Question 11.
we assume:
(1) For every $t \geq 0, Y_t$ and $U_{t+1}$ are independent random variables.
(2) The covariance matrix of $\mathrm{Y}0$ is $\mathrm{W}_0$ and the covariance matrix of $\mathrm{U}{\mathrm{t}+1}$ is $\mathrm{V}$ for every $t \geq 0$.
(3) For every non-zero $\beta \in \mathrm{R}^{\mathrm{n}}, \beta^{\mathrm{T}} \mathrm{V} \beta>0$.
Q1. Show that for every $Y_t \geq 0$, Yt has the covariance matrix: $\mathrm{W}{\mathrm{t}}=\sum{j=0}^{t-1} \mathrm{~A}^j \mathrm{~V}\left(\mathrm{~A}^{\mathrm{T}}\right)^j+\mathrm{A}^{\mathrm{t}} \mathrm{W}0\left(\mathrm{~A}^{\mathrm{T}}\right)^{\mathrm{t}}$ Q2. A main issue in the analysis of time series is whether $\mathrm{W}{\mathrm{t}} \mathrm{con}$ verges as $t \rightarrow \infty$.
Throughout the question we focus on the case where $\mathrm{A}$ is diagonalizable in complex numbers: A $=$ PDP $^{-1}$ for some invertible complex matrix $P$ and diagonal matrix $\mathrm{D}$. The diagonal entries of $\mathrm{D}$ are denoted by $\lambda_1, \lambda_2, \ldots, \lambda_{\mathrm{n}}$. It is assumed that $\left|\lambda_j\right| \leq\left|\lambda_1\right|$ for every $j$ and $\left|\lambda_1\right|<1$. Show that there exists an $M>0$ and $\mathrm{b} \in(0,1)$ such that every matrix entry of $A^j V\left(A^T\right)^j$ has absolute value less than $\mathrm{Mb}^{\mathrm{j}}$ for every $\mathrm{j} \geq 0$.
Q3. Consider the following process:
$$x_{t+1}=a_1 x_t+a_2 x_t-1+a_3 x_{t-2}+\epsilon_{t+1}, \text { for } t=2,3,4, \ldots$$
Here $\mathrm{x}{\mathrm{t}}$ and $\epsilon{\mathrm{t}}$ are random variables for $\mathrm{t} \geq 0$ and $\epsilon_{\mathrm{t}+1}$ is independent of $\left(\mathrm{x}{\mathrm{t}}, \mathrm{x}{\mathrm{t}-1}, \mathrm{x}{\mathrm{t}}\right.$ ${ }{-2}$ ) for every $t \geq 2$. the above equation (1) may be rewritten as the following system of equations:
\begin{aligned} \mathrm{x}{\mathrm{t}+1} &=\mathrm{a}_1 \mathrm{x}{\mathrm{t}}+\mathrm{a}2 \mathrm{x}{\mathrm{t}-1}+\mathrm{a}3 \mathrm{x}{\mathrm{t}-2}+\epsilon_{\mathrm{t}+1} ; \ \mathrm{x}{\mathrm{t}} &=\mathrm{x}{\mathrm{t}}+0 \mathrm{x}{\mathrm{t}-1}+0 \mathrm{x}{\mathrm{t}-2}+0 ; \ \mathrm{x}{\mathrm{t}-1} &=0 \mathrm{x}{\mathrm{t}}+\mathrm{x}{\mathrm{t}-1}+0 \mathrm{x}{\mathrm{t}-2}+0 . \end{aligned}
the last two equations hold trivially for every $t \geq 2$. Using this trick, convert the above equation (1) to the form in equation (2) with appropriately chosen $\mathrm{A}, \mathrm{Y}{\mathrm{t}}$ and $\mathrm{U}{\mathrm{t}+1 \text { : }}$
$$Y_{t+1}=A Y_t+U_{t+1} \text {, for } t=0,1,2, \ldots .$$
Here $\mathrm{Y}{\mathrm{t}}$ and $\mathrm{U}{\mathrm{t}+1}$ are both $\mathrm{R}^{\mathrm{n}}$-valued random variables for every $\mathrm{t} \geq 0$ and $\mathrm{A}$ is an $\mathrm{n} \times \mathrm{n}$ matrix assumed to be known.

And then find the characteristic polynomial of your $\mathrm{A}$ in terms of $\mathrm{a}_1, \mathrm{a}_2$ and $\mathrm{a}_3$.

# Math 545 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Undergraduate background in real analysis (Math 431) and probability (Math 230 or 340)

#### Syllabus

• Brownian Motion and Stochastic Processes, Martingales
• Construction and properties of Brownian motion, Kolmogorov extension theorem, continuity theorem
• Ito Integrals with respect to Brownian motions and Ito processes
• Ito’s Formula and Quadratic Variation
• Stochastic Differential Equations: Existence and Uniqueness, Weak and strong solutions, Markov Property
• PDEs and SDE: Infinitesimal generators, optional stopping, stoping times, localization
• Connection with PDEs: Forward and  Backward Kolmogorov equation, Dirichlet Problems and hitting probabilities, Poisson Equations and Feynman-Kac formula
• Levy-Doob theorem,  martingales representation theory
• Kolmogorv-doob like inequalities for martingales
• Girsonov’s Theorem, Time changes
• One Dimensional Feller processes:  reachable and unreachable boundary points, nature scale, speed measures
• Bessel process
• Invariant measures
• Tanaka formula
• Examples of SDE models

#### References

•  Fima C. Klebaner, Introduction to stochastic calculus with applications
•  Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus
•  Richard Durrett, Stochastic calculus
•  Bernt Øksendal, Stochastic differential equations
•  H. P. McKean, Stochastic integrals
•  Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion