2018/08/31 by Franziska Kühn · 1 citation
Mathematics · #math.PR #math.OC #msc:60J25 #msc:60G51 #msc:60J35 #msc:35D40 #msc:49L25 #msc:47H20 #msc:47J35
paper · pdf · doi:10.30757/alea.v16-20
published as ALEA Latin American Journal for Proability and Mathematical Statistics 16 (2019), 531-559 · fixed some typos, updated bibliography
arxiv created 2019/03/06 · arxiv updated 2019/06/14
Using probabilistic methods we study the existence of viscosity solutions to non-linear integro-differential equations ∂t u(t,x) - supα∈ I ( bα(x) ⋅ ∇x u(t,x) + (1)/(2) tr(Qα(x) ⋅ ∇2x u(t,x)) +∫y ≠ 0 (u(t,x+y)-u(t,x)-∇x u(t,x) ⋅ h(y) ) να(x,dy) ) = 0 with initial condition u(0,x)= φ(x); here (bα(x),Qα(x),να(x,dy)), α∈ I, x ∈ ℝd, is a family of Lévy triplets and h is some truncation function. The solutions, which we construct, are of the form u(t,x) = Tt φ(x) for a sublinear Markov semigroup (Tt)t ≥ 0 with representation Tt φ(x) = Ex φ(Xt):= sup_ℙ ∈ \mathfrakPx ∫Ω φ(Xt) dℙ where (Xt)t ≥ 0 is a stochastic process and \mathfrakPx, x ∈ ℝd, are families of probability measures. The key idea is to exploit the connection between sublinear Markov semigroups and the associated Kolmogorov backward equation. In particular, we obtain new existence and uniqueness results for viscosity solutions to Kolmogorov backward equations associated with Lévy(-type) processes for sublinear expectations and Feller processes on classical probability spaces.