Geodesic
On explicit formulas for solutions of stochastic equations
Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 239-256

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The article is devoted to the proof of some criteria for the existence of a strong solution of a stochastic integral equation of the form $dx_t=\sigma(t,x_t)\,dw_t+b(t,x_t)\,dt$. One of the criteria appears as a Fredholm alternative; others are formulated in terms of the theory of differential equations of parabolic type. The proof of these criteria is based on finding formulas expressing $\mathsf M\{\varphi(x_t)|\mathscr F^w_t\}$ via multiple stochastic integrals, formulas which in the case $\varphi(x)\equiv x$ give an expression for $x_t$, if $x_t$ is a strong solution of the stochastic equation. Bibliography: 11 titles. 
@article{SM_1976_29_2_a8,
     author = {A. Yu. Veretennikov and N. V. Krylov},
     title = {On explicit formulas for solutions of stochastic equations},
     journal = {Sbornik. Mathematics},
     pages = {239--256},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_29_2_a8/}
}
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A. Yu. Veretennikov; N. V. Krylov. On explicit formulas for solutions of stochastic equations. Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 239-256. http://geodesic.mathdoc.fr/item/SM_1976_29_2_a8/