Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 77, pp. 1-14.
Title: The Kolmogorov equation with time-measurable coefficients
Author: Jay Kovats (Florida Institute of Technology, Melbourne, FL, USA)
Abstract:
Using both probabilistic and classical analytic techniques,
we investigate the parabolic Kolmogorov equation
$$
L_t v +\tfrac {\partial v}{\partial t}\equiv \frac 12 a^{ij}(t)v_{x^ix^j}
+b^i(t) v_{x^i} -c(t) v+ f(t) +\tfrac {\partial v}{\partial t}=0
$$
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.