Theory Of Stochastic Flows Research Articles

Nash Equilibrium of Stochastic Partial Differential Game with Partial Information via Malliavin Calculus

In this article, we consider the Nash equilibrium of stochastic differential game where the state process is governed by a controlled stochastic partial differential equation and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functionals are assumed to be random. We find an explicit strong solution of the linear stochastic partial differential equation with a generalized probabilistic representation for this solution with the benefit of Kunita’s stochastic flow theory. We use Malliavin calculus to derive a stochastic maximum principle for the optimal control and obtain the Nash equilibrium of this type of stochastic differential game problem.

Stochastic Interpretation of Quasilinear Parabolic Systems with Cross Diffusion

We construct a stochastic interpretation of the generalized solution of the Cauchy problem for a strongly coupled system of quasilinear parabolic equations. To this end we derive a system of stochastic equations and prove that its solution allows us to construct a probabilistic representation of the original Cauchy problem solution and investigate its properties. Our construction of a probabilistic representation of the generalized solution of the Cauchy problem is crucially based on results of stochastic flow theory.

Application of stochastic flows to the sticky Brownian motion equation

We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.

GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS AND DIFFUSION PROCESSES

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, we use results from Kunita's theory of stochastic flows.

Dissipative canonical flows in classical and quantum mechanics

A theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the overall canonical/symplectic structure on the algebra is preserved. We study both classical and quantum systems and the importance of physical interpretation in the Stratonovich interpretation is stressed. We find the natural formulation of quantum dissipative systems to be given in terms of quantum stochastic calculus. This treatment allows for a physically meaningful treatment of both constant and nonlinear dissipation. As an application, we quantize a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator.

Strong feller property for stochastic semilinear equations

A method from stochastic flow theory is used to obtain smoothing properties of the transition semigroup Pt of a class of stochastic differential equations on Hilbert space. The equations considered may have unbounded coefficients and include such stochastic partial differential equations as for Xt in L2 (0,π) In certain cases a formula for the Fréchet derivative of Ptf is given, exhibiting this smoothing property

The Wiener space derivative for functionals of diffusions on manifolds

The Clark-Haussmann formula, giving the integrand in the stochastic integral representation of a functional L(H( omega )), where H=(xt)t epsilon (0,T) is a diffusion process in Rn and L is a mapping from C((0, T),Rn) to R, is equivalent to a computation of the Wiener space derivative of the functional omega to L(H( omega )). It involves the Frechet derivative of L. The author is concerned with the case of diffusions on a manifold M and with a functional L:C((0, T),M) to R; now the Frechet derivative is no longer available. They show that a similar formula can be obtained involving a family of scalar measures nu x and 1-forms qx(t) associated with L. The proof uses stochastic flow theory and the manifold structure of the space C((0, T),M).

Diffusions conditionnelles. II. Générateur conditionnel. Application au filtrage

This paper is a continuation of “Diffusions conditionelles, I.” If ( x t , z t ) is a two-component diffusion process, it is shown that under appropriate conditions, the process x t ( t ⩽ T), given ( z s , s ⩽ T) is a nonhomogeneous strong Markov process, whose generator is explicitly found by using the theory of stochastic flows. The filtering equation is reduced to an ordinary partial differential equation.