Monte Carlo estimates of the solution of a parabolic equation and its derivatives made by solving stochastic differential equations
In this paper a method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial variables is proposed. The method uses a probability representation of a solution of a parabolic equation in the form of a functional of a diffus...
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| Published in | Communications in nonlinear science & numerical simulation Vol. 9; no. 2; pp. 177 - 185 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.04.2004
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| Subjects | |
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| Abstract | In this paper a method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial variables is proposed. The method uses a probability representation of a solution of a parabolic equation in the form of a functional of a diffusion process. This process is the solution of a system of stochastic differential equations (SDE) corresponding to the parabolic operator. To obtain the derivatives of the solution of the parabolic boundary value problem the differentiation of the SDE system with respect to the parameters or the initial data is applied. |
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| AbstractList | In this paper a method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial variables is proposed. The method uses a probability representation of a solution of a parabolic equation in the form of a functional of a diffusion process. This process is the solution of a system of stochastic differential equations (SDE) corresponding to the parabolic operator. To obtain the derivatives of the solution of the parabolic boundary value problem the differentiation of the SDE system with respect to the parameters or the initial data is applied. A method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial variables is proposed. The method uses a probability representation of a solution of a parabolic equation in the form of a functional of a diffusion process. This process is the solution of a system of stochastic differential equations (SDE) corresponding to the parabolic operator. To obtain the derivatives of the solution of the parabolic boundary value problem the differentiation of the SDE system with respect to the parameters or the initial data is applied. |
| Author | Gusev, S.A. |
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| Copyright | 2003 Elsevier B.V. |
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| References | Kazakov (BIB3) 1977 Cramer (BIB2) 1975 Tikhonov, Mironov (BIB4) 1977 Averina TA, Artemiev SS. Some problems of construction and using of numerical methods for solving SDE’s. Preprint no. 728, Comp Cent Sib Branch, USSR Acad Sci, Novosibirsk, 1987 [in Russian] Tikhonov (10.1016/S1007-5704(03)00108-4_BIB4) 1977 Cramer (10.1016/S1007-5704(03)00108-4_BIB2) 1975 Kazakov (10.1016/S1007-5704(03)00108-4_BIB3) 1977 10.1016/S1007-5704(03)00108-4_BIB1 |
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| Snippet | In this paper a method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial... A method of estimation of both the solution to a parabolic boundary value problem and its derivatives with respect to parameters and spatial variables is... |
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| SubjectTerms | Euler method Parabolic equation Parametric derivative Stochastic differential equation |
| Title | Monte Carlo estimates of the solution of a parabolic equation and its derivatives made by solving stochastic differential equations |
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