2011/05/04 by Wanyang Dai, Dai, Wanyang
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Electrical engineering #FOS: Mathematics #FOS: Physical sciences #Financial Risk and Volatility Modeling #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and financial applications #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1105.0881
openalex publication_date 2011/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.