Malliavin calculus for jump processes and studies on densities

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740083
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Osaka City University
• Principal Investigator: TAKEUCHI Atsushi 大阪市立大学, 大学院理学研究科, 准教授 (30336755)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: マリアヴァン解析 / ジャンプ過程 / 確率微分方程式 / 確率関数微分方程式 / 密度関数

Outline of Final Research Achievements

The Wiener-Poisson space is the family of right-continuous functions with left hand limits. I have applied the Malliavin calculus on the Wiener-Poisson space in order to study the properties on densities for jump processes. In the study, I could get the results on the logarithmic derivatives of densities with respect to some parameters, which characterize the process. And I have obtained the Greeks computations with numerical simulations on asset price dynamical models with jumps in mathematical finance. Furthermore, I studied the asymptotic behavior of densities for solutions to stochastic functional differential equations, whose current state depends on the past histories of the process. The most remarkable point is that the delay parameter of the stochastic functional differential equation plays a crucial role.

Free Research Field

確率解析