This review examines methods of terrain representation, creation and realism described in literature. In particular a combination of terrain synthesis using Fractional Brownian Motion, and erosion procedures is examined, as well as the procedural formation of rivers. Other techniques as examined, such as Triangulated Irregular Networks and ridges generation and discarded in terms of this projec...

The applicability of the Bene s approach to the \fractional Brownian storage", i.e. a storage model where the net input process is a fractional Brownian motion (FBM) with drift, is studied. This requires the analysis of the last exit time probability density of a drifted FBM that, in turn, motivates the proof of a general \localization theorem" for FBM. The resulting Bene s formula contains a u...

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaus...

We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of Holder-regularity (any Hurst parameter...

We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version o...

In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage-free under proportional transaction costs in the same spirit of Guasoni et al [14, 15]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. ...

We study the two-dimensional fractional Brownian motion with Hurst parameter H > 1 2. In particular, we show, using stochastic calculus , that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion.

Though the various models proposed in the literature for capturing the long-range dependent nature of network traac are all either exactly or asymptotically second order self-similar, their eeect on network performance can be very diierent. We are thus motivated to characterize the limiting distributions of these models so that they lead to parsimonious modeling and a better understanding of ne...

We prove that solutions of stochastic differential equations driven by fractional Brownian motion for H > 1/2 define flows of homeomorphisms on R. AMS Subject Classification: 60H05, 60H07