Grigori N. Milstein

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.

Symplectic Integration of Hamiltonian Systems with Additive Noise

Hamiltonian systems with additive noise possess the property of preserving symplectic structure. Numerical methods with the same property are constructed for such systems. Special attention is paid to systems with separable Hamiltonians and to second-order differential equations with additive noise. Some numerical tests are presented.

Numerical Integration of Stochastic Differential Equations with Nonglobally Lipschitz Coefficients

We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by

Numerical Algorithms for Forward-Backward Stochastic Differential Equations

\varepsilon +O(h^{p}),

Numerical Methods in the Weak Sense for Stochastic Differential Equations with Small Noise

where

Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises

\varepsilon

Solving parabolic stochastic partial differential equations via averaging over characteristics

can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments.

Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations

Efficient numerical algorithms are proposed for a class of forward-backward stochastic differential equations (FBSDEs) connected with semilinear parabolic partial differential equations. As in [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940-968], the algorithms are based on the known four-step scheme for solving FBSDEs. The corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach. The derivatives of the solution u of the semilinear equation are found by finite differences. The forward equation is simulated by mean-square methods of order 1/2 and 1. Corresponding convergence theorems are proved. Along with the algorithms for FBSDEs on a fixed finite time interval, we also construct algorithms for FBSDEs with random terminal time. The results obtained are supported by numerical experiments.

Numerical Analysis of Monte Carlo Evaluation of Greeks by Finite Differences

We propose a new approach to constructing weak numerical methods for finding solutions to stochastic systems with small noise. For these methods we prove an error estimate in terms of products

Regression methods in pricing American and Bermudan options using consumption processes

h^i\varepsilon ^j