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Math735课程简介
Course description
Stochastic analysis is a term that refers to stochastic integration and stochastic differential equations and related themes. Here is a list of topics we expect to cover. The amount of time devoted to the fundamentals in the beginning will depend on the level of background that the audience possesses.
Foundations of probability theory, especially conditional expectation
Generalities about stochastic processes, Brownian motion, Poisson process
Martingales
Stochastic integral with respect to Brownian motion (quick overview of the Math 635 stochastic integral)
Predictable processes and stochastic integral with respect to cadlag martingales and semimartingales
Itô’s formula
Stochastic differential equations
Local time for Brownian motion, Girsanov’s theorem
White noise integrals and a stochastic partial differential equation
Prerequisites
Prerequisites
This course has flexible prerequisites. The ideal background would be one or two semesters of graduate measure-theoretic probability theory, such as our 733 or 733-734. An essential prerequisite is a certain degree of mathematical maturity, so familiarity with advanced probability is not absolutely necessary. The course will rely on modern integration theory (measure theory covered in Math 629 and 721) and advanced probability, and we can cover some of these points quickly in the beginning.
Evaluation
Course grades will be based on take-home work and a possible in-class exam. Homework will be posted on Learn@UW. You can also see your score record on Learn@UW.
Lecture notes
The course is based on lecture notes written by the instructor, available on Learn@UW. No textbook purchase is required.
Math735 Stochastic Analysis HELP（EXAM HELP， ONLINE TUTOR）
Exercise 1 Let $\tau_1(\omega)$ and $\tau_2(\omega)$ be stopping times with respect to the filtration $\mathbb{F}=\left(\mathcal{F}_t: t \in T\right)$ taking values in $T$. Here $T$ could be either $\mathbb{R}^{+}$or $\mathbb{N}$.
Use the definition of stopping time to show that $\sigma(\omega)=\min \left(\tau_1(\omega), \tau_2(\omega)\right)$ is a $\mathbb{F}$-stopping time.
Solution 1 We have
$$
{\sigma(\omega) \leq t}=\left{\omega: \tau_1(\omega) \leq t \text { or } \tau_2(\omega) \leq t\right}=\left{\tau_1(\omega) \leq t\right} \cup\left{\tau_2(\omega) \leq t\right} \in \mathcal{F}_t
$$
so $\sigma$ is a stopping time.
Exercise 2 Let $\left(M_t: t \in \mathbb{R}^{+}\right)$be a $\mathbb{F}$-martingale, and $\tau$ a $\mathbb{F}$-stopping time.
Show that the stopped process $\left(M_{t \wedge \tau}: t \in \mathbb{R}^{+}\right)$
$$
M_t^\tau(\omega)=M_{t \wedge \tau}(\omega)=M_t(\omega) \mathbf{1}(t \leq \tau(\omega))+M_{\tau(\omega)}(\omega) \mathbf{1}(t>\tau(\omega))
$$
is a $\mathbb{F}$-martingale.
To show that $\left(M_{t \wedge \tau}: t \in \mathbb{R}^{+}\right)$ is a $\mathbb{F}$-martingale, we need to verify the two conditions of martingale:
- $\left(M_{t \wedge \tau}: t \in \mathbb{R}^{+}\right)$ is adapted to the filtration $\mathbb{F}$.
- For all $s, t \in \mathbb{R}^{+}$ with $s \leq t$, we have $\mathbb{E}\left(M_{t \wedge \tau} \mid \mathcal{F}{s}\right)=M{s \wedge \tau}$ almost surely.
First, we show that $\left(M_{t \wedge \tau}: t \in \mathbb{R}^{+}\right)$ is adapted to the filtration $\mathbb{F}$. Let $t \in \mathbb{R}^{+}$, then $M_{t \wedge \tau}$ is measurable with respect to the sigma-algebra $\mathcal{F}{t}$ since it is a function of $M{s}$ for $s \leq t$ and $\tau$, both of which are $\mathcal{F}{t}$-measurable. Hence, $\left(M{t \wedge \tau}: t \in \mathbb{R}^{+}\right)$ is adapted to the filtration $\mathbb{F}$.
Next, we show that for all $s, t \in \mathbb{R}^{+}$ with $s \leq t$, we have $\mathbb{E}\left(M_{t \wedge \tau} \mid \mathcal{F}{s}\right)=M{s \wedge \tau}$ almost surely.
Case 1: $s \leq \tau$ In this case, $t \wedge \tau \geq s \wedge \tau$, and we have \begin{aligned} \mathbb{E}\left(M_{t \wedge \tau} \mid \mathcal{F}{s}\right) &=\mathbb{E}\left(M{t \wedge \tau} \mid \mathcal{F}{s \wedge \tau}\right) \ &=\mathbb{E}\left(M{t} \mathbf{1}(t \leq \tau)+M_{\tau} \mathbf{1}(t>\tau) \mid \mathcal{F}{s \wedge \tau}\right) \ &=\mathbb{E}\left(M{t} \mathbf{1}(t \leq \tau)+M_{\tau} \mathbf{1}(t>\tau) \mid \mathcal{F}{s}\right) \ &=\mathbb{E}\left(M{t} \mathbf{1}(t \leq \tau) \mid \mathcal{F}{s}\right)+\mathbb{E}\left(M{\tau} \mathbf{1}(t>\tau) \mid \mathcal{F}{s}\right) \ &=M{s} \mathbf{1}(s \leq \tau)+M_{\tau} \mathbf{1}(s>\tau) \ &=M_{s \wedge \tau}. \end{aligned}
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