| Summary: | Using both probabilistic and classical analytic techniques, we investigate the parabolic Kolmogorov equation $$ L_t v +frac {partial v}{partial t}equiv frac 12 a^{ij}(t)v_{x^ix^j} +b^i(t) v_{x^i} -c(t) v+ f(t) +frac {partial v}{partial t}=0 $$ in $H_T:=(0,T) imes E_d$ and its solutions when the coefficients are bounded Borel measurable functions of $t$. We show that the probabilistic solution $v(t,x)$ defined in $ar H_T$, is twice differentiable with respect to $x$, continuously in $(t,x)$, once differentiable with respect to $t$, a.e. $t in [0,T)$ and satisfies the Kolmogorov equation $L_t v +frac {partial v}{partial t}=0$ a.e. in $ar H_T$. Our main tool will be the Aleksandrov-Busemann-Feller Theorem. We also examine the probabilistic solution to the fully nonlinear Bellman equation with time-measurable coefficients in the simple case $bequiv 0,,cequiv 0$. We show that when the terminal data function is a paraboloid, the payoff function has a particularly simple form.
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