A mathematically-offensive introduction to stochastic calculus and Brownian motion

Alexandre René

INM-6 Book club • 14 Jan 2022

Based on

Outline

White noise is weird
Basic concepts
- Brownian motion (aka Wiener process)
- Stochastic integrals
- Stochastic differentials
- Solving an SDE
- Change of variables (Itô’s lemma)
Examples
- Ornstein-Uhlenbeck process
- (Multiplicative noise)

Also discussed:

Itô vs Stratonovitch interprations

White noise is weird

“White” ⇒ no correlation; $\langle ξ(t) ξ(t')\rangle = \delta(t-t')$
Same power at all frequencies

Fractal: looks the same at all scales
$ξ(t)$ is ill-defined: all values are explored in any finite interval. However,

⇒

statistics can be defined;
$\int_0^T ξ(t) dt$ is a well defined random variable.

Closest thing:

$ξ(t) := \frac{1}{T} \int_0^T ξ(t) dt$

Wiener process

$ W(T) := \int_0^T ξ(t) dt$ is a Brownian motion.

\begin{aligned} \mathop{\mathrm{Var}} \left[ \int_0^T ξ(t) dt \right] &= \mathop{\mathrm{Var}} \biggl[ \underbrace{\int_0^{T/2} ξ(t) dt}_{=: W_1} + \underbrace{\int_{T/2}^T ξ(t) dt}_{=: W_2} \biggr] \\ &= 2 \mathop{\mathrm{Var}} \left[ \int_0^{T/2} ξ(t) dt \right] \\ \therefore \qquad\qquad\qquad &= n \mathop{\mathrm{Var}} \left[ \int_0^{T/n} ξ(t) dt \right] \quad \forall n \in \mathbb{N} \end{aligned}

Let’s assume $\mathbb{E} \left[ \int_0^T ξ(t) dt \right] = 0$. What should $\mathrm{Var} \left[ \int_0^T ξ(t) dt \right] $ be ?

with $W_1, W_2$ i.i.d.

\displaystyle \therefore \mathop{\mathrm{Var}} \left[ \int_0^T ξ(t) dt \right] = DT

$D=1$ ⇒

Wiener process

Diffusion constant

Wiener process

\begin{aligned} W(T) = \int_0^T ξ(t) dt &= \sum_{k=1}^n \underbrace{\int_{(k-1)T/n}^{kT/n} ξ(t) dt}_{:=W_k(kT/n)} \quad \forall n \in \mathbb{N} \end{aligned}

What should the distribution of $W(T) := \int_0^T ξ(t) dt $ be ?

with $W_k$ i.i.d.

\begin{alignedat}{4} W(T) &= \int_0^T ξ(t) dt &&\sim \mathcal{N}(0, T) \\ \\ X(T) &= &&\sim \mathcal{N}(0, DT) \end{alignedat}

(Wiener process)

Central limit theorem ⇒ $W(t)$ must* be a Gaussian RV:

(Brownian motion)

Wiener process

Brownian motion was described by Einstein in one of his four annus mirabilis papers.
- (More precisely: he computed D for particles in a fluid)
No stochastic calculus is needed.

A solution to a stochastic diff. equation therefore take the form of a time-dependent random variable.

Stochastic integrals

we use the Riemann-Stieltjes definition of an integral:

To compute objects of the form

\displaystyle X(t) = \int_0^t f(s) dW(s) \,,

\begin{aligned} \int f(s) ds &\approx \sum_i f(s_i) (s_{i+1} - s_i) \end{aligned}

\begin{aligned} \int f(s) dW(s) &\approx \sum_i f(s_i) (W(s_{i+1}) - W(s_i)) \end{aligned}

\begin{aligned} \int f(s) dW(s) &\approx \sum_i \frac{f(s_i) + f(s_{i+1})}{2} (W(s_{i+1}) - W(s_i)) \end{aligned}

(Itô)

(Stratonovich)

Stochastic differentials

For our purposes, we consider only drift-diffusion processes

\displaystyle dX(t) = f(t, X) dt + g(t, X) dW

$X$ is Markovian
$X$ is continuous

the stochastic increment must be $dW$

(Gillespie 1996)

Reason: self-consistency to first order

(i.e. $\sim \mathcal{N}(0, dt)$).

then

Stochastic differentials

\displaystyle dX(t) = f(t, X) dt + g(t, X) dW

the stochastic increment must be $dW$

\begin{aligned} X(t + 2dt) &\stackrel{!}{=} X(t) + 2dX + \mathcal{O}(dt\,dW) \end{aligned}

\begin{aligned} X(t) + 2 f(t, X(t)) dt + 2g(t, X(t)) dW_{2dt}(t) + \mathcal{O}(dt\,dW) \end{aligned}

\begin{aligned} &= X(t) + f(t, X(t))dt + g(t, X(t))\,dW \\ &\quad+ f(t, X(t+dt))dt + g(t, X(t+dt))\, dW(t+dt) \end{aligned}

\begin{aligned} &= X(t) + f(t, X(t))dt + f(t, X(t))dt + \frac{\partial f}{\partial x} dX(t)dt \\ &\quad+ g(t, X(t))\,dW(t) + g(t, X(t))dW(t+dt) + \frac{\partial f}{\partial x} dX(t)dW(t+dt)\\ \end{aligned}

\begin{aligned} &= X(t) + 2f(t, X(t))dt + g(t, X(t))\,(dW(t) + dW(t+dt)) + \mathcal{O}(dt\,dW) \end{aligned}

\displaystyle dW_{2dt}(t) \stackrel{!}{=} dW_{dt}(t) + dW_{dt}(t+dt)

Stochastic differentials – update schemes

\displaystyle dW_{2dt}(t) \stackrel{!}{=} dW_{dt}(t) + dW_{dt}(t+dt)

It is generally safer to think of $dt$ and $dW$ as small finite differences, rather than true differentials

A drift-diffusion SDE thus provides an update scheme:

X(t+dt) = X(t) + f(t,X(t))\, dt + g(t,X(t)) \sqrt{dt}\, \mathcal{N}(0,1)

X(t+dt) = X(t) + f(t,X(t))\, dt + \frac{g(t,X(t)) + g(t,X(t+dt))}{2} \sqrt{dt}\, \mathcal{N}(0,1)

Itô interpretation, explicit

Stratonovich interpretation, implicit

(Euler-Maruyama scheme)

Stochastic differentials – calculus rules

In the Itô interpretation, we have the following

\begin{aligned} \langle dW \rangle &= 0 \\ \langle X(t) dW(t) \rangle &= \langle X(t) \rangle \langle dW(t) \rangle \\ \langle dW(t) dW(t') \rangle &= \begin{cases} dt & \text{if $t = t'$} \\ 0 & \text{if $t \ne t'$} \end{cases} \end{aligned}

←Itô only

Some authors (e.g. Kurt Jacobs) also write

but this is a leaky abstraction – it can lead to wrong results.

\begin{aligned} dW(t) dW(t') &= \begin{cases} dt & \text{if $t = t'$} \\ 0 & \text{if $t \ne t'$} \end{cases} \end{aligned}

\langle dW dt \rangle = \langle dt^2 \rangle = 0

Stochastic differentials – Itô’s lemma

\begin{aligned} dh = \left[\frac{\partial h}{\partial t} + \frac{\partial h}{\partial x}f(t, X) + \frac{1}{2}\frac{\partial^2 h}{\partial x^2}g(t,X)\right]dt + \frac{\partial h}{\partial x} g(t,X) dW \end{aligned}

Given a function $h(t, X(t))$, we have

(Itô)

\begin{aligned} dh = \left[\frac{\partial h}{\partial t} + \frac{\partial h}{\partial x}f(t, X) \hphantom{+ \frac{1}{2}\frac{\partial^2 h}{\partial x^2}g(t,X)}\right]dt + \frac{\partial h}{\partial x} g(t,X) dW \end{aligned}

(Stratonovich)

Ornstein-Uhlenbeck (O-U) process

dX(t) = -μ X(t) dt + σ dW

Remark: If $X(t)$ is Gaussian, then $X(t+dt)$ is also Gaussian.

(A known initial condition – $p(0, x) = δ(x-x_0)$ – counts as Gaussian.)

⇒ It suffices to solve for mean and variance.

Strategy: Obtain differential equations for the moments.

\begin{aligned} d \langle X(t) \rangle &= \langle dX(t) \rangle \\ &= \langle - μ X(t) dt \rangle + \langle σ dW \rangle \\ &= - μ \langle X(t) dt \rangle + σ \langle dW \rangle \\ &= -μ\langle X(t) \rangle dt \\ \end{aligned}

\langle X(t) \rangle = e^{-μt} \langle X(0) \rangle

\displaystyle \frac{d}{dt}\langle X(t) \rangle = -μ \langle X(t) \rangle

Ornstein-Uhlenbeck (O-U) process

dX(t) = -μ dt + σ dW

\begin{aligned} d \langle X(t)^2 \rangle &= \langle X(t+dt)^2 \rangle - \langle X(t)^2 \rangle \\[1.5ex] &= \langle [X(t) - μdt + σdW ]^2 \rangle - \langle X(t)^2 \rangle \\[1.5ex] &= \langle X(t)^2 \rangle - \langle X(t)^2 \rangle \\ &\quad -2μ \langle X(t) dt \rangle + 2σ \langle X(t)dW \rangle - 2μσ \langle dt\,dW \rangle \\ &\quad+ μ \langle dt^2 \rangle + σ^2 \langle dW^2 \rangle\\[1.5ex] &= \left[-2μ \langle X(t)\rangle + σ^2 \right]dt \\ \end{aligned}

$= 0$ under Itô

\langle X(t)^2 \rangle = e^{-2μt} \langle X(0)^2 \rangle + \left[1 - e^{-2μt} \right] \frac{σ^2}{2μ}

\displaystyle \frac{d}{dt}\langle X(t)^2 \rangle = -2μ \langle X(t)\rangle + σ^2

Integration factor

Ornstein-Uhlenbeck (O-U) process

\begin{aligned} X(t) &\sim \mathcal{N}(m(t), Σ(t)) \end{aligned}

\begin{aligned} m(t) &= e^{-μt} \langle X(0) \rangle \\ Σ(t) &= e^{-2μt} \langle X(0)^2 \rangle + \left[1 - e^{-2μt} \right] \frac{σ^2}{2μ} - m(t)^2\\ \end{aligned}

Full solution:

where

In particular, we have the stationary solution

\begin{aligned} X(\infty) &\sim \mathcal{N}\left(0, \frac{σ^2}{2μ}\right) \end{aligned}

Final remarks

With mild assumptions compatible with a physical process, the correct white noise limit is the Stratonovich one.
- Thus one often hears that the Stratonovich form is the correct one to use for physical systems.
- A better approach may be to check for yourself how the limit works out in your case.
Itô and Stratonovich SDE’s are easily converted from one form to the other.
Multivariate processes are conceptually similar, but do require additional care
- (non-communativity of matrices)
- (sums of n-d Gaussians are not necessarily Gaussian)
Langevin form