Geodesic
Solving SDE's numerically to estimate parametric derivatives of the solution to a parabolic boundary value problem with a Neumann boundary condition
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 3, pp. 237-246 
S. A. Gusev. Solving SDE's numerically to estimate parametric derivatives of the solution to a parabolic boundary value problem with a Neumann boundary condition. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 3, pp. 237-246. http://geodesic.mathdoc.fr/item/SJVM_2007_10_3_a1/
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     author = {S. A. Gusev},
     title = {Solving {SDE's} numerically to estimate parametric derivatives of the solution to a~parabolic boundary value problem with {a~Neumann} boundary condition},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, a parabolic boundary value problem with a Neumann boundary condition is considered. The diffusion process with reflection from the boundary corresponds to the boundary problem. A statistical method to estimate the solution and parametric derivatives of the considered problem is proposed. This method is based on solving SDE's by the Euler method. The order of convergence of the obtained estimates is established. The results of numerical computations are presented.

[1] Skorokhod A. V., “Stokhasticheskie uravneniya dlya diffuzionnykh protsessov v ogranichennoi oblasti. Chast 1”, Teoriya veroyatnostei i ee primeneniya, 6 (1961), 264–274 ; Часть 2, Теория вероятностей и ее применения, 7 (1962), 3–23 | Zbl | Zbl

[2] Tanaka H., “Stochastic differential equations with reflected boundary condition in convex regions”, Hiroshima Math. J., 9 (1979), 163–177 | MR | Zbl

[3] Lions P. L., Sznitman A. S., “Stochastic differential equations with reflected boundary conditions”, Comm. Pure Appl. Math., 37 (1984), 511–537 | DOI | MR | Zbl

[4] Saisho Y., “Stochastic differential equations for multi-dimensional domain with reflecting boundary”, Probab. Theory Rel. Fields, 74 (1987), 455–477 | DOI | MR | Zbl

[5] Slominski L., “On approximation of solutions of multidimensional stochastic differential equations with reflecting boundary conditions”, Stochastic Processes and their Applications, 50 (1994), 197–220 | DOI | MR

[6] Slominski L., “Some remarks on approximation of solutions of SDE's with reflecting boundary conditions”, Mathematics and Computers in Simulation, 38 (1995), 109–117 | DOI | MR | Zbl

[7] Milshtein G. N., “Primenenie chislennogo integrirovaniya stokhasticheskikh uravnenii dlya resheniya kraevykh zadach s granichnymi usloviyami Neimana”, Teoriya veroyatnostei i ee primeneniya, 41:1 (1996), 210–218 | MR | Zbl

[8] Bossy M., Gobet E., Talay D., “Symmetrized Euler scheme for an efficient approximation of reflected diffusions”, J. Appl. Probab., 41:3 (2004), 877–889 | DOI | MR | Zbl

[9] Costantini C., Pacchiarotti P., Sartoretto F., “Numerical approximation for functionals for reflected diffusion processes”, SIAM J. Appl. Math., 58:1 (1998), 73–102 | DOI | MR | Zbl

[10] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[11] Gobet E., “Weak approximation of killed diffusion using Euler schemes”, Stochastic Processes and their Applications, 87 (2000), 167–197 | DOI | MR | Zbl