The Kolmogorov equation with time-measurable coefficients
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Using both probabilistic and classical analytic techniques, we investigate the parabolic Kolmogorov equation Ltv + ∂v/∂t ≡ 1/2αij (t)v xixj + bi(t)v xi - c(t)v + f(t) + ∂v/∂t = 0 in HT : = (0, T) x Ed and its solutions when the coefficients are bounded Borel measurable functions of t. We show that the probabilistic solution v(t, x) defined in ĦT, is twice differentiable with respect to x, continuously in (t, x), once differentiable with respect to t, a.e. t ∈ [0, T) and satisfies the Kolmogorov equation Ltv + ∂v/∂t = 0 a.e. in ĦT. 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 b ≡ 0, c ≡ 0. We show that when the terminal data function is a paraboloid, the payoff function has a particularly simple form.
CitationKovats, J. (2003). The Kolmogorov equation with time-measurable coefficients. Electronic Journal of Differential Equations, 2003(77), pp. 1-14.
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