数学代写|STAT6540 Stochastic Process
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STAT6540 Stochastic Process课程简介
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PREREQUISITES
Course Content: Dynamical processes throughout science and economics are often influenced by random fluctuations. Mathematically, a dynamical model that explicitly includes random fluctuation is a stochastic process. Math 4320 will introduce you to both the theory and the applications of stochastic processes. We will first review probability theory before examining new material. In particular, we will discuss background in probability theory with emphasis on conditional expectations and conditional distributions. Then we will cover more advanced topics such as discrete-time Markov chains, Poisson process, continuous-time Markov chains. Grading \& Make-up Policy/Assignment \& Exam Details: Please consult your instructor’s syllabus regarding any and all grading/assignment guidelines.
STAT6540 Stochastic Process HELP（EXAM HELP， ONLINE TUTOR）
Theorem 4.2.16 Let $\left{X_n\right}_{n \geq 1}$ be a sequence of integrable random variables and let $X$ be some random variable. The following are equivalent:
(a) $\left{X_n\right}_{n \geq 1}$ is uniformly integrable and $X_n \stackrel{P r .}{\rightarrow} X$ as $n \rightarrow \infty$.
(b) $X$ is integrable and $X_n \stackrel{L^1}{\rightarrow} X$ as $n \rightarrow \infty$.
Proof. (a) implies (b): Since $X_n \stackrel{P r}{\rightarrow} X$, there exists a subsequence $\left{X_{n_k}\right}_{k \geq 1}$ such that $X_{n_k} \stackrel{a . s}{\rightarrow} X$. By Fatou’s lemma,
$$
\mathrm{E}[|X|] \leq \liminf k \mathrm{E}\left[\left|X{n_k}\right|\right] \leq \sup {n_k} \mathrm{E}\left[\left|X{n_k}\right|\right] \leq \sup n \mathrm{E}\left[\left|X_n\right|\right]<\infty $$ Therefore $X \in L^1$. Also for fixed $\varepsilon>0$ $$ \begin{aligned} \mathrm{E}\left[\left|X_n-X\right|\right] & \leq \int{\left{\left|X_n-X\right|<\varepsilon\right}}\left|X_n-X\right| \mathrm{d} P+\int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}\left|X_n\right| \mathrm{d} P+\int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}|X| \mathrm{d} P \ & \leq \varepsilon+\int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}\left|X_n\right| \mathrm{d} P+\int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}|X| \mathrm{d} P \end{aligned} $$ Recall (Remark 4.2.11) that adding an integrable random variable to a uniformly integrable collection retains uniformly integrability. Apply (b) of Theorem 4.2.12 to the uniformly integrable family $\left{X_n\right}_{n \geq 0}$ where $X_0:=X$, denoting by $\delta^{\prime}$ the corresponding $\delta$. By hypothesis, $P\left(\left|X_n-X\right| \geq \varepsilon\right) \leq \delta^{\prime}$ for large enough $n$. By (b) of Theorem 4.2.12 with $A:=\left{\left|X_n-X\right| \geq \varepsilon\right}$, for large enough $n, \int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}\left|X_n\right| \mathrm{d} P \leq \varepsilon$ and $\int_{\left{\left|X_n-X\right| \geq \varepsilon\right}}|X| \mathrm{d} P \leq \varepsilon$. Therefore, $\mathrm{E}\left[\left|X_n-X\right|\right] \leq 3 \varepsilon$ for large enough $n$, thus proving convergence in $L^1$ (b) implies (a): Let $\varepsilon>0$ be given and let $n_0$ be such that $\mathrm{E}\left[\left|X_n-X\right|\right] \leq \varepsilon$ for all $n \geq n_0$. The random variables $X, X_1, \ldots, X_{n_0}$ being integrable, there exists a $\delta>0$ such that if $P(A) \leq \delta, \int_A|X| \mathrm{d} P \leq \frac{\varepsilon}{2}$ and $\int_A\left|X_n\right| \mathrm{d} P \leq \frac{\varepsilon}{2}$ for $n \leq n_0$. If $n \geq n_0$, by the triangle inequality,
$$
\int_A\left|X_n\right| \mathrm{d} P \leq \int_A|X| \mathrm{d} P+\int_A\left|X_n-X\right| \mathrm{d} P \leq 2 \varepsilon,
$$
and therefore (b) of Theorem 4.2.12 is satisfied. Whereas (a) of Theorem 4.2.12 is satisfied since $\mathrm{E}\left[\left|X_n\right|\right] \leq \mathrm{E}\left[\left|X_n-X\right|\right]+\mathrm{E}[|X|]$.
Lemma 4.3.8 Let $\mathcal{A}$ be an algebra generating the $\sigma$-field $\mathcal{F}$ and let $P$ be a probability on $\mathcal{F}$. With any event $B \in \mathcal{F}$ and any $\varepsilon>0$, one can associate an event $A \in \mathcal{A}$ such that $P(A \triangle B) \leq \varepsilon$
Proof. The collection of sets
$$
\mathcal{G}:={B \in \mathcal{F} ; \forall \varepsilon>0, \exists A \in \mathcal{A} \text { with } P(A \triangle B) \leq \varepsilon}
$$
contains $\mathcal{A}$. It is moreover a $\sigma$-field, as we now show. First, $\Omega \in \mathcal{A} \subseteq \mathcal{G}$ and the stability of $\mathcal{G}$ under complementation is clear. For the stability of $\mathcal{G}$ under countable unions, let $B_n(n \geq 1)$ be in $\mathcal{G}$ and let $\varepsilon>0$ be given. Also, by definition of $\mathcal{G}$, there exist $A_n$ ‘s in $\mathcal{A}$ such that $P\left(A_n \triangle B_n\right) \leq 2^{-n-1} \varepsilon$. Therefore, for all $K \geq 1$,
$$
P\left(\left(\cup_{n=1}^K A_n\right) \Delta\left(\cup_{n=1}^K B_n\right)\right) \leq \sum_{n=1}^K 2^{-n-1} \varepsilon \leq \sum_{n \geq 1} 2^{-n-1} \varepsilon=2^{-1} \varepsilon .
$$
By the sequential continuity property of probability, there exists an integer $K=K(\varepsilon)$ such that $P\left(\cup_{n \geq 1} B_n-\cup_{n=1}^K B_n\right) \leq 2^{-1} \varepsilon$. Therefore, for such an integer,
$$
P\left(\left(\cup_{n=1}^K A_n\right) \Delta\left(\cup_{n \geq 1} B_n\right)\right) \leq \varepsilon
$$
The proof of stability of $\mathcal{G}$ under countable unions is completed since $\mathcal{A}$ is an algebra and therefore $\cup_{n=1}^K A_n \in \mathcal{A}$. Therefore $\mathcal{G}$ is a $\sigma$-field containing $\mathcal{A}$ and consequently contains the $\sigma$-field $\mathcal{F}$ generated by $\mathcal{A}$.
We now proceed to the proof of Theorem 4.3.7.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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