# Thread: Geometric Brownian Motion with Markov Switching Volatility

1. ## Geometric Brownian Motion with Markov Switching Volatility

Hi all,

I hope this is the right forum for this -- if not, it would be nice if someone can tell me where I should post this.

Suppose $\displaystyle dx_{t}/x_{t}=\mu dt+\sigma_{t}dB_{t}$ , where B is the standard Brownian motion, and where $\displaystyle \sigma_{t}\in\{\sigma_{H},\sigma_{L}\}$
with transition probability matrix between t and t+dt:
$\displaystyle P=\left(\begin{array}{cc}p & 1-p\\1-q & q\end{array}\right)dt$

- what is $\displaystyle E[x_{t}|\sigma_{0}=\sigma_{H}]$?
- what is $\displaystyle E[x_{t}^{\alpha}|\sigma_{0}=\sigma_{H}]$?

Thanks!
Dan

2. ## Re: Geometric Brownian Motion with Markov Switching Volatility

Originally Posted by danlfnce
Hi all,

I hope this is the right forum for this -- if not, it would be nice if someone can tell me where I should post this.

Suppose $\displaystyle dx_{t}/x_{t}=\mu dt+\sigma_{t}dB_{t}$ , where B is the standard Brownian motion, and where $\displaystyle \sigma_{t}\in\{\sigma_{H},\sigma_{L}\}$
with transition probability matrix between t and t+dt:
$\displaystyle P=\left(\begin{array}{cc}p & 1-p\\1-q & q\end{array}\right)dt$

- what is $\displaystyle E[x_{t}|\sigma_{0}=\sigma_{H}]$?
- what is $\displaystyle E[x_{t}^{\alpha}|\sigma_{0}=\sigma_{H}]$?

Thanks!
Dan
this might help.

at first glance of the wiki page on geometric brownian motion it appears the first moment is independent of sigma so your markov switching of the volatility won't affect that.

not so for the higher moments.

further there is an initial value parameter that needs to be specified.

good luck

3. ## Re: Geometric Brownian Motion with Markov Switching Volatility

Thanks!

The links doesn't seem to offer much help on this.

My guess is that $\displaystyle E[x_{t}|\sigma_{0}=\sigma_{H}]$ is just $\displaystyle x_0 e^{\mu t}$ (assuming the initial value is just $\displaystyle x_0$).
$\displaystyle E[x_{t}^{\alpha}|\sigma_{0}=\sigma_{H}]$ would probably depend on $\displaystyle \sigma$, potentially the steady-state $\displaystyle \sigma$.
Under constant volatility, I know how to apply Ito's lemma to get the answer. But I'm not sure how to approach the case with stochastic volatility....