Stochastic Processes and Financial Mathematics
(part one)
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\DeclareMathOperator {\var }{var}\)
\(\DeclareMathOperator {\cov }{cov}\)
\(\def \ra {\Rightarrow }\)
\(\def \to {\rightarrow }\)
\(\def \iff {\Leftrightarrow }\)
\(\def \sw {\subseteq }\)
\(\def \wt {\widetilde }\)
\(\def \mc {\mathcal }\)
\(\def \mb {\mathbb }\)
\(\def \sc {\setminus }\)
\(\def \v {\textbf }\)
\(\def \p {\partial }\)
\(\def \E {\mb {E}}\)
\(\def \P {\mb {P}}\)
\(\def \R {\mb {R}}\)
\(\def \C {\mb {C}}\)
\(\def \N {\mb {N}}\)
\(\def \Q {\mb {Q}}\)
\(\def \Z {\mb {Z}}\)
\(\def \B {\mb {B}}\)
\(\def \~{\sim }\)
\(\def \-{\,;\,}\)
\(\def \|{\,|\,}\)
\(\def \qed {$\blacksquare $}\)
\(\def \1{\unicode {x1D7D9}}\)
\(\def \cadlag {c\`{a}dl\`{a}g}\)
\(\def \p {\partial }\)
\(\def \l {\left }\)
\(\def \r {\right }\)
\(\def \F {\mc {F}}\)
\(\def \G {\mc {G}}\)
\(\def \H {\mc {H}}\)
\(\def \Om {\Omega }\)
\(\def \om {\omega }\)
Chapter 8 Further theory of stochastic processes \(\msconly \)
In this chapter we develop some further tools for analysing stochastic processes: the dominated convergence theorem, the optional stopping theorem and the strong Markov property. In chapters 8 and 9 we will write
\[ \min (s,t)=s\wedge t,\hspace {2pc} \max (s,t)=s\vee t. \]
This is common notation in the field of stochastic processes.
Note that this whole chapter is marked with a \(\msconly \) – it is for independent study in MAS61023 but it is not part of MAS352.
8.1 The dominated convergence theorem \(\msconly \)
The monotone convergence theorem, from Section 6.2, applies to sequences of increasing random variables. However, most
sequences are not so well behaved, and in these cases we need a more powerful theorem. Note that this section is marked with \(\msconly \), meaning that it is for independent study in MAS61023, but it is not
included in MAS352.
The random variable \(Y\) is often known as the dominating function or dominating random variable. You can find a proof of the theorem in MAS31002/61022.
-
Let \(Z\sim N(\mu ,\sigma ^2)\). Let \(X\in L^1\) be any random variable
and let
\[X_n=X+\frac {Z}{n}.\]
We can think of \(X_n\) as a noisy measurement of the random variable \(X\), where the noise term \(\frac {Z}{n}\) becomes smaller as \(n\to \infty \).
Let us check the first condition of the theorem. Note that \(|X_n-X|=\frac {|Z|}{n}\), which tends to zero as \(n\to \infty \) because \(|Z|<\infty \). Hence \(X_n\stackrel {a.s.}{\to } X\) as
\(n\to \infty \), which by Lemma 6.1.2 implies \(X_n\stackrel {\P }{\to }X\).
Let us now check the second condition, with \(Y=|X|+|Z|\). Then \(\E [Y]=\E [|X|]+\E [|Z|]\), which is finite since \(X\in L^1\) and \(Z\in L^1\). Hence, \(Y\in L^1\). We have \(|X_n|\leq |X|+\frac
{1}{n}|Z|\leq Y\).
Therefore, we can apply the dominated convergence theorem and deduce that \(\E [X_n]\to \E [X]\) as \(n\to \infty \). Of course, we can also calculate \(\E [X_n]=\E [X]+\frac {1}{n}\mu \), and check
that the result really is true.
In the above example, we could calculate \(\E [X_n]=\E [X]+\frac {\mu }{n}\) and notice that \(\E [X_n]\to \E [X]\), without using any theorems at all. The real power of the dominated convergence
theorem is in situations where we don’t specify, or can’t easily calculate, the means of \(X_n\) or \(X\). See, for example, exercises 8.1 and 8.2, or the applications in
Section 8.2.
If our sequence of random variables \((X_n)\) has \(|X_n|\leq c\) for some deterministic constant \(c\), then the dominating function can be taken to be (the deterministic random variable) \(c\). This case is
the a very common application, see e.g. exercise 8.1.