Simulating Brownian Motion

Euler-Maruyama Method

We can simulate Brownian Motion (BM) via the discrete approximation of its independent increments.

Let $\delta t>0,n\in \mathbb{Z}$ and $T=n\delta t$. Let $t_k=k\delta t$ for $k=0,1,2,\dots ,n$ and $t_n=T$. The objective is construct a discrete time process

$$X(t_k)=X_k$$

such that $X_k\approx B(t_k)$ for a standard BM, $B(t)$. The scaling factor $\sqrt{\delta t}$ arises directly from the property that Brownian increments are normally distributed with variance $\delta t$.

Algorithm: (Euler, with notation as above)

1. Set $X_0=0$
2. For $i=0$ to $n-1$,
   1. Generate $Z\sim \mathcal{N}(0,1)$
   2. Set $X_{i+1}=X_i+\sqrt{\delta t}Z$
3. Output $X_0,X_1,\dots ,X_n$ to get $X(0),X(t_1),\dots ,X(t_n)$

This should look like path (similar to what a stock ticker would look like). We can do linear interpolation to connect $(t_k,X(t_k))$ to get approximate sample path $X(t)$.

BM with Drift

Let $X(t)=ut+\sqrt{2D}\cdot B(t)$ where $B(t)$ is a BM. Note that $u$ represents the deterministic drift velocity, and $D$ represents the diffusion coefficient. We can example the distribution of $X(t)$:

1. $\mathbb{E}\left[X(t)\right]=ut$, since $\mathbb{E}\left[B(t)\right]=0$ by properties.
2. $\text{Var}(X(t))=\mathbb{E}\left[(X(t)-ut{)}^2\right]=\mathbb{E}\left[2D\cdot B(t{)}^2\right]=2Dt$ Thus the marginal distribution is $X(t)\sim \mathcal{N}(ut,2Dt)$.

SDE Notation

Recall the SDE notation. We can extend this with drift:

$$\underset{\text{increment of X}}{\underbrace{dX}}=\underset{\text{deterministic change in t}}{\underbrace{udt}}+\underset{\text{scaled Brownian increment}}{\underbrace{\sqrt{2D}\cdot dB}}$$

We can “integrate” to get $X(t)=ut+\sqrt{2D}B(t)$.

Stochastic integration is different in general! You need different frameworks, the Riemann-Stieltjes Integral fails due to infinite total variation.

Euler’s Method For BM with Drift

Let $\delta t>0,n\in \mathbb{Z},t_k=k\delta t$ for $k=0,1,\dots ,n$.

• Set $X_0=0$
• For $i=0$ to $n-1$,
  • Generate $Z\sim \mathcal{N}(0,1)$
  • $X_{i+1}=X_i+u\delta t+\sqrt{2D\delta t}\cdot Z$
• Output $X_0,X_1,\dots ,X_n$ $X_i$ approximates true solution to $X(t_i)$.

Remarks on Euler’s Method

• The continuous SDE is the formal limit of the discrete update rule as $\delta t\to 0$: $$dX=u\cdot dt+\sqrt{2D}\cdot dB(t)$$
• In Euler’s Method, the discrete update computes the next state by adding the deterministic drift and the stochastic diffusion evaluated over the interval $\delta t$: $$X_{i+1}=X_i=u\delta t+\sqrt{2D\delta t}\cdot Z$$
• We can think of a SDE solution $X(t)$ as a “continuum limit” of the solution to a discrete equation.
• Intuitively, $dB$ is like $\sqrt{dt}\cdot Z$. Indeed, since $dB(t)=B(t+\delta t)=B(t)\sim \mathcal{N}(0,dt)$, then $\mathbb{E}\left[dB\right]=0$ and $\mathbb{E}\left[d{B}^2\right]=dt$.
• Langevin notation: divide by $dt$: $$\frac{dX}{dt}=u+\sqrt{2D}\cdot \xi (t)$$ here, $\xi (t)$ is “white noise”. It is not a function! From above we get $dB/dt\sim Z/\sqrt{dt}$.

Moments of BM with Drift

Evaluate the first moment by applying the expectation operator to the SDE.

$$dX=u\cdot dt+\sqrt{2D}\cdot dB$$

Taking the Expectation yields $\mathbb{E}\left[dX\right]=udt+\sqrt{2D}\cdot \mathbb{E}\left[dB\right]$. Since $\mathbb{E}\left[dB\right]=0$, we can isolate the deterministic component. The expectation operator commutes with the differential, generating an ordinary differential equation (ODE) for the mean

$$d\mathbb{E}\left[X\right]=udt⟹\frac{d}{dt}\mathbb{E}\left[X\right]=u$$

Integrating this yields $\mathbb{E}\left[X(t)\right]=ut$. From the discrete Euler approximation, the expectation of the difference equation strictly produces the same continuous limit:

$$X_{i+1}-X_i=u\delta t+\sqrt{2D\delta t}\cdot Z⟹\mathbb{E}\left[X_{i+1}\right]-\mathbb{E}\left[X_i\right]=u\delta t+\sqrt{2D\delta t}\cdot \mathbb{E}\left[Z\right]$$

Since $\mathbb{E}\left[Z\right]=0$, the relation reduces to

$$\mathbb{E}\left[X_{i+1}\right]=\mathbb{E}\left[X_i\right]+u\delta t$$

Dividing by $\delta t$ and evaluating the limit as $\delta t\to 0$ recovers the exact ODE:

$$\begin{array}{rl}\underset{\delta t\to 0}{\lim }\frac{\mathbb{E}\left[X_{i+1}\right]-\mathbb{E}\left[X_i\right]}{\delta t}& =u\\ \frac{d}{dt}\mathbb{E}\left[X\right]& =u\end{array}$$