機率研討會

主講者: 1.吳慶堂教授(交通大學) 2.2. Professor Ren-Raw Chen (Fordham University, NY)
講題: 1.Müntz Linear Transforms of Brownian Motion 2.Non-parametric bounds for European option prices
時間: 2009-10-12 (Mon.)  14:10 - 17:00
地點:
Abstract:

(1)A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges. Moreover, calculations are shown to be more explicit for the class of M\"untz transforms such as the kernels, the control of the order to be infinite and the ergodic properties.

(2) We consider the problem of choosing an optimal portfolio to minimize the probability that the growth rate of the wealth process falling below a given level. In finance, such problems arise in risk management. The mathematical problem involved is not a conventional optimization problem and a solution is not readily found. We show a duality relation of this problem and the portfolio optimization problem with risk sensitive criterion. The latter has been studied in several recent papers. For some factor models, it can be reformulated as a risk sensitive stochastic control problem (hence the name for such portfolio optimization problem). The dynamic programming approach can be used to derive the Bellman equation which is a nonlinear partial differential equation. As in the theory of stochastic control, a candidate of optimal portfolio can be derived from a solution of the Bellman equation. Therefore, the duality relation mentioned above suggests a possibility to indirectly use the theory of stochastic control in such an unconventional optimization problem discussed here. Such duality relation has origin in the theory of large deviations. A difficult part in the analysis involving the idea of changing the probability measures that is used often in the theory of large deviations. How to choose a suitable new probability measure seems to require a nontrivial insight, since in our problem we need to deal with the control process which is absent in the theory of large deviations.

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