On explicit formulas for solutions of stochastic equations

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.
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