We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set Θ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in Θ. The nonlinear Lévy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-differential equation.Let X = (X t ) t∈R + be an R d -valued process with càdlàg paths and X 0 = 0, defined on a measurable space (Ω, F) which is equipped with a nonlinear expectation E(·). For our purposes, this will be a sublinear operatorwhere P is a set of probability measures on (Ω, F) and E P [ · ] is the usual expectation, or integral, under the measure P . In this setting, if Y andfor all bounded Borel functions ϕ, and if Y and Z are of the same dimension, they are said to be identically distributed iffor all bounded Borel functions ϕ. We note that both definitions coincide with the classical probabilistic notions if P is a singleton. Following [8, Definition 19], the process X is a nonlinear Lévy process under E(·) if it has stationary and independent increments; that is, X t − X s and X t−s are identically distributed for all 0 ≤ s ≤ t, and X t − X s is independent of (X s 1 , . . . , X sn ) for all 0 ≤ s 1 ≤ · · · ≤ s n ≤ s ≤ t. The particular case of a classical Lévy process is recovered when P is a singleton. Let Θ be a set of Lévy triplets (b, c, F ); here b is a vector, c is a symmetric nonnegative matrix, and F is a Lévy measure. We recall that each Lévy triplet characterizes the distributional properties and in particular the infinitesimal generator of a classical Lévy process. More precisely, the asso-where, e.g., h(z) = z1 |z|≤1 . Our goal is to construct a nonlinear Lévy process whose local characteristics are described by the set Θ, in the sense that the analogue of the Kolmogorov equation will be the fully nonlinear (and somewhat nonstandard) partial integro-differential equation v t (t, x) − sup (b,c,F )∈Θ bv x (t, x) + 1 2 tr[cv xx (t, x)] (1.2) + [v(t, x + z) − v(t, x) − v x (t, x)h(z)] F (dz) = 0.

We establish the duality formula for the superreplication price in a setting of volatility uncertainty which includes the example of "random G-expectation." In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.

Given a càdlàg process X on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let P sem be the set of all probability measures P under which X is a semimartingale. We construct processes (B P , C, ν P ) which are jointly measurable in time, space, and the probability law P , and are versions of the semimartingale characteristics of X under P for each P ∈ P sem . This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.

We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi-closed form. Moreover, we provide a saddle point analysis describing a worst-case model.

In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.