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

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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Jay Kovats

Department of Mathematical Sciences

Florida Institute of Technology

Melbourne, FL 32901, USA

email: jkovats@zach.fit.edu

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