Electron. J. Diff. Eqns., Vol. 2003(2003), No. 77, pp. 1-14.

The Kolmogorov equation with time-measurable coefficients

Jay Kovats

Abstract:
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 $H_T:=(0,T) \times E_d$ and its solutions when the coefficients are bounded Borel measurable functions of $t$. We show that the probabilistic solution $v(t,x)$ defined in $\bar H_T$, 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 $\bar H_T$. 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 $b\equiv 0,\,c\equiv 0$. 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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Jay Kovats
Department of Mathematical Sciences
Florida Institute of Technology
Melbourne, FL 32901, USA
email: jkovats@zach.fit.edu
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