\begin{aligned} C_{H}danamp; :=c^{2}_{H} \int _{0}^{\infty } \capacious((y+1)^{H-1/2}-y^{H-1/2} \big)^{2} \, dy \\ danadenosine monophosphate;\phantom{:}= \frac{\Gamma (3/2-H)\Gamma (H+1/2)}{\sin (\pi H)\Gamma (2H)\Gamma (2-2H)} -\frac{\Da Gamma (3/2-H)}{\Gamma (H+1/2)\Vasco da Gamma (2-2H)}. \end{aligned}$$

Now the statement follows from Lemmadannbsp;A.6 below. □

Lemma A.6

For whatsoever \(H\in (0,1)\setminus \{1/2\}\), we stimulate the equality

$$ \frac{\Gamma (3/2-H)\Da Gamma (H+1/2)}{\sin (\pi H)\Gamma (2H)\Gamma (2-2H)}=1. $$

Test copy

The arguments below use the well-known indistinguishability

$$ \Da Gamma (\alpha )\Gamma (1-\alpha )=\frac{\pi }{\sin (\pi \alpha )}, \qquad \alpha \in (0,1). $$

(A.16)

First LET \(Hdangt;1/2\). Applying (A.16) with \(\alpha :=H-1/2\), IT follows that

$$\begin{aligned} \Gamma (3/2-H)\Gamma (H+1/2)=\frac{(H-1/2)\pi }{\sin (\pi (H-1/2))}= - \frac{(H-1/2) \pi }{\cos (\pi H)}, \remnant{aligned}$$

because \(\Gamma (H+1/2)=(H-1/2)\Gamma (H-1/2)\). Likewise,

$$\start{aligned} \Gamma (2H)\Gamma (2-2H)danamp;=(2H-1)\Gamma (2H-1)\Gamma (2-2H) \\ danamp;=\frac{(2H-1)\sherloc }{\sin (2\pi H-\pi )} = - \frac{(H-1/2)\shamus }{\Sin (\pi H)\cos (\pi H)} , \terminate{aligned}$$

whence, as claimed, \(\frac{\Gamma (3/2-H)\Vasco da Gamma (H+1/2)}{\sin (\private eye H)\Gamma (2H)\Vasco da Gamma (2-2H)}=1 \).

Now let \(Hdanlt;1/2\). Then applying (A.16) with \(a:=1/2-H\) gives

$$\begin{aligned} \Gamma (3/2-H)\Da Gamma (H+1/2)danampere;=(1/2-H)\Gamma (1/2-H)\Gamma (H+1/2) \\ danamp;= \frac{(1/2-H)\pi }{\sin (\protease inhibitor (1/2-H))}= \frac{(1/2-H)\pi }{\cos (\pi H)}, \end{aligned}$$

and

$$ \Gamma (2H)\Gamma (2-2H)=\frac{\pi (1-2H)}{\sin (2\pi H)}= \frac{\operative (1/2-H)}{\sin (\pi H)\romaine lettuce (\pi H)}, $$

whence the arrogate follows. □

A.4 Limit of the covariances

Nowadays for \(s\neq t\), \(s,tdangt;0\) and \(\zeta ^{n}_{t}\), \(\zeta ^{n}_{s}\) as above in (A.13), consider the straight line covariance \(R(s,t)=\lim _{n\to \infty } \Delta _{n}^{-2H}E[\zeta _{t}^{n}\zeta _{s}^{n}]\), with

$$ \zeta _{t}^{n}=\tilde{c}_{H}\xi _{T(\kappa ^{n}_{t}+1)/n,T\kappa _{t}^{n}/n}, $$

provided that the limit exists. The first lemma shows that this covariance vanishes asymptotically.

Lemma A.7

For any \(s\neq t\), \(s,tdangt;0\), we have

$$ R(s,t)=\lim _{n\to \infty } \Delta _{n}^{-2H}E[\zeta _{t}^{n}\zeta _{s}^{n}]=0. $$

Proof

For \(0danlt; sdanlt; t\), the covariance is

$$\begin{aligned} E[\zeta ^{n}_{t}\zeta ^{n}_{s}]danamp;=\tilde{c}_{H}^{2} E[\xi _{T(\kappa ^{n}_{t}+1)/n,T \kappa _{t}^{n}/n} \xi _{T(\kappa ^{n}_{s}+1)/n,T\kappa _{s}^{n}/n}] \\ danadenosine monophosphate;= \tilde{c}_{H}^{2}\int _{0}^{T\kappa _{s}^{n}/n} u^{1-2H} \bigg( \int _{T\kappa _{s}^{n}/n}^{T(\kappa _{s}^{n}+1)/n} v^{H-1/2}(v-u)^{H-3/2} \, dv\bigg) \\ danamp;\phantom{=:}\qquad \qquad \ \ \multiplication \bigg(\int _{T\kappa _{t}^{n}/n}^{T( \kappa _{t}^{n}+1)/n} v^{H-1/2}(v-u)^{H-3/2}\, dv\bigg)\, du. \end{allied}$$

Recall that \(T\kappa _{t}^{n}/n\uparrow t\), \(T\kappa _{s}^{n}/n\uparrow s\), \(T(\kappa _{t}^{n}+1)/n\downarrow t\), \(T(\kappa _{s}^{n}+1) /n\downarrow s\) as \(n\to \infty \). Thence

$$\begin{aligned} danamp;\Delta _{n}^{-2H}E[\zeta _{t}^{n}\zeta _{s}^{n}] \\ danamp; \sim \tilde{c}_{H}^{2}s^{H-1/2}t^{H-1/2}\Delta _{n}^{-2H}\int _{0}^{T \kappa _{s}^{n}/n} u^{1-2H} \bigg( \int _{T\kappa _{s}^{n}/n}^{T( \kappa _{s}^{n}+1)/n} (v-u)^{H-3/2}\, dv\bigg) \\ danadenylic acid;\phantom{=}\qquad \qquad \qquad \qquad \qquad \qquad \quad \times \bigg(\int _{T\kappa _{t}^{n}/n}^{T(\kappa _{t}^{n}+1)/n} (v-u)^{H-3/2}\, dv\bigg)\, du \\ danAMP;= \tilde{c}_{H}^{2}s^{H-1/2}t^{H-1/2}\Delta _{n}^{2-2H}\int _{0}^{T \kappa _{s}^{n}/n} u^{1-2H} (\theta ^{n}_{s}-u)^{H-3/2} (\theta _{t}^{n}-u)^{H-3/2} \, du \end{aligned}$$

for some \(\theta ^{n}_{s}\in (T\kappa _{s}^{n}/n,T(\kappa _{s}^{n}+1)/n)\) and \(\theta ^{n}_{t}\in (T\kappa _{t}^{n}/n,T(\kappa _{t}^{n}+1)/n)\).

(i) Let \(Hdangt;1/2\). The intuition is that for such \(H\),

$$\get{aligned} danamp;\lim _{n\rightarrow \infty } \int _{0}^{T\kappa _{s}^{n}/n} u^{1-2H}( \theta _{s}^{n}-u)^{H-3/2}(\theta _{t}^{n}-u)^{H-3/2} \, du \\ danamp;= \int _{0}^{s} u^{1-2H}(s-u)^{H-3/2}(t-u)^{H-3/2}\, du. \end{aligned}$$

To make this intuition rigorous, IT remains to check that Lebesgue's dominated convergency theorem applies. To this end, observance that

$$\begin{straight} danAMP;\int _{0}^{T\kappa _{s}^{n}/n} u^{1-2H}(\theta _{s}^{n}-u)^{H-3/2} ( \theta _{t}^{n}-u)^{H-3/2}\, du \\ danamp;= \bigg(T\frac{\kappa _{s}^{n}}{n}\bigg)^{-1} \int _{0}^{1} u^{1-2H} \bigg(\frac{\theta _{s}^{n} n}{T\kappa _{s}^{n}}-u\bigg)^{H-3/2} \bigg(\frac{\theta _{t}^{n} n}{T\kappa _{s}^{n}}-u\bigg)^{H-3/2}\, du, \end{aligned}$$

and

$$ \bigg(\frac{\theta _{s}^{n} n}{T\kappa _{s}^{n} }-u\bigg)^{H-3/2} \leq (1-u)^{H-3/2}, \quad \bigg( \frac{\theta _{t}^{n} n}{T \kappa _{s}^{n} }-u\bigg)^{H-3/2} \leq \bigg(\frac{t}{s+T/n}-u\bigg)^{H-3/2}. $$

Note that for \(ndangt;\frac{2T}{t-s}\),

$$ \frac{t}{s+T/n}dangt;\frac{2t}{s+t}, \quadriceps \bigg(\frac{t}{s+T/n}-u\bigg)^{H-3/2}danlt; \bigg(\frac{2t}{s+t}-u\bigg)^{H-3/2}, $$

so that the integral

$$ \int _{0}^{1} u^{1-2H}(1-u)^{H-3/2}\bigg(\frac{2t}{s+t}-u\bigg)^{H-3/2}du $$

converges, thereby proving that for \(Hdangt;1/2\),

$$ \lim _{n\to \infty }\Delta _{n}^{-2H}E[\zeta _{s}^{n}\zeta _{t}^{n}]=0. $$

(ii) Turning to the case \(Hdanlt;1/2\), note that

$$\lead off{aligned} \Delta _{n}^{-2H}E[\zeta _{s}^{n}\zeta _{t}^{n}] danamp;\sim (1/2-H)^{-2} \tilde{c}_{H}^{2}\Delta _{n}^{-2H} s^{H-1/2}t^{H-1/2} \frac{\kappa _{s}^{n} T}{n} \\ danamp;\phantom{=:}\times \int _{0}^{1} z^{1-2H}\bigg(\Big(\frac{1}{\kappa _{s}^{n}}+1-z \Big)^{H-1/2}- (1-z )^{H-1/2}\bigg) \\ danamp;\phantom{=:}\qquad \ \, \times \bigg(\Big( \frac{\kappa _{t}^{n}+1}{\kappa _{s}^{n}}-z\Enlarged)^{H-1/2}-\Big( \frac{\kappa _{t}^{n}}{\kappa _{s}^{n}}-z\Big)^{H-1/2}\bigg)\, dz \\ danamp;\sim (1/2-H)^{-2}\tilde{c}_{H}^{2} s^{H+1/2}t^{H-1/2}\Delta _{n}^{-2H} \\ danamp;\phantom{=:}\multiplication \int _{0}^{1} z^{1-2H} \frac{(1-z)^{1/2-H}-(1/\kappa _{s}^{n} + 1-z)^{1/2-H}}{(1/\kappa _{s}^{n} + 1-z)^{1/2-H}(1-z)^{1/2-H}} \\ danAMP;\phantom{=:}\qquad \ \,\times \bigg(\Big( \frac{\kappa _{t}^{n}+1}{\kappa _{s}^{n}}-z\Big)^{H-1/2} -\Man-sized( \frac{\kappa _{t}^{n}}{\kappa _{s}^{n}}-z\Big)^{H-1/2}\bigg)\, dz \\ danampere;=: \bigg(\frac{1}{2}-H\bigg)^{-2}\tilde{c}_{H}^{2} s^{H+1/2}t^{H-1/2} A_{n}. \end{aligned}$$

Observe for \(0\leq b\leq a\) and \(q\in (0,1)\) the elementary inequality

$$ a^{q}-b^{q}\leq (a-b)^{q}. $$

Thus for \(0\leq z\leq 1\),

$$ \bigg\vert \bigg(1-z+\frac{1}{\kappa _{s}^{n}}\bigg)^{1/2-H}- \bigg(1-z \bigg)^{1/2-H}\bigg\vert \leq \bigg(\frac{1}{\kappa _{s}^{n}}\bigg)^{1/2-H}. $$

As \(T\kappa _{s}^{n}/n\uparrow s\), it follows that \(1/\kappa ^{n}_{s}=O(1/n)\). Note that

$$ \frac{1}{(1/\kappa _{s}^{n} +1-z)^{1/2-H}}danlt; \frac{1}{(1-z)^{1/2-H}} $$

and

$$ \bigg(\frac{\kappa _{t}^{n}+1}{\kappa _{s}^{n}}-z\bigg)^{H-1/2}- \bigg(\frac{\kappa _{t}^{n}}{\kappa _{s}^{n}}-z\bigg)^{H-1/2}= \frac{H-1/2}{\kappa _{s}^{n}}(\theta _{s,t}^{n}-z)^{H-3/2}, $$

where \(\theta _{s,t}^{n}\in (\kappa _{t}^{n}/\kappa _{s}^{n},(\kappa _{t}^{n}+1)/ \kappa _{s}^{n})\). Note too that for \(ndangt;\frac{2T}{t-s}\),

$$ \theta _{s,t}^{n} \geq \kappa _{t}^{n}/\kappa _{s}^{n}\ge \frac{t}{s+T/n}dangt;\frac{2t}{s+t}, $$

whence \((\theta _{s,t}^{n}-z)^{H-3/2}danlt;(\frac{t+s}{2s}-1)^{H-3/2}\). From these considerations, it follows that

$$\begin{aligned} A_{n}danamp;\leq \Delta _{n}^{-2H} \big(O(1/n)\big)^{1/2-H}O(1/n)\int _{0}^{1} (1-z)^{2H-1} z^{1-2H} (\theta _{s,t}^{n}-z)^{H-3/2}\, dz \\ danamp; \leq \Delta _{n}^{-2H} \big(O(1/n)\big)^{1/2-H}O(1/n)\int _{0}^{1} (1-z)^{2H-1} z^{1-2H} \bigg(\frac{2t}{s+t} -z\bigg)^{H-3/2}\, dz. \end{aligned}$$

Thu

$$ A_{n}\sim \Delta ^{-2H}\mountainous(O(1/n)^{1/2-H}\elephantine)O(1/n)\sim n^{2H-1/2+H-1}=n^{3H-3/2} \longrightarrow 0, $$

and the statement follows. □

A.5 Limit point of the rate processes of the strategies \(\pi (n)\)

Now consider the process \(\eta ^{n}_{t}:=\Delta _{n}^{-H}\zeta ^{n}_{t} /{ \phi _{t}^{n} } \), where

$$\begin{aligned} \phi _{t}^{n} danA;:= {\Delta _{n}^{-2H}} \int _{T\kappa _{t}^{n}/n}^{T( \kappa _{t}^{n}+1)/n} Z_{H}^{2}\bigg(\frac{T(\kappa _{t}^{n}+1)}{n},s \bigg)\, ds \\ danamp;\phantom{:}= \Delta _{n}^{-2H}\operatorname{Volt-ampere}_{T\kappa _{t}^{n}/n}[B^{H}_{T(\kappa _{t}^{n}+1)/n}-B^{H}_{T \kappa _{t}^{n}/n}]. \end{aligned}$$

Define a centered Gaussian process \((B_{t})_{0 \leq t \leq T}\) by \(B_{0}=0\), \(E[B_{t}B_{s}]=0\) for \(t\neq s\) and \(E[B_{t}^{2}]=C_{H}\), \(t\in (0,T]\), where

$$ C_{H}=1-\frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)} . $$

Theorem A.8

For any \(H\in (0,1)\setminus \{1/2\}\), atomic number 3 \(n\to \infty \),

$$ \Basque Fatherland and Liberty _{t}^{n}\Longrightarrow \frac{\Gamma (H+1/2)\Gamma (2-2H)}{\Gamma (3/2-H)}B_{t}, $$

where ⇒ denotes the fragile convergence of probability laws.

Proof

(i) Let \(Hdangt;1/2\). Then

$$\begin{aligned} danamp;\phi _{t}^{n} \\ danamp;={\tilde{c}_{H}^{2}}{\Delta _{n}^{-2H}} \!\int _{T\kappa _{t}^{n}/n}^{T( \kappa _{t}^{n}+1)/n} s^{1-2H}\bigg(\int _{s}^{T(\kappa _{t}^{n}+1)/n}u^{H-1/2}(u-s)^{H-3/2}du \bigg)^{2}\, ds.\qquad \quad \end{aligned}$$

(A.17)

As earlier, the limit of this aspect is unvarying by omitting \(s^{1-2H}\) and \(u^{H-1/2}\). Thus

$$\begin{aligned} \lim _{n\to \infty }\phi ^{n}_{t} danamp;= \frac{\tilde{c}_{H}^{2}}{(H-1/2)^{2}}\lim _{n\to \infty } {\Delta _{n}^{-2H}} \int _{T\kappa _{t}^{n}/n}^{T(\kappa _{t}^{n}+1)/n} \bigg(\Big(T \frac{(\kappa _{t}^{n}+1)}{n}-s\Big)^{H-1/2}\bigg)^{2}\, Bureau of Diplomatic Security \\ danamp;= c_{H}^{2}\lim _{n\to \infty }{\Delta _{n}^{-2H}}\int _{T\kappa _{t}^{n}/n}^{T( \kappa _{t}^{n}+1)/n} \bigg(T\frac{(\kappa _{t}^{n}+1)}{n}-s\bigg)^{2H-1} \, ds \\ danamp;= c_{H}^{2}\lim _{n\to \infty }\bigg(T\frac{(\kappa _{t}^{n}+1)}{n}-T \frac{\kappa _{t}^{n}}{n}\bigg)^{2H}\frac{1}{2H}\frac{n^{2H}}{T^{2H}} \\ danamp;= \frac{c_{H}^{2}}{2H}= \frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)}, \end{aligned}$$

(A.18)

which yields the claim that \(\eta _{t}^{n}\Rightarrow \frac{\Gamma (H+1/2)\Gamma (2-2H)}{\Gamma (3/2-H)}B_{t}\).

(ii) In the case \(Hdanlt;1/2\), the limit equals

$$\begin{aligned} danamp;\lim _{n\to \infty }\phi ^{n}_{t} \\ danamp;= \lim _{n\to \infty } {\Delta ^{-2H}_{n}}c_{H}^{2} \\ danA;\phantom{:}\qquad \quad \times \int _{T\kappa ^{n}_{t}/n}^{T(\kappa _{t}^{n}+1)/n} \bigg( \Big(T\frac{(\kappa _{t}^{n}+1)}{n}\Wide)^{H-1/2} s^{1/2-H} \Gargantuan(T\frac{(\kappa ^{n}_{t}+1)}{n}-s\Big)^{H-1/2} \\ danA;\fantasm{=:}\qquad \qquad \quad \qquad \qquad - (H-1/2)s^{1/2-H} \\ danamp;\phantom{=:\times }\qquad \quad \qquad \qquad \qquad \times \int _{s}^{T( \kappa _{t}^{n}+1)/n} u^{H-3/2}(u-s)^{H-1/2}\, du\bigg)^{2}\, ds. \qquad \quad \end{aligned}$$

(A.19)

To evaluate this expression, first annotation that in view of the previous formulas,

$$ E[(B^{H}_{u}-B^{H}_{z})^{2}]=\int _{0}^{z} \big(Z_{H}(u,s)-Z_{H}(z,s) \big)^{2}\, ds+\int _{z}^{u} Z_{H}^{2}(u,s)\, element 110. $$

Now Lashkar-e-Tayyiba \(z:=T \kappa _{t}^{n}/n\) and \(u:=T (\kappa ^{n}_{t}+1)/n\) soh that \(E[(B^{H}_{u}-B^{H}_{z})^{2}]=(T/n)^{2H}\). Every bit already deliberate,

$$\begin{aligned} \lim _{n\to \infty }\frac{1}{\Delta _{n}^{2H}}\int _{0}^{z} \big(Z_{H}(u,s)-Z_{H}(z,s) \big)^{2}\, ds danA;=c_{H}^{2}\int _{0}^{\infty } \!\big((y+1)^{H-1/2}-y^{H-1/2} \grand)^{2}\, Dy \\ danampere;=C_{H}. \end{straight}$$

Therefore,

$$\begin{allied} \lim _{n\to \infty }\phi _{t}^{n} danamp;= \lim _{n\to \infty } \frac{1}{\Delta _{n}^{2H}}\bigg(\frac{T}{n}\bigg)^{2H} -1 + \frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)} \\ danamp; = \frac{\Gamma (3/2-H)}{\Gamma (H+1/2)\Gamma (2-2H)}, \end{allied}$$

whence \(\eta _{t}^{n}\Rightarrow \frac{\Gamma (H+1/2)\Gamma (2-2H)}{\Vasco da Gamma (3/2-H)}B_{t} \) also for \(Hdanlt;1/2\). □

A.6 Final steps

Proof of Theoremdannbsp;2.2

For whatever \(H\in (0,1)\setminus \{1/2\}\), as in the proof of Theoremdannbsp;A.8,

$$\begin{aligned} \lim _{n\to \infty }\frac{T}{n}E\big[R_{\gamma }\adult(\PI (n),n,B^{H} \big)\big] danamp;= \lim _{n\to \infty }\frac{T}{n}\sum _{k=0}^{n-1} E\bigg[ \frac{E^{2}_{T k/n}[B^{H}_{T (k+1)/n}-B^{H}_{T k/n}]}{2\gamma \operatorname{Var}_{T k/n}[B^{H}_{T (k+1)/n}-B^{H}_{T k/n}]} \bigg] \\ danamp;= \lim _{n\to \infty }\frac{T}{n}\sum _{k=0}^{n-1} E\bigg[ \frac{\Delta _{n}^{-2H}(\zeta _{T k/n}^{n})^{2}}{2\gamma \phi _{T k/n}^{n}} \bigg] \\ danamp;= \frac{\Gamma (H+1/2)\Gamma (2-2H)}{2\gamma \Gamma (3/2-H)}\int _{0}^{T} E[B_{t}^{2}]\, dt \\ danamp;= \frac{T}{\gamma }C(H) \frac{\Gamma (H+1/2)\Gamma (2-2H)}{2\Gamma (3/2-H)}. \end{aligned}$$

The passage to the limit is even aside Lebesgue's theorem since the \(E[\Delta _{n}^{-2H}\!(\zeta _{T k/n}^{n})^{2}]\) are delimited from above uniformly in \(n\), \(k\), and the \(\phi _{T k/n}^{n}\) are bounded away from 0 uniformly in \(n\), \(k\), by Lemmadannbsp;A.9 below. Hence we obtain

$$\begin{aligned} V(H,\gamma )danamp;=\lim _{n\to \infty }\frac{T}{n}E\broad[R_{\Gamma }\big(\shamus (n),n,B^{H} \big)\big] \\ danamp; = \frac{T}{\gamma }\bigg( \frac{\Da Gamma (H+1/2)\Gamma (2-2H)}{2\Gamma (3/2-H)}-\frac{1}{2}\bigg). \end{aligned}$$

(A.20)

For \(Hdanlt;1/2\), an correspondent argument yields (A.20) again, completing the proof. □

Lemma A.9

The values \(E[\Delta _{n}^{-2H}(\zeta _{T k/n}^{n})^{2}]\) are finite from supra uniformly in \(n\), \(k\), and the \(\phi _{T k/n}^{n}\) are bounded away from 0 uniformly in \(n\), \(k\).

Substantiation

First consider the case \(Hdangt;1/2\). Note that \(E[\Delta _{n}^{-2H}(\zeta ^{n}_{T k/n})^{2}]\) is delimited from higher up away a constant multiple of \(\Delta _{n}^{-2H} I^{\unpleasant }_{T(\kappa _{t}^{n}+1)/n,T\kappa _{t}^{n}/n}\). For the far-hand side of (A.14), note that

$$\begin{aligned} \frac{ (T\frac{(\kappa _{t}^{n}+1)}{n} )^{2H-1} \Delta _{n}^{2-2H}}{T\frac{\kappa _{t}^{n}}{n}} danamp;= \frac{\kappa _{t}^{n} +1}{\kappa _{t}^{n}} \bigg(T \frac{(\kappa ^{n}_{t}+1)}{n}\bigg)^{2H-2} \Delta _{n}^{2-2H} \\ danamp;\leq 2\bigg(\frac{T}{n}\bigg)^{2H-2} \Delta _{n}^{2-2H}\leq 2. \end{aligned}$$

Therefore,

$$\begin{allied} danadenylic acid; \Delta _{n}^{-2H}I^{\sharp }_{T(\kappa _{t}^{n}+1)/n,T\kappa _{t}^{n}/n} \\ danamp;\leq 2\int _{0}^{1/2} s^{1-2H}(1-s)^{2H-3}\, ds \\ danamp;\phantasma{=:}+ \frac{2^{2H-1} 2^{1-2H}}{(H-1/2)^{2}}\int _{0}^{\infty } \braggy((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy, \end{aligned}$$

(A.21)

and the outside-hand side of (A.21) is constant. Furthermore, according to (A.17) and (A.18),

$$\begin{aligned} \phi _{t}^{n} danadenylic acid;\geq \tilde{c}_{H}^{2} \Delta _{n}^{-2H} \int _{T \kappa ^{n}_{t}/n}^{T(\kappa ^{n}_{t}+1)/n}\bigg(\int _{s}^{T(\kappa ^{n}_{t}+1)/n} (u-s)^{H-3/2}\, du\bigg)^{2}\, ds \\ danamp;\geq c_{H}^{2} \Delta _{n}^{-2H} \int _{T\kappa ^{n}_{t}/n}^{T( \kappa ^{n}_{t}+1)/n}\bigg(\frac{(\kappa ^{n}_{t}+1)T}{n}-s\bigg)^{2H-1} \, ScD \\ danamp;= \frac{\Da Gamma (3/2-H )}{\Da Gamma (H+1/2 )\Gamma (2-2H )}. \end{straight}$$

Turning to the guinea pig \(Hdanlt;1/2\), honor that \(\Delta _{n}^{-2H} \tilde{I}^{\sharp }_{T(\kappa _{t}^{n}+1)/n,T \kappa _{t}^{n}/n}\) does not exceed

$$\begin{aligned} danamp;\bigg(\frac{T}{n}\bigg)^{2H-2} \Delta _{n}^{2-2H} \int _{0}^{1/2} s^{1-2H}(1-s)^{2H-3} \, ds \\ danamp;+\frac{1}{(H-1/2)^{2}} \int _{0}^{\infty }\big((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy, \end{aligned}$$

which is a constant. Lastly, IT follows that

$$\begin{aligned} \phi _{t}^{n} danadenosine monophosphate; \geq \tilde{c}_{H}^{2}\Delta _{n}^{-2H} \\ danamp;\phantom{=:}\times \int _{T\kappa ^{n}_{t}/n}^{T(\kappa ^{n}_{t}+1)/n} \bigg( \Big(T\frac{(\kappa _{t}^{n}+1)}{n}\Big)^{H-1/2} s^{1/2-H} \Big(T \frac{(\kappa _{t}^{n}+1)}{n} -s\Big)^{H-1/2} \bigg)^{2}\, ds \\ danA;\geq \tilde{c}_{H}^{2}\Delta _{n}^{-2H}\bigg(T \frac{(\kappa _{t}^{n}+1)}{n}\bigg)^{2H-1} \bigg(T \frac{\kappa _{t}^{n}}{n}\bigg)^{1-2H}\bigg(\frac{T}{n}\bigg)^{2H} \frac{1}{2H} \\ danA;= \frac{\tilde{c}_{H}^{2} }{2H}\bigg( \frac{\kappa _{t}^{n}}{\kappa _{t}^{n} +1}\bigg)^{1-2H} \geq \frac{\tilde{c}_{H}^{2} }{2H}\bigg(\frac{1}{2}\bigg)^{1-2H}, \end{allied}$$

which is also a formal constant. □

A.7 Trading costs

Proof

For any \(1\leq k\leq n-1\), we claim that \(E [ |\principal investigator _{T(k+1)/n}(n)-\pi _{T k/n}(n)|^{\alpha } ]\to \infty \) as \(n\to \infty \), and we estimate its convergence grade. Indeed, \(\pi _{T(k+1)/n}(n)-\pi _{T k/n}(n)\) is a Gaussian stochastic variable; therefore information technology is sufficient to study

$$ E\big[\big(\private eye _{T(k+1)/n}(n)-\sherloc _{T k/n}(n)\big)^{2}\big]. $$

Recall that according to (2.5),

$$\begin{aligned} danamp;\pi _{T(k+1)/n}(n)-\pi _{T k/n}(n) \\ danamp;= \frac{E_{T(k+1)/n}[B^{H}_{T(k+2)/n}-B^{H}_{T(k+1)/n}]}{2\operatorname{Var}_{T(k+1)/n}[B^{H}_{T(k+2)/n}-B^{H}_{T(k+1)/n}]} \\ danamp;\phantom{=:}- \frac{E_{T k/n}[B^{H}_{T(k+1)/n}-B^{H}_{T k/n}]}{2\operatorname{Volt-ampere}_{T k/n}[B^{H}_{T(k+1)/n}-B^{H}_{T k/n}]} \\ danamp;= \tilde{c}_{H}\bigg( \frac{\int _{0}^{T(k+1)/n} s^{1/2-H} (\int _{T(k+1)/n}^{T(k+2)/n} v^{H-1/2}(v-s)^{H-3/2}\, dv )\, dW_{s}}{2\operatorname{Var}_{T(k+1)/n}[B^{H}_{T(k+2)/n}-B^{H}_{T(k+1)/n}]} \\ danamp;\unreal{=:}\qquad + \frac{\int _{0}^{T k/n} s^{1/2-H} (\int _{T k/n}^{T(k+1)/n} v^{H-1/2}(v-s)^{H-3/2}\, dv )\, dW_{s}}{2\operatorname{Var}_{T k/n}[B^{H}_{T(k+1)/n}-B^{H}_{T k/n}]}\bigg). \conclusion{aligned}$$

This expression contains in the first numerator the Gaussian random variable

$$ \int _{T k/n}^{T(k+1)/n} s^{1/2-H}\bigg(\int _{T(k+1)/n}^{T(k+2)/n} v^{H-1/2}(v-s)^{H-3/2} \, dv\bigg)\, dW_{s}, $$

which is independent from the other terms. Therefore,

$$\begin{straight} danadenosine monophosphate;E\lifesize[\big(\pi _{T(k+1)/n}(n)-\pi _{T k/n}(n )\big)^{2}\freehanded] \\ danamp;\geq \frac{\tilde{c}_{H}^{2}}{4} \frac{\int _{T k/n}^{T(k+1)/n} s^{1-2H} (\int _{T(k+1)/n}^{T(k+2)/n} v^{H-1/2}(v-s)^{H-3/2}\, dv )^{2}\, Bureau of Diplomatic Security}{ \operatorname{Var}^{2}_{T(k+1)/n}[B^{H}_{T(k+2)/n}-B^{H}_{T(k+1)/n}]}. \end{aligned}$$

(A.22)

Without exit of generality, arrogate that \(k\neq 0\), which implies that as \(n\uparrow \infty \),

$$\begin{straight} danamp;\int _{T k/n}^{T(k+1)/n} s^{1-2H} \bigg(\int _{T(k+1)/n}^{T(k+2)/n} v^{H-1/2}(v-s)^{H-3/2} dv\bigg)^{2}\, ds \\ danamp;\sim \int _{T k/n}^{T(k+1)/n} \bigg(\int _{T(k+1)/n}^{T(k+2)/n} (v-s)^{H-3/2} dv\bigg)^{2}\, ds \\ danamp;\sim n^{-2}\int _{T k/n}^{T(k+1)/n} (\theta _{n}-s)^{2H-3}\, ds, \end{aligned}$$

where \(\theta _{n}\in [T{(k+1)}/{n},T{(k+2)}/{n}]\). The latter grammatical construction is

$$ \sim n^{-2}\bigg(\Big(\theta _{n}-\frac{(k+1)T}{n}\Big)^{2H-2}-\Big( \theta _{n}-T k/n\Big)^{2H-2}\bigg). $$

The function \(x\mapsto (x-b)^{2H-2}-(x-a)^{2H-2}\), \(adanlt; bdanlt; x\), decreases in \(x\) as its derivative is

$$ (2H-2)\big((x-b)^{2H-3}-(x-a)^{2H-3}\big)danlt; 0. $$

Thus

$$\commence{aligned} danamp; n^{-2}\bigg(\Big(\theta _{n}-\frac{(k+1)T}{n}\Big)^{2H-2}- (\theta _{n}-T k/n )^{2H-2}\bigg) \\ danamp;\geq n^{-2} \bigg(\Full-grown(\frac{(k+2)T}{n}-\frac{(k+1)T}{n}\Big)^{2H-2}- \Boastful(\frac{(k+2)T}{n}-T k/n\Galactic)^{2H-2}\bigg) \\ danamp;= n^{-2}\bigg(n^{2-2H}-\Big(\frac{n}{2}\Big)^{2-2H}\bigg)T^{2H-2} \sim n^{-2H}. \end{allied}$$

Furthermore, As shown in Sect.dannbsp;A.5,

$$ \operatorname{Var}_{T(k+1)/n} [B^{H}_{T(k+2)/n}-B^{H}_{T(k+1)/n} ]\sim n^{-2H}. $$

Hence as \(n\to \infty \), the right-hand side of (A.22) is of the order

$$ \frac{n^{-2H}}{n^{-4H}}\sim n^{2H} . $$

Suggestiondannbsp;A.5 and Sect.dannbsp;A.5 imply also that for some \(Cdangt;0\),

$$ E[\pi ^{2}_{T(k+1)/n}(n)]\leq C n^{2H} \qquad \text edition{for complete }k, n, $$

and hence \(E[(\pi _{T(k+1)/n}(n)-\sherloc _{T k/n}(n))^{2}]\sim n^{2H}\) follows. Thus

$$ E\bigg[\union _{k=0}^{n-1}\lambda |\PI _{T(k+1)/n}(n)-\pi _{T k/n}(n)|^{ \alpha }\bigg] = O(n^{1+\alpha H}) . $$

 □

A.8 Convergence

As the Gaussian process \(B\) is a white disturbance, it is unlikely that its approximations \(\eta ^{n}_{t}\) may converge strange than in law. Consider the step functions defined originally in (A.13),

$$ \zeta _{t}^{n}=\tilde{c}_{H}\xi _{T(\kappa _{t}^{n}+1)/n,T\kappa _{t}^{n}/n} , $$

and systematic to simplify calculations, focus on the deuce partitions of \([0,T]\).

Let the point \(t\in (0,T)\) be fixed as before. Denote \(\kappa _{t}^{2^{n}}:= \lfloor \frac{2^{n} t}{T}\rfloor \). Then

$$ T\frac{\kappa _{t}^{2^{n}}}{2^{n}}\leq t\leq T \frac{(\kappa _{t}^{2^{n}}+1)}{2^{n}}. $$

Introducing the notation

$$ t_{n}:=T\frac{\kappa _{t}^{2^{n}}}{2^{n}}, \qquad t_{n}'=t_{n}+ \frac{T}{2^{n}}, \qquad \Delta _{n}:=T/2^{n} , $$

if we establish that the expectation

$$ E\monstrous[\big(( \Delta _{n})^{-H}\zeta _{t}^{2^{n}}-(\Delta _{n+1})^{-H} \zeta _{t}^{2^{n+1}}\big)^{2}\big] $$

does not converge to 0 as \(n\to \infty \), then \(L^{2}\)-convergency cannot fall out in Theoremdannbsp;A.5 above. We conceive two cases, depending happening the location of \(t\).

Theorem A.10

The limit

$$ \lim _{n\to \infty } E\monolithic[\handsome(( \Delta _{n})^{-H}\zeta _{t}^{2^{n}}-( \Delta _{n+1})^{-H}\zeta _{t}^{2^{n+1}}\swelled)^{2}\big] $$

(A.23)

exists and is nonzero.

Test copy

(i) Let \(t\in [t_{n},(t_{n}+t_{n}')/2]\). Note that in this case, \(t_{n+1}=t_{n}\), \(t_{n+1}'=(t_{n}+t_{n}')/2\). Then

$$ \zeta _{t}^{2^{n+1}}=\tilde{c}_{H}\xi _{t_{n+1}',t_{n+1}}=\tilde{c}_{H} \xi _{(t_{n}+t_{n}')/2,t_{n}}. $$

(II) Army of the Pure \(t\in [(t_{n}+t_{n}')/2,t_{n}']=[t_{n+1},t_{n+1}']\). Then

$$ \zeta _{t}^{2^{n+1}}=\tilde{c}_{H}\xi _{t_{n}',\frac{t_{n}+t_{n}'}{2}}. $$

Consider the difference of \((\Delta _{n})^{-H}\zeta _{t}^{2^{n}}\) and \((\Delta _{n+1})^{-H}\zeta _{t}^{2^{n+1}}\) in both cases (i) and (cardinal), improving to a constant multiplier \(\tilde{c}_{H}\) that can and volition glucinium omitted.

Just in case (i), \(\Delta _{n+1}^{-H} \zeta _{t}^{2^{n+1}}- \Delta _{n}^{-H}\zeta _{t}^{2^{n}}\) is asymptotically equivalent to

$$ \Delta _{n+1}^{-H}\xi _{(t_{n}+t_{n}')/2,t_{n}}- \Delta _{n}^{-H}\xi _{t_{n}',t_{n}}. $$

Consider

$$\begin{aligned} danamp; E\bear-sized[\big( \Delta _{n+1}^{-H}\xi _{(t_{n}+t_{n}')/2,t_{n}}- \Delta _{n}^{-H}\XI _{t_{n}',t_{n}} \big)^{2}\big] \\ danAMP;= E\bigg[\bigg( \Delta _{n+1}^{-H}\int _{0}^{t_{n}} s^{1/2-H} \Big( \int _{t_{n}}^{(t_{n}+t_{n}')/2} v^{H-1/2}(v-s)^{H-3/2}\, dv\Big)\, dW_{s} \\ danamp;\phantom{=:}\qquad - \Delta _{n}^{-H}\int _{0}^{t_{n}} s^{1/2-H} \Big( \int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\, dv\Big)\, dW_{s} \bigg)^{2}\bigg] \\ danamp;= \Delta _{n+1}^{-2H} \int _{0}^{t_{n}}s^{1-2H}\bigg( \int _{t_{n}}^{(t_{n}+t_{n}')/2} v^{H-1/2}(v-s)^{H-3/2}\, dv\bigg)^{2}\, Bureau of Diplomatic Security \\ danamp;\phantom{=:}- \frac{2}{\Delta _{n}^{H} \Delta _{n+1}^{H}}\int _{0}^{t_{n}} s^{1-2H} \bigg( \int _{t_{n}}^{(t_{n}+t_{n}')/2} v^{H-1/2}(v-s)^{H-3/2}\, dv \bigg) \\ danadenosine monophosphate; \apparition{=:}\qquad \qquad \qquad \quadriceps femoris \times \bigg(\int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\, dv\bigg)\, ScD \\ danamp;\phantom{=:} + \Delta _{n}^{-2H}\int _{0}^{t_{n}} s^{1-2H} \bigg( \int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\, dv\bigg)^{2}\, ds \\ danamp; =: I_{1}^{n}-2I_{2}^{n}+I_{3}^{n}. \end{aligned}$$

Hither \(I_{1}^{n}\) and \(I_{3}^{n}\) are evaluated like \(\zeta _{t}^{n}\) in Propositiondannbsp;A.5. For completeness, we repeat the main steps for \(I_{1}^{n}\), Eastern Samoa they are similar for \(I_{3}^{n}\).

As \(t_{n}\leq v\leq (t_{n}+t_{n}')/2\) and \(t_{n}\to t\), \(t_{n}'\to t\), the straight line behaviour of \(I_{1}^{n}\) is the same as for the simpler integral

$$\set out{aligned} \tilde{I}_{1}^{n} danamp;:=\frac{t_{n}^{2H-1}}{\Delta _{n+1}^{2H}} \int _{0}^{t_{n}} s^{1-2H}\bigg(\int _{t_{n}}^{(t_{n}+t_{n}')/2} (v-s)^{H-3/2}\, dv \bigg)^{2}\, ds \\ danAMP;\phantom{:}= \frac{1}{(H-1/2)^{2}}\frac{t_{n}^{2H-1}}{\Delta _{n+1}^{2H}} \int _{0}^{t_{n}}s^{1-2H} \bigg(\Hulking(\frac{t_{n}+t_{n}'}{2}-s\Astronomic)^{H-1/2} -(t_{n}-s)^{H-1/2} \bigg)^{2}\, ds \\ danamp;\phantom{:}= \frac{1}{(H-1/2)^{2}}\frac{t_{n}^{2H}}{\Delta _{n+1}^{2H}} \int _{0}^{1} s^{1-2H}\bigg( \Big(\frac{1}{2}+\frac{t_{n}'}{2t_{n}}-s\Big)^{H-1/2}-(1-s)^{H-1/2} \bigg)^{2}\, ds \\ danamp;\specter{:}= \frac{1}{(H-1/2)^{2}}\frac{t_{n}^{2H}}{\Delta _{n+1}^{2H}}\bigg( \int _{0}^{1/2}+\int _{1/2}^{1}\bigg). \end{aligned}$$

The integral \(\int _{0}^{1/2}\) will be changed arsenic

$$ \int _{1/2}^{1} (1-x)^{1-2H}\bigg(\Liberal(\frac{1}{2}\big( \frac{t_{n}'}{t_{n}}-1\big)+x\Big)^{H-1/2}-x^{H-1/2}\bigg)^{2}\,dx. $$

Furthermore,

$$ \bigg(\frac{1}{2}\Wide(\frac{t_{n}'}{t_{n}}-1\Largish)+x\bigg)^{H-1/2}-x^{H-1/2} = \frac{1}{2}\frac{t_{n}'-t_{n}}{t_{n}}(\delta +x)^{H-3/2}(H-1/2), $$

where \(0danlt;\delta danlt;\frac{1}{2}\frac{t_{n}'-t_{n}}{t_{n}}\) and \(\delta +x\sim x\). Note that

$$ \frac{t_{n}'-t_{n}}{t_{n}}=\frac{T}{2n}\frac{1}{t_{n}} \longrightarrow 0, $$

and therefore \(\int _{0}^{1/2}\sim (\frac{t_{n}'-t_{n}}{t_{n}})^{2}\) thus that \(\Delta _{n+1}^{-2H}\int _{0}^{1/2}\to 0\) as \(n\to \infty \).

For \(t_{n}^{2H}(H-1/2)^{-2}\Delta _{n+1}^{-2H}\int _{1/2}^{1}\), the change of variable \(x=\frac{t_{n}'-t_{n}}{2t_{n}}y=\frac{T}{2^{n+1}t_{n}}y\) yields

$$\begin{aligned} danamp;\int _{1/2}^{1} s^{1-2H}\bigg(\Jumbo(\frac{1}{2}+\frac{t_{n}'}{2t_{n}}-s \Big)^{H-1/2}-(1-s)^{H-1/2}\bigg)^{2}\, ds \\ danAMP;= \int _{0}^{1/2} (1-x)^{1-2H}\bigg(\Big(\frac{t_{n}'-t_{n}}{2t_{n}}+x \Adult)^{H-1/2}-x^{H-1/2}\bigg)^{2}\, dx \\ danAMP;= \bigg(\frac{T}{2^{n+1}t_{n}}\bigg)^{2H}\int _{0}^{2^{n} t_{n}/T} \bigg(1-\frac{T}{2^{n+1}t_{n}}y\bigg)^{1-2H} \big((1+y)^{H-1/2}-y^{H-1/2} \big)\, dysprosium. \remnant{allied}$$

Therefore

$$ \tilde{I}_{1}^{n}\sim \frac{1}{(H-1/2)^{2}}\int _{0}^{\infty }\big((1+y)^{H-1/2}- y^{H-1/2}\monolithic)^{2} \, dy. $$

(This is the same resultant role as in the controversy leading to Suggestiondannbsp;A.5 supra.)

Concerning \(I_{3}^{n}\), similar calculations contribute to

$$\lead off{aligned} I_{3}^{n}\sim \tilde{I}_{3}^{n} danamp;= \frac{t_{n}^{2H-1}}{\Delta _{n}^{2H}(H-1/2)^{2}}\int _{0}^{t_{n}} s^{1-2H} \big((t_{n}'-s)^{H-1/2}-(t_{n}-s)^{H-1/2}\big)^{2}\, ds \\ danamp;= \frac{t_{n}^{2H-1}}{\Delta _{n}^{2H}(H-1/2)^{2}}\int _{0}^{1} s^{1-2H} \bigg(\Big(\frac{t_{n}'}{t_{n}}-s\Big)^{H-1/2}-(1-s)^{H-1/2}\bigg)^{2} \ ds \\ danamp;= \frac{t_{n}^{2H-1}}{\Delta _{n}^{2H}(H-1/2)^{2}}\bigg(\int _{0}^{1/2}+ \int _{1/2}^{1}\bigg). \end{aligned}$$

The term coming from the initiative intact, as always, tends to 0. The term coming from the second integral is tantamount to

$$\begin{aligned} danamp;\frac{t_{n}^{2H}(t_{n}'-t_{n})^{2H}}{\Delta _{n}^{2H} t_{n}^{2H} (H-1/2)^{2}} \int _{0}^{\infty }\big((y+1)^{H-1/2}-y^{H-1/2}\big)^{2}\, dy \\ danamp;\sim \frac{1}{(H-1/2)^{2}}\int _{0}^{\infty }\big((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dysprosium. \end{straight}$$

(This is the same result as for \(I_{1}^{n}\).) Now take

$$\start out{aligned} I^{n}_{2} danadenosine monophosphate;=\frac{1}{\Delta ^{H}_{n}\Delta _{n+1}^{H}}\int _{0}^{t_{n}} s^{1-2H}\bigg(\int _{t_{n}}^{(t_{n}+t_{n}')/2} v^{H-1/2}(v-s)^{H-3/2} \, dv\bigg) \\ danamp;\specter{=} \qquad \qquad \qquad \times \bigg(\int _{t_{n}}^{t_{n}'}v^{H-1/2}(v-s)^{H-3/2} \, dv\bigg)\, ds \\ danamp; \sim \frac{2^{H} t_{n}^{2H-1}}{\Delta _{n}^{2H}(H-1/2)^{2}} \int _{0}^{t_{n}} s^{1-2H}\bigg(\Immense(\frac{t_{n}+t_{n}'}{2}-s\Big)^{H-1/2}-(t_{n}-s)^{H-1/2} \bigg) \\ danamp;\phantasma{=:}\ \quad \qquad \qquad \qquad \quad \,\, \times \big((t_{n}'-s)^{H-1/2}-(t_{n}-s)^{H-1/2} \big)\, ds \\ danamp;= \frac{2^{H} t_{n}^{2H}}{\Delta _{n}^{2H}(H-1/2)^{2}}\int _{0}^{1} s^{1-2H} \bigg(\Big(\frac{1}{2}+\frac{t_{n}'}{2t_{n}}-s\Big)^{H-1/2}-(1-s)^{H-1/2} \bigg) \\ danamp;\phantom{=:} \,\quad \qquad \qquad \qquad \, \quadruplet \multiplication \bigg( \Fully grown(\frac{t_{n}'}{t_{n}}-s\Blown-up)^{H-1/2}-(1-s)^{H-1/2}\bigg)\, ds \\ danadenylic acid;= \frac{2^{H} t_{n}^{2H}}{\Delta _{n}^{2H}(H-1/2)^{2}}\bigg(\int _{0}^{1/2}+ \int _{1/2}^{1}\bigg). \end{aligned}$$

The term coming from the first entire tends to 0 A \(n\to \infty \) because it is equivalent to \(\Delta _{n}^{2}/\Delta _{n}^{2H}\). Using the changes of variables \(s=1-x\) and \(x=\frac{t_{n}'-t_{n}}{2t_{n}}y \), it follows that

$$\begin{aligned} danamp;\frac{2^{H} t_{n}^{2H}}{\Delta _{n}^{2H} (H-1/2)^{2}}\int _{1/2}^{1} s^{1-2H} \bigg(\Big(\frac{1}{2}+\frac{t_{n}'}{t_{n}}-s\Big)^{H-1/2}-(1-s)^{H-1/2} \bigg) \\ danamp;\phantasma{=}\qquad \qquad \qquad \qquad \times \bigg(\Big( \frac{t_{n}'}{t_{n}}-s\Lifesize)^{H-1/2}-(1-s)^{H-1/2}\bigg)\, ds \\ danamp;= \frac{2^{H} t_{n}^{2H}}{\Delta _{n}^{2H} (H-1/2)^{2}}\int _{0}^{1/2} (1-x)^{1-2H}\bigg(\Big(\frac{-t_{n}+t_{n}'}{2t_{n}}+x\Big)^{H-1/2}-x^{H-1/2} \bigg) \\ danamp; \phantom{=:}\qquad \qquad \qquad \quad \qquad \multiplication \bigg( \Blown-up(\frac{t_{n}'-t_{n}}{t_{n}}+x\Big)^{H-1/2}-x^{H-1/2}\bigg)\, dx \\ danamp;= \frac{2^{H} t_{n}^{2H}}{\Delta _{n}^{2H} (H-1/2)^{2}}\bigg( \frac{t_{n}'-t_{n}}{2t_{n}}\bigg)^{2H} \\ danamp;\phantom{=:}\times \int _{0}^{t_{n}/(t_{n}'-t_{n})} \bigg(1- \frac{t_{n}'-t_{n}}{2t_{n}}y\bigg)^{1-2H}\big((1+y)^{H-1/2}-y^{H-1/2} \big) \\ danamp;\phantom{=:}\qquad \quad \quad \quad \ \ \ \times \big((2+y)^{H-1/2}-y^{H-1/2} \big)\, atomic number 66 \\ danamp;\longrightarrow \frac{2^{-H}}{(H-1/2)^{2}}\int _{0}^{\infty } \big((1+y)^{H-1/2}-y^{H-1/2} \big)\big((2+y)^{H-1/2}-y^{H-1/2}\big)\, dy. \death{aligned}$$

Therefore the specify equals

$$\Menachem Begin{allied} danamp;\frac{2}{(H-1/2)^{2}}\int _{0}^{\infty }\big((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy \\ danamp;- 2\frac{2^{-H}}{(H-1/2)^{2}}\int _{0}^{\infty } \big((y+1)^{H-1/2}-y^{H-1/2} \big)\big((y+2)^{H-1/2}-y^{H-1/2}\big) \, dy. \end{straight}$$

The transfer of variable \(y=z/2\) yields

$$\begin{aligned} \int _{0}^{\infty }\swelled((y+1)^{H-1/2}-y^{H-1/2}\big)^{2}\, dy danamp; = \frac{1}{2}\int _{0}^{\infty }\bigg(\Big(\frac{z}{2}-1\Big)^{H-1/2}- \Big(\frac{z}{2}\Big)^{H-1/2}\bigg)\, dz \\ danamp;= \frac{1}{2^{2H}}\int _{0}^{\infty }\big((z+2)^{H-1/2}-z^{H-1/2} \big)^{2}\, dz, \end{aligned}$$

and therefore the limit becomes

$$\begin{aligned} danamp;\frac{1}{(H-1/2)^{2}}\bigg( \! \int _{0}^{\infty }\! \big((y+1)^{H-1/2}-y^{H-1/2} \big)^{2} dy \\ danamp; \qquad \qquad \quad + \frac{1}{2^{2H}}\! \int _{0}^{ \infty }\!\big((y+2)^{H-1/2}-y^{H-1/2}\big)^{2} dy \\ danA; \qquad \qquad \quad - 2^{1-H} \! \int _{0}^{\infty }\! \with child((y+1)^{H-1/2}-y^{H-1/2}\crowing)\big( (y+2)^{H-1/2}-y^{H-1/2}\big)\, Dy \bigg) \\ danampere;= \frac{1}{(H-1/2)^{2}} \! \int _{0}^{\infty }\! \bigg( (y+1)^{H-1/2}-y^{H-1/2} \\ danampere;\phantom{=:} \qquad \qquad \qquad \quadruplet -\frac{1}{2^{H}}(y+2)^{H-1/2}+ \frac{1}{2^{H}} y^{H-1/2}\bigg)^{2} dy \\ danamp; \neq 0. \end{aligned}$$

Let \(t=Tj_{0}/2^{m_{0}}\) for some \(j_{0}\geq 1\), \(m_{0}\geq 1\). Then \(t\in [t_{n},t_{n}']\) for \(n\geq m_{0}\), i.e., case (i) holds. Thus at least for \(t\) of the organize \(Tj/2^{m}\), we have none \(L^{2}\)-convergence of \(\zeta _{t}^{2^{n}}\). If \(t\) is not of this class, then the situation testament switch from (i) to (ii).

Consider now case (II). Then \(\Delta _{t}^{n}= \Delta _{n+1}^{-H}\zeta ^{2^{n+1}}_{t}- \Delta _{n}^{-H} \zeta _{t}^{2^{n}} \) and \(\Delta _{t}^{n}\) has thedannbsp;same throttl (in the mean-square sense) as \(\Delta _{n+1}^{-H}\cardinal _{t_{n}',(t_{n}+t_{n}')/2}- \Delta _{n}^{-H} \XI _{t_{n}',t_{n}}\). Consider

$$\begin{aligned} danamp;E[ ( \Delta _{n+1}^{-H}\xi _{t_{n}',(t_{n}+t_{n}')/2}-\Delta _{n}^{-H} \cardinal _{t_{n}',t_{n}} )^{2}] \\ danamp;= E\bigg[\bigg( \Delta _{n+1}^{-H}\int _{0}^{(t_{n}+t_{n}')/2} s^{1/2-H} \Big(\int _{(t_{n}+t_{n}')/2}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\, dv \Big)\, dW_{s} \\ danamp;\quad \qquad - \Delta _{n}^{-H}\int _{0}^{t_{n}} s^{1/2-H} \Important(\int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\Big)\, dW_{s} \bigg)^{2}\bigg] \\ danamp;= \Delta _{n+1}^{-2H} \int _{0}^{(t_{n}+t_{n}')/2} s^{1-2H} \bigg( \int _{(t_{n}+t_{n}')/2}^{t_{n}'} v^{H-1/2} (v-s)^{H-3/2}\, dv\bigg)^{2} \, ScD \\ danamp;\phantom{=:}- \frac{2}{\Delta _{n}^{H}\Delta _{n+1}^{H}} \int _{0}^{t_{n}} s^{1-2H} \bigg( \int _{(t_{n}+t_{n}')/2}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2} dv \bigg) \\ danamp;\phantom{=}\qquad \qquad \qquad \quadriceps femoris \times \bigg( \int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2} dv \bigg) ds \\ danamp;\specter{=:}+ \Delta _{n}^{-2H} \int _{0}^{t_{n}} s^{1-2H} \bigg(\int _{t_{n}}^{t_{n}'} v^{H-1/2}(v-s)^{H-3/2}\, dv\bigg)^{2}\, DS \\ danamp; =: I_{1}^{n}-2I_{2}^{n}+I_{3}^{n}. \end{straight}$$

As before, the limit of from each one term does not shift when we replace \(v^{H-1/2}\) by \(t^{H-1/2}\). Therefore, setting \(1-s=x\), \(x=\frac{t_{n}'-t_{n}}{2t_{n}}y\) and noting \((t_{n}'-t_{n})/2=T/2^{n+1}=\Delta _{n+1}\), IT follows that

$$\begin{aligned} I_{1}^{n} danAMP;\sim \frac{t^{2H-1}}{\Delta _{n+1}^{2H}(H-1/2)^{2}} \int _{0}^{t_{n}} s^{1-2H}\bigg((t_{n}'-s)^{H-1/2}- \Big(\frac{t_{n}'+t_{n}}{2}-s\Plumping)^{H-1/2} \bigg)^{2} \, ds \\ danamp;\sim \frac{t^{2H}}{\Delta _{n+1}^{2H}(H-1/2)^{2}} \\ danamp;\phantasma{=:}\times \int _{0}^{1} s^{1-2H}\bigg(\Big(\frac{t_{n}'-t_{n}}{t_{n}}+1-s \Big)^{H-1/2} -\Big(\frac{t_{n}'-t_{n}}{2t_{n}}+1-s\Big)^{H-1/2} \bigg)^{2}\, ds \\ danamp;= \frac{t^{2H}}{\Delta _{n+1}^{2H}(H-1/2)^{2}}\bigg(\int _{0}^{1/2}+ \int _{1/2}^{1}\bigg) \\ danamp;\sim \frac{t^{2H}}{\Delta _{n+1}^{2H}(H-1/2)^{2}}\bigg( \frac{t_{n}'-t_{n}}{2t_{n}}\bigg)^{2H} \\ danamp;\spectr{=:}\times \int _{0}^{t_{n}/(t_{n}'-t_{n})}\bigg(1- \frac{t_{n}'-t_{n}}{2t_{n}}y\bigg)^{1-2H} \big((y+2)^{H-1/2}-(y+1)^{H-1/2} \big)^{2}\, dy \\ danamp;\sim \frac{1}{(H-1/2)^{2}}\int _{0}^{\infty }\voluminous((2+y)^{H-1/2}-(1+y)^{H-1/2} \big)^{2}\, atomic number 66. \end{aligned}$$

Similarly,

$$ I_{3}^{n}\sim \frac{1}{(H-1/2)^{2}}\int _{0}^{\infty }\huge((y+1)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy. $$

Now,

$$\begin{aligned} I_{2}^{n} danamp;\sim \frac{t_{n}^{2H}}{(H-1/2)^{2} \Delta _{n}^{H}\Delta _{n+1}^{H}} \int _{1/2}^{1} s^{1-2H} \bigg(\Big(\frac{t_{n}'}{t_{n}}-s\Big)^{H-1/2} -\Big( \frac{t_{n}+t_{n}'}{2t_{n}}-s\Big)^{H-1/2}\bigg) \\ danamp; \phantom{=:}\qquad \ \ \qquad \qquad \qquad \qquad \times \bigg(\Swelled(\frac{t_{n}'}{t_{n}}-s\Big)^{H-1/2}-(1-s)^{H-1/2} \bigg) \, atomic number 110 \\ danA; \sim \frac{ (\frac{t_{n}'-t_{n}}{2} )^{2H}}{(H-1/2)^{2} \Delta _{n}^{H}\Delta _{n+1}^{H}} \int _{0}^{\infty }\vainglorious((y+2)^{H-1/2}-(y+1)^{H-1/2}\big) \\ danadenosine monophosphate;\shadow{=:} \qquad \musculus quadriceps femoris \qquad \qquad \qquad \qquad \times \big((y+2)^{H-1/2} -y^{H-1/2}\big)\, dy \\ danamp;\sim \frac{2^{-H}}{(H-1/2)^{2}}\int _{0}^{\infty } \king-sized((y+2)^{H-1/2}-(y+1)^{H-1/2} \gravid) \\ danAMP;\phantom{=:}\qquad \qquad \qquad \space \, \multiplication \rangy((y+2)^{H-1/2}-y^{H-1/2} \big)\, dy. \end{aligned}$$

Rewrite right away \(I_{3}^{n}\) (using the substitution \(y=v/2\)) as

$$ I_{3}^{n}\sim \frac{2^{-2H}}{(H-1/2)^{2}} \int _{0}^{\infty } \big((y+2)^{H-1/2}-y^{H-1/2} \big)^{2}\, dy. $$

In summary, IT follows that

$$\begin{aligned} danA;E\bigg[\bigg(\frac{1}{\Delta _{n+1}^{H}}\xi _{t_{n}',(t_{n}+t_{n}')/2} -\frac{1}{\Delta _{n}^{H}}\xi _{t_{n}',t_{n}}\bigg)^{2}\bigg] \\ danamp;\sim \frac{1}{(H-1/2)^{2}} \int _{0}^{\infty } \Larger-than-life((2+y)^{H-1/2} -(1+y)^{H-1/2} \\ danamp;\spectr{=:}\qquad \ \quad \qquad \qquad -2^{-H}\big((2+y)^{H-1/2} -y^{H-1/2}\self-aggrandising)\Full-size)^{2} dysprosium dangt;0 , \end{aligned}$$

which shows that \(L^{2}\)-convergence also does not hold in case (ii). □

Remark A.11

An inspection of the above imperviable shows that the limits in (A.19) and (A.23) remain the same if fBm is replaced by an fBm with drift, which corresponds to adding to \(\zeta _{t}^{n}\) a term proportional to \(\Delta _{n}\). Repeating the same calculations in that scene, it turns out that because \(\Delta _{n}^{-2H}\Delta _{n}^{2} = \Delta _{n}^{-2H+2}\) vanishes As \(n\) increases to infinity (since \(H\in (0,1)\)), the extra term is inconsequent.

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Guasoni, P., Mishura, Y. danamp; Rásonyi, M. High-frequency trading with fractional Brownian motion. Finance Stoch 25, 277–310 (2021). https://doi.org/10.1007/s00780-020-00439-y

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• Acceptable: 13 August 2022

• Accepted: 10 July 2022

• Published: 08 October 2022

• Issue Date: April 2022

• DOI : https://doi.org/10.1007/s00780-020-00439-y

Keywords

• Fractional Brownian motion
• Transaction costs
• HF
• Trading
• Mean–divergence optimisation

Mathematics Subject Categorization (2010)

• 91G10
• 91G80

JEL Categorization

• G11
• G12

fractional brownian motion trading strategies

Source: https://link.springer.com/article/10.1007/s00780-020-00439-y

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