Generalized Solutions of Linear Parabolic Stochastic Partial Differential Equations

Existence and uniqueness theorems for parabolic stochastic partial differential equations with space—time white noise are proved. The method is a combination of the characterization theorem for Hida distributions with the Feynman—Kac and Girsanov formulae. Accepted 3 December 1996     

Homogenization problem for stochastic partial differential equations of Zakai type

We discuss homogenization for stochastic partial differential equations (SPDEs) of Zakai type with periodic coefficients appearing typically in nonlinear filtering problems. We prove such homogenization by two different approaches. One is rather analytic and the other is comparatively probabilistic.     

Shape Optimization in Viscous Compressible Fluids

   Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

Shape Optimization in Viscous Compressible Fluids

Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

Schrödinger Equations with Fractional Laplacians

It is shown that the unique solution of } can be represented as { } where X=(X _t , t≥ 0) is a stable process whose generator is (-Δ) ^α/2 with X ₀ =0 . Accepted 24 July 2000. Online publication 13 November 2000.     

Linear Forward—Backward Stochastic Differential Equations

The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Accepted 29 April 1997     

Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Large Deviation Principle for Optimal Filtering

   Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Asset Pricing with Stochastic Volatility

In this paper we study the asset pricing problem when the volatility is random. First, we derive a PDE for the risk-minimizing price of any contingent claim. Secondly, we assume that the volatility process \si _t is observed through an observation process Y _t subject to random error. A price formula and a PDE are then derived regarding the stock price S _t and the observation process Y _t as parameters. Finally, we assume that S _t is observed. In this case we have a complete market and any contingent claim is then priced by an arbitrage argument instead of by risk-minimizing. Accepted 15 August 2000. Online publication 8 December 2000.     

Homogenization of divergence-form operators with lower-order terms in random media

The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non-degeneracy and boundedness. Received: 27 August 1999 / Revised version: 27 October 2000 / Published online: 26 April 2001     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Solvability of the Navier—Stokes System with L 2 Boundary Data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L ² boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part. Accepted 15 March 1999     

Dynamics for Linear Feedback Controlled Two-Dimensional Bénard Equations with Distributed Controls

The long-time behavior of solutions for some feedback distributed control problems associated with the Bénard equations is studied. Some linear feedback solutions for the Bénard equations are constructed. Then we prove that these feedback solutions possess the decay (in time) properties. Accepted 5 March 2001. Online publication 20 June 2001.     

Heat Equations with Fractional White Noise Potentials

This paper is concerned with the following stochastic heat equations: where w ^H is a time independent fractional white noise with Hurst parameter H=(h ₁ , h ₂ ,..., h _d ) , or a time dependent fractional white noise with Hurst parameter H=(h ₀ , h ₁ ,..., h _d ) . Denote |H|=h ₁ +h ₂ +...+h _d . When the noise is time independent, it is shown that if ? <h _i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ? <h _i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h ₀ -1 ) , the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S _ρ , ρ∈ RR , is introduced so that every chaos of an element in S _ρ is in L ₂ . The Lyapunov exponents in S _ρ of the solution are also estimated. Accepted 10 October 2000. Online publication 19 February 2001.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par