Nine volumes ago, in S eminaire de Probabilit es XXXIII, a series of advanced courses was started; nine such courses have appeared since. Two of them are due to Antoine Lejay, including his Introduction to rough paths in v- umeXXXVII. Thisunrepentantrecidivistnowstrikesagain, withYetanother introduction to rough paths, which sheds a more algebraic light on the same matter. The various contributions which constitute the rest of the volume ex- plify the r ole the S eminaire intends to play on the probabilistic stage: junior authors go side by side with older contributors, with a predominance from French or francophile ones; short notes mix with real research articles; and the themes are well in the traditional spirit of the S eminaire, ranging over the broad spectrum of interest of the readership of the S eminaire. Catherine Donati-Martin, Michel Emery, Alain Rouault, Christophe Stricker vii Contents Yet Another Introduction to Rough Paths Antoine Lejay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Monotonicity of the Extremal Functions for One-dimensional Inequalities of Logarithmic Sobolev Type Laurent Miclo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Non-monotone Convergence in the Quadratic Wasserstein Distance Walter Schachermayer, Uwe Schmock, and Josef Teichmann. . . . . . . . . . 131 On the Equation ? =S t t Fangjun Xu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Shabat Polynomials and Harmonic Measure Philippe Biane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Radial Dunkl Processes Associated with Dihedral Systems Nizar Demni. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Matrix Valued Brownian Motion and a Paper by P olya Philippe Biane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 On the Laws of First Hitting Times of Points for One-dimensional Symmetric Stable L evy Processes Kouji Yano, Yuko Yano, and Marc Yor. . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

Stochastic processes are as usual the main subject of the Seminaire, with contributions on Brownian motion (fractional or other), Levy processes, martingales and probabilistic finance. Other probabilistic themes are also present: large random matrices, statistical mechanics. The contributions in this volume provide a sampling of recent results on these topics. All contributions with the exception of two are written in English language.

Who could have predicted that the S eminaire de Probabilit es would reach the age of 40? This long life is ?rst due to the vitality of the French probabil- tic school, for which the S eminaire remains one of the most speci?c media of exchange. Another factor is the amount of enthusiasm, energy and time invested year after year by the R edacteurs: Michel Ledoux dedicated himself tothistaskuptoVolumeXXXVIII,andMarcYormadehisnameinseparable from the S eminaire by devoting himself to it during a quarter of a century. Browsing among the past volumes can only give a faint glimpse of how much is owed to them; keeping up with the standard they have set is a challenge to the new R edaction. In a changing world where the status of paper and ink is questioned and where, alas, pressure for publishing is increasing, in particular among young mathematicians, we shall try and keep the same direction. Although most contributions are anonymously refereed, the S eminaire is not a mathema- cal journal; our ?rst criterion is not mathematical depth, but usefulness to the French and international probabilistic community. We do not insist that everything published in these volumes should have reached its ?nal form or be original, and acceptance-rejection may not be decided on purely scienti?c grounds.

Problems of stochastic optimization and various mathematical aspects of risk are the main themes of this contributed volume. The readers learn about the recent results and techniques of optimal investment, risk measures and derivative pricing. There are also papers touching upon credit risk, martingale theory and limit theorems.

Forefront researchers in probability and financial mathematics have contributed to this volume paying tribute to Yuri Kabanov, an eminent researcher in probability and mathematical finance, on the occasion of his 60th birthday. The volume gives a fair overview of these topics and the current approaches.