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Efficient numerical schemes for the approximation of expectations of functionals of the solution of A.S.D.E., and applications. (English) Zbl 0542.93077
Filtering and control of random processes, Proc. Colloq., Paris 1983, Lect Notes Control Inf. Sci. 61, 294-313 (1984).
[For the entire collection see Zbl 0527.00042.]
The aim of the paper is to develop a method of numerical approximation of such quantities as: \[ e_ t=E\{g(X_ t)\exp \{\int^{t}_{0}a(X_ s)ds+\int^{t}_{0}b(X_ s)dW_ s\}\} \] where a,b,g are smooth functions and \(X_ t\) is a multidimensional diffusion process. One proposes to discretize the S.D.E. whose \(X_ t\) is the solution, in order to get an approximate solution, \(\bar X_ t\). Then, by simulating a large number of paths of \(\bar X_ t\) and applying a Monte-Carlo method, one gets an approximate value of \(e_ t\), say \(\bar e_ t\). The problem is: what is the right discretization scheme? The error \(e_ t- \bar e_ t\) induced by classical schemes is of order h, if h is the step of the discretization. One gives sufficient conditions to be fulfilled by a discretization scheme to induce an error of order \(h^ 2\), and proposes useful schemes satisfying these properties (the results and the proofs are a simplified version of another paper of the author [Discrétisation d’une E.D.S. et calcul approché d’éspérane de fonctionnelles de la solution (to appear)]). Then one shows the possible application to the numerical computation of expectations arising in nonlinear filtering (Kallianpur-Striebel formula).

MSC:
93E25 Computational methods in stochastic control (MSC2010)
65C05 Monte Carlo methods
93E11 Filtering in stochastic control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93C10 Nonlinear systems in control theory
65L05 Numerical methods for initial value problems