# Estimate parameters of two correlated geometric Brownian motions

I would like estimate the parameters of the following set of Geometric Brownian Motions:

$d P(t) = \mu_P P(t) d t + \sigma_P P(t) d Z_P(t)$

$d X(t) = \mu_X X(t) d t + \sigma_X X(t) d Z_X(t)$

where $\mu_P,\mu_X$ are the drift parameters; $\sigma_P,\sigma_X$ are volatility parameters. Furthermore, $Z_P, Z_X$ are correlated Wiener processes with the instantaneous correlation $\rho dt.$ I would like to estimate $\mu_P,\mu_X,\sigma_P,\sigma_X,$ and $\rho.$

If these two Brownian motions were assumed to be not correlated, i.e., $\rho=0,$ one can use

EstimatedProcess[tsP["Values"], GeometricBrownianMotionProcess[μ, σ, α]]


to estimate $\mu_p,\sigma_P,$ and $P(0).$ Here, tsP is a time series that holds the observed data on $P(t).$

I am looking for a similar approach for estimating the system given above.

• I don't think general ItoProcesses can be used in EstimatedProces. I tried to see if Likelihood or LogLikelihood work with general SDEs, but didn't have any luck either. There's a good possibility that you'll have to write your own code for this problem. – Sjoerd Smit Aug 9 '17 at 8:42
• @SjoerdSmit Thanks, I guess there is no "package" that can do this then. – emper Aug 9 '17 at 17:09