# FxPaul

Math in finance or vice versa

## Maximum Likelihood Estimation of Stochastic Process Parameters

Maximum Likelihood estimation (MLE) is a method of parameter estimations of statistical model. The base idea is to establish joint density probability for observations and to maximize its value by model’s parameters. To say it differently, we are looking for the most probable explanation of observed data.

### Problem setup

Assume that we’re given a one-dimensional stochastic process:
$dS_t = \mu dt + \sigma dW_t$
where $\mu$ and $\sigma$ are some functions of arguments $\theta$.

We observe this process by measuring $\latex S_i(t_i)$ where $i=1..N$. For sake of simplicity assume that observations are equidistant in time, i.e. $\Delta t = t_{i-1} - t_i = const$.

So, let’s estimate parameters.

Written by fxpaul

November 2, 2011 at 08:00