Exponential twist of probability measures: drift correction in term of a generalized gradient. Complete version
Thibaut Bourdais, Nadia Oudjane, Francesco Russo
TL;DR
This work provides a comprehensive, general framework for the exponential twist of a Markov reference measure via a path-integral cost, linking large-deviation variational formulas to stochastic control. The authors prove that, under broad Markovian conditions, the twisted measure $\mathbb{Q}$ remains Markov and identify its generator as $a^{\mathbb{Q}}(\phi)=a^{\mathbb{P}}(\phi)+\Gamma^v(\phi)/v$, with $v(t,x)=\mathbb{E}^{\mathbb{P}}[e^{-\int_t^T f(r,X_r)dr - g(X_T)}|X_t=x]$ and $\Gamma^v(\phi)$ expressed via $a^{\mathbb{P}}(v\phi) - v a^{\mathbb{P}}(\phi) - \phi a^{\mathbb{P}}(v)$. A key innovation is the decomposition of $\Gamma^v$ into continuous and jump parts, yielding a drift correction that can be interpreted as a generalized gradient, even when the reference drift is distributional. The framework is instantiated in jump-diffusions, diffusions, and SDEs with distributional drift, connecting to a path-integral control perspective and providing a principled method to compute Markovian drift corrections in diverse settings.
Abstract
In this paper we study the exponential twist, i.e. a path-integral exponential change of measure, of a Markovian reference probability measure $¶$. 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 $¶$, 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} $v$.