Mathematics and Financial Economics

, Volume 12, Issue 1, pp 111–134 | Cite as

Backward nonlinear expectation equations

  • Christoph Belak
  • Thomas Seiferling
  • Frank Thomas Seifried
Article
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Abstract

Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.

Keywords

Backward stochastic differential equation Nonlinear expectation Random G-expectation Recursive utility Volatility uncertainty 

Mathematics Subject Classification

60G20 60H30 91B16 

JEL Classification

D81 D91 

1 Introduction

We study backward nonlinear expectation equations (BNEEs) of the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
(1.1)
Here \({\mathcal {E}}_t\) is a nonlinear expectation operator; g is a generator function; \(\mu \) is a finite positive measure; and \(\xi \) is the terminal value. Equation (1.1) is the continuous-time analog of the discrete-time nonlinear aggregation
$$\begin{aligned} X^\varDelta _{k} \triangleq {\mathcal {E}}_{t_k} \Bigl [ \mu \bigl ((t_k,t_{k+1}]\bigr )\, g\bigl (t_k,{\mathcal {E}}_{t_k} [X^\varDelta _{k+1}]\bigr ) + X^\varDelta _{k+1} \Bigr ], \quad k=N-1, \ldots , 0, \end{aligned}$$
(1.2)
where \(\varDelta : 0= t_0< t_1< \cdots < t_N = T\). We establish existence, uniqueness and stability for solutions of (1.1). In addition, we demonstrate that (1.1) emerges in the continuous-time limit of the discrete aggregations (1.2).
Our main motivation to study the BNEE (1.1) and the discrete-time aggregation (1.2) is to provide a rigorous justification of continuous-time recursive utility specifications under ambiguity. If c is a consumption plan and \(\xi \) denotes the utility of a terminal lump-sum payment at time T, the continuous-time recursive utility of \((c,\xi )\) under ambiguity is defined as \(U_0\), where U solves
$$\begin{aligned} U_t = {{\mathrm{{\mathrm {ess}}}}}\inf _{Q\in {\mathcal {Q}}_t} {{\mathrm{{\mathrm {E}}}}}_t^{Q} \Bigl [{\textstyle \int _t^T} f(c_s,U_s) {\mathrm {d}}s + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
(1.3)
for some aggregator function f. The set of propability measures \({\mathcal {Q}}_t\) can be thought of as time-t conditional probabilistic models. Under suitable regularity assumptions on \({\mathcal {Q}}_t\), the operator \({\mathcal {E}}_t \triangleq {{\mathrm{{\mathrm {ess}}}}}\inf _{Q\in {\mathcal {Q}}_t} {{\mathrm{{\mathrm {E}}}}}_t^{Q}\) defines a dynamic nonlinear expectation, leading in a natural way to the investigation of the BNEE (1.1). Specific choices of \({\mathcal {Q}}_t\) have been considered in Chen and Epstein [7] (drift uncertainty) and Epstein and Ji [16] (drift and volatility uncertainty), among others. In the special case where \({\mathcal {Q}}_t\) is a singleton (i.e., \({\mathcal {E}}_t\) reduces to a classical linear expectation), we recover the stochastic differential utility model introduced in the seminal work of Duffie and Epstein [12].
To justify (1.3) from an economic perspective, one argues that the recursive utility process U emerges as the continuous-time limit of a sequence of discrete-time recursive utility processes \(U^\varDelta \) given by \(U_N^\varDelta = \xi \) and
$$\begin{aligned} U^\varDelta _{k} \triangleq W\bigl (t_{k+1} - t_k, c_k^\varDelta ,{\mathcal {E}}_{t_k}[U_{k+1}^\varDelta ]\bigr ),\quad k=N-1,\ldots ,0; \end{aligned}$$
(1.4)
see, e.g., Kreps and Porteus [25]. In (1.4), W is a discrete-time aggregator and \(c^\varDelta \) is a suitable discretization of the consumption plan c. With a suitable specification of W, the recursion (1.4) reduces to the discrete-time aggregation (1.2), and hence we are lead to the question of convergence of the discrete-time aggregation to the continuous-time BNEE. In the linear case without ambiguity, i.e. if \({\mathcal {Q}}_t\) is a singleton, the convergence has been proved in Kraft and Seifried [24]. One objective of this paper is to generalize this to models with ambiguity, thus providing a rigorous economic underpinning for the models of, e.g., Chen and Epstein [7] and Epstein and Ji [16].
Loosely speaking, the BNEE (1.1) may be thought of as a backward stochastic differential equation (BSDE) in a nonlinear expectation framework. Indeed, in the special case of a classical linear conditional expectation \({\mathcal {E}}_t = {{\mathrm{{\mathrm {E}}}}}_t\), a generator g that depends on X only through its current value, and \(\mu \) being the Lebesgue measure, the BNEE (1.1) rewrites as the BSDE
$$\begin{aligned} \textstyle X_t&= \textstyle {{\mathrm{{\mathrm {E}}}}}_t\Bigl [\int _t^T g(s,X_s){\mathrm {d}}s + \xi \Bigr ], \quad t\in [0,T],\\&\textstyle \text {or equivalently}\quad {\mathrm {d}}X_t = g(t,X_t){\mathrm {d}}t + {\mathrm {d}}M_t,\quad X_T=\xi \end{aligned}$$
where M is a martingale. Existence and uniqueness results for this class of BSDEs first appeared in Duffie and Epstein [12]. In a Brownian setting, BSDEs had previously been studied in Bismut [4] and Pardoux and Peng [31]. Subsequently, BSDEs have been widely used in a range of applications including option pricing, risk measures, stochastic differential utility, and more; see, e.g., El Karoui et al. [14] or Delong [10] for an overview.

As indicated above, the prototype of a nonlinear expectation \(\mathcal E_t\) is a dynamic version of the worst-case expectation over a set \({\mathcal {Q}}\) of probability measures, i.e. a dynamic, time-consistent analogue of \({\mathcal {E}}\triangleq \inf _{Q\in {\mathcal {Q}}}{{\mathrm{{\mathrm {E}}}}}^Q\). As each \(Q\in {\mathcal {Q}}\) may be thought of as one probabilistic model, the BNEE (1.1) can be interpreted as a BSDE under model uncertainty. Indeed, the most prominent nonlinear expectations in the literature fall into this class: g-expectations studied in Peng [32, 33] capture drift uncertainty, see Chen and Epstein [7]; G-expectations as put forward by Peng [35, 36] model volatility uncertainty with constant bounds on the volatility process; and random G-expectations as introduced and analyzed by Nutz [27, 28] can be understood as a specification of volatility uncertainty with random bounds on the volatility process. Another class of nonlinear expectations are dynamic risk measures, see e.g. Riedel [40]. Thus, in light of the broad range of applications of BSDEs in financial and economic models, our fundamental existence, uniqueness, and stability results for BNEEs contribute to the study of financial and economic models in the presence of uncertainty.

In the literature, BNEEs have previously appeared only in special cases, and for specific choices of the nonlinear expectation: Peng [33, 34] considers the case of g-expectations; the case of G-expectations has been investigated in, among others, Peng [37], and Hu et al. [21, 22], the latter building on the G-martingale representation results of Soner et al. [43]. A related class of equations known as second-order BSDEs (2BSDEs) has been introduced by Cheridito et al. [9] and Soner et al. [45, 46]; see also Soner et al. [44] for related results. In the economics literature, BNEEs have appeared in the context of dynamic robust risk preferences; see, e.g., Chen and Epstein [7], Hayashi [20], and Epstein and Ji [16]. The general framework of the present paper includes g- and G-expectations as well as the models of Chen and Epstein [7] and Epstein and Ji [16]. We also discuss how BNEEs link to the theory of 2BSDEs, and we show that our general framework makes it possible to study BSDEs in a random G-expectation setting.

Our results contribute to the existing literature in several directions:
  1. 1.

    We develop an abstract framework that makes it possible to study a big class of nonlinear expectations in a unified framework,

     
  2. 2.

    establish new existence, uniqueness, and stability results for BSDEs in a random G-expectation setting (BNEEs),

     
  3. 3.

    prove convergence of the discrete-time aggregations (1.2), and

     
  4. 4.

    provide a rigorous justification of the continuous-time recursive utility models with ambiguity of Chen and Epstein [7], Epstein and Ji [16].

     
The remainder of this article is organized as follows: Sect. 2 introduces the nonlinear expectation framework. In Sect. 3 we provide existence, uniqueness, and stability results for BNEEs and establish convergence of the associated discrete-time nonlinear aggregations. In Sect. 4 we establish a regularity result that embeds a class of random G-expectations (and, in particular, classical G-expectations) into the framework of this article. In Sect. 5 we present an application of our results to discrete- and continuous-time recursive utility under ambiguity by showing that the discrete-time utility process \(U^\varDelta \) in (1.4) converges to the continuous-time utility process U in (1.3). Finally, Appendix A gathers some auxiliary results and Appendix B provides a simplified proof of regularity for classical G-expectations.

2 Nonlinear expectation framework

In this section we formulate an abstract framework for the analysis of backward nonlinear expectation equations that subsumes g-, G- and random G-expectations.

2.1 Nonlinear expectations and appropriate domains

To begin with, we introduce a family of spaces \(\{{\mathrm {L}}_t\}_{t\in [0,T]}\) on which the nonlinear expectation is to be defined. Although one may intuitively think of \({\mathrm {L}}_t\) as a space of equivalence classes of random variables that are measurable with respect to the information available at time t, at this point the definition of \({\mathrm {L}}_t\) is deliberately kept abstract so that it also subsumes nonlinear expectations where different specifications are necessary. In particular, \({\mathrm {L}}_t\) does in general not carry any probabilistic structure.

To make this more precise, let \(\varOmega \ne \emptyset \) be a set and let \({\mathcal {N}}\subset 2^\varOmega \) be a non-empty collection of subsets of \(\varOmega \) that is closed under countable unions and does not contain \(\varOmega \). Then \({\mathcal {N}}\) induces an equivalence relation \(\sim _{{\mathcal {N}}}\) on \(\varOmega ^{\mathbb {R}}\) via
$$\begin{aligned} f\sim _{{\mathcal {N}}} g\Longleftrightarrow f(\omega ) = g(\omega )\text { for all }\omega \in \varOmega {\setminus } N\text { and some } N\in {\mathcal {N}}. \end{aligned}$$
The set \(\{f\in \varOmega ^{\mathbb {R}}: f\sim _{\mathcal {N}}0\}\) forms a linear subspace of \(\varOmega ^{\mathbb {R}}\), and the corresponding quotient space is denoted by \(\varOmega ^{\mathbb {R}}/ {\mathcal {N}}\). By definition, pointwise operations on \(\varOmega ^{\mathbb {R}}/{\mathcal {N}}\) are well-behaved: If \(\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(f\sim _{{\mathcal {N}}} g\), then \(\varphi \circ f\sim _{{\mathcal {N}}} \varphi \circ g\). Moreover, \(\varOmega ^{\mathbb {R}}/{\mathcal {N}}\) inherits the pointwise partial order on \(\varOmega ^{\mathbb {R}}\) via
$$\begin{aligned} X\le Y\text { in }\varOmega ^{\mathbb {R}}/{\mathcal {N}}\Longleftrightarrow f(\omega )\le g(\omega )\text { for all }\omega \in \varOmega \text { and some }f\in X,g\in Y, \end{aligned}$$
and the canonical injection \(i : {\mathbb {R}}\rightarrow \varOmega ^{\mathbb {R}}/{\mathcal {N}}\), \(x\mapsto [\varOmega \rightarrow {\mathbb {R}}, \omega \mapsto x]_{\sim _{\mathcal {N}}}\) is order preserving.

Definition 2.1

(Lebesgue family) A family \({\mathscr {L}}= \{({\mathrm {L}}_t,\Vert {\cdot }\Vert _{\mathrm {L}})\}_{t\in [0,T]}\) of separable Banach spaces is called a Lebesgue family if the following statements hold true for all \(t\in [0,T]\):
  1. (L1)

    \({\mathrm {L}}_t\) is a linear subspace of \(\varOmega ^{\mathbb {R}}/{\mathcal {N}}\),

     
  2. (L2)

    \({\mathrm {L}}_s\subset {\mathrm {L}}_t\) for all \(0\le s\le t\),

     
  3. (L3)

    \(|X|\in {\mathrm {L}}_t\) whenever \(X\in {\mathrm {L}}_t\),

     
  4. (L4)

    \({\mathrm {L}}_0 = i({\mathbb {R}})\) and \(i:{\mathbb {R}}\rightarrow {\mathrm {L}}_0\) is an isometry, and

     
  5. (L5)

    \({\mathrm {L}}_t^+ \triangleq \{X \in {\mathrm {L}}_t : X\ge 0\}\) is closed.\(\diamond \)

     

The canonical example of a Lebesgue family is as follows: Suppose \((\varOmega ,\mathcal A,\{{\mathcal {F}}_t\}_{t\in [0,T]},P)\) is a complete filtered probability space, \({\mathcal {F}}_0\) is P-trivial, and \({\mathbb {L}}^1(\varOmega ,\mathcal A,P)\) is separable. Then the family \({\mathscr {L}}= \{{\mathbb {L}}^1(\varOmega ,{\mathcal {F}}_t,P) : t\in [0,T]\}\) of Banach spaces of equivalence classes of integrable random variables, equipped with the usual norm \(\Vert {\cdot }\Vert _{\mathrm {L}}\triangleq {{\mathrm{{\mathrm {E}}}}}[|{\cdot }|]\), is a Lebesgue family. In this case \({\mathcal {N}}\) is chosen to be the collection of P-null sets.

Lebesgue families provide the natural structure to define dynamic nonlinear expectations.

Definition 2.2

(Nonlinear expectation) Let \({\mathscr {L}}= \{({\mathrm {L}}_t,\Vert {\cdot }\Vert _{\mathrm {L}})\}_{t\in [0,T]}\) be a Lebesgue family. A family \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) of operators \({\mathcal {E}}_t: {\mathrm {L}}_T \rightarrow {\mathrm {L}}_t\), \(t\in [0,T]\), is a (time-consistent) nonlinear expectation on \(\mathscr {L}\) if the following conditions are satisfied: For all \(X, Y \in {\mathrm {L}}_T\) and \(t \in [0,T]\) the map \({\mathcal {E}}_t\) is
  1. (M)

    monotone, i.e. \(X\le Y\) implies \({\mathcal {E}}_t[X]\le {\mathcal {E}}_t[Y]\),

     
  2. (SI)

    shift-invariant, i.e. \({\mathcal {E}}_t[X +Y] = X + {\mathcal {E}}_t[Y]\) if \(X\in {\mathrm {L}}_t\),

     
  3. (TC)

    time-consistent, i.e. \({\mathcal {E}}_t[{\mathcal {E}}_s[X]] = {\mathcal {E}}_t[X]\) for every \(s\in [t,T]\), and

     
  4. (N)

    normalized, i.e. \({\mathcal {E}}_t[0] = 0\).

     
A nonlinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\) is called sublinear if each \({\mathcal {E}}_{t}^{{\mathrm {sub}}}\) is
  1. (PH)

    positively homogeneous, i.e. \({\mathcal {E}}_{t}^{{\mathrm {sub}}}[\lambda X] = \lambda {\mathcal {E}}_{t}^{{\mathrm {sub}}}[X]\) for all \(\lambda > 0\), and

     
  2. (SUB)

    sub-additive, i.e. \({\mathcal {E}}_{t}^{{\mathrm {sub}}}[X+Y] \le {\mathcal {E}}_{t}^{{\mathrm {sub}}}[X] + {\mathcal {E}}_{t}^{{\mathrm {sub}}}[Y]\).\(\diamond \)

     
From (SI) and (N) it is obvious that a nonlinear expectation also
  1. (PC)

    preserves constants, i.e. \({\mathcal {E}}_t[X]= X\) for all \(X\in {\mathrm {L}}_t\) and \(t\in [0,T]\).1

     
For a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\) we set \({\mathcal {E}}_{}^{{\mathrm {sub}}} \triangleq {\mathcal {E}}_{0}^{{\mathrm {sub}}}\) and define a semi-norm on \({\mathrm {L}}_T\) via
$$\begin{aligned} \rho : {\mathrm {L}}_T \rightarrow [0, \infty ), \quad X \mapsto {\mathcal {E}}_{}^{{\mathrm {sub}}} [|X|]. \end{aligned}$$
(2.1)
If \(\rho \) is consistent with \(\Vert {\cdot }\Vert _{\mathrm {L}}\), then we refer to the Lebesgue family as appropriate.

Definition 2.3

(Appropriate domain) A Lebesgue family \({\mathscr {L}}\) is called an appropriate domain for \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\) if \(\rho = \Vert {\cdot }\Vert _{\mathrm {L}}\) on \({\mathrm {L}}_T\). \(\diamond \)

In this article, we focus on nonlinear expectations that are defined on appropriate domains. This restriction is necessary if one insists on a linear structure for the domain of the nonlinear expectation. For instance, Nutz and van Handel [30] show that it is not possible to construct G-expectations on a linear space containing all Borel functions on Wiener space.

In general, nonlinear expectations \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) that are not sublinear cannot generate their own norm via (2.1) above. To accommodate such nonlinear expectations, we consider expectation operators that are dominated by a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) in the sense of the following definition:

Definition 2.4

Let \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) be a nonlinear expectation such that
$$\begin{aligned} {\mathcal {E}}_t[X] - {\mathcal {E}}_t[Y] \le {\mathcal {E}}_{t}^{{\mathrm {sub}}}[X-Y]\quad \text {for all }X,Y\in {\mathrm {L}}_T\ \text {and }t\in [0,T] \end{aligned}$$
for a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\). Then \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is said to be dominated by \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\).\(\diamond \)
Dominated nonlinear expectations satisfy the triangle inequality
$$\begin{aligned} |{\mathcal {E}}_t[X] - {\mathcal {E}}_t[Y]| \le {\mathcal {E}}_{t}^{{\mathrm {sub}}} [|X-Y|] \quad \text {for all }X,Y\in {\mathrm {L}}_T \end{aligned}$$
(2.2)
and thus \(|{\mathcal {E}}_t[X]| \le {\mathcal {E}}_{t}^{{\mathrm {sub}}} [|X|]\). From this we obtain the projection property
$$\begin{aligned} \Vert {\mathcal {E}}_t[X]\Vert _{\mathrm {L}}\le \Vert X\Vert _{{\mathrm {L}}} \quad \text {for all } t \in [0,T] \text { and } X \in {\mathrm {L}}_T \end{aligned}$$
(2.3)
if \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) is defined on an appropriate domain. Finally, note that a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) dominates itself.

2.2 Example: g-expectation

Before we develop our general framework further, we illustrate the definitions of the previous section in the context of g-expectations; for additional specifications we refer to Seiferling [42]. In Sect. 4 we consider the more general notion of classical and random G-expectations in more detail.

Let \((\varOmega ,\mathcal A,P)\) be a complete probability space supporting a d-dimensional Brownian motion W and, with \(E\triangleq {\mathbb {R}}^l{\setminus }\{0\}\), a Poisson random measure \(\varGamma :\varOmega \times {\mathcal {B}}([0,T]\times E)\rightarrow {\mathbb {N}}_0\cup \{\infty \}\).2 We assume that the compensator \(\Lambda \) of \(\varGamma \) takes the form \(\Lambda ({\mathrm {d}}t,{\mathrm {d}}e) = \gamma ({\mathrm {d}}e){\mathrm {d}}t\) for a \(\sigma \)-finite measure \(\gamma \) on \((E,{\mathcal {B}}(E))\) with \(\int _E(1\wedge |e|^2)\gamma ({\mathrm {d}}e) < \infty \), and we denote by \(\tilde{\varGamma }\triangleq \varGamma -\Lambda \) the compensated jump measure and by \(\{{\mathcal {F}}_t\}_{t\in [0,T]}\) the augmented filtration generated by \((W,\varGamma )\).

In this setting we consider BSDEs of the form
$$\begin{aligned} \textstyle X_t = \xi + \int _t^T f(s,Y_s,Z_s)\, {\mathrm {d}}s - \int _t^T Y_s^\top \,{\mathrm {d}}W_s - \int _t^T\int _E Z_s(e)\tilde{\varGamma }({\mathrm {d}}s,{\mathrm {d}}e) \end{aligned}$$
(2.4)
for a driver \(f:\varOmega \times [0,T]\times {\mathbb {R}}^d\times {\mathbb {L}}^2(\gamma )\rightarrow {\mathbb {R}}\). Under standard measurability, integrability, and Lipschitz assumptions on f, the BSDE (2.4) admits a unique solution (XYZ) for every terminal condition \(\xi \in {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_T,P)\), and this solution satisfies
$$\begin{aligned} \textstyle {{\mathrm{{\mathrm {E}}}}}\bigl [\sup _{t\in [0,T]}|X_t|^2 + \int _0^T|Y_t|^2\,{\mathrm {d}}t + \int _0^T \Vert Z_t(e)\Vert ^2_{{\mathbb {L}}^2(\gamma )}\,{\mathrm {d}}t\bigr ] < \infty , \end{aligned}$$
(2.5)
see Tang and Li [47].
To construct the g-expectation we take as given two drivers g and h such that h is sublinear, \(g({\cdot },0,0) = 0 = h({\cdot },0,0)\), and h dominates g in that
$$\begin{aligned} g({\cdot },y_1,z_1) - g({\cdot },y_2,z_2) \le h({\cdot },y_1-y_2,z_1-z_2) \end{aligned}$$
for all \(y_1,y_2\in {\mathbb {R}}^d\), \(z_1,z_2\in {\mathbb {L}}^2(\gamma )\). Moreover, we assume that for both \(f=g\) and \(f=h\), the following comparison principle holds: Let \(\xi \in {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_T,P)\), let (XYZ) be the unique solution of (2.4), and suppose that \((\bar{X},\bar{Y},\bar{Z})\) satisfies the integrability condition (2.5). If \(\xi \ge \eta = \bar{X}_T\in {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_T,P)\) and
$$\begin{aligned} \textstyle {\mathrm {d}}\bar{X}_t = -\bigl [f(t,\bar{Y}_t,\bar{Z}_t) + \beta _t\bigr ]\,{\mathrm {d}}t + \bar{Y}_t^\top \,{\mathrm {d}}W_t + \int _E\bar{Z}_t(e)\tilde{\varGamma }({\mathrm {d}}t,{\mathrm {d}}e) \end{aligned}$$
with \(\beta \le 0\), then \(\bar{X}\le X\). Similarly, if \(\xi \le \eta \) and \(\beta \ge 0\), then \(\bar{X}\ge X\). Sufficient conditions for this to hold are provided by, e.g., El Karoui et al. [14, 41] and Royer [39]. Under these assumptions, the family of operators
$$\begin{aligned} {\mathcal {E}}_t^g : {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_T,P) \rightarrow {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_t,P),\quad {\mathcal {E}}_t^g[\xi ] \triangleq X_t, \end{aligned}$$
where X is the unique solution of (2.4) with \(f=g\), is easily seen to satisfy (M), (SI), (TC), and (N) of Definition 2.2; we refer to Royer [41] for details. In the same way, h generates a sublinear expectation \(\{{\mathcal {E}}_t^h\}_{t\in [0,T]}\), and since h dominates g, by comparison \(\{{\mathcal {E}}_t^h\}_{t\in [0,T]}\) dominates \(\{{\mathcal {E}}_t^g\}_{t\in [0,T]}\) in the sense of Definition 2.4. To obtain an appropriate domain for \(\{{\mathcal {E}}_t^g\}_{t\in [0,T]}\) and \(\{{\mathcal {E}}_t^h\}_{t\in [0,T]}\), one can take
$$\begin{aligned} {\mathrm {L}}_t\quad \text {as the closure of}\quad {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_t,P)\quad \text {with respect to}\quad \Vert {\cdot }\Vert _{\mathrm {L}}\triangleq {\mathcal {E}}_0^h[|{\cdot }|]. \end{aligned}$$
We refer to Seiferling [42] for further details and discussions.

2.3 Processes and integrals

By definition, a stochastic process is a family \(X = \{X_t\}_{t\in [0,T]}\) of random variables. In line with our notion of \({\mathrm {L}}_t\) as the space of time-t measurable, integrable random variables, the natural definition of processes in the context of nonlinear expectations is as follows.

Definition 2.5

(\({\mathscr {L}}\)Process) Let \({\mathscr {L}}= \{({\mathrm {L}}_t, \Vert \cdot \Vert _{\mathrm {L}})\}_{t \in [0,T]}\) be a Lebesgue family. An \({\mathscr {L}}\)-process is a function \(X: [0,T] \rightarrow {\mathrm {L}}_T\). It is measurable if X is \({\mathcal {B}}([0,T])\)-\({\mathcal {B}}({\mathrm {L}}_T)\)-measurable, and adapted if \(X_t \in {\mathrm {L}}_t\) for all \(t \in [0,T]\). We denote the set of all measurable and adapted \({\mathscr {L}}\)-processes by \(\mathcal X\). \(\diamond \)

Let \(\mu \) be a positive finite measure on \({\mathcal {B}}([0,T])\) and set
$$\begin{aligned} \Vert X\Vert _{{\mathrm {P}}} \triangleq {\textstyle \int _{[0,T]}} \Vert X_t\Vert _{\mathrm {L}}\, \mu ({\mathrm {d}}t) , \quad X \in \mathcal X. \end{aligned}$$
Then \(\Vert {\cdot }\Vert _{{\mathrm {P}}}\) is a seminorm on \({\mathcal {P}}\triangleq \{ X \in \mathcal X : \Vert X\Vert _{{\mathrm {P}}} < \infty \}\). Identifying \(X,Y\in {\mathcal {P}}\) if \(\Vert X-Y\Vert _{{\mathrm {P}}}=0\) we obtain the associated quotient space \({\mathrm {P}}\). It follows that \(({\mathrm {P}},\Vert {\cdot }\Vert _{{\mathrm {P}}})\) is a Banach space; see Lemma A.1 in the appendix for the proof. For every \(X\in {\mathrm {P}}\), one can define the Bochner integral
$$\begin{aligned} \textstyle \int _A X\, {\mathrm {d}}\mu \in {\mathrm {L}}_T,\quad A\in {\mathcal {B}}([0,T]) \end{aligned}$$
of X over A with respect to \(\mu \). Adopting a standard convention, we write \(\int _a^b X\,{\mathrm {d}}\mu \triangleq \int _{(a,b]} X\,{\mathrm {d}}\mu \) for \((a,b]\subset [0,T]\). To emphasize the integration variable, we also write \(\int _A X_t\,\mu ({\mathrm {d}}t) \triangleq \int _A X\, {\mathrm {d}}\mu \). Note that since \(1_{A} X\) is a measurable mapping \([0,T] \rightarrow {\mathrm {L}}_t\), the integral of each process \(X \in {\mathrm {P}}\) satisfies
$$\begin{aligned} {\textstyle \int _A} X_s \, \mu ({\mathrm {d}}s) = {\textstyle \int _{[0,T]}}1_{A}(s) X_s\, \mu ({\mathrm {d}}s) \in {\mathrm {L}}_t \quad \text {for all } A \in {\mathcal {B}}([0,t]). \end{aligned}$$
(2.6)
We conclude this section by introducing subspaces of \({\mathrm {P}}\) that represent natural domains for backward nonlinear expectation equations.

Definition 2.6

We define
$$\begin{aligned} {\mathrm {S}}\triangleq \bigl \{ X \in \mathcal X : \Vert X\Vert _{{\mathrm {S}}} <\infty \bigr \}\quad \text {where}\quad \Vert X\Vert _{{\mathrm {S}}} \triangleq {\textstyle \sup _{t \in [0,T]}} \Vert X_t\Vert _{{\mathrm {L}}}, \ X \in \mathcal X. \end{aligned}$$
Then \(({\mathrm {S}},\Vert {\cdot }\Vert _{{\mathrm {S}}})\) is a Banach space and
$$\begin{aligned} {\mathrm {D}}\triangleq \bigl \{X \in {\mathrm {S}}: [0,T] \rightarrow {\mathrm {L}}_T, \ t \mapsto X_t \text { is c}\grave{a}\text {dl}\grave{a}\text {g} \bigr \} \end{aligned}$$
is a closed subspace.\(\diamond \)

Since \(\Vert X\Vert _{{\mathrm {P}}} \le \mu ([0,T]) \Vert X\Vert _{{\mathrm {S}}}\) for \(X \in {\mathrm {S}}\) we can regard \({\mathrm {S}}\subset {\mathrm {P}}\) as a subspace. Moreover, the integral induces a continuous linear operator \({\mathrm {P}}\rightarrow {\mathrm {D}}\); see Lemma A.2 in the appendix.

3 Backward nonlinear expectation equations

Let \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) be a nonlinear expectation dominated by a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) on an appropriate domain \({\mathscr {L}}= \{ ({\mathrm {L}}_t, \Vert {\cdot }\Vert _{{\mathrm {L}}})\}_{t \in [0,T]}\). We are interested in solutions of backward nonlinear expectation equations (BNEEs) of the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \, \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
(3.1)
where X is an \({\mathscr {L}}\)-process and \((g,\xi )\) is a suitable parameter (see Definition 3.3 below). When \({\mathcal {E}}_t\) is a linear expectation, such backward equations have been studied extensively in the literature; see, e.g., Pardoux and Peng [31], Antonelli [1] and El Karoui et al. [14]. To the best of our knowledge, Chen and Epstein [7] and Peng [33] are the first to formulate equations of the form (3.1) under nonlinear expectations (more precisely: g-expectations). In the following, we provide a general theory of BNEEs in the framework set out in Sect. 2.
First, note that it is natural to require a priori that \(X\in {\mathrm {S}}\). Indeed, for the nonlinear expectation and the integral on the right-hand side of (3.1) to be well-defined, \(g({\cdot },X)\) must be an integrable \({\mathscr {L}}\)-process. On the other hand, a solution X of the equation is automatically adapted, so \(X\in \mathcal X\). Moreover, by the projection property,
$$\begin{aligned} {\textstyle \sup _{ t \in [0,T]}} \Vert X_t\Vert _{{\mathrm {L}}} \le {\textstyle \int _0^T} \Vert g(t,X)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) + \Vert \xi \Vert _{{\mathrm {L}}}< \infty . \end{aligned}$$
In summary, the BNEE (3.1) only makes sense for \(X\in {\mathrm {S}}\).

Remark 3.1

Note that the measurability condition implied by \(X\in {\mathrm {S}}\) is not satisfied automatically: For a general nonlinear expectation and \(\xi \in {\mathrm {L}}_T\), the mapping \(t\mapsto {\mathcal {E}}_t[\xi ]\) need not be well-behaved, so even the simplest BNEE
$$\begin{aligned} X_t = {\mathcal {E}}_t [\xi ], \quad t \in [0,T] \end{aligned}$$
may not admit a solution in \({\mathrm {S}}\). \(\diamond \)

Definition 3.2

A nonlinear expectation \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is measurable if for each \(\xi \in {\mathrm {L}}_T\) the mapping
$$\begin{aligned}{}[0,T]\rightarrow {\mathrm {L}}_T,\quad t\mapsto {\mathcal {E}}_t[\xi ]\quad \text { is } {\mathcal {B}}([0,T])-{\mathcal {B}}({\mathrm {L}}_T)-\text { measurable.} \end{aligned}$$
If it is càdlàg, then \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is called regular.\(\diamond \)

In Sect. 4 we establish regularity for a large class of random G-expectations (including classical G-expectations). Regularity of g-expectations follows from Lemma A.3 in the appendix with \(M\triangleq {\mathbb {L}}^2(\varOmega ,{\mathcal {F}}_T,P)\) using classical stability results for BSDEs; see, e.g., Proposition 2.2 in Barles et al. [3]. In particular, all of the following existence, uniqueness, stability, and discretization results apply to these classes of nonlinear expectations.

3.1 Existence and uniqueness

In view of Remark 3.1, we restrict attention to measurable nonlinear expectations for our analysis of BNEEs. For \(X \in \mathcal X\) and \(t \in [0,T]\) we write \(X^t \triangleq 1_{[t,T]} X \in \mathcal X\) as short-hand notation.

Definition 3.3

((Standard) BNEE parameter) Let \(\xi \in {\mathrm {L}}_T\) and \(g:[0,T] \times {\mathrm {S}}\rightarrow {\mathrm {L}}_T\) with \(g(\cdot ,0) \in {\mathrm {P}}\), \(g(\cdot , X) \in \mathcal X\) for all \(X \in {\mathrm {S}}\), and
$$\begin{aligned} \Vert g(t, X) - g(t, Y)\Vert _{{\mathrm {L}}} \le L \Vert X^t - Y^t\Vert _{{\mathrm {S}}} \quad \text { for } X , Y \in {\mathrm {S}},\ t \in [0,T]. \end{aligned}$$
(3.2)
Then the pair \((g,\xi )\) is called a BNEE parameter.
Similarly, if \(f:[0,T] \times {\mathrm {L}}_T \rightarrow {\mathrm {L}}_T\) is \({\mathcal {B}}([0,T]) \otimes {\mathcal {B}}({\mathrm {L}}_T)\)-\({\mathcal {B}}({\mathrm {L}}_T)\)-measurable with \(f(\cdot , 0 ) \in {\mathrm {P}}\), \(f(t, \eta ) \in {\mathrm {L}}_t\) whenever \(\eta \in {\mathrm {L}}_t\), \(t \in [0,T]\), and
$$\begin{aligned} \Vert f(t, \zeta ) - f(t, \eta )\Vert _{{\mathrm {L}}} \le L \Vert \zeta - \eta \Vert _{{\mathrm {L}}} \quad \text {for all } \zeta ,\eta \in {\mathrm {L}}_T \text { and } t \in [0,T], \end{aligned}$$
then \((f,\xi )\) is called a standard BNEE parameter. \(\diamond \)
Note that (3.2) and continuity of the integral imply
$$\begin{aligned} \Vert g(\cdot , X)\Vert _{{\mathrm {P}}} \le \mu ([0,T]) L \Vert X\Vert _{{\mathrm {S}}} + \Vert g (\cdot ,0)\Vert _{{\mathrm {P}}}\quad \text {for all }X\in {\mathrm {S}}. \end{aligned}$$
(3.3)
The BNEE associated to a BNEE parameter \((g,\xi )\) is given by
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \,\mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
The standard BNEE parameter \((f,\xi )\) canonically induces the BNEE parameter \((g,\xi )\) given by
$$\begin{aligned} g: [0,T] \times {\mathrm {S}}\rightarrow {\mathrm {L}}_T, \quad (t,X) \mapsto f(t, X_t), \end{aligned}$$
and the BNEE associated to \((f,\xi )\) takes the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} f(s, X_s) \,\mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
The remainder of this subsection is devoted to the proof of the following result:

Theorem 3.4

(Existence and uniqueness for BNEEs) Let \((g,\xi )\) be a BNEE parameter and suppose \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is measurable. Then the BNEE (3.1) has a unique solution \(X\in {\mathrm {S}}\). If \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is regular, then \(X \in {\mathrm {D}}\).

The proof is based on a classical fixed point (Picard iteration) approach. We first show that the corresponding iteration operator is well-defined.

Lemma 3.5

Under the assumptions of Theorem 3.4, the formula
$$\begin{aligned} (\varPhi X)_t \triangleq {\mathcal {E}}_t\Bigl [{\textstyle \int _t^T} g(s,X) \,\mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
(3.4)
defines an operator \( \varPhi : {\mathrm {S}}\rightarrow {\mathrm {S}}\). If \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is regular, then \(\varPhi ({\mathrm {S}}) \subset {\mathrm {D}}\).

Proof

Let \(X \in {\mathrm {S}}\). Then \(Y \triangleq g(\cdot , X) \in {\mathrm {P}}\) by (3.3) and \(\int _0^T Y_t \, \mu ({\mathrm {d}}t) \in {\mathrm {L}}_T\). Since \(\{{\mathcal {E}}_t\}_{ t\in [0,T]}\) is measurable, the \({\mathscr {L}}\)-process M given by
$$\begin{aligned} M_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _0^T} Y_s\, \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
is a member of \({\mathrm {S}}\) (and of \({\mathrm {D}}\) if \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) is regular). By Lemma A.2 the process I given by \(I_t \triangleq {\textstyle \int _0^t} Y_s\, \mu ({\mathrm {d}}s)\), \(t\in [0,T]\), is in \({\mathrm {D}}\). Thus \(\varPhi X \triangleq M-I \in {\mathrm {S}}\) (and \(\varPhi X\in {\mathrm {D}}\) if \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) is regular) and (3.4) follows from (SI). \(\square \)

Proof of Theorem 3.4

Let \(U, V\in {\mathrm {S}}\) and \(0 \le t \le s \le T\). By the triangle inequality (2.2) and the projection property (2.3)
$$\begin{aligned} \Vert (\varPhi U)_s - (\varPhi V)_s\Vert _{{\mathrm {L}}} \le \bigl \Vert {\textstyle \int _s^T} g(r, U) -g (r, V)\,\mu ({\mathrm {d}}r) \bigr \Vert _{{\mathrm {L}}}. \end{aligned}$$
Hence continuity of the integral and (3.2) imply
$$\begin{aligned} \Vert (\varPhi U)_s - (\varPhi V)_s\Vert _{{\mathrm {L}}} \le L {\textstyle \int _s^T} \Vert U^r - V^r\Vert _{{\mathrm {S}}} \,\mu ({\mathrm {d}}r) \le L {\textstyle \int _t^T} \Vert U^r - V^r\Vert _{{\mathrm {S}}}\, \mu ({\mathrm {d}}r) \end{aligned}$$
for all \(s \in [t,T]\). We therefore have for any \(t\in [0,T]\)
$$\begin{aligned} \Vert (\varPhi U)^t - (\varPhi V)^t \Vert _{{\mathrm {S}}} = \sup _{s\in [t,T]} \Vert (\varPhi U)_s - (\varPhi V)_s\Vert _{{\mathrm {L}}} \le L {\textstyle \int _t^T} \Vert U^s - V^s\Vert _{{\mathrm {S}}}\, \mu ({\mathrm {d}}s). \end{aligned}$$
Iterating this estimate yields
$$\begin{aligned} \Vert \varPhi ^n U - \varPhi ^n V\Vert _{{\mathrm {S}}}&\le L^n {\textstyle \int _0^T} {\textstyle \int _{t_1}^T} \cdots {\textstyle \int _{t_{n-1}}^T} \Vert U^{t_n} - V^{t_n}\Vert _{{\mathrm {S}}} \, \mu ({\mathrm {d}}t_n) \cdots \mu ({\mathrm {d}}t_2) \mu ({\mathrm {d}}t_1)\\&\le L^n \Vert U - V\Vert _{{\mathrm {S}}} \mu ([0,T])^n/n! \end{aligned}$$
so \(\varPhi \) is a contraction for sufficiently large \(n\in {\mathbb {N}}\). Thus \(\varPhi \) has a unique fixed point \(X \in {\mathrm {S}}\) that satisfies (3.1). Finally, if \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) is regular, then \(X=\varPhi X\in {\mathrm {D}}\) by Lemma 3.5. \(\square \)

3.2 Stability of BNEEs

In this and the following subsection we address the stability and discrete-time approximation of BNEEs. Related stability results for linear aggregation equations can be found in Antonelli [2], El Karoui et al. [14], Barles et al. [3], as well as Peng [33] and the references therein. In the context of nonlinear expectation equations, we have the following result.

Theorem 3.6

(Stability of BNEEs) Let \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) be measurable and let \((g^n,\xi ^n)\), \(n \in {\mathbb {N}}\), and \((g,\xi )\) be BNEE parameters. Suppose there is a constant \(L>0\) such that for all \(n\in {\mathbb {N}}\)
$$\begin{aligned} \Vert g^n(t, X) - g^n(t, Y)\Vert _{{\mathrm {L}}} \le L \Vert X^t - Y^t\Vert _{{\mathrm {S}}} \quad \text { for all } X , Y \in {\mathrm {S}},\ t \in [0,T]. \end{aligned}$$
Let \(X^n\), \(n\in {\mathbb {N}}\), and X denote the solutions of the associated BNEEs and suppose that
$$\begin{aligned} {\textstyle \int _0^T} \Vert g^n(t, X) - g(t, X)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) \rightarrow 0 \quad \text {and} \quad \xi ^n \rightarrow \xi \text { in }{\mathrm {L}}_T. \end{aligned}$$
Then \(X^n \rightarrow X\) in \({\mathrm {S}}\).

Proof

Let \(0 \le t \le s \le T\). By the triangle inequality (2.2), the projection property (2.3) and continuity of the integral, we have
$$\begin{aligned} \Vert X_s^n - X_s \Vert _{{\mathrm {L}}} \le {\textstyle \int _s^T} \Vert g^n(r,X^n) - g(r,X) \Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}r) + \Vert \xi ^n - \xi \Vert _{{\mathrm {L}}}. \end{aligned}$$
The Lipschitz condition (3.2) implies
$$\begin{aligned} \Vert g^n(r,X^n) - g(r,X)\Vert _{{\mathrm {L}}} \le L \Vert (X^n)^r - X^r \Vert _{{\mathrm {S}}} + \Vert g^n(r,X) - g(r,X) \Vert _{{\mathrm {L}}} \end{aligned}$$
for all \(r\in [s,T]\). Therefore
$$\begin{aligned} \Vert (X^n)^t - X^t\Vert _{{\mathrm {S}}} \le L {\textstyle \int _t^T} \Vert (X^n)^s - X^s\Vert _{{\mathrm {S}}}\, \mu ({\mathrm {d}}s) +\delta _n\quad \text {for all }t\in [0,T] \end{aligned}$$
where \(\delta _n \triangleq {\textstyle \int _0^T} \Vert g^n(t, X) - g(t, X)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) + \Vert \xi ^n -\xi \Vert _{{\mathrm {L}}} \rightarrow 0\). We conclude by Gronwall’s inequality. \(\square \)

3.3 Discretization of BNEEs

We now address the discrete-time approximation of BNEEs. In the linear case, related results can be found in, e.g., Zhang [48], Bouchard and Touzi [6] and Cheridito and Stadje [8]; see also Bouchard and Elie [5] and the references therein as well as Remark 3.9 below.

In the following, we fix a standard BNEE parameter \((f, \xi )\) and let
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} f(s, X_s)\, \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
(3.5)
denote the unique solution of the corresponding BNEE. To ease notation, we assume without loss of generality that \(\mu (\{0\})=\mu (\{T\})=0\). We are interested in a suitable discrete-time approximation of X that converges in the continuous-time limit of vanishing grid size. More specifically, given a partition \(\varDelta : \ 0 =t_0< t_1< \cdots < t_{N(\varDelta )} = T\) of [0, T] we set
$$\begin{aligned} |\varDelta | \triangleq \max _{k=1, \ldots ,N(\varDelta )}\bigl \{ \mu \bigl ((t_{k-1},t_k)\bigr ) + |t_k-t_{k-1}| \bigr \}. \end{aligned}$$
We consider the \(\varDelta \)-discretization scheme Open image in new window that is defined by a suitable approximation of the terminal value \(X^\varDelta _{N(\varDelta )}\triangleq \xi ^\varDelta \) and
$$\begin{aligned} X^\varDelta _{k} \triangleq {\mathcal {E}}_{t_k} \Bigl [ \mu \bigl ((t_k,t_{k+1}]\bigr ) f^\varDelta \bigl (t_k,{\mathcal {E}}_{t_k} [X^\varDelta _{k+1}]\bigr ) + X^\varDelta _{k+1} \Bigr ] \end{aligned}$$
(3.6)
for \(k=N(\varDelta )-1, \ldots , 0\). We are interested in the convergence \(X^\varDelta \rightarrow X\) for vanishing grid size \(|\varDelta |\).3 In (3.6) the mapping \(f^\varDelta \) is a standard BNEE parameter that may depend on the grid \(\varDelta \) and approximates f as \(|\varDelta |\rightarrow 0\) in the sense made precise in the following definition.

Definition 3.7

Let \(\{\varDelta ^n\}_{n \in {\mathbb {N}}}\) be a sequence of partitions \(\varDelta ^n: 0= t_0^n< t_1 ^n<\cdots < t^n_{N_n} = T\), \(n\in {\mathbb {N}}\), and let \(\{(f^n, \xi ^n)\}_{n \in {\mathbb {N}}}\) be a sequence of standard BNEE parameters. Suppose there is a constant \(L>0\) such that
$$\begin{aligned} \Vert f^n(t,\zeta ) - f^n(t,\eta ) \Vert _{{\mathrm {L}}} \le L \Vert \zeta -\eta \Vert _{{\mathrm {L}}} \quad \text {for all }\zeta ,\eta \in {\mathrm {L}}_T, n\in {\mathbb {N}}. \end{aligned}$$
(3.7)
If \(|\varDelta ^n| \rightarrow 0\), \(\xi ^n \rightarrow \xi \) in \({\mathrm {L}}_T\) and
$$\begin{aligned} {\textstyle \sum _{k=0}^{N_n-1}} {\textstyle \int _{[t_k^n,t_{k+1}^n)}} \Vert f(s, X_s) - f^n(t_k^n, X_s) \Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}s) \rightarrow 0, \end{aligned}$$
(3.8)
then \(\{(\varDelta ^n, f^n, \xi ^n)\}_{n\in {\mathbb {N}}}\) is said to be \((f,\xi )\)-exhausting. \(\diamond \)

Remark 3.8

If \([0,T]\rightarrow {\mathrm {L}}_T,\ t\mapsto f(t,\eta )\) is left-continuous for every \(\eta \in {\mathrm {L}}_T\) and \(\{\varDelta ^n\}_{n\in {\mathbb {N}}}\) is any sequence of partitions with \(|\varDelta ^n| \rightarrow 0\), then \(\{(\varDelta ^n,f,\xi )\}_{n\in {\mathbb {N}}}\) is \((f,\xi )\)-exhausting.\(\diamond \)

Remark 3.9

Note that in the discretization scheme (3.6) the aggregator \(f^{\varDelta }\) is evaluated at \({\mathcal {E}}_{t_k} [X^\varDelta _{k+1}]\), while in classical discretization schemes for linear BSDEs \(f^\varDelta \) is typically evaluated at either \(X^\varDelta _{k}\) or \(X^\varDelta _{k+1}\); see, e.g., Zhang [48] or Bouchard and Touzi [6]. Clearly, from a computational perspective, our scheme is less favorable since it involves an additional estimation of \({\mathcal {E}}_{t_k}[X^\varDelta _{k+1}]\) in each time step. The reason for our choice is that our main objective is proving convergence of recursive models (in which case our formulation is the natural one; see Sect. 5 below), rather than devising numerical methods for the computation of \(X^\varDelta \). \(\diamond \)

For simplicity of notation we write \(X^n \triangleq X^{\varDelta ^n}\) in the sequel. The main result of this subsection is the following convergence theorem:

Theorem 3.10

Suppose that \(\{{\mathcal {E}}_t\}_{t \in [0,T]}\) is regular. Let \((f,\xi )\) be a standard BNEE parameter and let \(\{(\varDelta ^n, f^n ,\xi ^n)\}_{n \in {\mathbb {N}}}\) be \((f,\xi )\)-exhausting. Let X denote the unique solution of (3.5) and let \(X^n = X^{\varDelta ^n}\) be given by (3.6). Then
$$\begin{aligned} \max _{k=0, \ldots , N_n} \Vert X^n_k - X_{t_k^n}\Vert _{{\mathrm {L}}} \rightarrow 0. \end{aligned}$$

Proof

It will be useful to introduce the following continuous-time interpolation of \(X^n\) (with a slight abuse of notation, also denoted by \(X^n\)): For \(n \in {\mathbb {N}}\) we put \(X_T^n \triangleq \xi ^n\) and for \(k=N_n-1, \ldots , 0\) we set
$$\begin{aligned} X_t^n \triangleq {\mathcal {E}}_{t} \Bigl [ \mu \bigl ( (t, t_{k+1}^n] \bigr )f^n\bigl (t_k^n, {\mathcal {E}}_{t_k^n}[X^n_{t_{k+1}^n}] \bigr ) + X^n_{t^n_{k+1}} \Bigr ],\quad t \in [t_k^n, t_{k+1}^n). \end{aligned}$$
(3.9)
Moreover, for each \(n\in {\mathbb {N}}\) we define the generator \(g^n:[0,T]\times {\mathrm {S}}\rightarrow {\mathrm {L}}_T\) by
$$\begin{aligned} g^n(\cdot ,Y) \triangleq {\textstyle \sum _{k=0}^{N_n-1}} 1_{[t_k^n, t_{k+1}^n)} f^n(t_k^n, {\mathcal {E}}_{t_k^n} [Y_{t_{k+1}^n}]),\quad Y\in {\mathrm {S}}. \end{aligned}$$
(3.10)
Lemma A.4 in the appendix shows that \((g^n,\xi ^n)\) is a BNEE parameter and, with the constant \(L>0\) from (3.7), we have for all \(n \in {\mathbb {N}}\)
$$\begin{aligned} \Vert g^n(t,U) - g^n(t,V)\Vert _{{\mathrm {L}}} \le L \Vert U^t - V^t\Vert _{{\mathrm {S}}} \quad \text { for all }U,V\in {\mathrm {S}},\ t \in [0,T]. \end{aligned}$$
Moreover, Lemma A.5 shows that \(X^n\in {\mathrm {D}}\) is the unique solution of the BNEE
$$\begin{aligned} X_t^n = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g^n(s, X^n) \,\mu ({\mathrm {d}}s) + \xi ^n \Bigr ], \quad t \in [0,T]. \end{aligned}$$
In view of Theorem 3.6, it thus remains to prove that
$$\begin{aligned} {\textstyle \int _0^T} \Vert g^n(t,X) - f(t,X_t)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) \rightarrow 0. \end{aligned}$$
For this, let \(t \in [0,T]\) and \(k_n \in {\mathbb {N}}\) with \(t\in [t_{k_n}^n, t_{k_n+1}^n)\), \(n \in {\mathbb {N}}\). We note that
$$\begin{aligned}&\Vert g^n(t,X)- f(t,X_t) \Vert _{{\mathrm {L}}}\\&\le \Vert g^n(t,X) - f^n(t_{k_n}^n,X_t)\Vert _{{\mathrm {L}}} + \Vert f^n(t_{k_n}^n, X_t) - f(t,X_t)\Vert _{{\mathrm {L}}}. \end{aligned}$$
By definition \(g^n(t,X) = f^n(t_{k_n}^n, {\mathcal {E}}_{t_{k_n}^n} [X_{t_{k_n+1}^n}])\) and hence by (3.7)
$$\begin{aligned} \Vert g^n(t, X) - f^n(t_{k_n}^n, X_{t})\Vert _{{\mathrm {L}}} \le L \Vert {\mathcal {E}}_{t_{k_n}^n} [X_{t_{k_n+1}^n}] - X_t\Vert _{{\mathrm {L}}}. \end{aligned}$$
The triangle inequality (2.2) and the projection property (2.3) thus imply
$$\begin{aligned}&\Vert g^n(t, X) - f^n(t_{k_n}^n, X_{t})\Vert _{{\mathrm {L}}}\nonumber \\&\le L \Vert {\mathcal {E}}_{t_{k_n}^n}[X_{t_{k_n+1}^n}] - {\mathcal {E}}_{t_{k_n}^n}[X_t]\Vert _{{\mathrm {L}}}+ L \Vert {\mathcal {E}}_{t_{k_n}^n}[X_t] - X_t\Vert _{{\mathrm {L}}}\nonumber \\&\le L \Vert X_{t_{k_n+1}^n} - X_t\Vert _{{\mathrm {L}}}+ L \Vert {\mathcal {E}}_{t_{k_n}^n} [X_{t}] - X_t\Vert _{{\mathrm {L}}} \le 4 L \Vert X\Vert _{{\mathrm {S}}}. \end{aligned}$$
(3.11)
Since \(\{(\varDelta ^n, f^n ,\xi ^n)\}_{n \in {\mathbb {N}}}\) is \((f,\xi )\)-exhausting, we have \(t_{k_n}^n, t_{k_n+1}^n \rightarrow t\) as \(n \rightarrow \infty \) for all \(t\in [0,T]\). As \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) is regular, for each \(t\in [0,T]\) the mappings
$$\begin{aligned}{}[0,T] \rightarrow {\mathrm {L}}_T, \ s \mapsto X_s \quad \text {and} \quad [0,T] \rightarrow {\mathrm {L}}_T, \ s \mapsto {\mathcal {E}}_s [X_t] \end{aligned}$$
are càdlàg. Hence we obtain \(\Vert X_{t_{k_n+1}^n}- X_t\Vert _{{\mathrm {L}}}\rightarrow 0\) for all \(t\in [0,T]\). On the other hand, since \(\Vert {\mathcal {E}}_{t_{k_n}^n} [X_{t}] - X_t\Vert _{{\mathrm {L}}}\rightarrow 0\) for all but countably many \(t\in [0,T]\) and \(t_{k_n}^n = t\) for all but finitely many \(n\in {\mathbb {N}}\) if \(\mu (\{t\})>0\), we also have \(\Vert {\mathcal {E}}_{t_{k_n}^n} [X_{t}] - X_t\Vert _{{\mathrm {L}}}\rightarrow 0\) for \(\mu \)-a.e. \(t\in [0,T]\). Thus using (3.11) and dominated convergence we obtain
$$\begin{aligned}&{\textstyle \int _0^T} \Vert g^n(t,X) - f(t,X_t)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t)\nonumber \\&\le {\textstyle \sum _{k=0}^{N_n -1}} {\textstyle \int _{[t_k^n,t_{k+1}^n)}} \Vert g^n(t, X) - f^n(t_{k}^n, X_{t})\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) \nonumber \\&\quad + {\textstyle \sum _{k=0}^{N_n -1}} {\textstyle \int _{[t_k^n,t_{k+1}^n)}} \Vert f^n(t_{k}^n, X_t) - f(t,X_t)\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}t) \rightarrow 0 \end{aligned}$$
by (3.8) since \(\{(\varDelta ^n ,f^n, \xi ^n)\}_{n\in {\mathbb {N}}}\) is \((f, \xi )\)-exhausting. \(\square \)

4 Regularity for random \(\varvec{G}\)-expectations

In this section we establish regularity for random G-expectations; as a consequence, the results of the previous section apply to this class of nonlinear expectations. Random G-expectations have been introduced by Nutz [27, 28]; see also Nutz and Soner [29]. Here we focus on random G-expectations defined in terms of non-Markovian control problems as in Nutz [27], where we assume that the coefficients of the dynamics of the state process are bounded. This class includes the classical, non-random G-expectations developed in Peng [35, 36] and Denis et al. [11] as a special case and is also of interest in economics; see, e.g., Epstein and Ji [16] and Sect. 5. This section focuses on the general setting of random G-expectations; since regularity of classical (non-random) G-expectations can be verified by simpler arguments, we also outline an alternative proof for this case in Appendix B.

4.1 Preliminaries

Following Nutz [27], we first briefly review the construction of the random G-expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\). For \(0\le t\le s\le T\) consider the Wiener spaces
$$\begin{aligned} \varOmega _s^t \triangleq \{\omega : [t,s] \rightarrow {\mathbb {R}}^d \,:\, \omega \text { is continuous}\} \end{aligned}$$
equipped with the supremum norm \(\Vert {\cdot }\Vert _\infty \) and write \(\varOmega ^t \triangleq \varOmega ^t_T\), \(\varOmega _s\triangleq \varOmega ^0_s\) and \(\varOmega \triangleq \varOmega _T\). Note that \(\varOmega _s^t\) canonically identifies with a subspace of \(\varOmega _r^u\) for \(0\le r\le s\le t\le u\le T\). Moreover let \(W^t_s(\omega ) = \omega _s\) be the canonical process on \(\varOmega ^t\), let \(P^t\) the Wiener measure on \(\varOmega ^t\) and let \({\mathcal {F}}^t\) be the raw filtration generated by \(W^t\). For \(t=0\) the index t is dropped.
Next let U be a non-empty Borel subset of \({\mathbb {R}}^m\) and fix bounded functions \(\mu :[0,T]\times \varOmega \times U\rightarrow {\mathbb {R}}^d\), \(\sigma :[0,T]\times \varOmega \times U\rightarrow {\mathbb {R}}^{d\times d}\) such that \((r,\omega ) \mapsto \mu (r,X(\omega ),\nu _r(\omega ))\) and \((r,\omega ) \mapsto \sigma (r,X(\omega ),\nu _r(\omega ))\) are progressively measurable whenever X is continuous and adapted and \(\nu \) is progressive. The functions \(\mu (r,{\cdot },u)\) and \(\sigma (r,{\cdot },u)\) are assumed to be Lipschitz continuous, uniformly in (ru). For \(\eta \in \varOmega \) and \(t\in [0,T]\), the conditioned coefficients are defined as
$$\begin{aligned} \mu ^{t,\eta }&: [0,T]\times \varOmega ^t\times U\rightarrow {\mathbb {R}}^d,&\mu ^{t,\eta }(r,\omega ,u) \triangleq \mu (r,\eta \otimes _t\omega ,u),\\ \sigma ^{t,\eta }&: [0,T]\times \varOmega ^t\times U\rightarrow {\mathbb {R}}^{d\times d},&\sigma ^{t,\eta }(r,\omega ,u) \triangleq \sigma (r,\eta \otimes _t\omega ,u), \end{aligned}$$
where \((\eta \otimes _t\omega )_r \triangleq \eta _r 1_{[0,t)}(r) + (\eta _t + \omega _r - \omega _t)1_{[t,T]}(r)\) for \(r\in [0,T]\).
Let \(\mathcal U^t\) denote the set of all \({\mathcal {F}}^t\)-progressively measurable, U-valued processes. Then for every \(\eta \in \varOmega \) and \(\nu \in \mathcal U^t\) and under each \(P^t\), the SDE
$$\begin{aligned} X_s = \eta _t + {\textstyle \int _t^s} \mu ^{t,\eta }(r,X,\nu _r){\mathrm {d}}r + {\textstyle \int _t^s} \sigma ^{t,\eta }(r,X,\nu _r) {\mathrm {d}}W^t_r,\quad s\in [t,T], \end{aligned}$$
admits a \(P^t\)-a.s. unique solution \(X = X(t,\eta ,\nu )\). We fix \(x\in {\mathbb {R}}^d\) and assume that \({\mathcal {F}}\subset {\mathcal {F}}^X\), where \({\mathcal {F}}^X\) is the P-augmentation of the filtration generated by \(\{X(0,x,\nu ): \nu \in \mathcal U\}\).4 For \(\eta \in \varOmega \) and \(\nu \in \mathcal U^t\) set
$$\begin{aligned} Q(t,\eta ,\nu ) \triangleq P^t\circ (X(t,\eta ,\nu )-\eta _t)^{-1}\quad \text {and}\quad \mathcal Q \triangleq \{Q(0,x,\nu ): \nu \in \mathcal U\}. \end{aligned}$$
(4.1)
Given this framework, we define the spaces \({\mathrm {L}}_t\) as follows: We set
$$\begin{aligned} \textstyle \Vert {\cdot }\Vert _{{\mathrm {L}}}\triangleq \sup _{Q\in \mathcal Q} {{\mathrm{{\mathrm {E}}}}}^Q\left[ |{\cdot }|\right] , \end{aligned}$$
and, identifying random variables that coincide \(\mathcal Q\)-q.s., we let \({\mathbb {L}}^1(\mathcal Q)\) denote the Banach space of \({\mathcal {F}}_T\)-measurable random variables X with \(\Vert X\Vert _{{\mathrm {L}}}<\infty \). Moreover, we define \({\mathrm {L}}_t\) as the \(\Vert {\cdot }\Vert _{{\mathrm {L}}}\)-closure of \(C_{b}^{\mathrm {unif}}(\varOmega _t)\subset {\mathbb {L}}^1(\mathcal Q)\), where \(C_{b}^{\mathrm {unif}}(\varOmega _t)\) denotes the set of bounded and uniformly continuous functions on \(\varOmega _t\). Below we will show that \({\mathscr {L}}= \{({\mathrm {L}}_t, \Vert {\cdot }\Vert _{{\mathrm {L}}})\}_{t \in [0,T]}\) is an appropriate domain.

4.2 Construction of the random G-expectation

Following Nutz [27], for \(\xi \in C_b^\mathrm {unif}(\varOmega )\) the random G-expectation is defined \(\omega \) by \(\omega \) as the value function
$$\begin{aligned} V_t(\xi ,\omega ) \triangleq {\textstyle \sup _{\nu \in \mathcal U^t}} {{\mathrm{{\mathrm {E}}}}}^{Q(t,x\otimes \omega ,\nu )}[\xi ^{t,x\otimes \omega }],\qquad (t,\omega )\in [0,T]\times \varOmega , \end{aligned}$$
where \(x\otimes \omega \triangleq x\otimes _0\omega \) and \(\xi ^{t,\omega }(\tilde{\omega }) \triangleq \xi (\omega \otimes _t\tilde{\omega })\) for \(\tilde{\omega }\in \varOmega \) and \(t\in [0,T]\).5 Lemma 4.3 in Nutz [27] implies that \(V_t:C_b^{\mathrm {unif}}(\varOmega )\rightarrow C_b^{\mathrm {unif}}(\varOmega _t)\) and that \(V_t\) is Lipschitz continuous with respect to \(\Vert {\cdot }\Vert _{{\mathrm {L}}}\). Clearly \(V_t\) is sublinear, monotone and normalized on \(C_b^{\mathrm {unif}}(\varOmega )\). Since \(\xi ^{t,\omega } = \xi (\omega )\) for \(\xi \in C_b^{\mathrm {unif}}(\varOmega _t)\) it follows that \(V_t\) is shift-invariant. Finally, time-consistency is guaranteed by a deep result of Nutz [27], see his Theorem 3.2. It follows that \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\), where
$$\begin{aligned} {\mathcal {E}}_{t}^{{\mathrm {sub}}}: {\mathrm {L}}_T \rightarrow {\mathrm {L}}_t\quad \text {is the continuous extension of}\quad V_t:C_b^{\mathrm {unif}}(\varOmega )\rightarrow C_b^{\mathrm {unif}}(\varOmega _t), \end{aligned}$$
defines a sublinear expectation on \({\mathscr {L}}\) in the sense of Definition 2.2.

4.3 Appropriate domain and regularity

We now show that \({\mathscr {L}}\) is an appropriate domain for \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\) in the sense of Definition 2.3.

Proposition 4.1

The family \({\mathscr {L}}= \{ ({\mathrm {L}}_t, \Vert {\cdot }\Vert _{{\mathrm {L}}})\}_{t \in [0,T]}\) is an appropriate domain for \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\).

Proof

Separability of \({\mathscr {L}}\) follows from the fact that \({\mathcal {Q}}\) is uniformly tight, see Lemma A.6 in the appendix, and the fact that C(K) is separable for every compact subset \(K\subset \varOmega \) by the Stone–Weierstrass theorem. The remaining conditions are satisfied by construction. \(\square \)

In the remainder of this section, we show that the sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) is regular.

Lemma 4.2

For every \(\xi \in C_b^{\mathrm {unif}}(\varOmega )\) the set
$$\begin{aligned} A \triangleq \Bigl \{\omega \in \varOmega \,:\, \lim _{q\uparrow t, q \in {\mathbb {Q}}} V_q(\xi , \omega ) \text { and } \lim _{q\downarrow t, q \in {\mathbb {Q}}} V_q(\xi , \omega ) \text { exist for all } t \in [0,T]\Bigr \} \end{aligned}$$
is closed. Moreover \(\sup _{Q \in {\mathcal {Q}}} Q (A^c) = 0\).

Proof

Let \(\omega \in A^c\) and \(s_1 < s_2\) be an upcrossing of \(V(\xi ,\omega )\) through [ab]. Proposition 2.5 in Nutz [27] yields a modulus of continuity \(\rho _\xi \) with
$$\begin{aligned} |V_t(\xi ,\omega ^1) - V_t(\xi , \omega ^2)| \le \rho _\xi (\Vert \omega ^1- \omega ^2\Vert _\infty ) \quad \text {for all } \omega ^1,\omega ^2 \in \varOmega . \end{aligned}$$
Choosing \(r>0\) such that \(\rho _\xi (r) < (b-a)/4\) we see that \(s_1 < s_2\) is an upcrossing of \(V(\xi , \bar{\omega })\) through \([a + \rho _{\xi }(r), b- \rho _{\xi }(r)]\) for every \(\bar{\omega }\) with \(\Vert \bar{\omega }- \omega \Vert _\infty <r\). Since \(V(\xi ,\omega )\) is bounded and \(\omega \in A^c\) it follows that \(V(\xi , \omega )\) has infinitely many upcrossings through some non-empty interval and hence so does \(V(\xi ,\bar{\omega })\) for every \(\bar{\omega }\) in the r-neighborhood of \(\omega \). This implies that \(A^c\) is open. Finally, by Lemma 4.7 in Nutz [27] the process \(V(\xi )\) is a Q-supermartingale for each \(Q \in {\mathcal {Q}}\), whence \(Q(A^c)=0 \) for every \(Q \in {\mathcal {Q}}\). \(\square \)

Lemma 4.3

For every monotone sequence \(\{t_n\}_{n\in {\mathbb {N}}}\subset [0,T]\) and every \(\xi \in C_b^\mathrm {unif}(\varOmega )\) the sequence \(\{V_{t_n}(\xi )\}_{n\in {\mathbb {N}}}\) is Cauchy in \({\mathrm {L}}_T\).

Proof

It suffices to show that \(\{V_{t_n}(\xi )\}_{n\in {\mathbb {N}}}\) is Cauchy in \({\mathrm {L}}_T\) for every strictly monotone sequence \(\{t_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {Q}}\cap [0,T]\). Since \(V(\xi )\) is bounded, for this it is enough to prove that
$$\begin{aligned} {\textstyle \sup _{Q \in {\mathcal {Q}}}} Q( |V_{t_n}(\xi ) - V_{t_m}(\xi )| \ge \eta ) \rightarrow 0 \quad \text {for all } \eta >0 \text { as }m,n\rightarrow \infty . \end{aligned}$$
Thus let \(\varepsilon >0\) and \(\omega \in A\) (where A is given in Lemma 4.2). Then we find \(N(\omega )\) such that \(|V_{t_n}(\xi , \omega ) -V_{t_m}(\xi ,\omega ) | < \eta /2\) for all \(m,n \ge N(\omega )\). We choose \(r >0\) such that \(\rho _\xi (r)< \eta /4\). Then for all \(\bar{\omega }\in \varOmega \) with \(\Vert \bar{\omega }- \omega \Vert _\infty < r\) and \(m,n\ge N(\omega )\) we obtain
$$\begin{aligned} |V_{t_n}(\xi , \bar{\omega }) - V_{t_m}(\xi , \bar{\omega }) | < \eta . \end{aligned}$$
By Lemma A.6 there is some compact set K with \(\sup _{Q \in {\mathcal {Q}}} Q(K^c) <\varepsilon \). Clearly the family of r-balls \(\{B_r(\omega ) : \omega \in A\}\) is an open covering of the compact set \(A \cap K\), and hence there exist \(\omega ^1, \ldots ,\omega ^M\in A\) such that \(A\cap K\subset \bigcup _{i=1}^M B_r(\omega ^i)\). Setting \(N\triangleq \max _{i=1, \ldots ,M} N(\omega ^i)\) we have
$$\begin{aligned} | V_{t_n}(\xi ,\omega ) - V_{t_m}(\xi ,\omega ) | < \eta \quad \text {for all } \omega \in A\cap K\text { and }m,n\ge N. \end{aligned}$$
Hence with \(A_{m,n} \triangleq \{|V_{t_n}(\xi ) - V_{t_m}(\xi ) | \ge \eta \}\) we have \(K \cap A \cap A_{m,n} = \emptyset \) for all \(m,n\ge N\), and it follows that
$$\begin{aligned} \textstyle \sup _{Q \in {\mathcal {Q}}} Q ( A_{m,n}) \le \sup _{Q \in {\mathcal {Q}}} Q(K^c) + \sup _{Q \in {\mathcal {Q}}} Q(A^c) < \varepsilon \quad \text {for all }m,n \ge N \end{aligned}$$
since \(\sup _{Q\in {\mathcal {Q}}}Q(A^c)=0\) by Lemma 4.2. \(\square \)

Theorem 4.4

(Regularity of random G-expectations) The random G-expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t \in [0,T]}\) constructed above is regular.

Proof

In view of Lemma A.3 it suffices to show that \([0,T]\rightarrow {\mathrm {L}}_T,\ t\rightarrow {\mathcal {E}}_{t}^{{\mathrm {sub}}} [\xi ] = V_t(\xi )\) is càdlàg for all \(\xi \in C_b^{\mathrm {unif}}(\varOmega )\). Lemma 4.3 shows that it is làdlàg, and \(V_t(\xi ) = \lim _{q \downarrow t, q \in {\mathbb {Q}}} V_q(\xi )\) q.s. by Theorem 5.1 in Nutz [27]. \(\square \)

It is well-known that BSDEs in the (non-random) G-expectation setting are intimately related to so-called second-order BSDEs (2BSDEs); see Cheridito et al. [9], Soner et al. [45, 46] for background on 2BSDEs and, e.g., Hu et al. [21] for BSDEs under G-expectations and their relation to 2BSDEs. The following remark briefly outlines the link between 2BSDEs and BNEEs in a random G-expectation setting.

Remark 4.5

Let \((f,\xi )\) be a BNEE standard parameter where f is induced by a Lipschitz function \([0,T]\times \varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) via \([0,T]\times {\mathrm {L}}_T \ni (t,\eta ) \mapsto f(t,{\cdot },\eta ({\cdot }))\in {\mathrm {L}}_T\), and let \(X\in {\mathrm {D}}\) be the unique solution of the BNEE
$$\begin{aligned} X_t = {\mathcal {E}}_{t}^{{\mathrm {sub}}} \Bigl [{\textstyle \int _t^T} f(s,X_s) {\mathrm {d}}s + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
(4.2)
An application of Theorem 5.1 in Nutz [27] shows that the process
$$\begin{aligned} {\mathcal {E}}_{t}^{{\mathrm {sub}}} \Bigl [{\textstyle \int _0^T} f(s,X_s) {\mathrm {d}}s + \xi \Bigr ], \quad t \in [0,T], \end{aligned}$$
admits a \({\mathcal {Q}}\)-modification Y that is càdlàg and \(\bar{{\mathcal {F}}}\)-adapted, where \(\bar{{\mathcal {F}}}\) denotes the minimal right-continuous filtration containing \({\mathcal {F}}\) augmented by the collection of all \(({\mathcal {Q}},{\mathcal {F}}_T)\)-polar sets. Moreover, Theorem 6.4 in Nutz [27] shows that there exists an \(\bar{{\mathcal {F}}}\)-predictable process Z such that (YZ) satisfies
$$\begin{aligned} Y_t = \left[ \xi + {\textstyle \int _0^T} f(s,X_s) {\mathrm {d}}s\right] - {\textstyle \int _t^T} Z_s {\mathrm {d}}M^{W,Q}_s + K_T^Q - K_t^Q,\quad t\in [0,T],\ Q\text {-a.s.} \end{aligned}$$
for each \(Q\in {\mathcal {Q}}\). One can show that X admits a càdlàg and \(\bar{{\mathcal {F}}}\)-adapted modification \(\bar{X}\) with
$$\begin{aligned} \bar{X}_t = Y_t - {\textstyle \int _0^t} f(s,\bar{X}_s){\mathrm {d}}s\quad Q\text {-a.s.}\ \text { for all } Q\in {\mathcal {Q}}\ \text {and }t\in [0,T], \end{aligned}$$
so \((\bar{X},Z)\) solves the 2BSDE
$$\begin{aligned} \bar{X}_t = \xi + {\textstyle \int _t^T} f(s,\bar{X}_s) {\mathrm {d}}s - {\textstyle \int _t^T} Z_s {\mathrm {d}}M^{W,Q}_s + K_T^Q - K_t^Q. \end{aligned}$$
(4.3)
Thus the unique solution X of the BNEE (4.2) induces a solution of the 2BSDE (4.3), which is unique in the sense of Theorem 6.4 in Nutz [27]. Finally, note that the 2BSDE (4.3) is not included in the class of 2BSDEs studied by Soner et al. [45, 46], as the domain of the conjugate of the nonlinear generator is possibly path-dependent. Conversely, Soner et al. [45, 46] also investigate aggregators with Z-dependence, which is not covered by our framework.\(\diamond \)

5 Recursive utility with nonlinear expectations

In this section, we use the general framework for time-consistent nonlinear expectations of this article to construct recursive utilities under ambiguity. Specifically, suppose that \(\{{\mathcal {E}}_t\}_{t\in [0,T]}\) is a regular nonlinear expectation dominated by a sublinear expectation \(\{{\mathcal {E}}_{t}^{{\mathrm {sub}}}\}_{t\in [0,T]}\) on an appropriate domain \({\mathscr {L}}= \{ ({\mathrm {L}}_t, \Vert {\cdot }\Vert _{\mathrm {L}})\}_{t \in [0,T]}\); \({\mathrm {P}}\) denotes the space of \({\mathrm {d}}t\)-integrable adapted \({\mathscr {L}}\)-processes as defined in Sect. 2.3. We consider consumption plans \((c,\xi )\) where \(\xi \in {\mathrm {L}}_T\) is a terminal payoff and \(c\in {\mathrm {P}}\) is a consumption rate process. We suppose that there exists an approximating sequence \(\{c^n\}_{n \in {\mathbb {N}}} \subset {\mathrm {P}}\) with \(c^n\) piecewise constant on \(\varDelta ^n\) and \(c^n\rightarrow c\) in \({\mathrm {P}}\), i.e. \(c^n_t\rightarrow c_t\) for \({\mathrm {d}}t\)-a.e. \(t\in [0,T]\). Here \(\{\varDelta ^n\}_{n\in {\mathbb {N}}}\) is a sequence of partitions with \(|\varDelta ^n|\rightarrow 0\) as in Sect. 3.3. Recursive utility is constructed via a discrete-time generator
$$\begin{aligned} W: [0,T] \times {\mathrm {L}}_T \times {\mathrm {L}}_T \rightarrow {\mathrm {L}}_T,\qquad W(\varDelta , c,u) \triangleq u + \varDelta f^\varDelta (c,u) \end{aligned}$$
that satisfies the following standard conditions:6
  1. (A1)
    There exists a modulus of continuity \(h:[0,T] \rightarrow {\mathbb {R}}\) such that
    $$\begin{aligned} \Vert f^\varDelta (c,u)- f^0(c,u)\Vert _{{\mathrm {L}}} \le h(\varDelta ) (1+ \Vert c\Vert _{{\mathrm {L}}} + \Vert u\Vert _{{\mathrm {L}}})\quad \text {for all } c,u \in {\mathrm {L}}_T. \end{aligned}$$
     
  2. (A2)
    There exists \(L>0\) such that \(f^0\) satisfies the Lipschitz property
    $$\begin{aligned} \Vert f^0(c,u_1) - f^0(c,u_2) \Vert _{{\mathrm {L}}} \le L \Vert u_1-u_2\Vert _{{\mathrm {L}}}\quad \text {for all }c,u_1,u_2\in {\mathrm {L}}_T. \end{aligned}$$
     
  3. (A3)

    \(f^0({\cdot },u)\) is continuous for every \(u\in {\mathrm {L}}_T\).

     
  4. (A4)
    There exists a constant \(K>0\) such that
    $$\begin{aligned} \Vert f^0(c,0)\Vert _{{\mathrm {L}}} \le K(1 + \Vert c\Vert _{{\mathrm {L}}}) \quad \text {for every }c\in {\mathrm {L}}_T. \end{aligned}$$
     
We assume moreover that \(f^0(c,u)\in {\mathrm {L}}_t\) whenever \(c,u\in {\mathrm {L}}_t\). For brevity we write \(f\triangleq f^0\) in the sequel. The mapping \(f:{\mathrm {L}}_T\times {\mathrm {L}}_T\rightarrow {\mathrm {L}}_T\) is called the continuous-time generator. Note that the preceding assumptions imply that for each consumption plan \((c,\xi )\) the function
$$\begin{aligned} f_c:[0,T]\times {\mathrm {L}}_T\mapsto {\mathrm {L}}_T,\quad (t,\eta )\mapsto f(c_t,\eta ) \end{aligned}$$
gives rise to a standard BNEE parameter \((f_c,\xi )\).

Definition 5.1

(Recursive and stochastic differential utility) Let \((c,\xi )\) be a consumption plan with an associated approximating sequence \(\{c^n\}_{n\in {\mathbb {N}}}\). The discrete-time recursive utility process \(U^n\) is defined on the time grid \(\varDelta ^n\) via7
$$\begin{aligned} U_{k}^n \triangleq W\bigl (\varDelta _{k}^n, c_{t_k^n}^n, {\mathcal {E}}_{t^n_k} [U^n_{k+1}] \bigr )\quad \text {where }U_{N_n}^n = \xi . \end{aligned}$$
(5.1)
The unique solution U of the BNEE
$$\begin{aligned} U_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} f(c_s, U_s)\, {\mathrm {d}}s + \xi \Bigr ] = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} f_c(s, U_s)\, {\mathrm {d}}s + \xi \Bigr ] \end{aligned}$$
is called the continuous-time stochastic differential utility process.\(\diamond \)

Definition 5.1 subsumes the notions of continuous-time recursive utility under ambiguity in Chen and Epstein [7] and Epstein and Ji [16]. Note that existence and uniqueness of a solution \(U\in {\mathrm {D}}\) are guaranteed by Theorem 3.4.

The main result of this section demonstrates that the recursive utility processes \(U^n\) converge to the stochastic differential utility process U:

Theorem 5.2

(Continuous-time limit of recursive utility) Let \(U^n\), \(n \in {\mathbb {N}}\), be the discrete-time recursive utility process and let U be the stochastic differential utility process from Definition 5.1. Then
$$\begin{aligned} \max _{k=0,\ldots ,N_n}\Vert U^n_{k}-U_{t_k^n}\Vert _{{\mathrm {L}}} \rightarrow 0. \end{aligned}$$

Proof

Introducing the standard BNEE parameters
$$\begin{aligned} f^n \triangleq f_{c^n} = {\textstyle \sum _{k=0}^{N_n-1}} 1_{[t_k^n, t_{k+1}^n)} f(c_{t_k^n}^n,{\cdot }),\quad n\in {\mathbb {N}}, \end{aligned}$$
(5.2)
we have that \(\{(\varDelta ^n, f^n, \xi )\}_{n \in {\mathbb {N}}}\) is \((f_c,\xi )\)-exhausting; see Lemma A.7 in the appendix. Therefore, Theorem 3.10 shows that
$$\begin{aligned} \max _{k=0, \ldots , N_n} \Vert X^n_k - U_{t_k^n}\Vert _{{\mathrm {L}}} \rightarrow 0, \end{aligned}$$
where \(X_{N_n}^n = \xi \) and
$$\begin{aligned} X^n_{k} \triangleq {\mathcal {E}}_{t_k^n} \Bigl [ \varDelta _k^ n f^n\bigl (t_k,{\mathcal {E}}_{t_k^n} [X^n_{k+1}]\bigr ) + X^n_{k+1} \Bigr ], \quad k=N_n-1, \ldots , 0. \end{aligned}$$
(5.3)
Thus, it remains to show that \(\max _{k=0,\ldots ,N_n} \Vert U_k^n-X_k^n\Vert _{{\mathrm {L}}}\rightarrow 0\) to conclude the proof. In view of (5.3) and the representation (A.2), the triangle inequality and the projection property imply
$$\begin{aligned}&\Vert U_k^n - X_k^n\Vert _{{\mathrm {L}}}\\&\quad \le \varDelta _k^n \Vert f^{\varDelta _k^n } (c_{t_k^n}^n, {\mathcal {E}}_{t_{k}^n} [U_{k+1}^n]) - f(c_{t_k^n}^n, {\mathcal {E}}_{t_{k}^n} [X_{k+1}^n]) \Vert _{{\mathrm {L}}}+ \Vert U_{k+1}^n - X_{k+1}^n\Vert _{{\mathrm {L}}}. \end{aligned}$$
Using the triangle inequality as well as (A1) and (A2) we get
$$\begin{aligned}&\Vert f^{\varDelta _k^n } (c_{t_k^n}^n, {\mathcal {E}}_{t_{k}^n} [U_{k+1}^n]) - f(c_{t_k^n}^n, {\mathcal {E}}_{t_{k}^n} [X_{k+1}^n]) \Vert _{{\mathrm {L}}}\\&\quad \le h(\varDelta _k^n) (1+ \Vert c_{t_k^n}^n\Vert _{{\mathrm {L}}} + \Vert U_{k+1}^n \Vert _{{\mathrm {L}}} ) + L \Vert U_{k+1}^n - X_{k+1}^n \Vert _{{\mathrm {L}}} \end{aligned}$$
by the projection property. Therefore, with \(K \triangleq C_1 (1+ \Vert \xi \Vert _{{\mathrm {L}}} + \Vert c\Vert _{{\mathrm {P}}}) +1 + \Vert c\Vert _{{\mathrm {P}}}\), the a priori estimate provided by Lemma A.8 yields
$$\begin{aligned} \Vert U_k^n - X_k^n\Vert _{{\mathrm {L}}} \le \varDelta _k^n h(\varDelta ^n) K + (1+ \varDelta _k^n L)\Vert U_{k+1}^n - X_{k+1}^n\Vert _{{\mathrm {L}}} \end{aligned}$$
for all but finitely many \(n\in {\mathbb {N}}\). Iterating this and using \(1+x\le e^x\) we obtain \(\Vert U_k^n-X_k^n\Vert _{{\mathrm {L}}}\le K T e^{KT} h(\varDelta ^n)\) and the claim follows. \(\square \)

In discrete time, recursive utility under multiple priors was investigated by Epstein and Schneider [17] and Hayashi [20]; see also Epstein and Wang [18] for a related time-additive model. In continuous time, Chen and Epstein [7] construct stochastic differential utility under g-expectations (drift uncertainty), and Epstein and Ji [16] propose a continuous-time version of stochastic differential utility in the setting of Nutz [27] (volatility uncertainty); note that Theorems 3.4 and 4.4 guarantee existence and uniqueness of the associated utility processes. Theorem 5.2 substantiates the axiomatic approach taken in these articles by a discrete-time foundation: The relevant continuous-time utility processes are identified as limits of discrete-time recursive utilities under ambiguity. In particular the recursive utility values \(U^n_0\in {\mathbb {R}}\) converge to the stochastic differential utility value \(U_0\in {\mathbb {R}}\).

6 Conclusion

This article has laid the theoretical groundwork for a class of backward stochastic differential equations under nonlinear expectations, referred to as backward nonlinear expectation equations. We have provided the basic theoretical results (existence, uniqueness, stability, discretization), and applied them to study the discrete-time foundations of continuous-time recursive preferences under ambiguity. Such utilities, and the associated backward nonlinear expectation equations, feature prominently in equilibrium asset pricing theory with multiple priors (see, e.g., Epstein and Ji [15]). In addition to applications in this area, future research based on the results of this paper may investigate backward nonlinear expectation equations in the context of valuation adjustments for derivative pricing in the presence of model uncertainty, portfolio optimization under ambiguity (see, e.g., Lin and Riedel [26]), and risk measures for stochastic processes (see Penner and Réveillac [38]).

Footnotes

  1. 1.

    For sub-additive operators the converse is also true, i.e. (PC) implies (SI) and (N).

  2. 2.

    Here and in the following, \({\mathcal {B}}(S)\) denotes the Borel \(\sigma \)-algebra on the topological space S.

  3. 3.

    Note that \(X^\varDelta \) is defined on the continuous-time uncertainty framework; in particular, the following analysis is not of Donsker type.

  4. 4.

    See Remarks 2.2 and 2.3 in Nutz [27] for a discussion of this assumption.

  5. 5.

    In the notation of Nutz [27], our value function is given by \(V_t^x(\xi ,\omega )\). Note that \(V_t(\xi ,\omega )=V_t(\xi ,\tilde{\omega })\) whenever \(\omega -\tilde{\omega }\) is constant.

  6. 6.

    See, e.g., Kraft and Seifried [24].

  7. 7.

    Here \(\varDelta ^n: 0=t_0^n<t_1^n<\cdots <t_{N_n}^n = T\) and \(\varDelta _k^n \triangleq t^n_{k+1}-t^n_{k}\), \(k=0,\ldots ,N_n-1\).

  8. 8.

    See Theorem 52 in Denis et al. [11]. Note that Denis et al. [11] focus on bounded functions. However, by weak compactness of \({\mathcal {Q}}\) every continuous function can be approximated by compactly supported ones. Further note that \({\mathrm {L}}_T\) is separable by the Stone–Weierstrass theorem.

  9. 9.

    Every \(\xi \in {\mathcal {H}}_T\) admits a representation \(\varphi ^{\xi }_{\mathrm {min}}\) with a minimal number of time points. An arbitrary representation \(\varphi \) and \(\varphi ^{\xi }_{\mathrm {min}}\) share the same Lipschitz constant.

Notes

Acknowledgements

We wish to thank Mete Soner, Holger Kraft, Rama Cont, Yuri Kabanov, and Keita Owari for comments and suggestions. Thomas Seiferling gratefully acknowledges financial support from Studienstiftung des Deutschen Volkes.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Christoph Belak
    • 1
  • Thomas Seiferling
    • 2
  • Frank Thomas Seifried
    • 1
  1. 1.Department IV – MathematicsUniversity of TrierTrierGermany
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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