Geodesic
Symmetric stochastic differential equations with nonsmooth coefficients
Sbornik. Mathematics, Tome 44 (1983) no. 4, pp. 527-534 Cet article a éte moissonné depuis la source Math-Net.Ru

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The concept of a solution of a symmetric stochastic equation $$ X_t=x+\int^t_0\sigma(s,X_s)\circ dB_s+\int^t_0b(s,X_s)\,ds,\qquad t\geqslant0, $$ is generalized to the case when the coefficient $\sigma=\sigma(t,x)$, $(t,x)\in\mathbf R_+\times\mathbf R$, is continuous and continuously differentiable with respect to $t$, i.e., $\sigma\in C^{1,0}$. Here $B_t$, $t\geqslant0$, is a one-dimensional Brownian motion, and the stochastic integral is understood in the symmetric sense (in the sense of Stratonovich). Sufficient conditions are obtained for the existence and uniqueness of a solution, and the stability of a solution under perturbations of the coefficients is investigated. Bibliography: 14 titles. 
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     title = {Symmetric stochastic differential equations with nonsmooth coefficients},
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     volume = {44},
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     url = {http://geodesic.mathdoc.fr/item/SM_1983_44_4_a10/}
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V. Mackevičius. Symmetric stochastic differential equations with nonsmooth coefficients. Sbornik. Mathematics, Tome 44 (1983) no. 4, pp. 527-534. http://geodesic.mathdoc.fr/item/SM_1983_44_4_a10/

[1] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR

[2] Liptser R. Sh., Shiryaev A. N., Statistika sluchainykh protsessov, Nauka, M., 1974 | MR | Zbl

[3] Stratonovich R. L., “Novaya forma zapisi stokhasticheskikh integralov i uravnenii”, Vestnik MGU, 1 (1964), 3–12 | Zbl

[4] Toronzhadze T. A., Chitashvili R. Ya., “Ob ekvivalentnosti silno i slabo regulyarnykh stokhasticheskikh differentsialnykh uravnenii”, Soobsch. AN Gruz. SSR, 98:1 (1980), 37–40 | MR

[5] Doss H., “Liens entre équations différentielles stochastiques et ordinaires”, Ann. Inst. Henri Poincare, XIII:2 (1977), 99–125 | MR

[6] Krener A. J., “A formal approach to stochastic integration and differential equations”, Stochastics, 3:2 (1979), 105–125 | MR | Zbl

[7] Itô K., “Stochastic differentials”, Appl. Math. Optimization, 1:4 (1975), 374–381 | DOI | MR | Zbl

[8] Itô K., “Stochastic calculus”, Int. Symp. Math. Probl. Theor. Phys. (Kyoto, 1975), Lecture Notes Phys., 39, 1975, 218–223 | MR | Zbl

[9] Itô K., “Extension of stochastic integrals”, Proceedings of the Internat. Symp. on SDE, Kyoto, 1976, 95–109 | MR

[10] Meyer P.-A., “Un cours sur les intégrales stochastiques”, Lecture Notes Math., 511 (1976), 245–400 | DOI | MR | Zbl

[11] Stroock D. W., Varadhan S. R. S., “Diffusion processes with continuous coefficients”, J. Comm. Pure Appl. Math., XXII (1969), 345–400 | DOI | MR

[12] Stroock D. W., Varadhan S. R. S., Multidimensional diffusion prossess, Springer-Verlag, Berlin, Heidelberg, New York, 1979 | MR | Zbl

[13] Sussmann H. J., “An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point”, Bull. of AMS, 83 (1977), 296–298 | DOI | MR | Zbl

[14] Sussmann H. J., “On the gap between deterministic and stochastic ordinary differential equations”, The Annals of Probab, 6:1 (1978), 19–41 | DOI | MR | Zbl