A semigroup approach to nonlinear Lévy processes

Abstract

We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-toone relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators (A λ ) λ∈Λ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE ∂tu = sup λ∈Λ A λ u. The results are illustrated with several examples.Proof. As J π is a sublinear Markovian convolution for all π ∈ P t , the same holds for S (t). For f ∈ D, x ∈ G and ε > 0 there exists π 0 ∈ P t such thatBy Lemma 5.1 it follows thatFrom this, we obtain that lim tց0 S (t)f − f ∞ = 0 for f ∈ BUC(G) with the same density argument as in the proof of Lemma 5.2. 18ROBERT DENK, MICHAEL KUPPER, AND MAX NENDEL Lemma 5.4. For t ≥ 0, let (π n ) n∈N be a sequence in P t such that π n ⊂ π n+1 for all n ∈ N, and lim n→∞ |π n | ∞ = 0.ThenProof. Fix f ∈ BUC(G). For t = 0, the statement is trivial. For t > 0 and x ∈ G we defineThen, J ∞ is a sublinear Markovian convolution. Since π n ⊂ π n+1 , it follows from (5.3) that J πn f ր J ∞ f, as n → ∞. By definition of S (t), it clearly holds J ∞ f ≤ S (t)f . As for the other inequality, let π = {t 0 , t 1 , . . . , t m } ∈ P t with m ∈ N and 0 = t 0 < t 1 < . . . < t m = t. Since |π n | ∞ ց 0 as n → ∞, we may w.l.o.g. assume that #π n ≥ m + 1 for all n ∈ N. Moreover, we can choose 0 = t n 0 < t n 1 < . . . < t n m = t with π ′ n := {t n 0 , t n 1 , . . . , t n m } ⊂ π n and lim n→∞ t n i = t i for all i = 1, . . . , m − 1. Then, by Lemma 5.2, we have thatTaking the supremum over all π ∈ P t , we thus get thatCorollary 5.5. Let t ≥ 0. Then, there exists a sequence (π n ) in P t such thatMoreover, S (t)f = sup n∈N J t n n f = sup n∈N J 2 −n t 2 n f = lim n→∞ J 2 −n t 2 n f, (5.6) for all f ∈ BUC(G), where the supremum is understood pointwise.Proof. Choose π n := kt 2 n : k ∈ {0, . . . , 2 n } in Lemma 5.4 to obtain the first statement. In particular, S (t)f = S (t)f = sup n∈N J πn f = sup n∈N J 2 −n t 2 n

Affine processes under parameter uncertainty

“…Examples in this direction are the notions of g-Brownian motion and G-Brownian motion referring to a Brownian motion with drift or volatility uncertainty, see Peng (1997Peng ( , 2007a and references therein. Most recently, this theory has been extended to more general approaches, so-called non-linear Lévy processes, see Neufeld and Nutz (2017) and Denk et al (2017) in this regard.…”

Viscosity solutions to Hamilton–Jacobi–Bellman equations associated with sublinear Lévy(-type) processes

“…Corollary 4.2 generalizes [23] where it was shown that u(t, x) = sup P∈Px E P f (X t ) is a viscosity solution to (16) under the additional assumptions that f is Lipschitz continuous and that sup α∈I ∫ y >1 y ν α (dy) < ∞. Recently, Denk et al [8] studied semigroups associated with Lévy processes for sublinear expectations and showed that (15) implies the existence of a viscosity solution to (16) For the particular case that the index set I consists of a finitely many elements, the tightness condition (13) is automatically satisfied, and therefore Corollary 4.2 gives the following result for classical Lévy processes.…”

“…Since the proof of the maximal inequality for Feller processes relies essentially on Dynkin's formula, we can extend it to our framework. ⋅)), x ∈ R d , α ∈ I, be a family of Lévy triplets which is uniformly bounded on compact sets, i. e. which satisfies (8). For a given truncation function h denote by…”

“…Let q α (x, ⋅) ∶ R d → C, x ∈ R d , α ∈ I, be a family of continuous negative definite functions with characteristics (b α (x), Q α (x), ν α (x, ⋅)), α ∈ I, x ∈ R d , which is uniformly bounded on compact sets, i. e. which satisfies (8).…”

“…Using a version of Kolmogorov's extension theorem for non-linear expectations, Denk et al [8] recently established under the weaker condition…”