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05014850 Stummer, Wolfgang Exponentials, diffusions, finance, entropy and information. Aachen: Shaker Verlag (ISBN 3-8322-3186-2/pbk). viii, 223~p. EUR~35.80 (2004). 2004 Aachen: Shaker Verlag EN Markov process diffusion process stochastic differential equations uniformly finite expectations uniform Novikov conditions statistical information measures exponential properties of diffusion processes arbitrage and contingent claim pricing European contingent claims

http://www.shaker.de/de/content/catalogue/index.asp?lang=de&ID=8&ISBN=978-3-8322-3186-6

This monograph is devoted to presenting in a unifying framework several topics that arise in different application fields (such as physics and finance, e.g.) but are modelled with the same mathematical stochastic tool: diffusion processes, finance, entropy and generalized entropies, decision and information theory, and potential theory. The book contains seven chapters, organized in three main parts: Part A: General Markov Processes (Chapter 2); Part B: Diffusion Processes -- Existence (Chapter 3); Part C: Diffusion Processes -- Properties and Applications; (Chapters 4--7). After the short, introductory Chapter 1, Chapter 2 (Exponentials of Additive Functionals) considers general Markov processes (while the rest of the monograph investigates special classes of Markov processes) which are solutions of stochastic differential equations (SDE) on the Euclidian state space $\Bbb R^d$. This chapter presents necessary and sufficient conditions for ordinary and Stieltjes exponentials of additive functionals to have uniformly finite expectations. Furthermore, sharp bounds for these expectations are obtained. Chapter 3 (Uniform Novikov Conditions) examines sufficient and (sometimes) equivalent conditions for the validity of the so-called Novikov Condition for the existence of a weak solution $X$ for the process characterizing SDEs. These conditions are either of stochastic form or of non-stochastic nature (thus handier to be verified in concrete situations). Several examples illustrate the wide range of the context. Chapter 4 (Relative Entropy and $\cal J_\alpha$-Divergences) investigates the so-called ${\cal J}_\alpha$-divergences (Cressie-Read measures, Tsallis measures, R\'enyi measures) which can be viewed as generalizations of the relative entropy. Finiteness results and bounds for the $\cal J_\alpha$-divergences are obtained using the basic facts from Chapter 3. The first application of the analyses of Chapter 4 deals with statistical decision and information theory in Chapter 5 (Some Statistical Information Measures). Finiteness statements and bounds on statistical information measures of some Bayesian dichotomous decision problems are derived about the drift of a diffusion process. The second application of the basic results in Chapter 4 is given in Chapter 6 (Some Exponential Properties of Diffusion Processes), which provides several sufficient conditions for the validity of a certain exponential-type property (connected with the time-average of the kinetic energy) of the diffusion process $X$ resulted as the solution of the considered SDEs. Applying some results of the previous chapter, the final Chapter 7 (Arbitrage and Contingent Claim Pricing) obtains some sufficient stochastic and non-stochastic conditions on a diffusion process $X$, which ensure the existence of an (unique) equivalent martingale measure by applying the Girsanov theorem in the opposite direction. There are achieved assertions on the absence of arbitrage opportunities, the valuation formula for European contingent claims, and call options on an underlying financial instrument whose price is a diffusion process $X$, and for which the ``Nobel-prize winning Black-Scholes formula appears as a special case. Several examples illuminate and support these findings. Silvia Curteanu (Ia\c si)