# Solving SDE:$\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t)$

I want so solve the following SDE. Specifically, I want to know if $y(t)$ is a Gaussian Process and if so the corresponding mean and covariance function.

$$\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t)$$ 11

where $W(t)$ is the Wiener process, $\epsilon(t) \sim \mathcal{N}(0,\sigma^2_e)$ for all $t$ and all the $e(t)$ random variables are mutually indepedent, $\sigma_w$ is a non-negative scalar and $c \in \mathbb{R}$.

I don't have any background in solving stochastic differential equations. Thus, my approach so far has been trying mathematica, which was not too successful yet.

• Is any information about $\epsilon (t)$ ? – Khosrotash Aug 5 '15 at 17:13
• Can you write it as :$dy(t)=c dt +\epsilon (t) dt +\sigma_w y(t)dw(t)$ ? – Khosrotash Aug 5 '15 at 17:16
• @Khosrotash I added more information about $\epsilon(t)$. If one wants to write it in your form I think it should be $$dy(t) = cdt + e(t)dt + \sigma_w y(t)W(t)dt$$ The wiener process is used a bit differently as it is used usually. – Julian Karls Aug 6 '15 at 12:07
• Is $\epsilon(t)$ a continuous process? – Calculon Aug 6 '15 at 12:37
• yes, $\epsilon(t)$ is actually a gaussian process with mean function $m(t)=0$ and kernel function $m(t,t')=\delta_{tt'}\sigma_\epsilon^2$ – Julian Karls Aug 6 '15 at 12:39