Stochastic Processes and Financial Mathematics
(part two)
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\DeclareMathOperator {\var }{var}\)
\(\DeclareMathOperator {\cov }{cov}\)
\(\DeclareMathOperator {\indeg }{deg_{in}}\)
\(\DeclareMathOperator {\outdeg }{deg_{out}}\)
\(\newcommand {\nN }{n \in \mathbb {N}}\)
\(\newcommand {\Br }{{\cal B}(\R )}\)
\(\newcommand {\F }{{\cal F}}\)
\(\newcommand {\ds }{\displaystyle }\)
\(\newcommand {\st }{\stackrel {d}{=}}\)
\(\newcommand {\uc }{\stackrel {uc}{\rightarrow }}\)
\(\newcommand {\la }{\langle }\)
\(\newcommand {\ra }{\rangle }\)
\(\newcommand {\li }{\liminf _{n \rightarrow \infty }}\)
\(\newcommand {\ls }{\limsup _{n \rightarrow \infty }}\)
\(\newcommand {\limn }{\lim _{n \rightarrow \infty }}\)
\(\def \ra {\Rightarrow }\)
\(\def \to {\rightarrow }\)
\(\def \iff {\Leftrightarrow }\)
\(\def \sw {\subseteq }\)
\(\def \wt {\widetilde }\)
\(\def \mc {\mathcal }\)
\(\def \mb {\mathbb }\)
\(\def \sc {\setminus }\)
\(\def \v {\textbf }\)
\(\def \p {\partial }\)
\(\def \E {\mb {E}}\)
\(\def \P {\mb {P}}\)
\(\def \R {\mb {R}}\)
\(\def \C {\mb {C}}\)
\(\def \N {\mb {N}}\)
\(\def \Q {\mb {Q}}\)
\(\def \Z {\mb {Z}}\)
\(\def \B {\mb {B}}\)
\(\def \~{\sim }\)
\(\def \-{\,;\,}\)
\(\def \|{\,|\,}\)
\(\def \qed {$\blacksquare $}\)
\(\def \1{\unicode {x1D7D9}}\)
\(\def \cadlag {c\`{a}dl\`{a}g}\)
\(\def \p {\partial }\)
\(\def \l {\left }\)
\(\def \r {\right }\)
\(\def \F {\mc {F}}\)
\(\def \G {\mc {G}}\)
\(\def \H {\mc {H}}\)
\(\def \Om {\Omega }\)
\(\def \om {\omega }\)
\(\def \Vega {\mc {V}}\)
14.3 Exercises on Chapter 14
On the Feymann-Kac formula
-
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*}
-
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*}
-
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.
-
(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