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The Kolmogorov equation with time-measurable coefficients. (English) Zbl 1047.35137
For $$t\in[0,T]$$, $$x\in E_d$$ and $$s\in[t,T]$$ the author considers a diffusion process $\xi_s(t,x)= x +\int_{t}^{s}\sigma(r) \,dw(r)+\int_{t}^{s}b(r) \,dr$ where $$\sigma\in L_2([0,T])$$, $$b \in L_1([0,T])$$ are respectively measurable bounded nonrandom $$d\times d_1$$-matrix and $$E_d$$-valued functions defined on $$[0,T]$$ and $$w(t)\in E_{d_1}$$ is a Wiener process. He proves that the function $$v(t,x)=Eg(\xi_T(t,x))$$ is differentiable with respect to $$t$$ for almost every $$t\in [0,T)$$ and satisfies the backward Kolmogorov equation almost everywhere in $$(0,T)\times E_d$$. The results are applied to the investigation of the Cauchy problem for the Bellmann equation.
##### MSC:
 35R60 PDEs with randomness, stochastic partial differential equations 60J60 Diffusion processes 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35K15 Initial value problems for second-order parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
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