Regularity properties of transition probabilities in infinite dimensions

In a Hilbert space H we consider a process X solution of a semilinear stochastic differential equation, driven by a Wiener process. We prove that, under appropriate conditions, the transition probabilities of X are absolutely continuous with respect to a properly chosen gaussian measure μ in H, and the corresponding densities belong to some Wiener-Sobolev spaces over (H,μ). In the linear caseX is a nonsymmetric Ornstein-Uhlenbeck process, with possibly degenerate diffusion coefficient. The general case is treated by the Girsanov. Theorem and the Malliavin calculus. Examples and applications to stochastic partial differential equations are given     

Some Solvable Stochastic Control Problems With Delay

We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages

A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems     

New approach to optimal control of stochastic Volterra integral equations

ABSTRACT

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus.

• We give conditions under which there exist unique solutions of such equations.

• Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus.

• As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

     

Equations différentielles stochastiques dans avec conditions aux bords

In this paper, we study affine boundary value problems for one dimensional stochastic differential equations. Under suitable conditions on the coefficients of the SDE, we prove existence and uniqueness results. Moreover, when the diffusion coefficient is linear, we give a necessary and sufficient condition insuring the solution is a Markov field.     

Adaptive control of linear stochastic evolution systems

An adaptive control problem for some linear stochastic evolution systems in Hilbert spaces is formulated and solved in this paper. The solution includes showing the strong consistency of a family of least squares estimates of the unknown parameters and the convergence of the average quadratic costs with a control based on these estimates to the optimal average cost. The unknown parameters in the model appear affinely in the infinitesimal generator of the C ₀ semigroup that defines the evolution system. A recursive equation is given for a family of least squares estimates and the bounded linear operator solution of the stationary Riccati equation is shown to be a continuous function of the unknown parameters in the uniform operator topology     

Theéeorème limite pour une equation differentielle stochastique anticipative

Ocone and Pardoux have introduced a stochastic differential equation in which the initial condition and the drift depend on the driving Brownian motion in an anticipative way. In this paper we prove a limit theorem for such equations when the Brownian motion is approximated by a sequence of piecewise linear processes     

On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

Evolution equations with white-noise boundary conditions

The paper is devoted to nonlinear evolution equations with nonhomogenous boundary conditions of white noise type. Necessary and sufficient conditions for the existence of solutions in the linear case are given. It is also shown that if the nonlinearity satisfies appropriate dissipativity conditions the nonlinear equation has a solution as well. The results are applied to equations with polynomial nonlinearities