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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.
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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