# Stochastic Control: Markovian restriction

Consider a stochastic control problem, $$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big]$$ where $$X_t$$ is a weak solution to the SDE $$dX_t = C_t dB_t, \quad X_0 = x \in (-1,1).$$ $$\tau:= \inf\{t: X_t \notin (-1,1)\} \wedge T$$.

$$C$$ is a $$\mathbb{R}_+$$ valued progressively measurable stochastic process for some space $$(\Omega, \mathbb{F}=(\mathcal{F}_t)_{t\ge 0}, \mathbb{P})$$ for which the SDE above has a solution for some standard brownian motion $$B_t$$.

$$f(\cdot,\cdot)$$ is a measurable function. Also, $$f(\cdot, c)$$ is continuous for any $$c$$. I would like to claim the following:

Claim: Given any admissible control $$C$$, $$\exists$$ a Markovian control $$C'$$ (i.e. $$C'_t = g(t,X_t)$$ )(possibly on a different probability space) such that $$v^{C'}(0,x) \ge v^C(0,x)$$.

Does this sound true? There are various results in the vicinity of this that I have found in a survey by Borkar but, not being a practitioner myself, they are a bit hard to follow. Any help would be appreciated. Thanks.