# Minimizer of a class of SDEs

## Setup

• Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic process adapted to $\mathcal{X}$ satisfying the SDE: $$dX_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t,$$ where $W_t$ is a cylindrical Brownian-motion.
• Fix $\epsilon \geq 0$ and let $\mathscr{A}$ be the set of all pairs of bijective $\phi \in C^2(\mathscr{H},\mathscr{H}')$, and $\mathscr{F}_t$-measurable functions $g:\mathscr{H}\mapsto \mathscr{H}$ satisfying $$\int_0^t P(g(X_s),\phi) ds \leq \epsilon,$$ where $P$ is a fixed smooth functional depending on $\phi$ and $\phi^{-1}\circ g(X_t)$.

Note: In practice, for most $\phi$ the identity $g=1_{\mathscr{H}}$ rarely forms a pair $(\phi,g)$ in the $\mathscr{A}$.

## Question

How can I solve the minimization problem: $$\operatorname{inf}_{(\phi,g)\in \mathscr{A}}\mathbb{E}[ D_{KL}( \phi\circ g(X_s),\phi(X_s))^2 ],$$ where $D_{KL}$ is the Kullback-Leibler divergence

(however if it simplifies the problem we may replace $D_{KL}(X,Y)$ with $(X-Y)$).