We survey some new progress on the pricing models driven by fractional Brownian motion or mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with transaction costs. We end the paper by giving some approximation results and indicating...

The wavelet transform of a function f(t) is deened by the formula: Wf(t; a) = W a f(t) = 1 p a Z f(s)g(t ? s a) ds where g(t) is a xed function, t 2 R and a 2 R +. This transform yields a joint timescale representation the original input function that has been of great recent interest. (See for example D1] D2] and HW]). In a recent correspondence, Flandrin F] proposed the use of the wavelet tra...

In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of o...

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is s...

Fractals are shapes inwhichparts of the shape resemble thewhole shape in some way. Brownian motion, a type of random walk, is a fractal. Fractional Brownian motion, a biased random walk in which the walker favors certain directions at each step, is also a fractal. Used to model a wide range of phenomena, from river levels and landscape topography to computer network traffic and stock market ind...

In [10] a “direct” stochastic transfer principle was introduced, which represented multiple integrals with respect to fractional Brownian motion in terms of multiple integrals with respect to standard Brownian motion. The method employed in [10] involved an operator Γ (n) H , mapping a class of functions LH to L 2. However, the operator does not map LH onto L 2. Hence Γ (n) H is not invertible....

Dynamic systems in many branches of science and industry are often perturbed by various types of environmental noise. Analysis of this class of models are very popular among researchers. In this paper, we present a method for approximating solution of fractional-order stochastic delay differential equations driven by Brownian motion. The fractional derivatives are considered in the Caputo sense...

Self-similar stochastic processes are used for stochastic modeling whenever it is expected that long range dependence may be present in the phenomenon under consideration. After discusing some basic concepts of self-similar processes and fractional Brownian motion, we review some recent work on parametric and nonparametric inference for estimation of parameters for linear systems of stochastic ...