Stochastics

About two years ago, I attended a seminar given by Dr. Sid Redner of the Santa Fe Institute titled, “Is Basketball Scoring a Random Walk?” I was certainly skeptical that such an exciting game shared similarities with coin flipping, but, nevertheless, Dr. Redner went on to convince me–and surely many other audience members–that basketball does indeed exhibit behavior akin to a random walk. At the very end of his lecture, Dr. Redner said something along the lines of, “the obvious betting applications are left as an exercise to the audience.” So, as enthusiastic audience members, let’s try to tackle this exercise. ...

Let $\{\xi_m\}_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables such that $\mathbb{E}[\xi_n] = 0$ and $\mathbb{E}[\xi_n^2] = 1$. Then, define $$S_0 = 0, \quad S_N = \sum_{i=1}^N \xi_i$$and by the Central Limit Theorem, rescaling $S_N$ by $\sqrt{N}$, we get that $$\frac{S_N}{\sqrt{N}} \xrightarrow{d} \mathcal{N}(0,1)$$ (the $\xrightarrow{d}$ means convergence in distribution) as $N \rightarrow \infty$. Using this, we can define a continuous random function $W^N_t$ on $t \in [0,1]$ such that $W_0^N = 0$ and ...

Random Walk Let $\{\xi_i\}$ be i.i.d. random variables such that $\xi_i = \pm 1$ with probability $1/2$. Then, define $$X_n = \sum_{k=1}^{n} \xi_k, \quad X_0 = 0.$$ $\{X_n\}$ is the familiar symmetric random walk on $\mathbb{Z}$. Let $W(m,n) = \mathbb{P}(X_N = m)$. It is easy to see that $$W(m,n) = {N \choose (N+m)/2} \left( \frac{1}{2} \right)^N$$ and that the mean and std are $$\mathbb{E}[X_N] = 0, \quad \sigma^2_{X_N} = N$$Diffusion Coefficient Definition 6.2: (Diffusion coefficient). The diffusion coefficient $D$ is defined as ...

Definition 5.9: A stochasitc process $\{X_t\}_{t \geq 0}$ is a Gaussian Process if its finite dimensional distributions are consistent Gaussian measures for any $0 \leq t_1 < t_2 < \ldots < t_k$. Recall that a Gaussian random vector $\mathbf{X} = (X_1, X_2,\ldots,X_n)^T$ is completely characterized by its first and second moments $$\mathbf{m} = \mathbb{E}[\mathbf{X}], \quad \mathbf{K} = \mathbb{E}[(\mathbf{X} - \mathbf{m}) (\mathbf{X} - \mathbf{m})^T]$$Meaning that the characteristic function is expressed only in terms of $\mathbf{m}$ and $\mathbf{K}$ ...

Markov processes in continuous time and space Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and the filtration $\mathbb{F} = (\mathcal{F}_t)_{t \geq 0}$, a stochastic process $X_t$ is called a Markov process wrt $\mathcal{F}_t$ if $X_t$ is $\mathcal{F}_t$-adapted For any $t \geq s$ and $B \in \mathcal{R}$, we have $$\mathbb{P}(X_t \in B | \mathcal{F}_s) = \mathbb{P}(X_t \in B | X_s)$$ Essentially, this is saying that history doesn’t matter, only the current state matters. We can associate a family of probability measures $\{\mathbb{P}^x\}_{x\in\mathbb{R}}$ for the processes starting at $x$ by defining $\mu_0$ to be the point mass at $x$. Then, we still have $$\mathbb{P}^x(X_t \in B | \mathcal{F}_s) = \mathbb{P}^x(X_t \in B | X_s), \quad t \geq s$$ and $\mathbb{E}[f(X_0)] = f(x)$ for any function $f \in C(\mathbb{R})$. ⚠️ I am not fully confident on what the above section is saying. Specifically, I am having trouble with understanding how we are defining $\mathbb{P}^x$. However, I can understand the strong markov property, so I think I should be okay moving forward. ...

Filtration Definition 5.3: (Filtration). Given a probability space, the filtration is a nondecreaseing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t \leq 0}$ such that $\mathcal{F}_s \subset \mathcal{F}_t \subset \mathcal{F}$ for all $0 \leq s < t$. Intuitively, the filtration is a sigma algebra of events that can be determined before time $t$ (we can’t lose information by foing forward in time). A stochastic process is called $\mathcal{F}_t$-adapted if it is measurable with respect to $\mathcal{F}_t$; that is, for all $B \in \mathcal{R}$, $X_t^{-1}(B) \in \mathcal{F}_t$. We can always assume that the $\mathcal{F}_t$ contains $F_t^{X}$ and all sets of measure zero, where $F_t^{X} = \sigma(X_s, s \leq t)$ is the sigma algebra generated by the process $X$ up to time $t$. ...

Definition of a stochastic process A stochastic process is a parameterized random variable $\{X_t\}_{t\in\mathbf{T}}$ defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ taking on values in $\mathbb{R}$. $\mathbf{T}$ can seemingly be any subset of $\mathbb{R}$. For any fixed $t \in \mathbf{T}$, we can define the random variable $$X_t: \Omega \rightarrow \mathbb{R}, \quad \omega \rightarrowtail X_t(\omega)$$Thinking of a simple random walk, this means that $X_t$ is a random variable that takes in some subset of $\Omega = \{H,T\}^\mathbb{N}$ and outputs a real valued number (the sum of the first $t$ values in $\omega$): $\{\omega_1, \omega_2, \ldots \} \rightarrow \sum_{n \leq t} X(\omega_n)$ ...

Here are my notes for E, Li, and Vanden-Eijnden’s Applied Stochastic Analysis Chapter 5 - Stochastic Processes 5.1 - Axiomatic Construction of Stochastic Process 5.2 - Filtration and Stopping Time 5.3 - Markov Processes 5.4 - Gaussian Processes Chapter 6 - Wiener Process 6.1 - The Diffusion Limit of Random Walks 6.2 - The Invariance Principle