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

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this paper is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell-Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels. 1 2 R. DENK, M. KUPPER, and M. NENDELthe help of Choquet's capacibility theorem [10] we obtain the uniqueness of such an extension in a certain class of expectations. Thus, our extension result can be viewed as a generalization of the Daniell-Stone extension theorem, which states that a linear expectation E which is continuous from above on a Riesz subspace M has a unique linear extensionĒ to L ∞ over the σ-algebra σ(M) generated by M. While for linear expectations the extension is still continuous from above, the same does not hold for convex expectations. Note that the continuity from above of a convex expectation E on L ∞ is a very strong condition which, in particular, implies that E(X) = E(Y ) whenever X = Y µ-almost surely for some probability measure µ, and that the representing probability measures in the dual representation of E are dominated by µ as well. However, nonlinear expectations are continuous from above on certain subspaces of L ∞ , see e.g. Cheridito et al. [7] and the references therein. Hence, nonlinear expectations can be constructed by defining them on a subspace M and extending them to L ∞ , the space of bounded σ(M)-measurable functions.In the second part of the paper we illustrate this extension procedure in a Kolmogorov type setting. That is, for an arbitrary index set I we construct nonlinear expectations on L ∞ (S I ), where S I is the I-th product of a Polish space S. To that end, we first consider a family of expectations E J on linear subsetsindexed by the set H of all finite subsets of I. In line with Peng [26], under the natural consistency condition E K (f ) = E J (f • pr JK ) for every f ∈ M K and all J, K ∈ H with K ⊂ J, where pr JK denotes the projection from M J to M K , the family (E J ) can be extended to the space M := {f • pr J : f ∈ M J , J ∈ H }. Moreover, if each E J is convex and continuous from above on M J the same also holds for the extension on M. Hence, by the general extension result, Theorem 3.10, from the first part, there exists a convex expectationĒ on L ∞ which is continuous from below, such thatĒ(f • pr J ) = E J (f ) for all f ∈ M J and J ⊂ I finite, see Theorem 4.6. The corresponding dual version in the sublinear case leads to Theorem 4.7, which is a robust version of Kolmogorov's extension theor...

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In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

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