Stochastic Processes and Financial Mathematics
(part two)
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14.3 Exercises on Chapter 14
On the Feymann-Kac formula
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14.1 Find an explicit formula for the solution of the PDE
\(\seteqnumber{0}{14.}{7}\)
\begin{align*}
\frac {\p f}{\p t}(t,x)-2t\,\frac {\p f}{\p x}(t,x)&=0\\ f(T,x)&=e^x.
\end{align*}
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14.2 Find an explicit formula for the solution of the PDE
\(\seteqnumber{0}{14.}{7}\)
\begin{align*}
\frac {\p f}{\p t}(t,x)+\frac {\p f}{\p x}(t,x)+\frac 12\frac {\p ^2 f}{\p x^2}(t,x)&=0\\ f(T,x)&=x^2.
\end{align*}
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14.3 Let \(T>0\). Let \(F\) be the solution of the PDE
\(\seteqnumber{0}{14.}{7}\)
\begin{align}
\frac {\p F}{\p t}(t,x)+\alpha (t,x)\frac {\p F}{\p x}(t,x)+\frac 12\beta (t,x)^2\frac {\p ^2 F}{\p x^2}(t,x)+\frac {\p \gamma }{\p t}(t)&=0\label {eq:fk_gamma_1}\\
F(T,x)&=\Phi (x).\label {eq:fk_gamma_2}
\end{align}
Here, \(\alpha (t,x)\), \(\beta (t,x)\), \(\gamma (t)\) and \(\Phi (x)\) are known functions.
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(a) Let \(X_t\) satisfy \(dX_u=\alpha (u,x)\,du+\beta (u,x)\,dB_u.\) Define \(Z_t=F(t,X_t)+\gamma (t)\). Use Ito’s formula to
find \(dZ_t\).
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(b) Show that \(F(t,x)=\E _{t,x}\l [\Phi (X_T)\r ]+\gamma (T)-\gamma (t).\)
Challenge Questions