Stochastic modeling: Underlying stochastic processes and model dynamics.

摘要

Two stochastic modeling problems of interrelated interest are investigated. In the first problem, the limiting distribution of the limit lognormal multifractal, first introduced by Mandelbrot (Statistical Models and Turbulence , M. Rosenblatt and C. Van Atta, eds., Lecture Notes in Physics 12, Springer, New York, 1972, p. 333) and constructed explicitly by Bacry et al. (Phys. Rev. E 64, 026103 (2001)), is investigated using its Laplace transform. A partial differential equation for the Laplace transform is derived and it is shown that multifractality alone does not determine the limiting distribution. The increments of the limit multifractal process are strongly stochastically dependent. The precise nature of this stochastic dependence structure of increments (SDSI) is the determining characteristic of the limiting distribution. The SDSI of the limit process is quantified by means of two integro-differential relations obtained by renormalization in the sense of Leipnik (J. Aust. Math. Soc. B 32 , 327--347 (1991)). One is interpreted as a counterpart of the star equation of Mandelbrot and the other is shown to be an analogue of the classical Girsanov theorem. In the weak intermittency limit an approximate single-variable equation for the Laplace transform is obtained and successfully tested numerically by simulation.; In the second problem, a new framework for modeling the term structure of interest rates is introduced. The framework is based on the language of "diagonal processes". The LIBOR "diagonal process" is shown to induce a positive, arbitrage-free economy. Within our framework we give a multi-factor extension of the Markov-Functional Model of Hunt et al. (Finance and Stochastics 4, 391--408 (2000)). The extension preserves low-dimensionality and exact fitting of an initial caplet volatility surface. In addition, a discrete subset of an initial swaption volatility surface is also fit, its size depending on the number of factors. Calibration involves numerical integration and root finding only. LIBOR derivatives are priced on a lattice of dimension equal to the number of factors. The model is flexible enough to admit multifractals as processes underlying its dynamics.