On Bellman's equations for mean and variance control of a Markov diffusion

Abstract

A controlled diffusion process is considered with cost functions of ‘mean and variance’ type. A regularization is proposed for computing the value of the cost function via Bellman's equations. The latter equations are in particular useful because they imply sufficiency of markovian strategies, at least, for regularized versions of processes. For the diffusion without control, a system of two well-posed linear partial differential equations (PDEs) is derived, similar to Kac's and Dynkin's moment equations, along with an equivalent single degenerate equation which turns out to be well-posed, too.

Keywords:

• degenerate Markov diffusion
• mean and variance control
• Bellman's equation
• PDE system

MSC::

• 60H10
• 93E20

Acknowledgements

The second author thanks the grant RFBR 08-01-00105a for support.

Notes

^1. [email protected]

Additional information

Notes on contributors

G. Aivaliotis

1. 1. [email protected]