|Principal Lecturer: Professor DAVID NUALART, University of Kansas|
- Derivative operator on a Gaussian space.
We define the derivative operator on smooth and cylindrical random variables, which are measurable with respect to an underlying Gaussian process. We establish an integration-by-parts formula and apply it to extend the derivative operator and construct the associated Sobolev spaces. We characterize the derivative operator on the Wiener chaos, and we establish some of its basic properties.
- Divergence operator.
The divergence operator is introduced as the adjoint of the derivative operator, and we derive its basic properties. The divergence operator on the Wiener chaos expansion behaves as an anticipating stochastic integral, the Skorohod integral. We show that the classical Ito integral is a particular case of the Skorohod integral. We establish the Clark-Ocone representation formula and we derive the local properties of the derivative and divergence operators.
- The Ornstein-Uhlenbeck semigroup.
We introduce the Ornstein-Uhlenbeck semigroup and we study the properties of its infinitesimal generator. In particular, we analyze the relation between this operator and the derivative operator. We derive the hypercontractivity of the Ornstein-Uhlenbeck semigroup.
- Sobolev spaces and equivalence of norms.
The equivalence of norms established by Meyer allows us to prove the continuity of the divergence operator. We introduce the space of generalized random variables as the adjoint of the space of smooth random variables.
- Regularity of probability laws.
We use the techniques of the Malliavin calculus to find an explicit formula for the density of a nondegenerate random variable. Some estimates for the density are derived. The general criteria for absolute continuity and regularity of the density, in terms of the nondegeneracy of the Malliavin matrix, will be established. We discuss the application of the Malliavin calculus to obtain some properties of the support to the law of a smooth random variable.
- Hypoellipticity and Hörmander's theorem.
The criterion for the smoothness of the density leads to a probabilistic proof of Hörmander's hypoellipticity theorem. The main ingredients of this proof are some estimates for semimartingales obtained by Norris.
- Anticipating stochastic calculus.
The divergence operator can be used as an extension of the Skorohod integral to anticipating integrands. We derive a change of variable formula (Ito's formula) for this integral, and we obtain a formula that relates the Ito and the Stratonovich integrals.
- Stochastic calculus with respect to the fractional
We apply the techniques of the Malliavin calculus to develop a stochastic calculus with respect to the fractional Brownian motion. We analyze the relation between the stochastic integral defined in terms of the divergence operator and the pathwise integrals.
- Central limit theorems for multiple stochastic integrals.
We establish necessary and sufficient conditions for the central limit theorem to hold for a sequence of multiple stochastic integrals of fixed order using the derivative operator. Applications to the asymptotic behavior of some functionals of the fractional Brownian motion will be discussed.
- Malliavin calculus in finance.
The integration by parts formula of Malliavin calculus provides formulas for the price sensitivities (Greeks) in the Black--Scholes model. We derive some of these explicit formulas, which are useful for numerical computations. On the other hand, we discuss the application of the Clark--Ocone formula to find hedging portfolios in the Black--Scholes model.