The recently developed theory of wavelets is used to estimate small ball probability for a class of Gaussian processes including d-dimensional fractional Brownian motion.

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian Bridge are explored, and implications fo...

In this paper we establish a large deviation principle for solution of perturbed reflected stochastic differential equations driven by fractional Brownian motion B^H with Hurst index H ∈ (0;1). The key is to prove uniform Freidlin-Wentzell estimates on the set continuous square integrable functions in dual Schwartz space . We have built whole interval (0;1) new approch different from that Y. In...

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d ≥ 3, started off either from a Poisson field or in equilibrium, when suitably normalised, converges to a Brownian motion in d ≥ 4. In d = 3, the limit process is fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussia...

We prove the existence of equilibrium in a continuous-time nance model; our results include the case of dynamically incomplete markets as well as dynamically complete markets. In addition, we derive explicitly the stochastic process describing securities prices. The price process depends on the risk-aversion characteristics of the utility function, as well as on the presence of additional sourc...

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecrea...

In this paper an arbitrage strategy is constructed for the modified Black-Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brown...

We derive a large deviation principle for the occupation time functional, acting on functions with zero Lebesgue integral, for both superBrownian motion and critical branching Brownian motion in three dimensions. Our technique, based on a moment formula of Dynkin, allows us to compute the exact rate functions, which differ for the two processes. Obtaining the exact rate function for the super-B...