# 随机控制和动态资产配置MATH11150 Stochastic Control and Dynamic Asset Allocation代写

and is called the optimal value of the problem. When an optimal policy $\pi^{}$ exists, we write
$$J^{}\left(x_{0}\right)=J_{\pi^{}}\left(x_{0}\right)=\min {n \in \Pi} J{n}\left(x_{0}\right) .$$
When there does not exist an optimal policy we may be interested in finding an $\varepsilon$-optimal policy, i.e., a policy $\bar{\pi}$ such that
$$J^{}\left(x_{0}\right) \leqslant J_{\bar{}}\left(x_{0}\right) \leqslant J^{*}\left(x_{0}\right)+\varepsilon,$$
where $\varepsilon$ is some small number implying an acceptable deviation from optimality.

## MATH11150 COURSE NOTES ：

In the above equations the minimizations indicated are over all functions $\mu_{k}$ such that $\mu_{k}\left(x_{k}\right) \in U_{k}\left(x_{k}\right)$ for all $x_{k}$ and $k$. In addition, the minimization is subject to the ever-present system equation constraint
$$x_{k+1}=f_{k}\left[x_{k}, \mu_{k}\left(x_{k}\right), w_{k}\right] .$$
Now we use the fact that for any function $F$ of $x, u$, we have
$$\inf {\mu \in M} F[x, \mu(x)]=\inf {u \in U(x)} F(x, u)$$
where $M$ is the set of all functions $\mu(x)$ such that $\mu(x) \in U(x)$ for all $x$.