Lecture

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 $$W_t^N = \frac{1}{\sqrt{N}}(\theta S_k + (1-\theta)S_{k+1}), \quad Nt \in (k,k+1], \quad k = 0,1,\ldots,N-1$$ where $\theta = \lceil Nt \rceil - Nt$....

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....

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 $\mathbf{K}$ $$\mathbb{E}\left[e^{i \mathbf{\xi} \cdot \mathbf{X}}\right] = e^{i \mathbf{\xi} \cdot \mathbf{m} - \frac{1}{2}\mathbf{\xi}^T \mathbf{K} \mathbf{\xi}} $$ This means that for any $0 \leq t_1 < t_2 < \ldots < t_k$, the measure $\mu_{t_1, t_2, \ldots, t_k}$ is uniquely determined by an $\mathbf{m} = (m(t_1), \ldots, m(t_k))$ and a covariance matrix $\mathbf{K}_{ij} = K(t_i, t_j)$....

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....

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 $B \in \mathcal{R}$, $X_t^{-1}(B) \in \mathcal{F}_t$....

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 $\omega$): $\{\omega_1, \omega_2, \ldots \} \rightarrow \sum_{n \leq t} X(\omega_n)$...

2.2.1 - Definition and first examples Definition 2.2: A symmetric monoidal structure on a preoirder $(X, \leq)$ consists of (i) a monoidal unit, $I \in X$ (ii) a monoidal product $\otimes: X \times X \rightarrow X$ And the monoidal product $\otimes(x_1,x_2) = x_1 \otimes x_2$ must also satisfy the following properties (assume all elements are in $X$) (a) $x_1 \leq y_1$ and $x_2 \leq y_2 \implies x_1 \otimes x_2 \leq y_1 \otimes y_2$ (b) $I \otimes x = x \otimes I = x$ (c) associativity (d) commutivity/symmetry (a) is called monotnoicity and (b) is unitality...