On strong solutions and explicit formulas for solutions of stochastic integral equations

Conditions are obtained under which the stochastic equation
$$ x_t=x+\int^t_0\sigma(s,x_s)\,dw_s+\int^t_0b(s,x_s)\,ds $$
has a strong solution. In particular, in the multidimensional case where the diffusion matrix $\sigma$ is the identity matrix and the drift vector $b$ is bounded, these conditions are satisfied.
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