ISSN:
1432-1122
Keywords:
Key words: Semimartingales, stochastic integrals, reverse Hölder inequalities, BMO space, weighted norm inequalities, Föllmer-Schweizer decomposition JEL classification:G10, G13 Mathematics Subject Classification (1991): 60G48, 60H05, 90A09
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Economics
Notes:
Abstract. Let $X$ be an ${\Bbb R}^d$ -valued special semimartingale on a probability space $(\Omega , {\cal F} , ({\cal F} _t)_{0 \leq t \leq T} ,P)$ with canonical decomposition $X=X_0+M+A$ . Denote by $G_T(\Theta )$ the space of all random variables $(\theta \cdot X)_T$ , where $\theta $ is a predictable $X$ -integrable process such that the stochastic integral $\theta \cdot X$ is in the space ${\cal S} ^2$ of semimartingales. We investigate under which conditions on the semimartingale $X$ the space $G_T(\Theta )$ is closed in ${\cal L} ^2(\Omega , {\cal F} ,P)$ , a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of $G_T(\Theta )$ in ${\cal L} ^2(P)$ . Most of these conditions deal with BMO-martingales and reverse Hölder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Föllmer-Schweizer decomposition.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s007800050021
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