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Penalising a process is to modify its distribution with a limiting procedure, thus defining a new process whose properties differ somewhat from those of the original one. We are presenting a number of examples of such penalisations in the Brownian and Bessel processes framework. The Martingale theory plays a crucial role. A general principle for penalisation emerges from these examples. In particular, it is shown in the Brownian framework that a positive sigma-finite measure takes a large class of penalisations into account.

Brownian motion processes --- Martingales (Mathematics) --- Mathematical Theory --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Brownian motion processes. --- Wiener processes --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Stochastic processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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Operational research. Game theory --- stochastische analyse --- kansrekening

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Brownian motion processes --- Markov processes --- Mouvement brownien, Processus de --- Markov, Processus de

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The Black-Scholes formula plays a central role in Mathematical Finance; it gives the right price at which buyer and seller can agree with, in the geometric Brownian framework, when strike K and maturity T are given. This yields an explicit well-known formula, obtained by Black and Scholes in 1973. The present volume gives another representation of this formula in terms of Brownian last passages times, which, to our knowledge, has never been made in this sense. The volume is devoted to various extensions and discussions of features and quantities stemming from the last passages times representation in the Brownian case such as: past-future martingales, last passage times up to a finite horizon, pseudo-inverses of processes... They are developed in eight chapters, with complements, appendices and exercises.

Finance --- Economics --- Operational research. Game theory --- Probability theory --- Mathematics --- kennis --- waarschijnlijkheidstheorie --- stochastische analyse --- financiën --- wiskunde --- kansrekening

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We call peacock an integrable process which is increasing in the convex order; such a notion plays an important role in Mathematical Finance. A deep theorem due to Kellerer states that a process is a peacock if and only if it has the same one-dimensional marginals as a martingale. Such a martingale is then said to be associated to this peacock. In this monograph, we exhibit numerous examples of peacocks and associated martingales with the help of different methods: construction of sheets, time reversal, time inversion, self-decomposability, SDE, Skorokhod embeddings… They are developed in eight chapters, with about a hundred of exercises.

Business mathematics. --- Distribution (Probability theory). --- Negative binomial distribution. --- Martingales (Mathematics) --- Finance --- Distribution (Probability theory) --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Mathematical models --- Peafowl. --- Blue peafowl, Indian --- Common peafowl --- Indian blue peafowl --- Indian peafowl --- Pavo cristatus --- Peacocks --- Mathematics. --- Economics, Mathematical. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Quantitative Finance. --- Stochastic processes --- Pavo --- Distribution (Probability theory. --- Finance. --- Funding --- Funds --- Economics --- Currency question --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Methodology