On some problems of stochastic analysis of Wiener integrals that constructed by fractional brownian motions
by
Yuriy Krvavych
Ph.D. student of Kyiv National University of Taras Shevchenko, Ukraine.

We solved three problems.

Problem 1: the differentiability of fractional integral \Phi = \int₀^t\phi(t, s) ds, with kernel
\phi(t, s)=K_H0(t, s)\int₀^s\alpha(u) dB_u^H, where B_s^H - fractional Brownian motion (FBM) with Hurst index H in (\frac12, 1) and K_H0(t, s)=(t-s)^\frac12-H₀\beta(\fracst),   0 < s < t,   H₀ in (\frac12, 1). For this problem the mean-square differentiability and path differentiability conditions of fractional integral \Phi are obtained. These results are used in the proof of the Girsanov theorem for FBM and diffusion processes that constructed by FBM.

Problem 2: maximal inequalities for moments of Wiener integral I_t=\int₀^tf(s) dB_s^H,   t > 0, where f is deterministic, measurable, positive function that satisfies condition
\int₀^\infty\int₀^\inftyf(s)f(t)|s-t|^2H-2 ds dt < \infty. In this point the upper and lower maximal estimations for moments of Wiener integral I_t are established. They are applied to the solutions of stochastic differential equations involving FBM.

Problem 3: the presence and absence of arbitrage conditions on the three types of (B, S) - market.

For the first type which defined as a "fractional model" of (B, S) - market the absence of equivalent martingale measure is proved. Further, the self-financing portfolio that represent the amount invested in the stock and allows arbitrage opportunity on the (B, S) - market of the considered type.

In the second case for modified "fractional model" of (B, S) - market we proved that our (B, S) - market is an arbitrage-free market.

For the third type of (B, S) - market with stock price process S_t=exp(\int₀^th(t-s) c(s) dW_s) we proved that (B, S) - market is an arbitrage-free market.

Date received: March 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-83.