Stochastic Calculus for Fractional Brownian Motion and Related Processes

Let α > 0 (and in most cases below α < 1 though this is not obligatory). Define the Riemann–Liouville left- and right-sided fractional integrals on (a, b) of order α by and respectively.

We say that the function \(f \in D\left( {I_{a + \left( {b - } \right)}^\alpha } \right)\) (the symbol \(D\left( \cdot \right)\) denotes the domain of the corresponding operator), if the respective integrals converge for almost all (a.a.) \(x \in \left( {a,b} \right)\) (with respect to (w.r.t.) Lebesgue measure).

The Riemann-Liouville fractional integrals on R are defined as and respectively.

The function \(f \in D\left( {I_ \pm ^\alpha } \right)\) if the corresponding integrals converge for a.a.\(x \in R\). According to (SKM93), we have inclusion \(L_p \left( R \right) \subset D\left( {I_ \pm ^\alpha } \right),1 \le p < \frac{1}{\alpha }.\). Moreover, the following Hardy–Littlewood theorem holds.

In this subsection we consider pathwise integrals \(\int\limits_0^T {f\left( t \right)dB_t^H } \) for processes f from the fractional Sobolev type spaces \(I_{a + }^\alpha \left( {L^p } \right)\) for some p > 1. This approach was developed by Zähle (Zah98), (Zah99), (Zah01).

Consider two nonrandom functions f and g defined on some interval \(\left[ {a,b} \right] \subset {\rm R}\) and suppose that the limits \(f\left( {u + } \right): = \lim _{\delta \downarrow 0} f\left( {u + \delta } \right)\) and \(g\left( {u - } \right): = \lim _{\delta \downarrow 0} g\left( {u - \delta } \right),a \le u \le b,\) exist. Put \(f_{a + } \left( x \right): = \left( {f\left( x \right) - f\left( {a + } \right)} \right)1_{\left( {a,b} \right)} \left( x \right),g_{b - } \left( x \right): = \left( {g\left( {b - } \right) - g\left( x \right)} \right)1_{\left( {a,b} \right)} \left( x \right).\) Suppose also that \(f_{a + } \in I_{a + }^\alpha \left( {L_p \left[ {a,b} \right]} \right),g_{b - } \in I_{b - }^{1 - \alpha } \left( {L_p \left[ {a,b} \right]} \right)\) for some \(p \ge 1,q \ge 1,1/p + 1/q \le 1,0 \le \alpha \le 1.\) Then evidently, \(D_{a + }^\alpha f_{a + } \in L_p \left[ {a,b} \right],D_{b - }^{1 - \alpha } g_{b - } \in L_q \left[ {a,b} \right].\)

Consider the function \(\sigma = \sigma \left( {t,x} \right):\left[ {0,T} \right] \times {\rm R} \to {\rm R}\) satisfying the assumptions: σ is differentiable in x, there exist and for any \(M > 0,0 < \gamma ,\kappa \le 1\) R > 0 there exists M _R > 0 such that

Let \(0 < \beta < 1/2,f \in W_0^\beta \left[ {0,T} \right],g \in W_1^{1 - \beta } \left[ {0,T} \right].\) We need some preliminary estimates, in addition to Lemmas 2.1.9 and 2.1.10.

Consider on the norm, equivalent to \(\left\| \cdot \right\|_{0,\beta } :\)

Consider the real-valued signal process X _t and the observation process Y _t defined by the following system of equations: where \(\left\{ {W^i ,1 \le i \le N} \right\}\) are independent Wiener processes,\(\left\{ {B^{H_j } ,1 \le j \le M} \right\}\) are independent fractional Brownian motions with Hurst indices \(H_j \in \left( {\frac{1}{2},1} \right),B^H \) is an fBm with Hurst index \(H \in \left( {\frac{1}{2},1} \right),\) all the processes are mutually independent, random initial conditions \(\left( {\eta ,\xi } \right)\) are independent of each other and independent of all the processes \(\left( {W^i ,B^{H_j } ,B^H } \right),\) the functions \(a,b,A:\left[ {0,T} \right] \times {\rm R} \to {\rm R,}c_j ,C:\left[ {0,T} \right] \to {\rm R}\) are measurable in their variables and satisfy the conditions that are sufficient for the existence of pathwise integrals w.r.t. corresponding fBms.

As mentioned in the paper (WTT99), long-range dependence in economics and finance has a long history and is an area of active research (e.g., see (Lo91), (CKW95)). The importance of long-range dependent processes as stochastic models lies in the fact that they provide an explanation and interpretation of an empirical law that is commonly referred to as the Hurst law or Hurst effect. In short, for a given set of observations \(\left\{ {X_{i,} i \ge 1} \right\}\) with partial sum \(Y\left( n \right) = \sum\limits_{i = 1}^n {X_i ,n \ge 1,} \) and sample variance \(S^2 \left( n \right) = n^{ - 1} \sum\limits_{i = 1}^n {\left( {X_i - n^{ - 1} Y\left( n \right)} \right)^2 ,n \ge 1,} \) the rescaled adjusted range statistic or R/S-statistic is defined by

Hurst in (Hur51) found that many naturally occurring empirical records appear to be well represented by the relation \(E\left( {\left( {R/S} \right)\left( n \right)} \right) \sim c_1 n^H \) as \(n \to \infty \) with typical values of the Hurst parameter \(H \in \left( {1/2,1} \right)\), and c ₁ a finite positive constant not depending on n. But in the case when the observations come from a short-range dependent model, then \(E\left( {R/S\left( n \right)} \right) \sim c_2 n^{1/2} \) as \(n \to \infty \), where c ₂ does not depend on n. The discrepancy between these two relations is called the Hurst effect or Hurst phenomenon. The analysis of the R/S-statistic, provided in (WTT99), (TTW95) and (TT97), leads to the recommendation to use a diverse portfolio of time-domain-based and frequency-domain-based graphics and statistical methods, including the graphical R/S-method, the modified R/S-statistic (Lo91) and Whittle’s approach. Also, another (possibly, surprising) recommendation is: in the case when statistical analysis cannot be expected to provide a definitive answer concerning the presence or absence of long-range dependence in asset price returns, a more revealing and also much more challenging approach to tackle this problem consists of providing a mathematically rigorous physical “explanation” for the presence or absence of the long-range dependence phenomenon in stock returns.

As we have seen in Subsection 5.2.2, the form of geometric fBm (5.2.6) depends on the kind of integral that is used in its calculations: if we use the Riemann–Stieltjes integral, then \(S_t^{\left( 1 \right)} = S_0^{\left( 1 \right)} \exp \left\{ {\mu t + \sigma B_t^H } \right\},\) and if the behavior of geometric process is guided by the Wick integral, then \(S_t^{\left( 2 \right)} = S_0^{\left( 2 \right)} \exp \left\{ {\mu t + \sigma B_t^H - \frac{1}{2}\sigma ^2 t^{2H} } \right\}\) So, the natural question arises: what trend actually has geometric fBm? This question was considered in the paper (KMV05), and here we present a solution of this problem. In what follows the notation \(X_n = o_P \left( 1 \right)\) means that \(X_n \rightarrow{P}0,X_n = O_P \left( 1 \right)\) means that \(\mathop {\lim }\limits_{C \to \infty } \mathop {\lim \sup }\limits_n P\left\{ {\left| {X_n } \right| \ge C} \right\} = 0.\) Assume that \(H \in \left( {1/2,1} \right)\). For a fixed \(\mu \in {\rm R}\) let \(P_{\mu ,\sigma } \) be the distribution of the process in the space \(C_{\left[ {0,T} \right]} \) of continuous functions. Similarly, \(P_{\mu ,\sigma } \) is the distribution of the process in the space \(C_{\left[ {0,T} \right]} \)