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

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

We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, set $\Theta$ of triplets, we construct sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and generator corresponding to supremum all generators classical triplets in $\Theta$. The yields tractable model Knightian uncertainty about distribution jumps expectations Markovian functionals can be calculated by means partial...

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

We introduce multilevel Picard (MLP) approximations for McKean-Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that MLP algorithm approximates solution of SDE in $L^2$-sense without curse dimensionality. The latter means its computational cost grows at most polynomially both dimension and reciprocal prescribed error tolerance. In two numerical experiments, demonstrate applicability by...

Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform rigorous strong error analysis for SGD algorithms. particular, prove every arbitrarily small $\varepsilon \in (0,\infty)$ and large $p\in that the considered algorithm converges $L^p$-sense with order $\frac{1}{2}-\varepsilon$ to global minimum objective function stochastic approximation problem under standard convexity-type assumptions on...

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call nonlinear processes. This is done as follows: given set Θ parameters for the process, construct corresponding expectation on path space continuous By general dynamic programming principle, link this to variational form Kolmogorov equation, where generator single process replaced by supremum over all generators with in Θ. yields tractable model Knightian especially modelling interest rates...

Abstract We introduce a general framework for Markov decision problems under model uncertainty in discrete‐time infinite horizon setting. By providing dynamic programming principle, we obtain local‐to‐global paradigm, namely solving local, that is, one time‐step robust optimization problem leads to an optimizer of the global (i.e., time‐steps) stochastic optimal control problem, as well corresponding worst‐case measure. Moreover, apply this portfolio involving data . present two different...

We study a robust stochastic optimization problem in the quasi-sure setting discrete-time. show that under linearity-type condition admits maximizer. This is implied by no-arbitrage models of financial markets. As corollary, we obtain existence utility maximizer frictionless market model, markets with proportional transaction costs and also more general convex costs, like case impact.

Abstract In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, super‐replication price coincides with model‐free price. Namely, knowledge model does not reduce provide two families models: stochastic volatility models and rough models. Moreover, give several computational examples. Our approach is purely probabilistic.

We show that when the price process [Formula: see text] represents a fully incomplete market, optimal super-replication of any Markovian claim with being nonnegative and lower semicontinuous is buy-and-hold type. Since both (unbounded) stochastic volatility models rough are examples markets, one can interpret property super-replicating claims as natural phenomenon in markets.

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested the computation of model-free upper lower bounds for their arbitrage-free prices. work completely realistic setting, that only assume knowledge traded prices other single- multi-asset even allow presence bid–ask spread these provide fundamental theorem asset pricing this market model, as well superhedging duality result, allows to transform abstract maximization problem over...