Integral representations of some functionals of fractional Brownian motion

Abstract

We prove change of variables formulas [Itô formulas] for functions of both arithmetic and geometric averages of geometric fractional Brownian motion. They are valid for all convex functions, not only for smooth ones. These change of variables formulas provide us integral representations of functions of average in the sense of generalized Lebesgue-Stieltjes integral.

Aspects of Stochastic Integration with Respect to Processes of Unbounded p-variation

“…The above mentioned false argument was also applied in [22] to generalise the results of [2] to more general class of Gaussian processes, although by examining the proof in [22] it is clear that the proof is correct provided that µ SC ([−ǫ, ǫ]) = 0 for small enough ǫ. We also note that similar techniques was applied in [23], where the author studied average of geometric fractional Brownian motion and proved certain type of Itô formula in that case. To obtain the result the author in [23] proved that for a given functional X t and the approximating sequence…”

“…We also note that similar techniques was applied in [23], where the author studied average of geometric fractional Brownian motion and proved certain type of Itô formula in that case. To obtain the result the author in [23] proved that for a given functional X t and the approximating sequence…”

The existence of Riemann-Stieltjes integrals with applications to fractional Brownian motion

“…Under the conditions of Theorem 3.4, it allows to show that (2) holds for f ps, tq " Cpt ´sq H{q`H t ´H{q , s ď t, Xptq " B H ptq and Y ptq " F pB H ptqq, t ě 0, where C ą 0 do not depend on ps, tq. In fact, the latter is true for any X " Xptq such that }Xptq ´Xpsq} p ď C p |t ´s| H for all s, t P r0, T s and p ě 1 (e.g., Xptq " e B H ptq , see Lemma 3.6 in [13]).…”

Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

“…The lemma helps shows that on the average, singularities do not matter too much, if X has a bounded probability density function. The following lemma has appeared in [28,Lemma 3.2]. For the reader's convenience, we include a short proof of the result below.…”

“…Unfortunately, some proofs of the key results in [2] contain flaws that appear difficult to fix without incorporating additional assumptions. Tikanmäki [28] extended the analysis to nonlinear integral functionals of fractional Brownian motions. Another closely related paper is [4], where the rate of convergence of forward Riemann-Stieltjes sums over uniform partitions were studied for fractional Brownian motions.…”