Egbert Dettweiler
Characteristic Processes associated with a Discontinuous Martingale
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 56, 81-115(2002)

MSC:

60G44 Martingales with continuous parameter

Abstract: In \cite{det-My} the following embedding theorem was proved (cf.~\cite{det-De} for a
detailed and complete proof). Let $(M_k)_{k\ge 0}$ be an $L^4$-martingale on a
probability space $\WR$. Then there exists an extension
$(\bar{\Omega},\bar{\C{F}},\bar{\p})$ of $\Omega$, a Brownian
motion $(B_t)_{t\ge 0}$ relative to a filtration $(\C{G}_t)_{t\ge 0}$ on
$\bar{\Omega}$, and an increasing sequence $(T_n)_{n\ge 1}$ of
$(\C{G}_t)$-stopping times, such that the following properties hold:\\[0.5 cm]
(i) $\;\;\;M_n\:=\:B_{T_n}\;\;$ for all $n\ge 1$,\\[0.5 cm]
(ii) $\;\;\E\big\{T_n\,|M_0,M_1,\cdots\big\}\;=$ \\[0.3 cm]
\hspace*{1 cm}
$\frac{1}{3}\sum^n_{k=1}(M_k-M_{k-1})^2\:+\:\frac{2}{3}\sum^n_{k=1}\E\big\{(M_k-M_{k-1})^2\:|M_0,\cdots,M_{k-1}\big\}\,,$
\\[0.5 cm]
(iii) $\;{\rm{Var}}\big\{T_n\,|M_0,M_1,\cdots\big\}\;=$ \\[0.3 cm]
\hspace*{1.5 cm}
$\frac{2}{45}\sum^n_{k=1}(M_k-M_{k-1})^4\:+\,\frac{8}{45}\sum^n_{k=1}\E\big\{(M_k-M_{k-1})^4\:|M_0,\cdots,M_{k-1}\big\}$
\\[0.3 cm]
\hspace*{1 cm}
$+\;(\frac{4}{9}-c)\sum^n_{k=1}\big(\E\big\{(M_k-M_{k-1})^2\:|M_0,\cdots,M_{k-1}\big\}\big)^2$
\\[0.3 cm]
\hspace*{1 cm}
$+\;c\sum^n_{k=1}(M_k-M_{k-1})^2\E\big\{(M_k-M_{k-1})^2\:|M_0,\cdots,M_{k-1}\big\}$
\\[0.3 cm]
\hspace*{1 cm}
$-\;c\sum^n_{k=1}(M_k-M_{k-1})\E\big\{(M_k-M_{k-1})^3\:|M_0,\cdots,M_{k-1}\big\}\,.$
\\[0.5 cm]
The constant $c>0$ in (iii) depends on the embedding and can be explicitely
computed (cf. \cite{det-De} and \cite{det-My}).
Keywords: Martingales with continuous parameter