Nonlinear Expectations Research Articles

Dynamic programming principle and Hamilton-Jacobi-Bellman equation under nonlinear expectation

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under G-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution to a type of fully nonlinear HJB equation.

Gittins’ theorem under uncertainty

We study dynamic allocation problems for discrete time multi-armed bandits under uncertainty, based on the the theory of nonlinear expectations. We show that, under independence assumption on the bandits and with some relaxation in the definition of optimality, a Gittins allocation index gives optimal choices. This involves studying the interaction of our uncertainty with controls which determine the filtration. We also run a simple numerical example which illustrates the interaction between the willingness to explore and uncertainty aversion of the agent when making decisions.

<i>P</i>-DISTRIBUTION ALMOST PERIODIC SOLUTIONS OF SEMI-LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH <i>G</i>-BROWNIAN MOTION

As a class of recurrence, almost periodicity has been studied in stochastic differential equations (SDEs) under the framework of linear expectation. However, in the framework of nonlinear expectation, there are few literatures on Poisson stable solutions for SDEs and (pseudo) almost periodic solutions for SDEs with exponential dichotomy. This paper is devoted to the existence and asymptotical stability of <i>P</i>-distribution Poisson stable solutions for nonhomogeneous linear and semi-linear SDEs driven by <i>G</i>-Brownian motion satisfying exponential stability. Moreover, some existence results of (pseudo) almost periodic solution in <i>P</i>-distribution are established for semilinear SDEs driven by <i>G</i>-Brownian motion satisfying exponential dichotomy. Meanwhile, some examples are given to validate the obtained theoretical results.

Infinite horizon BSDEs under consistent nonlinear expectations

This paper considers backward stochastic differential equations (BSDEs for short) on the infinite horizon under consistent nonlinear expectations which are dominated by consistent sublinear expectations. We derive the existence and uniqueness result by discretization and approximation method. Besides, we show the existence of the solution of Markovian ergodic BSDEs under G˜-expectations and state some applications.

Stein's method under sublinear expectations

Stein's method is widely used for the error estimates of normal approximation. This is a review article about Stein's method under sublinear expectations, introducing the basic ideas of Stein's method under nonlinear expectation, and how to use it to give error estimates for the central limit theorem and the law of large numbers under sublinearexpectations.

Periodic dynamics for nonlocal Hopfield neural networks with random initial data

In this paper, a class of nonlocal Hopfield neural networks with random initial data is introduced, where the randomness may be of probability uncertainty. Sufficient conditions are derived to ensure the existence and globally exponential convergence of periodic solution for the addressed system in the frame of nonlinear expectation and linear expectation, respectively. Moreover, numerical examples are given to show the effectiveness of the obtained results.

$\alpha$-Hypergeometric Uncertain Volatility Models and their Connection to 2BSDEs

In this article we propose a $\alpha$-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connexion between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using asymptotic analysis for the G-HJB equation and the equivalent 2BSDE representation, we derive a limit model that provides an accurate description of the worst-case price scenario in cases when the bounds of the UV model are slowly varying. The analytical results are tested by numerical simulations using a deep learning based approximation of the underlying 2BSDE.

Optimal stopping under model ambiguity: A time‐consistent equilibrium approach

AbstractAn unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity‐averse an agent is. This inclusion of ambiguity attitude, via an ‐maxmin nonlinear expectation, renders the stopping problem time‐inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one‐dimensional diffusion with drift and volatility uncertainty, we show that any initial stopping policy will converge to an equilibrium through a fixed‐point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst‐case (or best‐case) analysis. In a concrete example of real options valuation under model ambiguity, all equilibrium stopping policies, as well as the best one among them, are fully characterized under appropriate conditions. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity‐averse, the more eager to stop—so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.

Robust deep hedging

We study pricing and hedging under parameter uncertainty for a class of Markov processes which we call generalized affine processes and which includes the Black–Scholes model as well as the constant elasticity of variance (CEV) model as special cases. Based on a general dynamic programming principle, we are able to link the associated nonlinear expectation to a variational form of the Kolmogorov equation which opens the door for fast numerical pricing in the robust framework. The main novelty of the paper is that we propose a deep hedging approach which efficiently solves the hedging problem under parameter uncertainty. We numerically evaluate this method on simulated and real data and show that the robust deep hedging outperforms existing hedging approaches in highly volatile periods.

Time-Consistent Conditional Expectation Under Probability Distortion

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

Robust Deep Hedging

We study pricing and hedging under parameter uncertainty for a class of Markov processes which we call generalized affine processes and which includes the Black-Scholes model as well as the constant elasticity of variance (CEV) model as special cases. Based on a general dynamic programming principle, we are able to link the associated nonlinear expectation to a variational form of the Kolmogorov equation which opens the door for fast numerical pricing in the robust framework. The main novelty of the paper is that we propose a deep hedging approach which efficiently solves the hedging problem under parameter uncertainty. We numerically evaluate this method on simulated and real data and show that the robust deep hedging outperforms existing hedging approaches, in particular in highly volatile periods.

Nonlinear expectations of random sets

Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting.We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

On nonlinear expectations and Markov chains under model uncertainty

The aim of this work is to give an overview on nonlinear expectations and to relate them to other concepts that describe model uncertainty or imprecision in a probabilistic framework. We discuss imprecise versions of stochastic processes with a particular interest in imprecise Markov chains. First, we focus on basic properties and representations of nonlinear expectations with additional structural assumptions such as translation invariance or convexity. In a second step, we discuss how stochastic processes under nonlinear expectations can be constructed via primal and dual representations. We illustrate the concepts by means of imprecise Markov chains with a countable state space, and show how families of Markov chains give rise to imprecise versions of Markov chains. We discuss dual representations and differential equations related to the latter.

Robust identification of investor beliefs

This paper develops a method informed by data and models to recover information about investor beliefs. Our approach uses information embedded in forward-looking asset prices in conjunction with asset pricing models. We step back from presuming rational expectations and entertain potential belief distortions bounded by a statistical measure of discrepancy. Additionally, our method allows for the direct use of sparse survey evidence to make these bounds more informative. Within our framework, market-implied beliefs may differ from those implied by rational expectations due to behavioral/psychological biases of investors, ambiguity aversion, or omitted permanent components to valuation. Formally, we represent evidence about investor beliefs using a nonlinear expectation function deduced using model-implied moment conditions and bounds on statistical divergence. We illustrate our method with a prototypical example from macrofinance using asset market data to infer belief restrictions for macroeconomic growth rates.

Markov chains under nonlinear expectation

AbstractIn 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.

A representation for filtration-consistent nonlinear expectations and its application

In this paper, we consider filtration-consistent nonlinear expectations which satisfy a general domination condition. We show that filtration-consistent nonlinear expectations can be represented by g-expectations defined by the solutions of backward stochastic differential equations, whose generators are independent of y and uniformly continuous in z. As an application, we establish a concentration inequality for time-consistent risk measures.

Exit times for semimartingales under nonlinear expectation

Let Eˆ be the upper expectation of a weakly compact but possibly non-dominated family P of probability measures. Assume that Y is a d-dimensional P-semimartingale under Eˆ. Given an open set Q⊂Rd, the exit time of Y from Q is defined by τQ≔inf{t≥0:Yt∈Qc}.The main objective of this paper is to study the quasi-continuity properties of τQ under the nonlinear expectation Eˆ. Under some additional assumptions on the growth and regularity of Y, we prove that τQ∧t is quasi-continuous if Q satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we obtain the quasi-continuity of exit times for multi-dimensional G-martingales, which nontrivially generalizes the previous one-dimensional result of Song (2011).

Asymptotic moment estimation for stochastic Lotka–Volterra model driven by G-Brownian motion

A stochastic Lotka–Volterra model disturbed by G-Brownian motion (G-LVM for short) in the framework of non-linear expectation is proposed in this paper. This model takes into account the uncertainty of variance of the noise. We prove the G-LVM exists a unique solution and the solution does not tend to infinity when the time is finite under some constraints, and obtain many asymptotic moment estimations which depend on the variance of G-Brownian motion by capacity theory, exponential martingale inequality and analytical skills.

Uncertainty and filtering of hidden Markov models in discrete time

We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter. We also investigate a simple control problem in this context.

Nonlocal stochastic functional differential equations driven by G‐Brownian motion and mean random dynamical systems

In this paper, we consider a class of nonlocal stochastic functional differential equations driven by G‐Brownian motion (GNSFDEs) at phase space whose coefficients are dependent on the p‐th moment. Existence and uniqueness of solutions for GNSFDEs are investigated by virtue of theory of nonlinear expectation. Furthermore, we show that the solution map of GNSFDEs can generate a p‐mean random dynamical system in nonlinear expectation framework. In addition, two examples are provided to illustrate the effectiveness of the obtained results