Stochastic Differential Equations

• ODEs are equations defined using derivatives of variables. In the simplest form, how a physical object position as a function of time $x(t)$ is updated can be defined using some function $f(t,x(t))$.
• What does this equation mean?
• Write the differential equation in a shorter way, and explain it.
  • $dx(t)=f(t,x(t))dt$
  • This states that infinitesimal change in position $x(t)$ is equal to the function $f(t,x(t))$ scaled by infinitesimal difference $dt$
• An equation of the form $y=f(x)$ gives the value of y according to function f at any point inside the support of the function. While a differential equation is a vector field that gives the change of position at any point and at any time t.
• The way to solve a differential equation is to take an initial point, and sum the infinitesimal changes over time. $x(T)=x(0)+{\int }_0^{T}dx(t)=x(0)+{\int }_0^{T}f(t,x(t))dt$
• To make the integral tractable, above formulation can be discretized to $\Delta x(t)$ and $\Delta t$, and turned into a summation.

Before talking about SDEs, let’s talk about Stochasticity over time: Weiner Process.

• Weiner Process models position of a particle with a purely random trajectory in a Euclidean space. Mathematically, we can represent the position as: $dx(t)=ϵdt$, where $ϵ$ is a probability distribution.
• The above equation is a differential equation with a non-deterministic function that follows some distribution, and we equate the infinitesimal change in position to random move in space scaled times the infinitesimal time that move is followed.
• Since $ϵ$ is a random variable, each position also follows the same distribution and, we treat each position in time itself as a random variable, governed by equation above.
• We can choose any distribution for $ϵ$, but Good ol’ friend Gaussian doesn’t like that, and is staring right at us through the Central limit hole! We model the probability of random movement with a standard Gaussian, $ϵ\sim \mathcal{N}(0,I)$.
• We choose amount of infinitesimal time to be $\sqrt{dt}$. We’ll derive the reason why.
• Thirdly, we want to the starting position of the differential equation $x_0=0$.

$$\begin{array}{c}{x}_T=x_0+{\int }_{t=0}^{T}d{x}_t={\int }_0^Tϵ\sqrt{dt}\end{array}$$

• Since $ϵ$ is a random variable, we don’t know its integral. Rather we look at the first two moments:
• $\mathbb{E}[x_T]=\mathbb{E}\left[{\int }_{t=0}^{T}d{x}_t\right]=0$
• $\mathbb{V}[x_T]=\mathbb{V}\left[\int x_T\right]={\int }_0^{T}dt\mathbb{V}[ϵ]=T$.
• So, all paths stay around initial position 0, and start to spread further proportional to the length of the path.
• The process defined so far is an elementary stochastic process called Wiener process $W_t$ that defines the path of infinitesimal movement of particle.

add a brownian motion picture

Let’s define the properties of the Wiener process:

• Initial position: $W_0=0$
• Independent increments: $\text{Cov}[W_{t+u}-W_s,W_s]=0,u\ge 0,s\le t$. Any future motion is independent of past motion. This can be seen as continuous time analog of iid steps in random walks.
• Stationary Gaussian increments: $W_{t+u}-W_t\sim \mathcal{N}(0,u)$. The displacement of the particle over an interval is Gaussian distributed with mean 0 and variance proportional to the interval. We just proved this above.
• Continuous time paths but nowhere differentiable. Paths are continuous, but this being a stochastic process, Wiener process is not differentiable because $W_{t+u}-W_t\sim \mathcal{N}(0,u)$, so ${\lim }_{u\to 0}\frac{W_{t+u}-W_t}{u}=\frac{\sqrt{u}}{u}\to \infty$.
• Other properties include the covariance structure: $\mathbb{E}[W_s{W}_t]=\min (s,t)$, Markov property: $p(W_t|\text{past})=p(W_t|W_s)$ for $s<t$, Martingale property: $\mathbb{E}[W_t|{\mathcal{F}}_s]=W_s$.

SDE

An SDE is just combination of ODE and stochastic process. We defined a stochastic differential equation which is the infinitesimal change in random variable $X_t$ as sum of deterministic change ${\mu }_{t}dt$ and stochastic process (infinitesimal Wiener process) ${\sigma }_{t}d{W}_t\sim \mathcal{N}(0,{\sigma }_t^{2}dt)$

$$d{X}_t=\underset{\text{deterministic}}{\underbrace{{\mu }_{t}dt}}+\underset{\text{stochastic}}{\underbrace{{\sigma }_{t}d{W}_t}}$$

This is also known as Ito drift-diffusion process. Solving the SDE, we’ll assume constant ${\mu }_t=\mu$ and ${\sigma }_t=\sigma$.

$$\begin{array}{rl}{X}_T& ={\int }_{t=0}^{T}d{X}_t\\ & ={\int }_{t=0}^T\mu dt+\sigma d{W}_t\\ & =\mu T+\sigma {\int }_0^{T}d{W}_t\\ & =\mu T+\sigma W_T\end{array}$$

$W_T\sim \mathcal{N}(0,T)$ is a random variable. Let’s calculate analytical mean and variance of the process.

$$\begin{array}{r}\mathbb{E}[X_T]=\mathbb{E}[\mu T+\sigma W_T]=\mu T+\sigma \mathbb{E}[W_T]=\mu T\\ \mathbb{V}[X_T]=\mathbb{V}[\mu T+\sigma W_T]={\sigma }^2\mathbb{V}[W_T]={\sigma }^{2}T\end{array}$$

• Ito’s Process: Ito’s lemma that tells how to differentiate an SDE: Ito’s (Di)Lemma
• Ornstein-Uhlenbeck process: Solving (Some) SDEs
• Fokker-Planck Equation
  • FokkerPlanck_Equation

Time reversal of diffusion/reverse SDE

• Why do we want to reverse an SDE?

Score estimation using SDE

Stochastic Calculus

Introduction to Stochastic Calculus | Ji-Ha’s Blog Itô Processes and The Fundamental Theorem of Stochastic Calculus ItoSDE_Tutorial LW - Blog