Numerical experiments, Tips, Tricks and Gotchas

## Brownian Motion and Geometric Brownian Motion

### Brownian motion

Brownian Motion or the Wiener process is an idealized continuous-time stochastic process, which models many real processes in physics, chemistry, finances, etc [1]. One can see a random "dance" of Brownian particles with a magnifying glass. More details can be seen with a microscope. But with further zooming, one can see that the particles fly freely between their collision with molecules . The randomness ends on this time (and space) scale. Unlike the real diffusion, the Wiener process remain stochastic at arbitrarily small scales.

### Brownian motion stochastic differential equation

The Brownian motion is described by the following stochastic differential equation (SDE): $$dX_{t}=\sigma dW_{t}\label{eq:BrM}$$ Here $W_{t}$ is the Wiener process and $\sigma$ is the diffusion coefficient. The solution is well known: $X_{t}\sim\mathit{\mathcal{N}}(0,\sigma^{2}t)$ where $\mathit{\mathcal{N}}(\mu, v)$ is the normal or the Gaussian variable with mean $\mu$ and variance $v$.

### Simulating Brownian motion

The usual recipe for simulation of the Brownian motion is $$\Delta X=\sigma\Delta W\label{eq:recipe}$$ with $$\Delta W=\sqrt{\Delta t}\,\mathcal{N}(0,1)\label{eq:gauss0}$$ where $\mathcal{N}(0,1)$ is a normal distribution with zero mean and unit variance. The variance of $\Delta X$ is $$\left\langle \Delta X^{2}\right\rangle =\sigma^{2}\Delta t\label{eq:var1}$$

### Simulation with random walk

Now we use the scaling property and divide $\Delta t$ in $n$ intervals of size $\tau$: $$\Delta t=n\tau\label{eq:Dt}$$ Then we replace $\Delta W$ with a random walk with steps $h_{i}=\pm h$: $$\Delta W=\sum_{i=1}^{n}h_{i}\label{eq:walk}$$ Now the variance of $\Delta X$ is $$\left\langle \Delta X^{2}\right\rangle =\sum_{i,j=1}^{n}\left\langle h_{i}h_{j}\right\rangle =nh^{2}=\sigma^{2}\Delta t\frac{h^{2}}{\tau}\label{eq:var2}$$ Here we used Eq. (\ref{eq:Dt}) and the fact that the random walk steps are uncorrelated $\left\langle h_{i}h_{j}\right\rangle =\delta_{i,j}h^{2}$ Comparing (\ref{eq:var1}) and (\ref{eq:var2}) we get the relation between the time and space steps: $$\tau=h^{2}\label{eq:tauh2}$$ Note that the distribution of distances (\ref{eq:walk}) follows the binomial distribution but it converges to the normal distribution at large $n$ according to the central limit theorem (CLT). Eq. (\ref{eq:recipe}) with $\Delta W$ (\ref{eq:walk}) and condition (\ref{eq:tauh2}) can be used directly for Brownian motion simulations. However $\Delta W$ (\ref{eq:gauss0}) does not rely on CLT and allows large steps.

### Brownian motion with drift

The Eq. (\ref{eq:BrM}) can be easy generalized for taking drift into account: $$dX_{t}=\mu dt+\sigma dW_{t}\label{eq:BrMdr}$$ The solution is also known: $$X_{t}\sim\mathit{\mathcal{N}}(\mu t,\sigma^{2}t)\label{eq:gauss}$$ The simulation formula $$\Delta X=\mu\Delta t+\sigma\Delta W\label{eq:recepemu}$$ Here both (\ref{eq:gauss0}) and (\ref{eq:walk}) can be used for $\Delta W$ .

### Geometric Brownian motion

Geometric Brownian motion with drift is described by the following stochastic differential equation: $$dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t}\label{eq:GBM}$$ To find the solution for (\ref{eq:GBM}) we consider a small difference $\Delta S=S(\Delta t)-S(0)$ . We again use Eq. (\ref{eq:Dt}) so that \begin{eqnarray*} S_{0} & = & S(0)\\ S_{1} & = & S(\tau)\\ \cdots & \cdot & \cdots\\ S_{n} & = & S(n\tau)=S(\Delta t) \end{eqnarray*} After the fist step \begin{eqnarray*} S_{1} & = & S_{0}+\mu S_{0}\tau+\sigma S_{0}h_{1}\\ & = & S_{0}\left(1+\mu\tau+\sigma h_{1}\right) \end{eqnarray*} After $n$ steps $$S(\Delta t)=S_{0}\prod_{i=1}^{n}\left(1+\mu\tau+\sigma h_{i}\right)\label{eq:SDt}$$ This expression is sufficient for the simulation of the geometric Brownian motion.

### The logarithm version of the equation

Now we take logarithm of the both sides of Eq. (\ref{eq:SDt}) $$ln\, S(\Delta t)=ln\, S_{0}+ln\left[\prod_{i=1}^{n}\left(1+\mu\tau+\sigma h_{i}\right)\right]\label{eq:lnSDt}$$ or $\Delta ln\, S\equiv ln\, S(\Delta t)-ln\, S(0)=\sum_{i=1}^{n}ln\left(1+\mu\tau+\sigma h_{i}\right)$ Using the expansion $ln(1+x)\approx x-\frac{x^{2}}{2}$ for $x\ll1$ we get \begin{eqnarray*} \Delta ln\, S & \approx & \sum_{i=1}^{n}\left[\left(\mu\tau+\sigma h_{i}\right)-\frac{1}{2}\left(\mu\tau+\sigma h_{i}\right)^{2}\right]\\ & = & \sum_{i=1}^{n}\left[\mu\tau+\sigma h_{i}-\frac{\sigma^{2}h^{2}}{2}-\mu\tau\sigma h_{i}-\frac{\mu^{2}\tau^{2}}{2}\right]\\ & = & \mu n\tau+\sigma\sum_{i=1}^{n}h_{i}-\frac{\sigma^{2}n\, h^{2}}{2}-\sigma\mu\tau\sum_{i=1}^{n}h_{i}-\frac{\mu^{2}n\,\tau^{2}}{2} \end{eqnarray*} Taking into account (\ref{eq:Dt}, \ref{eq:walk}) we get $\Delta ln\, S=\mu\Delta t+\sigma\Delta W-\frac{\sigma^{2}}{2}\Delta t-\frac{\sigma\mu\Delta t\Delta W}{n}-\frac{\mu^{2}(\Delta t)^{2}}{2n}$ The last two terms are of higher order and can be omitted: $$\Delta ln\, S\approx\left(\mu-\frac{\sigma^{2}}{2}\right)\Delta t+\sigma\Delta W\label{eq:DlnS}$$ Note that both that terms tend to zero when $n\rightarrow\infty$ . The above equation (\ref{eq:DlnS}) with $\Delta W$ (\ref{eq:walk}) can be used for the simulation of geometric Brownian motion.

### SDE and Ito's lemma

Turing from differences (\ref{eq:DlnS}) to differentials we get this SDE for $X=ln(S)$ $$dX_{t}=\left(\mu-\frac{\sigma^{2}}{2}\right)dt+\sigma dW_{t}\label{eq:DlnSDt}$$ Note that Ito's lemma (the term $-\frac{1}{2}\sigma^{2}dt$) is automatically taken into account. Eq. (\ref{eq:DlnSDt}) is actually Brownian motion (\ref{eq:BrMdr}) with modified drift. Therefore the solution for $X$ is (\ref{eq:gauss}).

### References

1. Gardiner C. W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Science, Berlin: Springer-Verlag, 1983.