### 统计代写|随机控制代写Stochastic Control代考|MATH4406

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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|Main Result

In the case $\varphi=\varphi(x)$, we consider the $x$-marginal of $\pi_{t}^{\varepsilon}(\varphi)$,
$$\pi_{t}^{\varepsilon, x}(\varphi) \equiv \int \varphi(x) \pi_{t}^{\varepsilon}(d x, d z)$$
If $X_{t}^{\varepsilon}$ takes values in $\mathbb{R}^{m}$ and $Z_{t}^{\varepsilon}$ in $\mathbb{R}^{n}$ with $m \leq n$, then it would be advantageous to consider the reduced (homogenized) filter equation
$$\pi_{t}^{0}(\varphi) \equiv \mathbb{E}{\mathbb{Q}}\left[\varphi\left(X{t}^{0}\right) \mid \mathscr{Y}{t}^{\varepsilon}\right] .$$ Yet the result $X{t}^{\varepsilon} \Rightarrow X_{t}^{0}$ does not necessarily imply $\pi_{t}^{\varepsilon, x} \rightarrow \pi_{t}^{0}$. In [16], convergence of the $x$-marginal to the homogenized filter is shown for the non-correlated case, $\alpha=0$. Specifically, it is proved that for any $T>0$, the difference between the $x$ -marginal of the original filter and the filter for the coarse-grain dynamics goes to zero as $\varepsilon \rightarrow 0$ at the rate $\sqrt{\varepsilon}$
$$\mathbb{E}{\mathbb{Q}}\left[d\left(\pi{T}^{\varepsilon, x}, \pi_{T}^{0}\right)\right] \leq \sqrt{\varepsilon} C$$
where $d$ denotes a suitable distance on the space of probability measures that generates the topology of weak convergence. Kushner [19] presents the next closest result to what we desire for two-timescale filtering problems, which is covered in great detail there, but does not obtain rates of convergence.

It is of interest to understand if a similar result holds in the correlated sensor noise case. For example, in our motivating problem of atmospheric or climate problems, sensors in those environments (e.g. floats, drifters, balloons) are coupled to their noisy environment. Further, as discussed in $[14,34]$, the correlated noise problem also occurs whenever a filter is based on a discrete time model that is derived from a continuous time model.

In this section, we provide the main ideas and tools used to show the following result,

## 统计代写|随机控制代写Stochastic Control代考|Zakai Equation

To show the above result, we make use of probabilistic representations of stochastic partial differential equations, that then allows us to get estimates giving a rate of convergence. To begin, we perform a standard Girsanov change of measure using the exponential martingale $D_{t}^{\varepsilon}$,
$$\left.D_{t}^{\varepsilon} \equiv \frac{d \mathbb{P}^{\mathbb{Q}}}{d \mathbb{Q}}\right|{\mathscr{F}{t}}=\exp \left(-\int_{0}^{t} h^{}\left(X_{s}^{\varepsilon}, Z_{s}^{\varepsilon}\right) d B_{s}-\frac{1}{2} \int_{0}^{t}\left|h\left(X_{s}^{\varepsilon}, Z_{s}^{\varepsilon}\right)\right|^{2} d s\right),$$ where $d B_{t} \equiv \alpha d W_{t}+\gamma d U_{t}$. If $\alpha \alpha^{}+\gamma \gamma^{*}=$ Id., then $B_{t}$ is a standard BM; this we assume for now. Then by the Kallianpur-Striebel formula, we can express the normalized condition measure $\pi_{t}^{\varepsilon}$ in terms of an unnormalized condition measure $\rho_{t}^{\varepsilon}$,

$$\pi_{t}^{\varepsilon}(\varphi)=\frac{\mathbb{E}{\mathbb{P} \varepsilon}\left[\varphi\left(X{t}^{\varepsilon}, Z_{t}^{\varepsilon}\right) \widetilde{D}{t}^{\varepsilon} \mid Y{t}^{\varepsilon}\right]}{\mathbb{E}{\mathbb{P}^{\varepsilon}}\left[\widetilde{D}{t}^{\varepsilon} \mid Y_{t}^{\varepsilon}\right]}=\frac{\rho_{t}^{\varepsilon}(\varphi)}{\rho_{t}^{\varepsilon}(1)},$$
where $\widetilde{D}{t}^{\varepsilon}=\left(D{t}^{\varepsilon}\right)^{-1}$. The advantage of working with $\rho_{t}^{\varepsilon}$ is that it’s evolution is defined by linear dynamics, whereas $\pi_{t}^{\varepsilon}$ is nonlinear. Similarly, define $\pi_{t}^{0}(\varphi)=$ $\rho_{t}^{0}(\varphi) / \rho_{t}^{0}(1)$ and the $x$-marginals,
$$\pi_{t}^{\varepsilon, x}(\varphi)=\rho_{t}^{\varepsilon, x}(\varphi) / \rho_{t}^{\varepsilon, x}(1), \quad \rho_{t}^{\varepsilon, x}(\varphi) \equiv \int \varphi(x) \rho_{t}^{\varepsilon}(d x, d z) .$$

## 统计代写|随机控制代写Stochastic Control代考|Main Result

$$\pi_{t}^{\varepsilon, x}(\varphi) \equiv \int \varphi(x) \pi_{t}^{\varepsilon}(d x, d z)$$

$$\pi_{t}^{0}(\varphi) \equiv \mathbb{E} \mathbb{Q}\left[\varphi\left(X t^{0}\right) \mid \mathscr{Y} t^{\varepsilon}\right] .$$

$$\mathbb{E} \mathbb{Q}\left[d\left(\pi T^{\varepsilon, x}, \pi_{T}^{0}\right)\right] \leq \sqrt{\varepsilon} C$$

## 统计代写|随机控制代写Stochastic Control代考|Zakai Equation

$$D_{t}^{\varepsilon} \equiv \frac{d \mathbb{P}^{\mathbb{Q}}}{d \mathbb{Q}} \mid \mathscr{F}{t}=\exp \left(-\int{0}^{t} h\left(X_{s}^{\varepsilon}, Z_{s}^{\varepsilon}\right) d B_{s}-\frac{1}{2} \int_{0}^{t}\left|h\left(X_{s}^{\varepsilon}, Z_{s}^{\varepsilon}\right)\right|^{2} d s\right),$$

$$\pi_{t}^{\varepsilon}(\varphi)=\frac{\mathbb{E P} \varepsilon\left[\varphi\left(X t^{\varepsilon}, Z_{t}^{\varepsilon}\right) \widetilde{D} t^{\varepsilon} \mid Y t^{\varepsilon}\right]}{\mathbb{E} \mathbb{P}^{\varepsilon}\left[\widetilde{D} t^{\varepsilon} \mid Y_{t}^{\varepsilon}\right]}=\frac{\rho_{t}^{\varepsilon}(\varphi)}{\rho_{t}^{\varepsilon}(1)},$$

$$\pi_{t}^{\varepsilon, x}(\varphi)=\rho_{t}^{\varepsilon, x}(\varphi) / \rho_{t}^{\varepsilon, x}(1), \quad \rho_{t}^{\varepsilon, x}(\varphi) \equiv \int \varphi(x) \rho_{t}^{\varepsilon}(d x, d z)$$

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