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?

  • $\begingroup$ What is your source? $\endgroup$
    – 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.

  • $\begingroup$ Yes, but this in theory, are there contracts like that? $\endgroup$
    – Medan
    Aug 24 '16 at 17:02
  • $\begingroup$ And if I use the discounted version of expectation, would both terms get multiplied by $e^{-r(T-t)} or only the terminal payoff? $\endgroup$
    – Medan
    Aug 24 '16 at 17:04
  • $\begingroup$ it will be $h(B_T)e^{-r(T-t)}+\int_t^Te^{-r(s-t)}g(B_s)ds$, such that $\partial_t u + \partial_{xx}u + g -r u = 0$ $\endgroup$
    – MJ73550
    Aug 24 '16 at 17:06
  • $\begingroup$ many contracts like MTN are structured with a periodic payment up to maturity where you are paid the performance of some index $\endgroup$
    – MJ73550
    Aug 24 '16 at 17:07
  • $\begingroup$ I see, thanks, and If I were to google, MTN stands for medium term notes in fixed income? $\endgroup$
    – Medan
    Aug 24 '16 at 18:02

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