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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机控制代写Stochastic Control代考|An equivalent formulation

We recall a variational formula from [1] (Theorem $3.3$ of ibid. specialized to the constant control case). Define
$$\mathscr{G}:=\left{\zeta(d x, d y)=\zeta_{0}(d x) \zeta_{1}(d y \mid x) \in \mathscr{P}\left(S_{0} \times S_{0}\right): \int_{S_{0}} \zeta_{0}(d x) \zeta_{1}(d y \mid x)=\zeta_{0}(d y)\right}$$
that is, $\zeta_{0}(d x)$ is invariant under the transition kernel $\zeta_{1}(d y \mid x)$. Then Theorem $3.3$ of [1] states that
$$\Gamma_{\phi}=\max {\zeta \in \mathscr{G}}\left[\int \zeta(d x, d y)\left(c{\phi}(x)-D\left(\zeta_{1}(d y \mid x) | q_{\phi}(d y \mid x)\right)\right)\right]$$
where $D(\cdot | \cdot)$ is the Kullback-Leibler divergence or ‘relative entropy’ defined by
\begin{aligned} D\left(\mu | \mu^{\prime}\right) &:=\int \mu(d z) \log \left(\frac{d \mu}{d \mu^{\prime}}(x)\right) \text { if } \mu \ll \mu^{\prime} \ &=\infty \quad \text { otherwise. } \end{aligned}

## 统计代写|随机控制代写Stochastic Control代考|Remarks on computational schemes

Problem $P^{*}$ has linear constraints that are separate in the two variables $\gamma$, $\zeta$, but the reward is not separable. The reward function is in fact strictly concave in each variable when the other variable is kept constant, but it is not jointly concave, which makes the problem hard. Strict concavity implies that the maximizer in either variable with the other kept fixed is unique and depends continuously on the latter. Using this, it is easy to see that alternating maximization will lead to a local maximum.
Note that once $\zeta$ is fixed, the constrained maximization with respect to $\gamma$ is a linear program for an ergodic control problem, whereas once $\gamma$ is fixed, the constrained maximization with respect to $\zeta$ is a concave maximization problem which too can be made into a linear program by considering RSMP in place of SMP which means $\mathscr{P}\left(\mathscr{P}\left(S_{0}\right)\right)$-valued controls. Thus alternating maximization amounts to alternating linear programs. In principle one could replace these linear programs by alternative computational schemes such as policy or value iteration, see, e.g., [8], [11]. The difficulty here is that while this is possible for control of $\left{X_{n}\right}$ with $\zeta^{1}$ frozen, it is not so easy for control of $\left{Y_{n}\right}$ with $\phi$ fixed, because there is no irreducibility type condition available that would be required for justifying such schemes. In fact this is so even when $S_{0}$ is finite, because the control space is not. If one approximates the control space by a finite set as well, then one has the extended linear and dynamic programming formulations that cover the general case ([13], Chapter 9). All this is for a fixed value of $\Lambda$, the Lagrange multiplier which is unknown.

To solve the overall constrained optimization problem, one has to also recursively learn the Lagrange multiplier using a ‘primal-dual’ philosophy. That is, run the iteration
$$\Lambda_{n+1}=\left[\Lambda_{n}-s\left(\eta+\Gamma_{n}\right)\right]^{+} .$$

# 随机控制代写

## 统计代写|随机控制代写Stochastic Control代考|An equivalent formulation

\left 的分隔符缺失或无法识别

$$\Gamma_{\phi}=\max \zeta \in \mathscr{G}\left[\int \zeta(d x, d y)\left(c \phi(x)-D\left(\zeta_{1}(d y \mid x) \mid q_{\phi}(d y \mid x)\right)\right)\right]$$

$$D\left(\mu \mid \mu^{\prime}\right):=\int \mu(d z) \log \left(\frac{d \mu}{d \mu^{\prime}}(x)\right) \text { if } \mu \ll \mu^{\prime} \quad=\infty \quad \text { otherwise. }$$

## 统计代写|随机控制代写Stochastic Control代考|Remarks on computational schemes

$$\Lambda_{n+1}=\left[\Lambda_{n}-s\left(\eta+\Gamma_{n}\right)\right]^{+} .$$

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