The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a ...The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested;2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.展开更多
In this paper the author constructs a solution of parabolic stochastic partial differential equation with random initial conditions by Kolmogorov's criterion.
In this paper we develop a new technique to prove existence of solutions of Fokker–Planck equations on Hilbert spaces for Kolmogorov operators with nontrace-class second order coefficients or equivalently with an ass...In this paper we develop a new technique to prove existence of solutions of Fokker–Planck equations on Hilbert spaces for Kolmogorov operators with nontrace-class second order coefficients or equivalently with an associated stochastic partial differential equation(SPDE)with non-trace-class noise.Applications include stochastic 2D and 3D-Navier–Stokes equations with non-trace-class additive noise.展开更多
In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighte...In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.展开更多
This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectatio...This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.展开更多
文摘The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested;2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.
基金Project supported by the National Natural Science Foundation of China (No.10301011)the 973 Project of the Ministry of Science and Technology of China.
文摘In this paper the author constructs a solution of parabolic stochastic partial differential equation with random initial conditions by Kolmogorov's criterion.
基金Support by the De Giorgi Centre and the DFG through SFB 701 is gratefully acknowledged.The last named author would also like to thank the Scuola Normale Superiore and the University of Pisa for the support and hospitality during several very pleasant visits during which large parts of this work were done.
文摘In this paper we develop a new technique to prove existence of solutions of Fokker–Planck equations on Hilbert spaces for Kolmogorov operators with nontrace-class second order coefficients or equivalently with an associated stochastic partial differential equation(SPDE)with non-trace-class noise.Applications include stochastic 2D and 3D-Navier–Stokes equations with non-trace-class additive noise.
文摘In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.
基金Project supported by the National Natural Science Foundation of China(No.10131040).
文摘This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.
