On nonlinear expectations and Markov chains under model uncertainty

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Abstract

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.

Introduction

Model uncertainty appears in many scientific disciplines, when, for example, due to statistical estimation methods, only a confidence interval for certain parameters of a model is known, or when certain aspects of a model cannot be determined exactly. In this context, one often speaks of imprecision or polymorphic uncertainty. In mathematical finance, model uncertainty or ambiguity is a frequent phenomenon since financial markets usually do not allow for repetition, whereas, in other disciplines, experiments can be repeated under similar conditions arbitrarily often. The most prominent example for ambiguity in mathematical finance is uncertainty with respect to certain parameters (drift, volatility, etc.) of the stochastic process describing the value of an underlying asset. This leads to the task of modeling stochastic processes under model uncertainty.

In mathematical finance, model uncertainty is often being described via nonlinear expectations, introduced by Peng [32]. Some of the most prominent examples of nonlinear expectations include the g-expectation, see Coquet et al. [11], describing a Brownian Motion with uncertainty in the drift parameter, and the G-expectation or G-Brownian Motion introduced by Peng [33], [34], describing a Brownian Motion with uncertain volatility. There is a close connection between g-expectations, backward stochastic differential equations (BSDEs) and semilinear partial differential equations. We refer to Coquet et al. [11] and Pardoux and Peng [30] for more details on this topic. We also refer to Cheridito et al. [6] and Soner et al. [35], [36] for the connection between G-expectations, 2BSDEs and fully nonlinear partial differential equations. Moreover, there is a one-to-one relation between sublinear expectations and coherent monetary risk measures as introduced by Artzner et al. [1] and Delbaen [14], [15]. Another related concept is the concept of a (Choquet) capacity (see e.g. Dellacherie-Meyer [16]) leading to Choquet integrals (see Choquet [7]).

On the other side, there is a large community working on similar questions related to model uncertainty in the field of imprecise probability. Here, the central objects are upper and lower previsions introduced by Walley [40]. In the sublinear case, there is a one-to-one relation between sublinear expectations and coherent upper previsions, which creates a huge intersection between the communities working on nonlinear expectations and upper/lower previsions. Within the field of imprecise probability, many work has been done in the direction of defining, axiomatizing, and computing transition operators of, both, discrete-time and continuous-time imprecise Markov chains, see e.g. De Bock [12], De Cooman et al. [13], Krak et al. [26], and Škulj [37], [38]. Concepts that are related to imprecise Markov chains include Markov set-chains, see Hartfiel [25], and, in the field of mathematical finance, BSDEs on Markov chains, see Cohen and Elliott [9], [10], and Markov chains under nonlinear expectations, see Nendel [27] and Peng [32].

The aim of this paper is to link and compare the concepts and results obtained in the fields of imprecise probability and mathematical finance. Since Markov chains under model uncertainty form the largest intersection between both communities, we put a special focus on the latter.

The paper is organized as follows: In Section 2, we start by introducing nonlinear expectations, and discussing basic properties and relations to upper/lower previsions, monetary risk measures and Choquet integrals. In Section 3, we present extension procedures for pre-expectations due to Denk et al. [17]. Here, we focus on two different extensions, one in terms of finitely additive measures, and the other in terms of countably additive measures. In Section 4, we discuss Kolmogorov-type extension theorems and the existence of stochastic processes under nonlinear expectations due to Denk et al. [17]. We conclude, in Section 5, by constructing imprecise versions of transition operators for families of time-homogeneous continuous-time Markov chains with countable state space. Here, we use an approach due to Nisio [29], which has been used in various contexts to construct imprecise versions of Markov processes, such as Lévy processes, Ornstein-Uhlenbeck processes, geometric Brownian Motions, and finite-state Markov chains, see Denk et al. [18], Nendel [27] and Nendel and Röckner [28]. Finally, we compare the Nisio approach to the methods used for continuous-time imprecise Markov chains in the field of imprecise probability. In Section 6, we conclude by summarizing the main insights and connections that are illustrated in the paper.

Section snippets

Nonlinear expectations and related concepts

In this section, we give an introduction into the theory of nonlinear expectations, and discuss related concepts. Throughout this section, let Ω be a nonempty set, and F2Ω be an arbitrary σ-algebra on Ω, where 2Ω denotes the power set of Ω. We emphasize that, throughout this section, F=2Ω is a possible choice for F. We denote the space of all bounded F-B(R)-measurable random variables X:ΩR by L(Ω,F), where B(R) denotes the Borel σ-algebra on R. The space L(Ω,F) and subspaces thereof are

Extension of pre-expectations

Let ML(Ω,F) with RM. Given a pre-expectation E:MR, we are looking for extensions of E to an expectation on L(Ω,F). Here, the main challenge is to preserve monotonicity. We start with the extension of linear pre-expectations.

Remark 3.1

Let ML(Ω,F) be a linear subspace of L(Ω,F) with 1M. We denote by ba+1(M) the space of all linear pre-expectations on M. A natural question is if the mappingba+1(Ω,F)ba+1(M),νν|M is bijective. The following theorem by Kantorovich shows that this mapping is

Stochastic processes under nonlinear expectation

In this section, we apply the extension results of the previous chapter to a Kolmogorov-type setting. That is, given a consistent family of finite-dimensional marginal expectations, we are looking for an expectation on a suitable path space with these marginals. As in the linear case, the formulation of Kolmogorov's consistency condition requires the distribution for finite-dimensional projections, cf. Remark 2.15. Our formulation of the consistency condition is very much in the spirit of the

Continuous-time Markov chains under nonlinear expectation

In this section, we consider time-homogeneous continuous-time Markov chains with a countable state space S (endowed with the discrete topology 2S). We identify (measurable) functions SR via sequences of the form (ui)iSRS, and use the notation :=L(S,2S). We call a (possibly nonlinear) map P: a kernel if

  • (i)

    PuPv for all u,v with uv,

  • (ii)

    Pα=α for all αR.

Here and throughout, we use the notation Pu=P(u) for a kernel P and u. In this section, we consider an arbitrary family of linear

Conclusion

In the present paper, we have discussed and compared the concept of nonlinear expectations to other concepts that are being used to describe model uncertainty or imprecision in a mathematical framework. In Section 2, we investigated basic properties of nonlinear expectations, their acceptance sets, and their convex conjugates. Moreover, we indicated how these concepts relate to concepts in the field of imprecise probability such as upper previsions, Choquet integrals, sets of desirable gambles,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    Financial support through the German Research Foundation via CRC 1283 is gratefully acknowledged.

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