last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

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14.3 Exercises on Chapter 14

On the Feymann-Kac formula
  • 14.1 Find an explicit formula for the solution of the PDE

    \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*}

  • 14.2 Find an explicit formula for the solution of the PDE

    \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*}

  • 14.3 Let \(T>0\). Let \(F\) be the solution of the PDE

    \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.

    • (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 \(dZ_t\).

    • (b) Show that \(F(t,x)=\E _{t,x}\l [\Phi (X_T)\r ]+\gamma (T)-\gamma (t).\)

Challenge Questions
  • 14.4 Give an example of a stochastic process that is adapted to the generated filtration \((\mc {F}_t)\) of a Brownian motion \((B_t)\), but which does not satisfy the Markov property.