Sensitivity results in stochastic analysis

This thesis consists of two quite distinct topics. In the first and bigger part we show that the Mandelbrot-van Ness representation of fractional Brownian motion is almost surely smooth in the Hurst parameter H. This dependence result is transferred to the solution of a stochastic differential equation driven by fractional Brownian motion if the stochastic differential equation is one-dimensional or H>1/2. In the multidimensional case of H in (1/3, 1/2] we use rough path theory to make sense of the differential equations. However, despite it being possible to lift fractional Brownian motion as well as its derivative in H to a rough path via the limit of dyadic approximations, they cannot be lifted jointly in the same way. Nevertheless, we obtain that the solution to a rough stochastic differential equation driven by fractional Brownian motion is locally Lipschitz continuous in H. In the last part of the thesis we define a directional Malliavin derivative connected to a continuous linear operator. We show that this directional Malliavin derivative being zero is equivalent to some measurability or independence condition on the random variable. Using this result, we obtain that two random variables, whose classical Malliavin derivatives live in orthogonal subspaces, are independent. We also extend the chain rule to directional Malliavin derivatives and a broader class of functions with weaker regularity assumptions

The Multiparameter Fractional Brownian Motion

We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the L\'evy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Brownian motion along increasing paths is analysed.Comment: 9 page

Fractional constant elasticity of variance model

This paper develops a European option pricing formula for fractional market models. Although there exist option pricing results for a fractional Black-Scholes model, they are established without accounting for stochastic volatility. In this paper, a fractional version of the Constant Elasticity of Variance (CEV) model is developed. European option pricing formula similar to that of the classical CEV model is obtained and a volatility skew pattern is revealed.Comment: Published at http://dx.doi.org/10.1214/074921706000001012 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The fractional volatility model : no-arbitrage, leverage and completeness

When the volatility process is driven by fractional noise one obtains a model which is consistent with the empirical market data. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behaviors. Here, the no-arbitrage and completeness properties of the models are rigorously studied

On the linear fractional self-attracting diffusion

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\frac12<H<\frac3{4}$.Comment: 14 Pages. To appear in Journal of Theoretical Probabilit