A stochastic approach to a new type of parabolic variational inequalities

2012/03/21 by Tianyang Nie, Nie, Tianyang
Computer Science · Mathematics · #49J40 #60H10 #60H30 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR) #math.PR #msc:49J40 #msc:60H10 #msc:60H30

paper · pdf · doi:10.48550/arxiv.1203.4840

38 pages

openalex publication_date 2012/03/21 · arxiv created 2012/03/23 · arxiv updated 2012/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the following quasilinear partial differential equation with two subdifferential operators: (∂ u)/(∂ s)(s,x) + (Lu)(s,x,u(s,x),(∇ u(s,x))^∗σ(s,x,u(s,x))) + f(s,x,u(s,x),(∇ u(s,x))^∗σ(s,x,u(s,x))) ∈ ∂φ(u(s,x)) + , (s,x) ∈[0,T]× Domψ, u(T,x) =g(x), x∈ Domψ. where for u∈ C^1,2([0,T]× Domψ) and (s,x,y,z)∈ [0,T]× Domψ× Domφ×ℝ^{1× d}, (Lu)(s,x,y,z) := 1/2∑_i,j=1ⁿ (σσ^∗)_i,j(s,x,y)\frac∂²u∂ x_i∂ x_j(s,x) +∑_i=1ⁿ b_i(s,x,y,z)(∂ u)/(∂ x_i)(s,x). The operator ∂ψ (resp. ∂φ) is the subdifferential of the convex lower semicontinuous function ψ:ℝⁿ→ (-∞,+∞] (resp. φ:ℝ→(-∞,+∞]). We define the viscosity solution for such kind of partial differential equations and prove the uniqueness of the viscosity solutions when σ does not depend on y. To prove the existence of a viscosity solution, a stochastic representation formula of Feymann-Kac type will be developed. For this end, we investigate a fully coupled forward-backward stochastic variational inequality.

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