Problem 1.

Suppose $M_{n}$ is a non-negative Martingale. Use Theorem $5.13$ to show that for all $\lambda>0$
$$\mathbb{P}\left(\max {0 \leq k \leq n} M{k}>\lambda\right) \leq \frac{\mathbb{E}\left(M_{n}\right)}{\lambda}$$
Note that the above inequality is an improvement of the well-known Markov’s inequality
$$\mathbb{P}(|X|>\lambda) \leq \frac{\mathbb{E}|X|}{\lambda}$$

Problem 2.

Suppose $M_{n}=X_{1}+\cdots+X_{n}$ with $M_{0}=0$ is a Martingale wrt $X_{n}$. Show that (a) for all $n \geq 1, \mathbb{E}\left(X_{n}\right)=0$.
(b) $\operatorname{Var}\left(M_{n}\right)=\operatorname{Var}\left(X_{1}\right)+\cdots+\operatorname{Var}\left(X_{n}\right)$

Problem 3.

Consider a discrete M.C. on $S={0,1,2, \ldots}$ whose transition probabilities are defined as follows: $p(k, \ell)=\mathbb{P}{Y+Z=\ell}$ where $Y \sim \operatorname{binom}(k, p)$ and $Z \sim \operatorname{Pois}(\lambda)$ are independent. Suppose further that $X_{0} \sim \operatorname{Pois}(\gamma)$
(a) What is the distribution of $X_{1}$ ?
(b) What is the distribution of $X_{n}$ ?
(c) Find the limiting distribution of $X_{n}$ as $n \rightarrow \infty$, i.e. $\lim {n \rightarrow \infty} \mathbb{P}\left(X{n}=k\right)$.
(d) Find the stationary distribution of the chain.
HINT: moment-gencrating function nethod.

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# AMATH 562 A: Advanced Stochastic Processes

• WINTER 2022

View in MyPlanView in Time ScheduleCourse WebsiteMeeting Time: MWF 11:30am – 12:20pmLocation: DEN 113SLN: 10242Joint Sections: AMATH 562 BInstructor:

Hong QianView profileCatalog Description: Stochastic dynamical systems aimed at students in applied math. Introduces basic concepts in continuous stochastic processes including Brownian motion, stochastic differential equations, Levy processes, Kolmogorov forward and backward equations, and Hamilton-Jacobi-Bellman partial differential equations. Presents theories with applications from physics, biology, and finance. Prerequisite: AMATH 561 or permission of instructor; recommended: undergraduate course in probability and statistics. Offered: W.Credits: 5.0Status: ActiveLast updated: January 4, 2022 – 10:13am

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