Exponential twist of probability measures: drift correction in term of a generalized gradient

TL;DR

This paper characterizes the exponential twist of a Markov measure as a solution to a martingale problem, revealing a drift correction involving a generalized gradient of a value function.

Contribution

It provides a full characterization of the exponential twist measure via martingale problems under general Markovian assumptions.

Findings

Exponential twist can be represented as a martingale problem solution.

The twisted measure retains the Markov property.

Drift correction involves a generalized gradient of a value function.

Abstract

In this paper we study the exponential twist, i.e. a path-integral exponential change of measure, of a Markovian reference probability measure $\P$. This type of transformation naturally appears in variational representation formulae originating from the theory of large deviations and can be interpreted in some cases, as the solution of a specific stochastic control problem. Under a very general Markovian assumption on $\P$, we fully characterize the exponential twist probability measure as the solution of a martingale problem and prove that it inherits the Markov property of the reference measure. The ''generator'' of the martingale problem shows a drift depending on a {\it generalized gradient} of some suitable {\it value function} $v$.