I am learning Stochastic Process and there is an exercise I could not figure out.
Let W1 and W2 be two independent Standard Brownian motion.
Take $S_{1,t} = e^{\alpha W_{1,t} + \beta W_{2,t}}$ and $S_{2,t} = e^{\gamma W_{1,t} + \delta W_{2,t}}$.
(1) Find the infinitesimal mean and covariance $$ \alpha(s_1,s_2)dt = \mathbb{E}[(dS_1,dS_2)|S_{1,t}=s_{1},S_{2,t}=s_{2}]$$ $$ \mu(s_1,s_2)dt = cov[(dS_1,dS_2)|S_{1,t}=s_{1},S_{2,t}=s_{2}].$$
(2) Is $S_{1,t}$ a one-dimensional Markov process? Is $(S_{1,t},S_{2,t}$) a two dimensional Markov process?
I don't really know how to get started. Since the exponential term basically prevent me from doing the regular calculation...I tried to get started, the following picture is where I got stopped.picture
And for the part(2), by definition Markov process is "independent increment", But in this case, the S relies on 2 Brownian motion. And there are certain relations between $S_{1}$ and $S_{2}$. So I don't know how to think of this question.
Looking forward to someone's help. Really appreciate.
With Ito’s rule,
$$dS_{1,t} = S_{1,t}[\alpha dW_{1,t} + \beta dW_{2,t} +\frac12 (\alpha^2+\beta^2)dt]$$
$$dS_{2,t} = S_{2,t}[\gamma dW_{1,t} + \delta dW_{2,t} +\frac12 (\gamma^2+\delta^2)dt]$$
Evaluate,
$$ \mathbb{E}[(dS_1,dS_2)|S_{1,t}=s_{1},S_{2,t}=s_{2}] =\frac12 ( s_1(\alpha^2+\beta^2), s_2(\gamma^2+\delta^2))dt$$
$$ cov[(dS_1,dS_2)|S_{1,t}=s_{1},S_{2,t}=s_{2}] =s_1s_2(\alpha\gamma+\beta\delta)dt$$
$S_{1,t}$ is a Markov process, so is $(S_{1,t},S_{2,t})$.