Let us consider the PDE

with terminal condition

In order to apply the representation theorem, we observe that \(r=0\), \((x t)=\), and , so that the underlying stochastic process boils down to a martingale described by

Therefore, conditional on \(X(t)=x\), integration over the time interval \(\left[\begin{array}{ll}t, & T\end{array}\right]\) yields 

where . Therefore,

It is easy to check that this function satisfies the PDE and the terminal condition.

Transcribed Image Text:

af t + 1 20 2 12 = 0,