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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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