2022/04/30 by Mingshang Hu, Hu, Mingshang, Lianzi Jiang +5 · 2 citations
Economics, Econometrics and Finance · Mathematics · Decision Sciences · #Stochastic processes and financial applications #Nonlinear Differential Equations Analysis #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2205.00203
This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for α∈(1,2), the i.i.d. sequence \ ( (1)/(√(n))∑i=1nXi,(1)/(n)∑ i=1nYi,\frac1√[α]n∑i=1nZi) \ n=1∞ converges in distribution to L1, where Lt=( ξt,ηt,ζt), t∈ [0,1], is a multidimensional nonlinear Lévy process with an uncertainty set Θ as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) \ ∂tu(t,x,y,z)-sup (Fμ,q,Q)∈ Θ \ ∫ℝdδλu(t,x,y,z)Fμ(dλ).
. +⟨ Dyu(t,x,y,z),q⟩+(1)/(2)tr[Dx2u(t,x,y,z)Q] \ =0,
u(0,x,y,z)=ϕ(x,y,z), ∀(t,x,y,z)∈ \lbrack 0,1]× ℝ3d, . with δλu(t,x,y,z):=u(t,x,y,z+λ)-u(t,x,y,z)-⟨ Dzu(t,x,y,z),λ⟩. To construct the limit process (Lt)t∈ \lbrack0,1], we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize (Lt)t∈ [0,1]. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.