Using both probabilistic and classical analytic techniques, we investigate the parabolic Kolmogorov equation
in and its solutions when the coefficients are bounded Borel measurable functions of . We show that the probabilistic solution defined in , is twice differentiable with respect to , continuously in , once differentiable with respect to , a.e. and satisfies the Kolmogorov equation a.e. in . 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 . We show that when the terminal data function is a paraboloid, the payoff function has a particularly simple form.
Submitted March 11, 2003. Published July 13, 2003.
Math Subject Classifications: 35K15, 35B65, 35K15, 60J60.
Key Words: Diffusion processes, Kolmogorov equation, Bellman equation.
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