Jorge A. León

On Lévy processes, Malliavin calculus and market models with jumps

Abstract. Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump–diffusion model are given.

On the Short-Time Behavior of the Implied Volatility for Jump-Diffusion Models With Stochastic Volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

An Anticipating Calculus Approach to the Utility Maximization of an Insider

In this paper we consider a financial market with an insider that has, at time t= 0, some additional information of the whole developing of the market. We use the forward integral, which is an anticipating integral, and the techniques of the Malliavin calculus so that we can take advantage of the privileged information to maximize the expected logarithmic utility from terminal wealth.

A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).

Synchronization of nonlinear fractional order systems

This paper deals with the master–slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown state variables. For solving this problem we introduce a Fractional Algebraic Observability (FAO) property which is used as a main tool in the design of the master system. As numerical examples we consider a fractional order Rossler hyperchaotic system and a fractional order Lorenz chaotic system and by means of some simulations we show the effectiveness of the suggested approach.

A CHAOS APPROACH TO THE ANTICIPATING CALCULUS FOR THE POISSON PROCESS

In this paper we use the Poisson-Ito chaos decomposition approach to define a variational derivative operator and its adjoint, which is an anticipating integral (i.e., it agrees with the martingale Poisson-Ito integral with respect to the compensated Poisson process for predictable integrands). Also an integration by parts formula and characterizations of these operators are given.Finally, we prove that, in the case where the basic probability space is the canonical Poisson space, our derivative operator is equal to the Carlen and Pardoux gradient operator that is defined by means of variation of the jump times

Stochastic Differential Equations with Random Coefficients

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.

Itô's formula for linear fractional PDEs

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

Abstract In this article, we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here, the coefficients are deterministic, the initial condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results.

A pathwise approach to backward and forward stochastic differential equations on the poisson space

We study the existence and uniqueness of pathwise solutions to backward and forward stochastic differential equations on the Poisson space. We obtain the structure of these pathwise solutions to give the relationship between them. Also, in the bilinear case, we calculate the explicit form of their chaos decompositions. *Partially supported by DGICYT grant PB93-0052, PB96-1182 and CIRIT grant 97-SGR00144.