19th Ave New York, NY 95822, USA

# 数学代写|ST302 Stochastic processes

## ST302课程简介

Availability

This course is compulsory on the BSc in Actuarial Science. This course is available on the BSc in Business Mathematics and Statistics, BSc in Data Science, BSc in Financial Mathematics and Statistics, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available as an outside option to students on other programmes where regulations permit and to General Course students.

Pre-requisites

Students must have completed either Probability, Distribution Theory and Inference (ST202) or Probability and Distribution Theory (ST206).

## Prerequisites

Teaching

This course will be delivered through a combination of classes, lectures, and Q&A sessions totalling a minimum of 29 hours across Michaelmas Term.

The course includes a reading week in Week 6 of Michaelmas Term.

Formative coursework

Compulsory written answers to two sets of problems.

Lecture notes will be provided. Relevant books include R Durrett, Essentials of Stochastic Processes; T Mikosch, Elementary Stochastic Calculus with Finance in View; Institute of Actuaries core reading notes.

## ST302 Stochastic processes HELP（EXAM HELP， ONLINE TUTOR）

Exercise 1. If $X \in L^1(\Omega, \mathscr{F}, \mathbb{P})$, show that the class
$${\mathbb{E}[X \mid \mathscr{A}]: \mathscr{A} \text { sub } \sigma \text {-algebra of } \mathscr{F}}$$
is Uniformly Integrable.
(1) Show that, for any $\varepsilon>0$, there exists $\delta>0$ such that
$$\mathbb{E}\left[|X| 1_A\right] \leq \varepsilon, \text { whenever } \mathbb{P}[A] \leq \delta .$$
(2) Show the conclusion.

Exercise 2. Customers arrive in a supermarket as a Poisson process with intensity $N$. There are $N$ aisles in the supermarket and each customer selects one of them at random, independently of the other customers. Let $X_t^N$ denote the proportion of aisles which remain empty by time $t$. Show that
$$X_t^N \rightarrow e^{-t}, \quad \text { in probability as } N \rightarrow \infty .$$

Exercise 3. Let $T_1, T_2, \ldots$ be independent exponential random variables of parameter $\lambda$.
(1) For all $n \geq 1$, the $\operatorname{sum} S=\sum_{i=1}^n T_i$ has the probability density function
$$f_S(x)=\frac{\lambda^n x^{n-1}}{(n-1) !} e^{-\lambda x}, \quad x>0 .$$
This is called the $\operatorname{Gamma}(n, \lambda)$ distribution.
(2) Let $N$ be an independent geometric random variable with
$$\mathbb{P}[N=n]=\beta(1-\beta)^{n-1}, \quad n=1,2, \ldots$$
Show that $T=\sum_{i=1}^N T_i$ has exponential distribution of parameter $\lambda \beta$.

Exercise 4. Let $\left(N^i\right){i \geq 1}$ be a family of independent Poisson processes with respective positive intensities $\left(\lambda_i\right){i \geq 1}$. Then
(1) Show that any two distinct Poisson processes in this family have no points in common.
(2) If $\sum_{i \geq 1} \lambda_i=\lambda<\infty$, then $N_t=\sum_{i \geq 1} N_t^i$ defines the counting process of a Poisson process with intensity $\lambda$.

MY-ASSIGNMENTEXPERT™可以为您提供LSE.AC.UK ST302 STOCHASTIC PROCESSES随机过程课程的代写代考和辅导服务！