Backward stochastic differential equations under super linear G-expectation and associated Hamilton-Jacobi-Bellman equations

2010/09/06 by Yuhong Xu, Xu, Yuhong
Decision Sciences · Economics, Econometrics and Finance · #35K05 #35K55 #49L25 #60H10 #60H30 #60J65 #Economic theories and models #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1009.1042

openalex publication_date 2010/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper first studies super linear G-expectation. Uniqueness and existence theorem for backward stochastic differential equations (BSDEs) under super linear expectation is established to provide probabilistic interpretation for the viscosity solution of a class of Hamilton-Jacobi-Bellman equations, including the well known Black-Scholes-Barrenblett equation, arising in the uncertainty volatility model in mathematical finance. We also show that BSDEs under super linear expectation could characterize a class of stochastic control problems. A direct connection between recursive super (sub) strategies with mutually singular probability measures and classical stochastic control problems is provided. By this result we give representation for solutions of Black-Scholes-Barrenblett equations and G-heat equations.

Citations

• 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
• Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes
• Stopping Times and Related Itô's Calculus with G-Brownian Motion

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