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# 数学代写|Math56a Stochastic Porcesses

## Math56a 课程简介

A stochastic process is a random process which evolves with time. The basic model is the Markov chain. This is a set of “states” together with transition probabilities from one state to another. For example, in simple epidemic models there are only two states: $S=$ “susceptible” and $I=$ “infected.” The probability of going from $S$ to $I$ increases with the size of $I$. In the simplest model The $S \rightarrow I$ probability is proportional to $I$, the $I \rightarrow S$ probability is constant and time is discrete (for example, events happen only once per day). In the corresponding deterministic model we would have a “difference equation.”

## Prerequisites

In a continuous time Markov chain, transition events can occur at any time with a certain probability density. The corresponding deterministic model is a first order differential equation. This includes the “general stochastic epidemic.”

The number of states in a Markov chain is either finite or countably infinite. When the collection of states becomes a continuum, e.g., the price of a stock option, we no longer have a “Markov chain.” We have a more general stochastic process. Under very general conditions we obtain a Wiener process, also known as Brownian motion. The mathematics of hedging implies that stock options should be priced as if they are exactly given by this process. Ito’s formula explains how to calculate (or try to calculate) stochastic integrals which give the long term expected values for a Wiener process.

## Math56a Stochastic Porcesses 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
$$\mathbb{E}\left[|X| 1_A\right] \leq \varepsilon, \quad \text { whenever } \mathbb{P}[A] \leq \delta$$
(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
$$X_t^N \rightarrow e^{-t}, \quad \text { in probability as } N \rightarrow \infty \text {. }$$

Exercise 3. Let $T_1, T_2, \ldots$ be independent exponential random variables of parameter $\lambda$.
(1) For all $n \geq 1$, the 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 $\lambda$.

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