G-Gaussian Processes under Sublinear Expectations and q-Brownian Motion in Quantum Mechanics

2011/05/05 by Shigē Péng, Shige Peng, Peng, Shige
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #46N50 #60G15 #60J65 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:46N50 #msc:60G15 #msc:60J65

paper · pdf · doi:10.48550/arxiv.1105.1055

arxiv created 2011/05/05 · openalex publication_date 2011/05/05 · arxiv updated 2011/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a general approach to construct a stochastic process with a given consistent family of finite dimensional distributions under a nonlinear expectation space. We use this approach to construct a generalized Gaussian process under a sublinear expectation and a q-Brownian motion. The later one is under a complex-valued linear expectation, with which a new type of Feynman-Kac formula can be derived to represent the solution of a Schrödinger equation.

Citations

• A New Central Limit Theorem under Sublinear Expectations
• Properties of hitting times for G-martingale
• Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures
• User’s guide to viscosity solutions of second order partial differential equations
• G-Lévy Processes under Sublinear Expectations
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
• Explicit solutions of G-heat equation with a class of initial conditions by G-Brownian motion
• Law of Large Numbers and Central Limit Theorem under Nonlinear Expectations
• Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes

Related

• G-Gaussian random fields and stochastic quantization under nonlinear expectation
• Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation
• G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type
• A Complex Version of G-Expectation and its Application to Conformal Martingale
• Quantum Stochastic Calculus and Quantum Gaussian Processes