Investigation of stochastic problems of mathematical physics

V. S. Parfenenkova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 212-221

Résumé

The paper is devoted to constructing approximations of the Brownian motion in models leading to stochastic differential equations. For fundamental problems of mathematical physics, namely, for the problem of small vibrations of a string and the problem of heat conduction in a rod, approaches to defining and formalizing random perturbations are shown. For each of these problems, a sequence of random variables is constructed that converges in distribution to the Brownian motion describing random perturbations. The constructed approximations can be used for finding approximate solutions of stochastic problems.

Détail

Keywords: Cauchy problem, Brownian motion, approximate solutions, continuous models, central limit theorem.
Mots-clés : binomial models

Bibliographie

[1] Alshanskii M. A., Melnikova I. V., “Regulyarizovannye i obobschennye resheniya beskonechnomernykh stokhasticheskikh zadach”, Mat. sb., 202:11 (2011), 3–30 | DOI | MR

[2] Gikhman I. I., Skorokhod A. V., Vvedenie v teoriyu sluchainykh protsessov, Nauka, M., 1965, 654 pp. | MR

[3] Ito K., Makkin G., Diffuzionnye protsessy i ikh traektorii, Mir, Moskva, 1968, 394 pp. | Zbl

[4] Kuznetsov D. F., Chislennoe integrirovanie stokhasticheskikh differentsialnykh uravnenii, Izd-vo S.-peterb. gos. un-ta, SPb., 2001, 712 pp.

[5] Petrov V. V., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Teoriya ver. i mat. stat., 39, Nauka, gl. red. fiz.-mat. lit., M., 1987, 31 pp. | MR | Zbl

[6] Rozovskii B. L., “O stokhasticheskikh differentsalnykh uravneniyakh v chastnykh proizvodnykh”, Mat. sb., 96(138):2 (1975), 314–341 | MR | Zbl

[7] Allen E. J., “Derivation of stochastical partial differential equations”, Stoch. Anal. Appl., 26:2 (2008), 357–378 | DOI | MR | Zbl

[8] Cabana E. M., “The vibrating theorem forced by white noise”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 111–130 | DOI | MR | Zbl

[9] Curtain R. F., Pritchard A. J., Infinite dimensional linear systems theory, Lect. Notes in Control and Information Sciences, 8, Springer-Verlag, New York, 1978, 297 pp. | DOI | MR | Zbl

[10] Follmer H., Schied A., Stochastic finance. An introduction in discrete time, Walter de Gruyter GmbH Co. KG, Berlin–New York, 2002, 422 pp. | MR

[11] Hamza K., Klebaner F. C., “On solutions of first order stochastic partial differential equations”, Far East J. Theor. Statist., 1:1 (2006), 13–25 | MR | Zbl

[12] Milstein G. N., Tretyakov M. V., Stochastic numerics for mathematical physics, Springer-Verlag, New York, 2004, 594 pp. | MR | Zbl

[13] Melnikova I. V., “Regularized solutions to Cauchy problems well posed in the extended sense”, Integral Transforms Spec. Funct., 17:2–3 (2006), 185–191 | DOI | MR | Zbl

[14] Melnikova I. V., Filinkov A. I., Alshansky M. A., “Abstract stochastic equations. II. Solutions in spaces of abstract stochastic distributions”, J. Math. Sci., 116:5 (2003), 3620–3656 | DOI | MR | Zbl

[15] Oksendal B., Stochastic differential equations: an introduction with applications, 5 ed., Springer-Verlag, New York, 2000, 352 pp.

[16] Da Prato G., Zabczyk J., Stochastic equations in infinite dimensions, Encycl. Math. Appl., 45, Cambridge Univ. Press, New York, 1992, 454 pp. | MR

[17] Shreve S. E., Stochastic calculus for finance, v. II, Continuous-time models, Springer-Verlag, New York, 2004, 550 pp. | MR

[18] Walsh J. B., “An introduction to stochastic partial differential equations”, Lect. Notes in Math., 1180, Springer, Berlin, 1986, 265–439 | DOI | MR