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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 $(X-Y)$).

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