Solving Stochastic Differential Equations Can anyone help me with the following SDE?
Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$ 
with $Y_0=0$.
Hint: Try a solution of the form $Z_tH_t$ where $Z_t = exp(cB_t+(a-\frac{1}{2}c^2t))$ and $dH_t=F(t)dt+G(t)dB_t$ for some adapted process F and G which need to be determined.
Thanks gt6989b for your input!
Please correct me if I'm wrong anywhere.
Since, $$Z_t = exp(cB_t+(a-\frac{1}{2}c^2t))$$
Hence we get: $dZ_t = Z_t(cdB_t-\frac{1}{2}c^2)dt$
and we also have $dH_t=F(t)dt+G(t)dB_t$
Then we apply Ito's Lemma on $Z_tH_t$, which yields
\begin{eqnarray*}
d(Z_tH_t) &=& Z_tdH_t+H_tdZ_t+dH_tdZ_t \\
&=& Z_t(F(t)dt+G(t)dB_t)+Z_tH_t(cdB_t-\frac{1}{2}c^2dt)+Z_tcG(t)dt \\
&=& Z_t(F(t)-\frac{1}{2}c^2H_t+cG(t))dt+(Z_tG(t)+cZ_tH_t)dB_t
\end{eqnarray*}
By letting $Y_t = Z_tH_t$, we compare between the expressions $dY_t$ and $d(Z_tH_t)$ in the $dt$ and $dB_t$ terms respectively. And we get the following:
\begin{eqnarray}
Y_t &=& Z_tH_t \\
G(t) &=& \frac{Z_t}{b(t)} \\
F(t) &=& (a+\frac{1}{2}c^2)H_t - c\frac{Z_t}{b(t)} 
\end{eqnarray}
Note: $F(t) = P$ and $G(t) =  Q$ for your P and Q respectively.
But now, I don't really understand how would this result help me in solving the original $dY_t$ SDE. 
 A: Here is an idea to get you started.
Note that
$$
dZ_t = cZ_t dB_t + \frac{c^2}{2}Z_t dt = Z_t \left(cdB_t + \frac{c^2}{2}dt\right)
$$
and let $dH_t = P dt + Q dB_t$.
Consider $A_t = Z_t H_t$. Then, by Ito's Lemma, and expansion for $dZ_t$,
$$\begin{split}
dA_t &= Z_t dH_t + H_t dZ_t + dZ_t dH_t \\
     &= Z_t (P dt + Q dB_t) + H_t Z_t \left(cdB_t + \frac{c^2}{2}dt\right) + cQdt \\
     &= dt\left[ ??? \right] + dB_t \left[ ??? \right]
\end{split}
$$
Now match this to $dY_t$, what can you say about $P,Q$?
A: A linear SDE $$\text{d}Y_{t} = (\alpha(t)+\beta(t)Y_{t})\,\text{d}t+(\gamma(t)+\delta(t)Y_{t})\,\text{d}W_{t}$$ has an explicit (strong) solution which can be found on Wikipedia . Here $\alpha$, $\beta$, $\gamma$ and $\delta$ denote deterministic functions and $W$ a one-dimensional Wiener process. The solution, $Y\ ,$ is expressed in terms of the stochastic exponential, $X\ ,$ which is the strong solution to the linear SDE $$\text{d}X_{t} = \beta(t)Y_{t}\,\text{d}t+\delta(t)Y_{t}\,\text{d}W_{t}\ .$$
