Fractional Hida-Malliavan derivatives and series representations of fractional conditional expectations

Abstract

Abstract. We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L 2 (P H ) space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.

Dyson type formula for pure jump Lévy processes with some applications to finance

“…It consists of a series reminiscent of the Dyson series in quantum mechanics. A similar representation was obtained in [6] for functionals of the fractional Brownian motion for H > 1/2. In both cases, the representation involves the Malliavin derivative.…”

A representation theorem for smooth Brownian martingales

“…We also sketch an alternate method, which we call the density method of proof of the exponential formula, which uses the denseness of stochastic exponentials in L 2 (Ω). The complete proof 4 goes along the lines of the proof of the exponential formula for fractional Brownian motion (fBm) with Hurst parameter H > 1/2, which we present in a separate paper [15]. We emphasize that it is most likely nontrivial to obtain the exponential formula in the Brownian case by a simple passage to the limit of the exponential formula for fBm when H tends to 1/2 from above.…”

“…By using a similar computation as in (3.10), Now let's introduce an application of (3.13) to some pricing problem. Recall that (see [23]) the bond price is affine with respect to r(t) and satisfies E[F |F t ] = exp(−C(t, T )r(t) − A(t, T )), (3.14) where C(t, T ) solves the time-dependent Riccati equation below: 15) and A satisfies ∂A(t,T ) ∂t = −dσ(t) 2 C(t, T ). By (3.13) and the Taylor expansion of the righthand side of (3.14), we have:…”