Wiener Processes

* A Brief Introduction to Stochastic Differential Equations *

Wiener Processes

Stochastic differential equations differ from ordinary differential equations because they are parametrized by Wiener processes in addition to time. A Wiener process W_t is a non-differentiable random function of time t obtained by sampling the normal probability density

$\begin{eqnarray} \frac{1}{\sqrt{2\pi t}} e^{-W_t^2/2t}\nonumber \end{eqnarray}$

at each time t > 0. Numerically, W_t is usually generated by sampling on some finite equidistant grid of points t_j = j $\Delta$ t, for j = 1,..., K, such that

$\begin{eqnarray} W_{t_j}&=&W_0+\sum_{l=1}^j\Delta W_t^l\nonumber \end{eqnarray}$

where $\Delta$ W_t^l is sampled from

$\begin{eqnarray} \frac{1}{\sqrt{2\pi\Delta t}} e^{-\Delta W_t^{l2}/2\Delta t}.\nonumber \end{eqnarray}$

Here $\Delta$ t is the spacing between times in the grid. The increments $\Delta$ W_t^l are random and therefore not equidistant, and have zero mean and variance $\Delta$ t (i.e. $\overline{{\Delta W_t^{l2}}}$ = $\Delta$ t). In practice it is simpler to sample a number u from the density

$\begin{eqnarray} \frac{1}{\sqrt{2\pi }} e^{-u^2/2}\nonumber \end{eqnarray}$

and construct $\Delta$ W_t^l through $\Delta W_t^l=u\sqrt{\Delta t}$ .

Top Next

* Innovative Stochastic Algorithms *