# parabolic pde with source term

I was wondering if someone is aware of the application when pdes of the form arise $$u_t+u_{xx}+g=0$$ i.e. there is a source term now. Any financial instruments that have this type of pde?

• What is your source? – user16651 Aug 24 '16 at 17:05

$$u(t,x)=\mathbb{E}\left[h(B_T)+\int_t^Tg(B_s)ds|B_t=x\right]$$ where $B$ is a brownian motion.
So if you enter a contract whose underlying asset is $B$, such that you pay every day $t$, $-g(B_t)dt$ up to time $T$ where you receive $h(B_T)$, then the value of this contract is $u$
$$\partial_t u + \frac{1}{2}\partial_{xx}u + g = 0$$ there is $\frac{1}{2}$ in front of $\partial_{xx}u$ same for my comment below.
• And if I use the discounted version of expectation, would both terms get multiplied by $e^{-r(T-t)} or only the terminal payoff? – Medan Aug 24 '16 at 17:04 • it will be$h(B_T)e^{-r(T-t)}+\int_t^Te^{-r(s-t)}g(B_s)ds$, such that$\partial_t u + \partial_{xx}u + g -r u = 0\$ – MJ73550 Aug 24 '16 at 17:06