Solve this simple Linear SDE?

Question

How do I solve the following BSDE? $$ \left\{ \begin{aligned} dX_t &=(rX_t+\theta Z_t ) \, dt + Z_t \, dW_t \\ X_T &=\xi \end{aligned} \right. $$

There appears to be nothing online about this, so maybe its super trivial?

Answer 1

Here we use Solution to General Linear SDE

\begin{align*} \mathrm{d}X_t = (a(t)X_t+ b(t)) \mathrm{d}t + (g(t)X_t+ h(t))\mathrm{d}B_t \end{align*}

for $g=0, a=r,b=\theta Z_{t},h(t)=Z_{t}$

\begin{align*} X_t = & X_0 e^{ \int_0^t a(s)\mathrm{d}s}+ e^{ \int_0^t a(s)\mathrm{d}s}\left( \int_0^t e^{ -\int_0^s a(r)\mathrm{d}r}b(s) \mathrm{d}s + \int_0^t e^{ -\int_0^s a(r)\mathrm{d}r}h(s) \mathrm{d}B_s\right)\\ = & X_0 e^{ t}+ e^{ t}\left( \int_0^t e^{ -s}\theta Z_{s} \mathrm{d}s + \int_0^t e^{ -s} Z_{s}\mathrm{d}B_s\right). \end{align*}