Stochastic Processes and Applications to Mathematical Finance; Proceedings of the 5th Ritsumeikan International Symposium
The International Colloquium on Stochastic Processes and Applications to Mathematical Finance was held at the Ritsumeikan University during March 2005. This volume contains the original papers presented at the colloquium that were devoted to applications of the theory of stochastic processes and stochastic analysis to financial problems.
The first paper by E. Barucci, P. Malliavin and M. E. Mancino describes how to compute the volatility of a semimartingale based on Fourier series. The method allows the computation of instantaneous and integrated volatility and is therefore useful for high frequency data. T. R. Bielecki, M. Jeanblanc and M. Rutkowski present alternative mathematical techniques used to derive hedging strategies for credit derivatives in models with totally unexpected defaults. In this paper, two alternative approaches are introduced: the stochastic calculus approach in order to establish abstract characterization results for hedgeable contingent claims in a general set-up and the partial differential equation approach in a Markovian setting. A. Kohatsu-Higa and A. Sulem define a forward integral and show that the forward integral is suitable for applications on a portfolio maximization problem. Y. Miyahara introduces the geometric Lévy processes and minimal entropy martingale measure pricing model as a pricing model for an incomplete financial market. The author proves useful properties of the model and presents several examples of applications of this model.
M. Yamazato explains important properties related to gamma processes, defines subclasses of the class of infinitely divisible distributions, which are generated by mixtures and convolutions of gamma distributions, and studies their properties. H. Hashimoto, T. Tsuchiya and T. Yamada treat Tanaka's equation in the case of symmetric stable processes and discuss a uniqueness result in the one-dimensional case. The second part of the paper delivers some results concerning the comparison problem and in the last part, a sufficient condition is proposed that guarantees pathwise uniqueness in the d-dimensional case. Watanabe presents a martingale representation theorem for the cases of discreet and continuous time and the Clark-Bismut-Ocone formula with a proof based on the Wiener chaos expansion. The volume contains a lot of interesting theoretical results together with their applications. It is a good source of information for research in the field of stochastic processes in financial mathematics.