9706760 Swiech The proposed research revolves around the notion of viscosity solution, which is a notion of weak solution for equations of Hamilton-Jacobi-Bellman-Isaacs (HJBI) type. It is both basic and applied and can be divided into two parts: partial differential equations in infinite dimensional spaces and applications to stochastic optimal control, and elliptic and parabolic equations. As regards the first part, the investigator plans to continue research in the development of the theory of viscosity solutions (i.e. generalized solutions) of second order HJBI equations in Hilbert space. The basic issues are the existence and uniqueness of solutions, their regularity, convergence properties, etc. HJBI equations are of great importance from the point of view of applications, since they are so-called dynamic programming equations, corresponding to problems of stochastic optimal control of dynamical systems driven by stochastic partial differential equations. The second part of the proposed research concentrates on some fundamental problems in the theory of fully nonlinear, second order, uniformly elliptic and parabolic partial differential equations. The major issue is the regularity and uniqueness of solutions of nonlinear, uniformly parabolic and elliptic equations which are not continuous in the space variable and, on a larger scale, the development of a theory of generalized solutions for fully nonlinear equations with measurable ingredients. Despite some recent results, uniqueness is in general still a major open problem, even for linear equations, where it is connected to the uniqueness of so-called martingale solutions of stochastic differential equations with discontinuous diffusion coefficient. Finally the investigator proposes to work on a notion of weak solution for degenerate elliptic and parabolic partial differential equations with measurable terms. Equations of this type appear, for instance, in problems involving motion driven by mean curva ture. The classical theory of viscosity solutions does not apply here. Its extension to such cases is needed. The notion of viscosity solution is widely used by applied mathematicians, and the proposed research has a potential of having far-reaching consequences in applied sciences, especially the part of it dealing with HJBI equations and control of dynamical systems driven by stochastic partial differential equations. In real life these are systems governed by partial differential equations that exhibit noisy behavior. Applications range from engineering to mathematical finance. One has to mention stochastic optimal control of nonlinear partially observed systems (so-called nonlinear filtering), distributed and boundary control of systems modelled by stochastic partial differential equations, minimax type control of infinite dimensional systems widely used in engineering (which is connected to infinite dimensional differential games), risk-sensitive control of small noise systems (important in control of adaptive tracking devices), and recent market models of interest rate dynamics (option pricing). The investigator is convinced that the proposed research will help understand the above problems and will provide a mathematical framework that can be used by people working in these areas.
