By Peter Bank, Hans Föllmer (auth.)

ISBN-10: 3540401938

ISBN-13: 9783540401933

The **Paris-Princeton Lectures in monetary Mathematics**, of which this is often the 1st quantity, will, on an annual foundation, submit state of the art examine in self-contained, expository articles from amazing - tested or upcoming! - experts. the purpose is to provide a chain of articles that may function an introductory reference for study within the box. It arises because of common exchanges among the finance and fiscal arithmetic teams in Paris and Princeton. the current quantity units criteria with articles through P. Bank/H. Föllmer, F. Baudoin, L.C.G. Rogers, and M. Soner/N. Touzi.

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43–94, 2003. e. F is complete and right-continuous and F is assumed to be trivial). We assume that the price process of a given contingent claim is a continuous d−dimensional and F-adapted square integrable local martingale (St )0≤t≤T . In addition, we shall assume that the quadratic covariation matrix of the d−dimensional process S which is denoted by S : S t = Si, Sj t 1≤i,j≤d is almost surely valued in the space of positive matrix, which means that S is nondegenerate. For a matrix M, M ∗ will denote the transpose of M and for a vector v ∈ Rd , diag (v) denotes the d × d matrix v1 0 .

2 Strong Information Modeling The theory of initial enlargement of ﬁltration has been developed by the French school of probability during the eighties (see [27], [28], [29], [47] and [46]). This theory has many deep applications, most of which have been worked out by T. Jeulin and M. Yor. In the past few years we have seen new interest in this theory because of its applications in mathematical ﬁnance in the topic of the asymmetry of information. Papers where applications of the enlargement of ﬁltration technique is applied to portfolio optimization of an insider include [2], [3], [18], [20], [24], [25], and [41].

E. y ∈ P , let us denote by Qy the conditional probability Q (· | Y = y) . From our assumption and Proposition 2, the process M is, under Qy , a local martingale. e. y ∈ P, Qy is locally absolutely continuous on F with respect to P. e. y ∈ P, dQy/Ft = ηty dP/Ft , t < T. e. e. y ∈ P, and hence Q= P P (· | Y = y) ν (dy) where ν is the law of Y under Q. e. s. Lemma 2. (See [3]) The process ZtY := 1 , t

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