### 统计代写|随机控制代写Stochastic Control代考|MATH4091

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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|Uniform ergodic properties

We recall some important definitions used in [3, Section 2.3].
Definition 1. We fix some convex function $\psi \in C^{2}(\mathbb{R})$ with the property that $\psi(t)$ is constant for $t \leq-1$, and $\psi(t)=t$ for $t \geq 0$. The particular form of this function is not important. But to aid some calculations we fix this function as
$$\psi(t):= \begin{cases}-\frac{1}{2}, & t \leq-1 \ (t+1)^{3}-\frac{1}{2}(t+1)^{4}-\frac{1}{2} & t \in[-1,0], \ t & t \geq 0 .\end{cases}$$
Let $\mathcal{J}={1, \ldots, m}$. With $\delta$ and $p$ positive constants, we define
$$\Psi(x):=\sum_{i \in \mathcal{J}} \frac{\psi\left(x_{i}\right)}{\mu_{i}}, \quad \text { and } \quad V_{p}(x):=\left(\delta \Psi(-x)+\Psi(x)+\frac{m}{\min {i \in \mathcal{J}} \mu{i}}\right)^{p} .$$
Note that the term inside the parenthesis in the definition of $V_{p}$, or in other words $V_{1}$, is bounded away from 0 uniformly in $\delta \in(0,1]$. The function $V_{p}$ also depends on the parameter $\delta$ which is suppressed in the notation.

For $x \in \mathbb{R}^{m}$ we let $x^{\pm}:=\left(x_{1}^{\pm}, \ldots, x_{m}^{\pm}\right)$. The results which follows is a corollary of Lemma $2.1$ in [3], but we sketch the proof for completeness.
Lemma 1. Assume $\beta>0$, and let $\delta \in(0,1]$ satisfy
$$\left(\max {i \in \mathcal{J}} \frac{\eta{i}}{\mu_{i}}-1\right)^{+} \delta \leq 1 .$$
Then, the function $V_{p}$ in Definition 1 satisfies, for any $p>1$ and for all $u \in \Delta$,
\begin{aligned} &\left\langle b(x, u), \nabla V_{p}(x)\right\rangle \leq p\left(\delta \beta+\frac{m}{2}(1+\delta)-\delta|x|_{1}\right) V_{p-1}(x) \quad \forall x \in \mathcal{K}{-} \ &\left\langle b(x, u), \nabla V{\tilde{p}}(x)\right\rangle \leq-p\left(\frac{\beta}{m}-\delta \beta-\delta \frac{m}{2}+\delta\left|x^{-}\right|_{1}\right) V_{\tilde{p}-1}(x) \quad \forall x \in \mathcal{K}_{+} \end{aligned}

## 统计代写|随机控制代写Stochastic Control代考|Ergodic properties of the limiting SDEs arising from queueing models with service interruptions

The limiting equations of multiclass $G / M / n+M$ queues with asymptotically negligible service interruptions under the $\sqrt{n}$-scaling in the Halfin-Whitt regime are Lévy-driven SDEs of the form
$$\mathrm{d} X_{t}=b\left(X_{t}, U_{t}\right) \mathrm{d} t+\sigma \mathrm{d} W_{t}+\mathrm{d} L_{t}, \quad X_{0}=x \in \mathbb{R}^{m}$$
Here, the drift $b$ is as in Section $2, \sigma$ is a nonsingular diagonal matrix, and $\left{L_{t}\right}_{t \geq 0}$ is a compound Poisson process, with a drift $\vartheta$, and a finite Lévy measure $\eta(\mathrm{d} y)$ which is supported on a half-line of the form ${t w: t \in[0, \infty)}$, with $\left\langle e, M^{-1} w\right\rangle>0$. This can be established as in Theorem 6 in Section 5, assuming that the control is of the form $U_{t}=v\left(X_{t}\right)$ for a map $v: \mathcal{K}{+} \rightarrow \Delta$, such that $b{v}(x)$ is locally Lipschitz, when the scaling is of order $\sqrt{n}$ (see also Section $4.2$ of [4]).

As we explain later, under any stationary Markov control, the SDE in (29) has a unique strong solution which is an open-set irreducible and aperiodic strong Feller process. Therefore, as far as the study of the process $\left{X_{t}\right}_{t \geq 0}$ is concerned, we do not need to impose a local Lipschitz continuity condition on the drift, but can allow the control to be any element of $\mathfrak{U}_{\mathrm{sm}}$.

There are two important parameters to consider. The first is the parameter $\theta_{c}$, which is defined by
$$\theta_{c}:=\sup \left{\theta \in \Theta_{c}\right}, \quad \text { with } \quad \Theta_{c}:=\left{\theta>0: \int_{\mathcal{B}^{c}}|y|^{\theta} \eta(\mathrm{d} y)<\infty\right}$$
The second is the effective spare capacity, defined as
$$\widetilde{\beta}:=-\left\langle e, M^{-1} \tilde{\ell}\right\rangle$$
where
$$\tilde{\ell}:= \begin{cases}\ell+\vartheta+\int_{\mathcal{B} c} y \eta(\mathrm{d} y), & \text { if } \int_{\mathcal{B} c}|y| \eta(\mathrm{d} y)<\infty \ \ell+\vartheta, & \text { otherwise. }\end{cases}$$

## 统计代写|随机控制代写Stochastic Control代考|Uniform ergodic properties

$$\psi(t):=\left{-\frac{1}{2}, \quad t \leq-1(t+1)^{3}-\frac{1}{2}(t+1)^{4}-\frac{1}{2} \quad t \in[-1,0], t \quad t \geq 0 .\right.$$

$$\Psi(x):=\sum_{i \in \mathcal{J}} \frac{\psi\left(x_{i}\right)}{\mu_{i}}, \quad \text { and } \quad V_{p}(x):=\left(\delta \Psi(-x)+\Psi(x)+\frac{m}{\min i \in \mathcal{J} \mu i}\right)^{p}$$

$$\left(\max i \in \mathcal{J} \frac{\eta i}{\mu_{i}}-1\right)^{+} \delta \leq 1 .$$

$$\left\langle b(x, u), \nabla V_{p}(x)\right\rangle \leq p\left(\delta \beta+\frac{m}{2}(1+\delta)-\delta|x|{1}\right) V{p-1}(x) \quad \forall x \in \mathcal{K}-\quad\langle b(x, u), \nabla V \tilde{p}(x)\rangle \leq-p$$

## 统计代写|随机控制代写Stochastic Control代考|Ergodic properties of the limiting SDEs arising from queueing models with service interruptions

$$\mathrm{d} X_{t}=b\left(X_{t}, U_{t}\right) \mathrm{d} t+\sigma \mathrm{d} W_{t}+\mathrm{d} L_{t}, \quad X_{0}=x \in \mathbb{R}^{m}$$

$$\tilde{\beta}:=-\left\langle e, M^{-1} \tilde{\ell}\right\rangle$$

$$\tilde{\ell}:=\left{\ell+\vartheta+\int_{\mathcal{B} c} y \eta(\mathrm{d} y), \quad \text { if } \int_{\mathcal{B} c}|y| \eta(\mathrm{d} y)<\infty \ell+\vartheta, \quad\right. \text { otherwise. }$$

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