Superhedging and Dynamic Risk Measures under ... - Columbia Math

Superhedging and Dynamic Risk Measures under Volatility Uncertainty Marcel Nutz

∗

†

H. Mete Soner

First version: November 12, 2010. This version: June 2, 2012.

Abstract We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

Keywords volatility uncertainty, risk measure, time consistency, nonlinear martin-

gale, superhedging, replication, second order BSDE, 𝐺-expectation AMS 2000 Subject Classications primary 91B30, 93E20, 60G44; secondary 60H30 JEL Classications D81, G11.

Acknowledgements

Research supported by the European Research Council

Grant 228053-FiRM, the Swiss National Science Foundation Grant PDFM2120424/1 and the ETH Foundation.

The authors thank two anonymous

referees for helpful comments.

1

Introduction

Coherent risk measures were introduced in [1] as a way to quantify the risk associated with a nancial position. Since then, coherent risk measures and sublinear expectations (which are the same up to the sign convention) have been studied by numerous authors; see [15, 29, 30] for extensive references. ∗ †

Department of Mathematics, Columbia University,

[email protected]

Department

and

of

Mathematics,

ETH

Zurich,

[email protected]

1

Swiss

Finance

Institute,

Most of these works consider the case where scenarios are probability measures absolutely continuous with respect to a given reference probability (important early exceptions are [14, 26]). The present paper studies dynamic sublinear expectations and superhedging under volatility uncertainty, which is naturally related to singular measures.

The concept of volatility uncer-

tainty was introduced in nancial mathematics by [2, 11, 21] and has recently

𝐺-expectations

received considerable attention due to its relation to

[27, 28]

and second order backward stochastic dierential equations [6, 32], called 2BSDEs for brevity. Any (static) sublinear expectation

ℰ0∘ ,

dened on the set of bounded

measurable functions on a measurable space

(Ω, ℱ),

has a convex-dual rep-

resentation

ℰ0∘ (𝑋) = sup 𝐸 𝑃 [𝑋]

(1.1)

𝑃 ∈𝒫

𝒫

for a certain set

of measures which are

𝜎 -additive

certain continuity properties (cf. [15, Section 4]).

as soon as

ℰ0∘

satises

𝒫

The elements of

can

be seen as possible scenarios in the presence of uncertainty and hence (1.1) corresponds to the worst-case expectation. In this paper, we take the canonical space of continuous paths and

𝒫

Ω

to be

to be a set of martingale laws

for the canonical process, corresponding to dierent scenarios of volatilities. For this case,

𝒫

is typically not dominated by a nite measure and (1.1)

was studied in [5, 10, 11] by capacity-theoretic methods. We remark that from the pricing point of view, the restriction to the martingale case entails no loss of generality in an arbitrage-free setting. An example with arbitrage was studied in [13]. While any set of martingale laws gives rise to a static sublinear expectation via (1.1), we are interested in

dynamic

sublinear expectations; i.e.,

conditional versions of (1.1) satisfying a time-consistency property. If dominated by a probability

𝑃∗ ,

𝒫

is

a natural extension of (1.1) is given by ′

ℰ𝑡∘,𝑃∗ (𝑋) = ess sup𝑃∗ 𝐸 𝑃 [𝑋∣ℱ𝑡∘ ] 𝑃∗ -a.s., 𝑃 ′ ∈𝒫(ℱ𝑡∘ ,𝑃∗ )

where

𝒫(ℱ𝑡∘ , 𝑃∗ ) = {𝑃 ′ ∈ 𝒫 : 𝑃 ′ = 𝑃∗

on

ltration generated by the canonical process.

ℱ𝑡∘ }

𝒫

(see [7]).

𝔽∘ = {ℱ𝑡∘ }

is the

Such dynamic expectations

are well-studied; in particular, time consistency of by a stability property of

and

ℰ ∘,𝑃∗

can be characterized

In the non-dominated case, we can

similarly consider the family of random variables

{ℰ𝑡∘,𝑃 (𝑋), 𝑃 ∈ 𝒫}.

Since

a reference measure is lacking, it is not straightforward to construct a single random variable

ℰ𝑡∘ (𝑋)

such that ′

ℰ𝑡∘ (𝑋) = ℰ𝑡∘,𝑃 (𝑋) := ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡∘ ] 𝑃 -a.s.

for all

𝑃 ∈ 𝒫.

(1.2)

𝑃 ′ ∈𝒫(ℱ𝑡∘ ,𝑃 )

This problem of aggregation has been solved in several examples. ticular, the

𝐺-expectations

and random

2

𝐺-expectations

In par-

[23] (recalled in

Section 2) correspond to special cases of (1.2).

The construction of

𝐺-

expectations is based on a PDE, which directly yields random variables dened for all

𝜔 ∈ Ω.

The random

𝐺-expectations

are dened pathwise

using regular conditional probability distributions. A general study of aggregation problems is presented in [31]; see also [4]. However, the study of aggregation is not an object of the present paper.

In view of the diverse

approaches, we shall proceed axiomatically and start with a given aggregated family

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]}.

𝐺-expectations,

Having in mind the example of (random)

this family is assumed to be given in the raw ltration

𝔽∘

and without any regularity in the time variable. The main goal of the present paper is to provide basic technology for the study of dynamic sublinear expectations under volatility uncertainty as

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]}, we construct a corresponding càdlàg process ℰ(𝑋), called the ℰ -martingale associated with 𝑋 , in a suitably enlarged ltration 𝔽 (Proposition 4.5). We use this

stochastic processes.

Given the family

process to dene the sublinear expectation at stopping times and prove an

ℰ -martingales (Theorem 4.10). Furthermore, ℰ(𝑋) into an integral of the canonical process

optional sampling theorem for we obtain a decomposition of

and an increasing process (Proposition 4.11), similarly as in the classical op-

ℰ -martingale yields the dynamic 𝑋 yields the superhedging price of the nancial claim 𝑋 and the integrand 𝑍 superhedging strategy. We also provide a connection between ℰ -martingales 𝑋 and 2BSDEs by characterizing (ℰ(𝑋), 𝑍 ) as the minimal solution of such a tional decomposition [19]. In particular, the

backward equation (Theorem 4.16). Our last result concerns the uniqueness of time-consistent extensions and gives conditions under which (1.2) is indeed the only possible extension of the static expectation (1.1). In particular, we introduce the notion of local strict monotonicity to deal with the singularity of the measures (Proposition 5.3). To obtain our results, we rely on methods from stochastic optimal control and the general theory of stochastic processes. Indeed, from the point of view of dynamic programming,

ℰ𝑡∘ (𝑋)

is the value process of a control

problem dened over a set of measures, and time consistency corresponds to Bellman's principle. Taking the control representation (1.2) as our starting point allows us to consider the measures

𝑃 ∈ 𝒫

separately in many

arguments and therefore to apply standard arguments of the general theory. The remainder of this paper is organized as follows.

In Section 2 we

detail the setting and notation. Section 3 relates time consistency to a pasting property.

In Section 4 we construct the

ℰ -martingale

and provide the

optional sampling theorem, the decomposition, and the characterization by a 2BSDE. Section 5 studies the uniqueness of time-consistent extensions.

3

2

Preliminaries

We x a constant

𝑇 >0

Ω = {𝜔 ∈ 𝐶([0, 𝑇 ]; R𝑑 ) : 𝜔0 = 0}

and let

be the

canonical space of continuous paths equipped with the uniform topology. We

𝐵 the canonical process 𝐵𝑡 (𝜔) = 𝜔𝑡 , by 𝑃0 the Wiener measure 𝔽∘ = {ℱ𝑡∘ }0≤𝑡≤𝑇 , ℱ𝑡∘ = 𝜎(𝐵𝑠 , 𝑠 ≤ 𝑡) the raw ltration generated

denote by and by by

𝐵.

As in [10, 23, 33, 32] we shall use the so-called strong formulation of

volatility uncertainty in this paper; i.e., we consider martingale laws induced by stochastic integrals of

𝐵

under

𝑃0 .

More precisely, we dene

𝒫𝑆

to be

the set of laws

𝑃 𝛼 := 𝑃0 ∘ (𝑋 𝛼 )−1 ,

𝑋𝑡𝛼 :=

where

(𝑃0∫) 𝑡

𝛼𝑠1/2 𝑑𝐵𝑠 ,

𝑡 ∈ [0, 𝑇 ]

(2.1)

0 and

𝛼

𝔽∘ -progressively measurable processes with values 𝑑×𝑑 denotes the set ∣𝛼𝑡 ∣ 𝑑𝑡 < ∞ 𝑃0 -a.s. Here 𝕊>0 𝑑 ⊂ ℝ

ranges over all

𝕊>0 𝑑 satisfying

∫𝑇 0

in of

strictly positive denite matrices and the stochastic integral in (2.1) is the Itô integral under

𝑃0 ,

constructed in

coincides with the set denoted by

𝔽∘

𝒫𝑆

(cf. [36, p. 97]). We remark that

𝒫𝑆

in [31].

𝒫 ⊆ 𝒫𝑆 which represents 𝑡 ∈ [0, 𝑇 ], we dene 𝐿1𝒫 (ℱ𝑡∘ ) to variables 𝑋 satisfying

The basic object in this paper is a nonempty set the possible scenarios for the volatility. For be the space of

ℱ𝑡∘ -measurable

random

∥𝑋∥𝐿1 := sup ∥𝑋∥𝐿1 (𝑃 ) < ∞, 𝒫

𝑃 ∈𝒫

∥𝑋∥𝐿1 (𝑃 ) := 𝐸[∣𝑋∣]. More precisely, we take equivalences classes with 𝒫 -quasi-sure equality so that 𝐿1𝒫 (ℱ𝑡∘ ) becomes a Banach space. (Two functions are equal 𝒫 -quasi-surely, 𝒫 -q.s. for short, if they are equal up to a 𝒫 -polar set. A set is called 𝒫 -polar if it is a 𝑃 -nullset for all 𝑃 ∈ 𝒫 .) 1 1 ∘ We also x a nonempty subset ℋ of 𝐿𝒫 := 𝐿𝒫 (ℱ𝑇 ) whose elements play the role of nancial claims. We emphasize that in applications, ℋ is typically 1 smaller than 𝐿𝒫 . The following is a motivating example for many of the where

respect to

considerations in this paper.

Example 2.1. 𝐺-expectation

(i) Given real numbers

(for dimension

𝑑 = 1)

0 ≤ 𝑎 ≤ 𝑎 < ∞,

corresponds to the choice

} 𝒫 = 𝑃 𝛼 ∈ 𝒫𝑆 : 𝑎 ≤ 𝛼 ≤ 𝑎 𝑃0 × 𝑑𝑡-a.e. , {

cf. [10, Section 3]. Here the symbol

𝐺(𝛾) := If

𝑋 = 𝑓 (𝐵𝑇 )

the associated

𝐺

(2.2)

refers to the function

1 sup 𝑎𝛾. 2 𝑎≤𝑎≤𝑎 𝑓 , then ℰ𝑡∘,𝐺 (𝑋) is dened equation −∂𝑡 𝑢 − 𝐺(𝑢𝑥𝑥 ) = 0 with

for a suciently regular function

via the solution of the nonlinear heat

4

𝑢∣𝑡=𝑇 = 𝑓 . In [27], the mapping ℰ𝑡∘,𝐺 is extended to random variables of the form 𝑋 = 𝑓 (𝐵𝑡1 , . . . , 𝐵𝑡𝑛 ) by a stepwise evaluation of the PDE and nally to the ∥ ⋅ ∥𝐿1 -completion ℋ of the set of all such 𝒫 random variables. For 𝑋 ∈ ℋ, the 𝐺-expectation then satises boundary condition

′

ℰ𝑡∘,𝐺 (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡∘ ] 𝑃 -a.s.

for all

𝑃 ∈ 𝒫,

𝑃 ′ ∈𝒫(ℱ𝑡∘ ,𝑃 )

ℋ coincides with the ∥⋅∥𝐿1 -completion 𝒫 𝐶𝑏 (Ω), the set of bounded continuous functions on Ω, and is strictly smaller 1 than 𝐿𝒫 as soon as 𝑎 ∕= 𝑎. (ii) The random 𝐺-expectation corresponds to the case where 𝑎, 𝑎 are

which is of the form (1.2). The space of

random processes instead of constants and is directly constructed from a set

𝒫

of measures (cf. [23]). In this case the space

of

UC𝑏 (Ω),

ℋ

∥ ⋅ ∥𝐿1 -completion 𝒫 functions on Ω. If 𝑎 is

is the

the set of bounded uniformly continuous

nite-valued and uniformly bounded,

3

ℋ

coincides with the space from (i).

Time Consistency and Pasting

𝒫 ⊆ 𝒫𝑆 ℋ ⊆ 𝐿1𝒫 is xed 𝔽∘ -stopping times

In this section, we consider time consistency as a property of the set and obtain some auxiliary results for later use. throughout.

Moreover, we let

𝒯 (𝔽∘ )

The set

be the set of all

taking nitely many values; this choice is motivated by the applications in the subsequent section. However, the results of this section hold true also if

𝒯 (𝔽∘ )

is replaced by an arbitrary set of

𝔽∘ -stopping

times containing

𝜎 ≡ 0;

in particular, the set of all stopping times and the set of all deterministic times. Given

𝒜 ⊆ ℱ𝑇∘

and

𝑃 ∈ 𝒫,

we use the standard notation

𝒫(𝒜, 𝑃 ) = {𝑃 ′ ∈ 𝒫 : 𝑃 ′ = 𝑃

on

𝒜}.

At the level of measures, time consistency can then be dened as follows.

Denition 3.1. ess sup𝑃 𝐸 𝑃 𝑃 ′ ∈𝒫(ℱ𝜎∘ ,𝑃 )

′

[

The set

ess sup𝑃

𝒫 ′

𝑃 ′′ ∈𝒫(ℱ𝜏∘ ,𝑃 ′ )

𝔽∘ -time-consistent on ℋ if ] ′′ ′ 𝐸 𝑃 [𝑋∣ℱ𝜏∘ ] ℱ𝜎∘ = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎∘ ] 𝑃 -a.s. ′ ∘ is

𝑃 ∈𝒫(ℱ𝜎 ,𝑃 )

(3.1) for all

𝑃 ∈ 𝒫, 𝑋 ∈ ℋ

and

𝜎≤𝜏

in

𝒯 (𝔽∘ ).

This property embodies the principle of dynamic programming (e.g., [12]). We shall relate it to the following notion of stability, also called mstability, fork-convexity, stability under concatenation, etc.

Denition 3.2. 𝜏 ∈𝒯

(𝔽∘ ),

𝒫 is stable under 𝔽∘ -pasting if for all 𝑃 ∈ 𝒫 , Λ∈ 𝑃1 , 𝑃2 ∈ 𝒫(ℱ𝜏∘ , 𝑃 ), the measure 𝑃¯ dened by [ ] 𝑃¯ (𝐴) := 𝐸 𝑃 𝑃1 (𝐴∣ℱ𝜏∘ )1Λ + 𝑃2 (𝐴∣ℱ𝜏∘ )1Λ𝑐 , 𝐴 ∈ ℱ𝑇∘ (3.2) The set

ℱ𝜏∘ and

is again an element of

𝒫. 5

As

𝔽∘

is the only ltration considered in this section, we shall sometimes

∘

omit the qualier  𝔽 .

Lemma 3.3. The set 𝒫𝑆 is stable under pasting. Proof.

𝑃, 𝑃1 , 𝑃2 , 𝜏, Λ, 𝑃¯ be as in Denition 3.2. Using the notation (2.1), 𝑖 𝛼 𝛼𝑖 = 𝑃 for 𝑖 = 1, 2. Setting let 𝛼, 𝛼 be such that 𝑃 = 𝑃 and 𝑃 𝑖 Let

𝛼 ¯ 𝑢 (𝜔) := [ ] 1[[0,𝜏 (𝑋 𝛼 )]] (𝑢)𝛼𝑢 (𝜔) + 1]]𝜏 (𝑋 𝛼 ),𝑇 ]] (𝑢) 𝛼𝑢1 (𝜔)1Λ (𝑋 𝛼 (𝜔)) + 𝛼𝑢2 (𝜔)1Λ𝑐 (𝑋 𝛼 (𝜔)) , we have

𝑃¯ = 𝑃 𝛼¯ ∈ 𝒫𝑆

by the arguments in [33, Appendix].

The previous proof also shows that the set appearing in (2.2) is stable under pasting. The following result is classical.

Lemma 3.4. Let 𝜏 ∈ 𝒯 (𝔽∘ ), 𝑋 ∈ 𝐿1𝒫 and 𝑃 ∈ 𝒫 . If 𝒫 is stable under pasting, then there exists a sequence 𝑃𝑛 ∈ 𝒫(ℱ𝜏∘ , 𝑃 ) such that ′

ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜏∘ ] = lim 𝐸 𝑃𝑛 [𝑋∣ℱ𝜏∘ ] 𝑛→∞

𝑃 ′ ∈𝒫(ℱ𝜏∘ ,𝑃 )

𝑃 -a.s.,

where the limit is increasing 𝑃 -a.s. Proof.

a.s. upward ltering (cf. [22,

′

{𝐸 𝑃 [𝑋∣ℱ𝜏∘ ] : 𝑃 ′ ∈ 𝒫(ℱ𝜏∘ , 𝑃 )} is 𝑃 ∘ Proposition VI-1-1]). Given 𝑃1 , 𝑃2 ∈ 𝒫(ℱ𝜏 , 𝑃 ),

It suces to show that the family

we set

and dene and

𝑃¯ ∈ 𝒫

{ } Λ := 𝐸 𝑃1 [𝑋∣ℱ𝜏∘ ] > 𝐸 𝑃2 [𝑋∣ℱ𝜏∘ ] ∈ ℱ𝜏∘ [ ] 𝑃¯ (𝐴) := 𝐸 𝑃 𝑃1 (𝐴∣ℱ𝜏∘ )1Λ + 𝑃2 (𝐴∣ℱ𝜏∘ )1Λ𝑐 . Then 𝑃¯ = 𝑃

on

ℱ𝜏∘

by the stability. Moreover,

¯

𝐸 𝑃 [𝑋∣ℱ𝜏∘ ] = 𝐸 𝑃1 [𝑋∣ℱ𝜏∘ ] ∨ 𝐸 𝑃2 [𝑋∣ℱ𝜏∘ ] 𝑃 -a.s., showing that the family is upward ltering. To relate time consistency to stability under pasting, we introduce the following closedness property.

Denition 3.5. 𝑃 ∈ 𝒫𝑆 If

𝒫

𝒫 is maximally chosen for ℋ ′ 𝐸 𝑃 [𝑋] ≤ sup𝑃 ′ ∈𝒫 𝐸 𝑃 [𝑋] for all 𝑋 ∈ ℋ.

We say that

satisfying

is dominated by a reference probability

with a subset of

𝐿1 (𝑃∗ )

𝑃∗ ,

then

𝒫

if

𝒫

contains all

can be identied

by the Radon-Nikodym theorem.

If furthermore

ℋ = 𝐿∞ (𝑃∗ ), the Hahn-Banach theorem implies that 𝒫 is maximally chosen 1 if and only if 𝒫 is convex and closed for weak topology of 𝐿 (𝑃∗ ). Along these lines, the following result can be seen as a generalization of [7, Theorem 12]; in fact, we merely replace functional-analytic arguments by algebraic ones.

6

Proposition 3.6. With respect to the ltration 𝔽∘ , we have: (i) If 𝒫 is stable under pasting, then 𝒫 is time-consistent on 𝐿1𝒫 . (ii) If 𝒫 is time-consistent on ℋ and maximally chosen for ℋ, then 𝒫 is stable under pasting. Proof. (i) This implication is standard; we provide the argument for later

𝑃 ′′ := 𝑃 ′ on arbitrary 𝑃 ∈ 𝒫

reference. The inequality  ≥ in (3.1) follows by considering the left hand side. To see the converse inequality, x an and choose a sequence

𝑃𝑛 ∈ 𝒫(ℱ𝜏∘ , 𝑃 ) ⊆ 𝒫(ℱ𝜎∘ , 𝑃 )

as in Lemma 3.4. Then

monotone convergence yields

𝐸

𝑃

[

𝑃

ess sup

𝐸

𝑃 ′ ∈𝒫(ℱ𝜏∘ ,𝑃 )

𝑃′



[𝑋∣ℱ𝜏∘ ] ℱ𝜎∘

]

= lim 𝐸 𝑃𝑛 [𝑋∣ℱ𝜎∘ ] 𝑛→∞

′

≤ ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎∘ ] 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ𝜎∘ ,𝑃 )

𝒫

(ii) Let

be time-consistent and let

tion 3.2. For any

𝑋 ∈ ℋ,

𝑃, 𝑃1 , 𝑃2 , 𝜏, Λ, 𝑃¯

be as in Deni-

we have

[ ] ¯ 𝐸 𝑃 [𝑋] = 𝐸 𝑃 𝐸 𝑃1 [𝑋∣ℱ𝜏∘ ]1Λ + 𝐸 𝑃2 [𝑋∣ℱ𝜏∘ ]1Λ𝑐 [ ] 𝑃 𝑃 𝑃 ′′ ∘ ≤𝐸 ess sup 𝐸 [𝑋∣ℱ𝜏 ] 𝑃 ′′ ∈𝒫(ℱ𝜏∘ ,𝑃 )

≤ sup 𝐸 𝑃

′

[

𝑃 ′ ∈𝒫

ess sup𝑃

′

𝑃 ′′ ∈𝒫(ℱ𝜏∘ ,𝑃 ′ )

] ′′ 𝐸 𝑃 [𝑋∣ℱ𝜏∘ ]

′

= sup 𝐸 𝑃 [𝑋], 𝑃 ′ ∈𝒫 where the last equality uses (3.1) with and

4

𝑃¯ ∈ 𝒫𝑆

𝜎 ≡ 0.

by Lemma 3.3, we conclude that

Since 𝒫 ¯ 𝑃 ∈ 𝒫.

is maximally chosen

ℰ -Martingales

As discussed in the introduction, our starting point in this section is a given

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]} of random variables which will serve as a raw version of the ℰ -martingale to be constructed. We recall that the sets 𝒫 ⊆ 𝒫𝑆 1 and ℋ ⊆ 𝐿𝒫 are xed. family

Assumption 4.1.

Throughout Section 4, we assume that

𝑋 ∈ ℋ and 𝑡 ∈ [0, 𝑇 ], ∘ variable ℰ𝑡 (𝑋) such that

(i) for all

there exists an

′

ℰ𝑡∘ (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡∘ ] 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ𝑡∘ ,𝑃 )

(ii) the set

𝒫

is stable under

𝔽∘ -pasting.

7

ℱ𝑡∘ -measurable for all

𝑃 ∈ 𝒫.

random

(4.1)

The rst assumption was discussed in the introduction; cf. (1.2). With the motivating Example 2.1 in mind, we ask for (4.1) to hold at deterministic times rather than at stopping times. The second assumption is clearly motivated by Proposition 3.6(ii), and Proposition 3.6(i) shows that

𝒫

is

time-consistent in the sense of Denition 3.1. (We could assume the latter property directly, but stability under pasting is more suitable for applications.) In particular, we have ′

ℰ𝑠∘ (𝑋) = ess sup𝑃 𝐸 𝑃 [ℰ𝑡∘ (𝑋)∣ℱ𝑠∘ ] 𝑃 -a.s.

for all

𝑃 ∈ 𝒫,

(4.2)

𝑃 ′ ∈𝒫(ℱ𝑠∘ ,𝑃 )

0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇 and 𝑋 ∈ ℋ. If we assume that ℰ𝑡∘ (𝑋) is again an element ∘ of the domain ℋ, this amounts to {ℰ𝑡 } being time-consistent (at determin∘ ∘ ∘ istic times) in the sense that the semigroup property ℰ𝑠 ∘ ℰ𝑡 = ℰ𝑠 is satis∘ ed. However, ℰ𝑡 (𝑋) need not be in ℋ in general; e.g., for certain random 𝐺-expectations. Inspired by the theory of viscosity solutions, we introduce the following extended notion of time consistency, which is clearly implied by (4.2).

Denition 4.2. A family (𝔼𝑡 )0≤𝑡≤𝑇 of mappings 𝔼𝑡 : ℋ → 𝐿1𝒫 (ℱ𝑡∘ ) is called 𝔽∘ -time-consistent at deterministic times if for all 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇 and 𝑋 ∈ ℋ, 𝔼𝑠 (𝑋) ≤ (≥) 𝔼𝑠 (𝜑)

for all

𝜑 ∈ 𝐿1𝒫 (ℱ𝑡∘ ) ∩ ℋ

such that

𝔼𝑡 (𝑋) ≤ (≥) 𝜑.

One can give a similar denition for stopping times taking countably many values. (Note that stopping time

𝔼𝜏 (𝑋)

is not necessarily well dened for a general

𝜏 .)

Remark 4.3. If Assumption 4.1 is weakened by requiring 𝒫 to be stable only under

𝔽∘ -pastings at deterministic times (i.e., Denition 3.1 holds with 𝒯 (𝔽∘ )

replaced by the set of deterministic times), then all results in this section remain true with the same proofs, except for Theorem 4.10, Lemma 4.15 and the last statement in Theorem 4.16.

4.1

Construction of the

ℰ -Martingale

Our rst task is to turn the collection

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]} of random variables

into a reasonable stochastic process. As usual, this requires an extension of the ltration. We denote by

𝔽+ = {ℱ𝑡+ }0≤𝑡≤𝑇 ,

∘ ℱ𝑡+ := ℱ𝑡+

the minimal right continuous ltration containing

∘ and ℱ𝑇 +

0≤𝑡𝑡 ℱ𝑠 𝒫 𝒩 of

∩

𝔽+ by the collection

to obtain the ltration

F = {ℱ𝑡 }0≤𝑡≤𝑇 ,

∘ := ℱ𝑡+

Then

F is right continuous and a natural analogue of the usual augmen-

tation that is standard in the case where a reference probability is given. More precisely, if

𝒫

is dominated by some probability measure, then one

can nd a minimal dominating measure a

𝑃∗ -nullset)

mark that

and then

𝑃∗

(such that every

𝒫 -polar

F is in general strictly smaller than the 𝒫 -universal augmentation

∘𝑃 𝑃 ∈𝒫 𝔽 , which seems to be too large for our purposes. Here ∘ the 𝑃 -augmentation of 𝔽 .

∩

Since

set is

F coincides with the 𝑃∗ -augmentation of 𝔽+ . We re-

𝔽

and

F+

𝔽∘

𝑃

denotes

𝒫 -polar sets, they can be identied for + ∘ that ℱ𝑇 = ℱ𝑇 = ℱ𝑇 𝒫 -q.s. We also recall

dier only by

most purposes; note in particular

the following result (e.g., [17, Theorem 1.5], [31, Lemma 8.2]), which shows that

F and 𝔽∘ dier only by 𝑃 -nullsets for each 𝑃 ∈ 𝒫 .

Lemma 4.4. Let 𝑃 ∈ 𝒫 . Then 𝔽∘ 𝑃 is right continuous and in particular contains F. Moreover, (𝑃, 𝐵) has the predictable representation property; i.e., for any right continuous (𝔽∘ 𝑃 , 𝑃 )-local martingale 𝑀 there exists an ∫ 𝑃 𝔽∘ -predictable process 𝑍 such that 𝑀 = 𝑀0 + (𝑃 ) 𝑍 𝑑𝐵 , 𝑃 -a.s. Proof.

We sketch the argument for the convenience of the reader. We dene

𝑎 ˆ𝑡 = 𝑑⟨𝐵⟩𝑡 /𝑑𝑡 taking values in 𝕊>0 𝑑 𝑃 × 𝑑𝑡-a.e., note −1/2 that (ˆ 𝑎) ∫ is square-integrable for 𝐵 by its very denition, and consider 𝑡 𝑎𝑢 )−1/2 𝑑𝐵𝑢 . Let 𝔽𝑊 be the raw ltration generated by 𝑊 . Since 𝑊𝑡 := (𝑃 ) 0 (ˆ a predictable process

𝑊 is a 𝑃 -Brownian motion by Lévy's characterization, the 𝑃 -augmentation 𝑃 𝔽𝑊 is right continuous and 𝑊 has the representation property. Moreover, 𝑃 𝑃 𝑃 as 𝑃 ∈ 𝒫𝑆 , [31, Lemma 8.1] yields that 𝔽𝑊 = 𝔽∘ . Thus 𝔽∘ is also right continuous and 𝐵 has the representation property since any integral of 𝑊 is also an integral of 𝐵 . We deduce from Lemma 4.4 that for is a (local) cess

𝐵.

𝒫 -polar

(F, 𝑃 )-martingale.

In particular, this applies to the canonical pro-

Note that Lemma 4.4 does not imply that

F and 𝔽∘ coincide up to

sets. E.g., consider the set

𝐴 :=

{

} ∘ lim sup 𝑡−1 ⟨𝐵⟩𝑡 = lim inf 𝑡−1 ⟨𝐵⟩𝑡 = 1 ∈ ℱ0+ . 𝑡→0

𝑡→0

Then the lemma asserts that number is the same for all for

𝑃 ∈ 𝒫 , any (local) (𝔽∘ , 𝑃 )-martingale

𝑃.

(4.3)

𝑃 (𝐴) ∈ {0, 1} for all 𝑃 ∈ 𝒫 , but not that this 𝛼 𝛼 Indeed, 𝑃 (𝐴) = 1 for 𝛼 ≡ 1 but 𝑃 (𝐴) = 0

𝛼 ≡ 2. We can now state the existence and uniqueness of the stochastic process

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]}. For brevity, we shall say that 𝑌 is an (F, 𝒫)-supermartingale if 𝑌 is an (F, 𝑃 )-supermartingale for all 𝑃 ∈ 𝒫 ; derived from

analogous notation will be used in similar situations.

9

Proposition 4.5. Let 𝑋 ∈ ℋ. There exists an F-optional process (𝑌𝑡 )0≤𝑡≤𝑇 such that all paths of 𝑌 are càdlàg and (i) 𝑌 is the minimal (F, 𝒫)-supermartingale with 𝑌𝑇 = 𝑋 ; i.e., if 𝑆 is a càdlàg (F, 𝒫)-supermartingale with 𝑆𝑇 = 𝑋 , then 𝑆 ≥ 𝑌 up to a 𝒫 -polar set. ∘ (𝑋) := lim ∘ (ii) 𝑌𝑡 = ℰ𝑡+ 𝑟↓𝑡 ℰ𝑟 (𝑋) 𝒫 -q.s. for all 0 ≤ 𝑡 < 𝑇 , and 𝑌𝑇 = 𝑋 . (iii) 𝑌 has the representation 𝑃 -a.s.

′

𝑌𝑡 = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡 ] 𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

for all 𝑃 ∈ 𝒫.

(4.4)

Any of the properties (i),(ii),(iii) characterizes 𝑌 uniquely up to 𝒫 -polar sets. The process 𝑌 is denoted by ℰ(𝑋) and called the (càdlàg) ℰ -martingale associated with 𝑋 . Proof.

We choose and x representatives for the classes

and dene the

𝑌𝑡 (𝜔) :=

ℝ ∪ {±∞}-valued lim sup

process

ℰ𝑟∘ (𝑋)(𝜔)

for

𝑌

ℰ𝑡∘ (𝑋) ∈ 𝐿1𝒫 (ℱ𝑡∘ )

by

0≤𝑡 𝑡, moreover, 𝒫(ℱ ∘ , 𝑃 ) = 𝒫(ℱ , 𝑃 ) since 𝒫(ℱ𝑡𝑛 , 𝑃 ) ⊆ 𝒫(ℱ𝑡+ 𝑛 𝑡 𝑡+ ∘ and ℱ coincide up to 𝒫 -polar sets. In view of (4.6), the inequality ℱ𝑡+ 𝑡

since

converse to (4.5) follows and (iii) is proved. To see the minimality property in (i), let with

𝑆𝑇 = 𝑋 .

𝑆

be an

′

𝑆𝑡 ≥ ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡 ] 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ

for all

𝑃 ∈ 𝒫.

𝑡 ,𝑃 )

By (iii) the right hand side is all

(F, 𝒫)-supermartingale

Exactly as in (4.5), we deduce that

𝑃 -a.s.

equal to

𝑌𝑡 .

Hence

𝑆𝑡 ≥ 𝑌𝑡 𝒫 -q.s.

for

𝑆 ≥ 𝑌 𝒫 -q.s. when 𝑆 is càdlàg 𝑌 and 𝑌 ′ are processes satisfying (i) or (ii) or (iii), then they 𝑃 -modications of each other for all 𝑃 ∈ 𝒫 and thus coincide up to a

𝑡

and

Finally, if are

𝒫 -polar

set as soon as they are càdlàg.

One can ask whether

ℰ(𝑋)

is a

𝒫 -modication

of

{ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]};

i.e., whether

ℰ𝑡 (𝑋) = ℰ𝑡∘ (𝑋) 𝒫 -q.s.

for all

0 ≤ 𝑡 ≤ 𝑇.

ℰ(𝑋) is a 𝒫 -modication as soon as there exists some 𝒫 -modication of the family {ℰ𝑡∘ (𝑋), 𝑡 ∈ [0, 𝑇 ]}, and this is the case 𝑃 ∘ only if 𝑡 7→ 𝐸 [ℰ𝑡 (𝑋)] is right continuous for all 𝑃 ∈ 𝒫 . We also

It is easy to see that càdlàg if and

remark that Lemma 4.4 and the argument given for (4.5) yield

ℰ𝑡 (𝑋) ≤ ℰ𝑡∘ (𝑋) 𝒫 -q.s.

for all

0≤𝑡≤𝑇

(4.7)

and so the question is only whether the converse inequality holds true as well. The answer is positive in several important cases; e.g., for the

11

𝐺-expectation

when

𝑋

is suciently regular [35, Theorem 5.3] and the sublinear expec-

tation generated by a controlled stochastic dierential equation [24, Theorem 5.1]. The proof of the latter result yields a general technique to approach this problem in a given example. However, the following (admittedly degenerate) example shows that the answer is negative in a very general case; this reects the fact that the set than the set

𝒫(ℱ𝑡∘ , 𝑃 )

𝒫(ℱ𝑡 , 𝑃 )

in the representation (4.4) is smaller

in (4.1).

Example 4.6. We shall consider a 𝐺-expectation dened on a set of irregular random variables.

∘ ) ℋ = 𝐿1𝒫 (ℱ0+

Let

𝑎 = 1, 𝑎 = 2

and let

𝒫

be as in (2.2).

We take

and dene

{ sup𝑃 ∈𝒫 𝐸 𝑃 [𝑋], 𝑡 = 0, ℰ𝑡∘ (𝑋) := 𝑋, 0 0. As noted after Lemma 3.3, the second part of Assumption 4.1 is also satised. Moreover, the càdlàg ℰ -martingale is given by for

ℰ𝑡 (𝑋) = 𝑋,

𝑡 ∈ [0, 𝑇 ].

𝑋 := 1𝐴 , where 𝐴 is dened as in (4.3). Then ℰ0∘ (𝑋) = 1 and ℰ0 (𝑋) = 1𝐴 are not equal 𝑃 2 -a.s. (i.e., the measure 𝑃 𝛼 for 𝛼 ≡ 2). In fact, 2 ∘ there is no càdlàg 𝒫 -modication since {ℰ𝑡 (𝑋)} coincides 𝑃 -a.s. with the deterministic function 𝑡 7→ 1{0} (𝑡).

Consider

We remark that the phenomenon appearing in the previous example is

𝒫

due to the presence of singular measures rather than the fact that

is

not dominated. In fact, one can give a similar example involving only two measures. Finally, let us mention that the situation is quite dierent if we assume that the given sublinear expectation is already placed in the larger ltration

𝔽

(i.e., Assumption 4.1 holds with

𝔽∘

replaced by

𝔽),

which would be in line

with the paradigm of the usual assumptions in standard stochastic analysis. In this case, the arguments in the proof of Proposition 4.5 show that is always a

𝒫 -modication.

This result is neat, but not very useful, since the

examples are typically constructed in

4.2

ℰ(𝑋)

𝔽∘ .

Stopping Times

The direct construction of

𝐺-expectations

at stopping times is an unsolved

problem. Indeed, stopping times are typically fairly irregular functions and it is unclear how to deal with this in the existing constructions (see also [20]). On the other hand, we can easily evaluate the càdlàg process stopping time tion at

𝜏.

𝜏

ℰ(𝑋)

at a

and therefore dene the corresponding sublinear expecta-

In particular, this leads to a denition of

12

𝐺-expectations at general

stopping times. We show in this section that the resulting random variable

ℰ𝜏 (𝑋)

indeed has the expected properties and that the time consistency ex-

tends to arbitrary

𝔽-stopping times; in other words, we prove an optional ℰ -martingales. Besides the obvious theoretical interest,

sampling theorem for the study of

ℰ(𝑋)

at stopping times will allow us to verify integrability con-

ditions of the type class (D); cf. Lemma 4.15 below. We start by explaining the relations between the stopping times of the dierent ltrations.

Lemma 4.7. (i) Let 𝑃 ∈ 𝒫 and let 𝜏 be an F-stopping time taking countably many values. Then there exists an 𝔽∘ -stopping time 𝜏 ∘ (depending on 𝑃 ) such that 𝜏 = 𝜏 ∘ 𝑃 -a.s. Moreover, for any such 𝜏 ∘ , the 𝜎-elds ℱ𝜏 and ℱ𝜏∘∘ dier only by 𝑃 -nullsets. (ii) Let 𝜏 be an F-stopping time. Then there exists an 𝔽+ -stopping time + 𝜏 such that 𝜏 = 𝜏 + 𝒫 -q.s. Moreover, for any such 𝜏 + , the 𝜎 -elds ℱ𝜏 and ℱ𝜏++ dier only by 𝒫 -polar sets. ∑ Proof. (i) Note that 𝜏 is of the form 𝜏 = 𝑖 𝑡𝑖 1Λ𝑖 for Λ𝑖 = {𝜏 = 𝑡𝑖 } ∈ ℱ𝑡𝑖 forming a partition of such that

Ω.

Λ𝑖 = Λ∘𝑖 𝑃 -a.s.

Since

F ⊆ 𝔽∘ 𝑃 by Lemma 4.4, we can nd Λ∘𝑖 ∈ ℱ𝑡∘

𝑖

and the rst assertion follows by taking

𝜏 ∘ := 𝑇 1(∪𝑖 Λ∘𝑖 )𝑐 +

∑

𝑡𝑖 1Λ∘𝑖 .

𝑖

𝐴 ∈ ℱ𝜏 . By the rst part, there exists an 𝔽∘ -stopping time (𝜏𝐴 )∘ ∘ ′ ∘ that (𝜏𝐴 ) = 𝜏𝐴 := 𝜏 1𝐴 + 𝑇 1𝐴𝑐 𝑃 -a.s. Moreover, we choose 𝐴 ∈ ℱ𝑇 ′ that 𝐴 = 𝐴 𝑃 -a.s. Then ( ) 𝐴∘ := 𝐴′ ∩ {𝜏 ∘ = 𝑇 } ∪ {(𝜏𝐴 )∘ = 𝜏 ∘ < 𝑇 }

Let such such

𝐴∘ ∈ ℱ𝜏∘∘ and 𝐴 = 𝐴∘ 𝑃 -a.s. A similar but simpler argument shows ∘ ′ ′ that for given Λ ∈ ℱ𝜏 ∘ we can nd Λ ∈ ℱ𝜏 such that Λ = Λ 𝑃 -a.s. + 𝑛 (ii) If 𝜏 is an F- (resp. 𝔽 -) stopping time, we can nd 𝜏 taking count𝑛 + ably many values such that 𝜏 decreases to 𝜏 and since F (𝔽 ) is right + + continuous, ℱ𝜏 𝑛 (ℱ𝜏 𝑛 ) decreases to ℱ𝜏 (ℱ𝜏 ). As a result, we may assume without loss of generality that 𝜏 takes countably many values. ∑ Let 𝜏 = 𝑖 𝑡𝑖 1Λ𝑖 , where Λ𝑖 ∈ ℱ𝑡𝑖 . The denition of F shows that there + + + exist Λ𝑖 ∈ ℱ𝑡 such that Λ𝑖 = Λ𝑖 𝒫 -q.s. and the rst part follows. The proof 𝑖 satises

of the second part is as in (i); we now have quasi-sure instead of almost-sure relations.

𝜎 is a stopping taking nitely many values (𝑡𝑖 )1≤𝑖≤𝑁 , we can ∑𝑁 time ∘ ∘ dene ℰ𝜎 (𝑋) := 𝑖=1 ℰ𝑡𝑖 (𝑋)1{𝜎=𝑡𝑖 } . We have the following generalization If

of (4.1).

Lemma 4.8. Let 𝜎 be an 𝔽∘ -stopping time taking nitely many values. Then ′

ℰ𝜎∘ (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎∘ ] 𝑃 ′ ∈𝒫(ℱ𝜎∘ ,𝑃 )

13

𝑃 -a.s.

for all 𝑃 ∈ 𝒫.

Proof.

𝑃 ∈ 𝒫 and 𝑌𝑡∘ := ℰ𝑡∘ (𝑋). Moreover, let (𝑡𝑖 )1≤𝑖≤𝑁 be the values ∘ of 𝜎 and Λ𝑖 := {𝜎 = 𝑡𝑖 } ∈ ℱ𝑡 . 𝑖 ′ (i) We rst prove the inequality  ≥. Given 𝑃 ∈ 𝒫 , it follows from (4.2) ∘ ′ ∘ that {𝑌𝑡 }1≤𝑖≤𝑁 is a 𝑃 -supermartingale in (ℱ𝑡 )1≤𝑖≤𝑁 and so the (discrete𝑖 𝑖 ∘ 𝑃′ ∘ time) optional sampling theorem [9, Theorem V.11] implies 𝑌𝜎 ≥ 𝐸 [𝑋∣ℱ𝜎 ] ′ ′ ∘ 𝑃 -a.s. In particular, this also holds 𝑃 -a.s. for all 𝑃 ∈ 𝒫(ℱ𝜎 , 𝑃 ), hence the Let

claim follows. (ii) We now show the inequality  ≤. Note that

(Λ𝑖 )1≤𝑖≤𝑁

form an

ℱ𝜎∘ -measurable

partition of

′

𝑌𝑡∘𝑖 1Λ𝑖 ≤ ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎∘ ]1Λ𝑖

Ω.

𝜎=

∑𝑁

𝑖=1 𝑡𝑖 1Λ𝑖 and that

It suces to show that

𝑃 -a.s.

for

1 ≤ 𝑖 ≤ 𝑁.

𝑃 ′ ∈𝒫(ℱ𝜎∘ ,𝑃 )

In the sequel, we x

𝑃¯ ∈

𝑖

and show that for each

𝒫(ℱ𝜎∘ , 𝑃 ) such that 𝑃¯ (𝐴 ∩ Λ𝑖 ) = 𝑃 ′ (𝐴 ∩ Λ𝑖 )

In view of (4.1) and

𝑃 ′ ∈ 𝒫(ℱ𝑡∘𝑖 , 𝑃 )

for all

𝐴 ∈ ℱ𝑇∘ .

(4.8)

′

′

there exists

𝐸 𝑃 [𝑋∣ℱ𝜎∘ ]1Λ𝑖 = 𝐸 𝑃 [𝑋∣ℱ𝑡∘𝑖 ]1Λ𝑖 𝑃 ′ -a.s.,

it will then

follow that

¯

′

𝑌𝑡∘𝑖 1Λ𝑖 = ess sup𝑃 𝐸 𝑃 [𝑋1Λ𝑖 ∣ℱ𝑡∘𝑖 ] ≤ ess sup𝑃 𝐸 𝑃 [𝑋1Λ𝑖 ∣ℱ𝜎∘ ] 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ𝑡∘ ,𝑃 ) 𝑖

𝑃¯ ∈𝒫(ℱ𝜎∘ ,𝑃 )

as claimed. Indeed, given

𝑃 ′ ∈ 𝒫(ℱ𝑡∘𝑖 , 𝑃 ),

we dene

𝑃¯ (𝐴) := 𝑃 ′ (𝐴 ∩ Λ𝑖 ) + 𝑃 (𝐴 ∖ Λ𝑖 ),

𝐴 ∈ ℱ𝑇∘ ,

(4.9)

Λ ∈ ℱ𝜎∘ , then Λ ∩ Λ𝑖 = Λ ∩ {𝜎 = 𝑡𝑖 } ∈ ℱ𝑡∘𝑖 𝑃 ′ (Λ ∩ Λ𝑖 ) = 𝑃 (Λ ∩ Λ𝑖 ). Hence 𝑃¯ = 𝑃 on ℱ𝜎∘ .

then (4.8) is obviously satised. If and

𝑃 ′ ∈ 𝒫(ℱ𝑡∘𝑖 , 𝑃 )

yields

Moreover, we observe that (4.9) can be stated as

] [ 𝑃¯ (𝐴) = 𝐸 𝑃 𝑃 ′ (𝐴∣ℱ𝑡∘𝑖 )1Λ𝑖 + 𝑃 (𝐴∣ℱ𝑡∘𝑖 )1Λ𝑐𝑖 ,

𝐴 ∈ ℱ𝑇∘ ,

which is a special case of the pasting (3.2) applied with

𝑃¯ ∈ 𝒫

by Assumption 4.1 and we have

𝑃¯ ∈ 𝒫(ℱ𝜎∘ , 𝑃 )

𝑃2 := 𝑃 .

Hence

as desired.

For the next result, we recall that stability under pasting refers to stopping times with nitely many values rather than general ones (Denition 3.2).

Lemma 4.9. The set 𝒫 is stable under 𝔽-pasting. Proof. Let 𝜏 ∈ 𝒯 (𝔽), then 𝜏 is of the form 𝜏=

∑

𝑡𝑖 1Λ𝑖 ,

Λ𝑖 := {𝜏 = 𝑡𝑖 } ∈ ℱ𝑡𝑖 ,

𝑖

∈ [0, 𝑇 ] are distinct and the sets Λ𝑖 form a partition of Ω. Moreover, Λ ∈ ℱ 𝜏 and 𝑃1 , 𝑃2 ∈ 𝒫(ℱ𝜏 , ]𝑃 ), then we have to show that the measure [ 𝑃 𝐸 𝑃1 ( ⋅ ∣ℱ𝜏 )1Λ + 𝑃2 ( ⋅ ∣ℱ𝜏 )1Λ𝑐 is an element of 𝒫 .

where 𝑡𝑖 let

14

(i) We start by proving that for any such that

𝐴=

𝐴′ holds

𝐴 ∈ ℱ𝜏

𝒫(ℱ𝜏 , 𝑃 )-q.s. Consider ∪ 𝐴 = (𝐴 ∩ Λ𝑖 ).

there exists

𝐴′ ∈ ℱ𝑇∘ ∩ ℱ𝜏

the disjoint union

𝑖

𝐴 ∩ Λ𝑖 ∈ ℱ𝑡𝑖 since 𝐴 ∈ ℱ𝜏 . As F ⊆ 𝔽∘ by Lemma 4.4, 𝐴𝑖 ∈ ℱ𝑡∘𝑖 and a 𝑃 -nullset 𝑁𝑖 , disjoint from 𝐴𝑖 , such that 𝑃

Here set

𝐴 ∩ Λ𝑖 = 𝐴𝑖 ∪ 𝑁𝑖 . (It is

not

there exist a

(4.10)

necessary to subtract another nullset on the right hand side.) We

𝐴′ := ∪𝑖 𝐴𝑖 , then 𝐴′ ∈ ℱ𝑇∘ and clearly 𝐴 = 𝐴′ 𝑃 -a.s. Let us check that ′ latter also holds 𝒫(ℱ𝜏 , 𝑃 )-q.s. For this, it suces to show that 𝐴 ∈ ℱ𝜏 .

dene the

Indeed, by the construction of (4.10),

{ 𝐴𝑖 ∈ ℱ𝑡∘𝑖 ⊆ ℱ𝑡𝑖 , 𝑖 = 𝑗, 𝐴𝑖 ∩ {𝜏 = 𝑡𝑗 } = ∅ ∈ ℱ𝑡𝑗 , 𝑗= ∕ 𝑖; i.e., each set

𝐴𝑖

is in

ℱ𝜏 .

Hence,

𝐴′ ∈ ℱ𝜏 ,

which completes the proof of (i).

∘ For later use, we dene the 𝔽 -stopping time

(𝜏𝐴 )∘ := 𝑇 1(𝐴′ )𝑐 +

∑

𝑡 𝑖 1𝐴 𝑖

𝑖

)∘

𝒫(ℱ𝜏 , 𝑃 )-q.s. (ii) Using the previous construction for 𝐴 = Ω, we see in ∘ ∘ ∘ there exist Λ𝑖 ∈ ℱ𝑡 such that Λ𝑖 = Λ𝑖 holds 𝒫(ℱ𝜏 , 𝑃 )-q.s. 𝑖 ∘ the 𝔽 -stopping time ∑ 𝜏 ∘ := 𝑇 1(∪𝑖 Λ∘𝑖 )𝑐 + 𝑡𝑖 1Λ∘𝑖

and note that

(𝜏𝐴

= 𝜏𝐴

holds

particular that We also dene

𝑖

𝜏∘ = 𝜏. ∘ (iii) We can now show that ℱ𝜏 ∘ and ℱ𝜏 may be identied (when 𝑃, 𝑃1 , 𝑃2 ′ are xed). Indeed, if 𝐴 ∈ ℱ𝜏 , we let 𝐴 be as in (i) and set ( ) 𝐴∘ := 𝐴′ ∩ {𝜏 ∘ = 𝑇 } ∪ {(𝜏𝐴 )∘ = 𝜏 ∘ < 𝑇 }.

which

𝒫(ℱ𝜏 , 𝑃 )-q.s.

satises

𝐴∘ ∈ ℱ𝜏∘∘ and 𝐴 = 𝐴∘ holds 𝒫(ℱ𝜏 , 𝑃 )-q.s. Conversely, given 𝐴∘ ∈ ℱ𝜏∘∘ , ∘ we nd 𝐴 ∈ ℱ𝜏 such that 𝐴 = 𝐴 holds 𝒫(ℱ𝜏 , 𝑃 )-q.s. We conclude that ] [ ] [ 𝐸 𝑃 𝑃1 ( ⋅ ∣ℱ𝜏 )1Λ + 𝑃2 ( ⋅ ∣ℱ𝜏 )1Λ𝑐 = 𝐸 𝑃 𝑃1 ( ⋅ ∣ℱ𝜏∘∘ )1Λ∘ + 𝑃2 ( ⋅ ∣ℱ𝜏∘∘ )1(Λ∘ )𝑐 . Then

The right hand side is an element of

𝒫

by the stability under

We can now prove the optional sampling theorem for particular, this establishes the

𝔽-time-consistency

stopping times.

15

of

{ℰ𝑡 }

𝔽∘ -pasting.

ℰ -martingales; along general

in

F-

Theorem 4.10. Let 0 ≤ 𝜎 ≤ 𝜏 ≤ 𝑇 be stopping times, ℰ(𝑋) be the càdlàg ℰ -martingale associated with 𝑋 . Then 𝑃 -a.s.

′

ℰ𝜎 (𝑋) = ess sup𝑃 𝐸 𝑃 [ℰ𝜏 (𝑋)∣ℱ𝜎 ] 𝑃 ′ ∈𝒫(ℱ𝜎 ,𝑃 )

𝑋 ∈ ℋ,

and let

for all 𝑃 ∈ 𝒫

(4.11)

and in particular 𝑃 -a.s.

′

ℰ𝜎 (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎 ] 𝑃 ′ ∈𝒫(ℱ𝜎 ,𝑃 )

for all 𝑃 ∈ 𝒫.

(4.12)

Moreover, there exists for each 𝑃 ∈ 𝒫 a sequence 𝑃𝑛 ∈ 𝒫(ℱ𝜎 , 𝑃 ) such that 𝑃 -a.s.

ℰ𝜎 (𝑋) = lim 𝐸 𝑃𝑛 [𝑋∣ℱ𝜎 ] 𝑛→∞

(4.13)

with an increasing limit. Proof.

Fix

𝑃 ∈𝒫

and let

𝑌 := ℰ(𝑋).

(i) We rst show the inequality  ≥ in (4.12).

𝑌

is an

(F, 𝑃 ′ )-supermartingale

By Proposition 4.5(i),

𝑃 ′ ∈ 𝒫(ℱ𝜎 , 𝑃 ).

for all

Hence the (usual)

optional sampling theorem implies the claim. (ii) In the next two steps, we show the inequality  ≤ in (4.12). In view of Lemma 4.7(ii) we may assume that

(𝜎 + 1/𝑛) ∧ 𝑇

𝜎

also assume that

𝜎

𝔽+ -stopping time, and then 𝑛 ≥ 1. For the time being, we

is an

∘ is an 𝔽 -stopping time for each

takes nitely many values. Let

𝐷𝑛 := {𝑘2−𝑛 : 𝑘 = 0, 1, . . . } ∪ {𝑇 } and dene

𝜎 𝑛 (𝜔) := inf{𝑡 ∈ 𝐷𝑛 : 𝑡 ≥ 𝜎(𝜔) + 1/𝑛} ∧ 𝑇. 𝔽∘ -stopping time taking nitely many values and 𝜎 𝑛 (𝜔) de𝑛 creases to 𝜎(𝜔) for all 𝜔 ∈ Ω. Since the range of {𝜎, (𝜎 )𝑛 } is countable, it ∘ follows from Proposition 4.5(ii) that ℰ𝜎 𝑛 (𝑋) → 𝑌𝜎 𝑃 -a.s. Since ∥𝑋∥𝐿1 < ∞,

Each

𝜎𝑛

is an

𝒫

the backward supermartingale convergence theorem [9, Theorem V.30] implies that this convergence holds also in

𝐿1 (𝑃 )

and that

𝑌𝜎 = lim 𝐸 𝑃 [ℰ𝜎∘𝑛 (𝑋)∣ℱ𝜎 ] 𝑃 -a.s., 𝑛→∞

where, by monotonicity, the a subsequence. sequence

𝑃 -a.s.

(4.14)

convergence holds without passing to

By Lemma 4.8 and Lemma 3.4, there exists for each

(𝑃𝑘𝑛 )𝑘≥1

in

𝒫(ℱ𝜎∘𝑛 , 𝑃 )

such that ′

𝑛

ℰ𝜎∘𝑛 (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎∘𝑛 ] = lim 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎∘𝑛 ] 𝑃 -a.s., 𝑘→∞

𝑃 ′ ∈𝒫(ℱ𝜎∘𝑛 ,𝑃 )

where the limit is increasing. Moreover, using that

{ ℱ𝜎+𝑛+1 = 𝐴 ∈ ℱ𝑇∘ : 𝐴 ∩ {𝜎 𝑛+1 < 𝑡} ∈ ℱ𝑡∘ 16

for

} 0≤𝑡≤𝑇 ,

𝑛

a

𝜎 𝑛 > 𝜎 𝑛+1 on {𝜎 𝑛 < 𝑇 } is seen to imply that ℱ𝜎+𝑛+1 ⊆ ℱ𝜎∘𝑛 . 𝑛+1 and Lemma 4.7(ii) we conclude that with 𝜎 ≤ 𝜎

the fact that Together

𝒫 -q.s.

ℱ𝜎 ⊆ ℱ𝜎𝑛+1 = ℱ𝜎+𝑛+1 ⊆ ℱ𝜎∘𝑛

and hence

𝒫(ℱ𝜎 , 𝑃 ) ⊇ 𝒫(ℱ𝜎∘𝑛 , 𝑃 ) (4.15)

for all

𝑛.

Now monotone convergence yields ′

𝑛

𝐸 𝑃 [ℰ𝜎∘𝑛 (𝑋)∣ℱ𝜎 ] = lim 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎 ]

≤ ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎 ] 𝑃 -a.s.

𝑘→∞

𝑃 ′ ∈𝒫(ℱ𝜎 ,𝑃 )

In view of (4.14), this ends the proof of (4.12) for

𝜎

taking nitely many

values. (ii') Now let

𝜎𝑛

𝜎

be general. We approximate

𝜎

by the decreasing sequence

:= inf{𝑡 ∈ 𝐷𝑛 : 𝑡 ≥ 𝜎} ∧ 𝑇 of stopping times with nitely many values. ℰ𝜎𝑛 (𝑋) ≡ 𝑌𝜎𝑛 → 𝑌𝜎 𝑃 -a.s. since 𝑌 is càdlàg. The same arguments as

Then

for (4.14) show that

𝑌𝜎 = lim 𝐸 𝑃 [ℰ𝜎𝑛 (𝑋)∣ℱ𝜎 ] 𝑃 -a.s.

(4.16)

𝑛→∞

By the two previous steps we have the representation (4.12) for Lemma 3.4, it follows from the stability under there exists for each

𝑛

a sequence

(𝑃𝑘𝑛 )𝑘≥1

in

𝜎𝑛.

As in

𝔽-pasting (Lemma 4.9) that 𝒫(ℱ𝜎𝑛 , 𝑃 ) ⊆ 𝒫(ℱ𝜎 , 𝑃 ) such

that ′

𝑛

ℰ𝜎𝑛 (𝑋) = ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎𝑛 ] = lim 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎𝑛 ] 𝑃 -a.s., 𝑘→∞

𝑃 ′ ∈𝒫(ℱ𝜎𝑛 ,𝑃 )

where the limit is increasing and hence ′

𝑛

𝐸 𝑃 [ℰ𝜎𝑛 (𝑋)∣ℱ𝜎 ] = lim 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎 ]

≤ ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝜎 ] 𝑃 -a.s.

𝑘→∞

𝑃 ′ ∈𝒫(ℱ𝜎 ,𝑃 )

Together with (4.16), this completes the proof of (4.12). (iii) We now prove (4.13). Since

𝜎

is general, the claim does not follow

from the stability under pasting. Instead, we use the construction of (ii'). Indeed, we have obtained

𝑃𝑘𝑛 ∈ 𝒫(ℱ𝜎𝑛 , 𝑃 )

such that

𝑛

𝑌𝜎 = lim lim 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎 ] 𝑃 -a.s. 𝑛→∞ 𝑘→∞

𝑛. Since 𝜎 𝑛 is an 𝔽-stopping time taking nitely many values and since ℱ𝜎 ⊆ ℱ𝜎𝑛 , it follows from the stability under 𝔽-pasting (applied to 𝜎 𝑛 ) that 𝑃′ ′ the set {𝐸 [𝑋∣ℱ𝜎 ] : 𝑃 ∈ 𝒫(ℱ𝜎 𝑛 , 𝑃 )} is 𝑃 -a.s. upward ltering, exactly as in the proof of Lemma 3.4. In view of 𝒫(ℱ𝜎 𝑛 , 𝑃 ) ⊆ 𝒫(ℱ𝜎 𝑛+1 , 𝑃 ), it follows (𝑁 ) ∈ 𝒫(ℱ that for each 𝑁 ≥ 1 there exists 𝑃 𝜎 𝑁 , 𝑃 ) such that Fix

𝐸𝑃

(𝑁 )

𝑛

[𝑋∣ℱ𝜎 ] = max max 𝐸 𝑃𝑘 [𝑋∣ℱ𝜎 ] 𝑃 -a.s. 1≤𝑛≤𝑁 1≤𝑘≤𝑛

17

Since

𝒫(ℱ𝜎𝑁 , 𝑃 ) ⊆ 𝒫(ℱ𝜎 , 𝑃 ),

this yields the claim.

ℰ𝜎 (𝑋) and ℰ𝜏 (𝑋) as essential for 𝜏 . The inequality  ≤ is then

(iv) To prove (4.11), we rst express suprema by using (4.12) both for

𝜎

and

immediate. The converse inequality follows by a monotone convergence argument exactly as in the proof of Proposition 3.6(i), except that the increasing sequence is now obtained from (4.13) instead of Lemma 3.4.

4.3

Decomposition and 2BSDE for

ℰ -Martingales

The next result contains the semimartingale decomposition of each

𝑃 ∈𝒫

ℰ(𝑋)

under

and can be seen as an analogue of the optional decomposition [19]

used in mathematical nance.

In the context of

𝐺-expectations,

such a

result has also been referred to as  𝐺-martingale representation theorem; see [16, 34, 35, 37]. Those results are ultimately based on the PDE description of the

𝐺-expectation

and are more precise than ours; in particular, they

(𝐾 𝑃 )𝑃 ∈𝒫 (but 1 see Remark 4.17). On the other hand, we obtain an 𝐿 -theory whereas those

provide a single increasing process

𝐾

results require more integrability for

rather than a family

𝑋.

Proposition 4.11. Let 𝑋 ∈ ℋ. There exist ∫ (i) an F-predictable process 𝑍 𝑋 with 0𝑇 ∣𝑍𝑠𝑋 ∣2 𝑑⟨𝐵⟩𝑠 < ∞ 𝒫 -q.s., (ii) a family (𝐾 𝑃 )𝑃 ∈𝒫 of 𝔽𝑃 -predictable processes such that all paths of 𝐾 𝑃 are càdlàg nondecreasing and 𝐸 𝑃 [∣𝐾𝑇𝑃 ∣] < ∞, such that (𝑃∫) 𝑡

ℰ𝑡 (𝑋) = ℰ0 (𝑋) +

𝑍𝑠𝑋 𝑑𝐵𝑠 − 𝐾𝑡𝑃

0

for all 0 ≤ 𝑡 ≤ 𝑇, 𝑃 -a.s.

(4.17)

for all 𝑃 ∈ 𝒫 . The process 𝑍 𝑋 is unique up to {𝑑𝑠 × 𝑃, 𝑃 ∈ 𝒫}-polar sets and 𝐾 𝑃 is unique up to 𝑃 -evanescence. Proof. We shall use arguments similar to the proof of [33, Theorem 4.5]. Let

𝑃 ∈ 𝒫.

It follows from Proposition 4.5(i) that

𝑌 := ℰ(𝑋)

is an

𝑃

(𝔽 , 𝑃 )-supermartingale. We apply the Doob-Meyer decomposition in the 𝑃 ltered space (Ω, 𝔽 , 𝑃 ) which satises the usual conditions of right continu𝑃 𝑃 and ity and completeness. Thus we obtain an (𝔽 , 𝑃 )-local martingale 𝑀 𝑃 𝑃 an 𝔽 -predictable increasing integrable process 𝐾 , càdlàg and satisfying 𝑀0𝑃 = 𝐾0𝑃 = 0, such that 𝑌 = 𝑌0 + 𝑀 𝑃 − 𝐾 𝑃 . (𝑃, 𝐵) has the predictable representation 𝑃 𝑃 such that exists an 𝔽 -predictable process 𝑍 (𝑃∫) 𝑌 = 𝑌0 + 𝑍 𝑃 𝑑𝐵 − 𝐾 𝑃 .

By Lemma 4.4, Hence there

18

property in

𝑃

𝔽

.

The next step is to replace calling that

∫

𝐵

𝑍𝑃

by a process

𝑍𝑋

independent of

is a continuous local martingale under each

𝑍 𝑃 𝑑⟨𝐵⟩𝑃 = ⟨𝑌, 𝐵⟩𝑃 = 𝐵𝑌 −

(𝑃∫)

𝑃,

𝑃.

Re-

we have

(𝑃∫)

𝐵 𝑑𝑌 −

𝑌− 𝑑𝐵

𝑃 -a.s.

(4.18)

(Here and below, the statements should be read componentwise.) The last two integrals are Itô integrals under

𝑃 , but they can also be dened pathwise

since the integrands are left limits of càdlàg processes which are bounded path-by-path.

This is a classical construction from [3, Theorem 7.14]; see

also [18] for the same result in modern notation.

To make explicit that

∫the resulting process is 𝔽-adapted, we recall the procedure for the example 𝑌− 𝑑𝐵 . One rst denes for each 𝑛 ≥ 1 the sequence of F-stopping times 𝑛 := inf{𝑡 ≥ 𝜏 𝑛 : ∣𝑌 − 𝑌 𝑛 ∣ ≥ 2−𝑛 }. Then one denes 𝐼 𝑛 by 𝜏0𝑛 := 0 and 𝜏𝑖+1 𝑡 𝜏𝑖 𝑖 𝐼𝑡𝑛 := 𝑌𝜏𝑘𝑛 (𝐵𝑡 − 𝐵𝜏𝑘𝑛 ) +

𝑘−1 ∑

𝑛 𝑌𝜏𝑖𝑛 (𝐵𝜏𝑖+1 − 𝐵𝜏𝑖𝑛 )

for

𝑛 𝜏𝑘𝑛 < 𝑡 ≤ 𝜏𝑘+1 ,

𝑘 ≥ 0;

𝑖=0 clearly

𝐼𝑛

is again

F-adapted and all its paths are càdlàg. Finally, we dene 𝐼𝑡 := lim sup 𝐼𝑡𝑛 ,

0 ≤ 𝑡 ≤ 𝑇.

𝑛→∞

Then

𝐼

is again

F-adapted and it is a consequence of the Burkholder-Davis-

Gundy inequalities that

∫ 𝑛 (𝑃 ) 𝑡 sup 𝐼𝑡 − 𝑌− 𝑑𝐵 → 0 𝑃 -a.s.

0≤𝑡≤𝑇 for each

𝑃.

0

Thus, outside a

exists as a limit uniformly in

𝒫 -polar 𝑡 and 𝐼

set, the limsup in the denition of has càdlàg paths. Since

𝒫 -polar

𝐼

sets

ℱ0 , we may redene 𝐼 := 0 on the exceptional set. Now ∫ 𝐼 is càdlàg F-adapted and coincides with the Itô integral (𝑃 ) 𝑌− 𝑑𝐵 up to 𝑃 -evanescence, for all 𝑃 ∈ 𝒫 . ∫ (𝑃 ) 𝐵 𝑑𝑌 and obtain a denition We proceed similarly with the integral

are contained in

for the right hand side of (4.18) which is

F-adapted, continuous and inde-

⟨𝑌, 𝐵⟩ simultaneously for all 𝑃 ∈ 𝒫 , 𝑎 ˆ = 𝑑⟨𝐵⟩/𝑑𝑡 be the (left) derivative in time of ⟨𝐵⟩, then 𝑎 ˆ is 𝔽∘ -predictable and 𝕊>0 𝑑 -valued 𝑃 × 𝑑𝑡-a.e. for all 𝑃 ∈ 𝒫 𝑋 by the denition of 𝒫𝑆 . Finally, 𝑍 := 𝑎 ˆ−1 𝑑⟨𝑌, 𝐵⟩/𝑑𝑡 is an F-predictable

pendent of

𝑃.

Thus we have dened

and we do the same for

⟨𝐵⟩.

Let

process such that

(𝑃∫)

𝑌 = 𝑌0 +

𝑍 𝑋 𝑑𝐵 − 𝐾 𝑃

We note that the integral is taken under to dene it for all

𝑃 ∈𝒫

simultaneously.

19

𝑃 -a.s. 𝑃;

for all

𝑃 ∈ 𝒫.

see also Remark 4.17 for a way

The previous proof shows that a decomposition of the type (4.17) ex-

(𝔽, 𝒫)-supermartingales,

ists for all càdlàg

and not just for

ℰ -martingales.

As a special case of Proposition 4.11, we obtain a representation for symmetric

ℰ -martingales.

The following can be seen as a generalization of the

corresponding results for

𝐺-expectations

given in [34, 35, 37].

Corollary 4.12. Let 𝑋 ∈ ℋ be such that −𝑋 ∈ ℋ. The following are equivalent: (i) ℰ(𝑋) is a symmetric ℰ -martingale; i.e., ℰ(−𝑋) = −ℰ(𝑋) 𝒫 -q.s. ∫ (ii) There exists an F-predictable process 𝑍 𝑋 with 0𝑇 ∣𝑍𝑠𝑋 ∣2 𝑑⟨𝐵⟩𝑠 < ∞ 𝒫 -q.s. such that 𝑡

∫

𝑍𝑠𝑋 𝑑𝐵𝑠

ℰ𝑡 (𝑋) = ℰ0 (𝑋) + 0

for all 0 ≤ 𝑡 ≤ 𝑇, 𝒫 -q.s.,

where the integral can be dened universally for all 𝑃 and 𝑍 𝑋 𝑑𝐵 is an (𝔽, 𝑃 )-martingale for all 𝑃 ∈ 𝒫 . In particular, any symmetric ℰ -martingale has continuous trajectories 𝒫 -q.s. ∫

Proof.

The implication (ii)⇒(i) is clear from Proposition 4.5(iii).

Con-

ℰ(𝑋) and −ℰ(𝑋) are 𝒫 -martingale. It follows that 𝑃 ≡ 0 and (4.17) becomes satisfy 𝐾

versely, given (i), Proposition 4.5(i) yields that both

𝒫 -supermartingales,

hence

ℰ(𝑋)

is a (true)

𝐾 𝑃 have to 𝑑𝐵 . ∫In particular, the stochastic setting 𝑍 𝑋 𝑑𝐵 := ℰ(𝑋) − ℰ0 (𝑋).

the increasing processes

(𝑃 )

∫

ℰ(𝑋) = ℰ0 (𝑋) +

dened universally by

Remark 4.13.

𝑍𝑋

integral can be

(a) Without the martingale condition in Corollary 4.12(ii),

the implication (ii)⇒(i) would fail even for

𝒫 = {𝑃0 },

in which case Corol-

lary 4.12 is simply the Brownian martingale representation theorem.

ℰ(𝑋) need not be a 𝒫 -modication of the fam𝑡 ∈ [0, 𝑇 ]}; in fact, the ℰ -martingale in Example 4.6 is symmetric.

(b) Even if it is symmetric,

∘ ily {ℰ𝑡 (𝑋),

However, the situation changes if the symmetry assumption is imposed di-

{ℰ𝑡∘ (𝑋)}. 𝑡 ∈ [0, 𝑇 ].

rectly on for all

∙

We call

{ℰ𝑡∘ (𝑋)} symmetric if ℰ𝑡∘ (−𝑋) = −ℰ𝑡∘ (𝑋) 𝒫 -q.s.

If {ℰ𝑡∘ (𝑋)} symmetric, then ℰ(𝑋) is a symmetric ℰ -martingale and a 𝒫 -modication of {ℰ𝑡∘ (𝑋)}.

Indeed, the assumption implies that each

𝑃 ∈𝒫

and so the process

is the usual càdlàg

ℰ(𝑋)

𝑃 -modication

Next, we represent the pair

of

{ℰ𝑡∘ (𝑋)}

is an

(𝔽∘ , 𝑃 )-martingale

for

of right limits (cf. Proposition 4.5(ii))

{ℰ𝑡∘ (𝑋)},

(ℰ(𝑋), 𝑍 𝑋 )

for all

𝑃.

from Proposition 4.11 as the

solution of a 2BSDE. The following denition is essentially from [32].

Denition 4.14. values in

ℝ × ℝ𝑑

Let

such

𝑋 ∈ 𝐿1𝒫 and consider a pair (𝑌, 𝑍) of processes with that 𝑌 is càdlàg F-adapted while 𝑍 is F-predictable 20

and

∫𝑇

∣𝑍𝑠 ∣2 𝑑⟨𝐵⟩𝑠 < ∞ 𝒫 -q.s.

0

Then

(𝑌, 𝑍)

is called a

𝑃 2BSDE (4.19) if there exists a family (𝐾 )𝑃 ∈𝒫 of 𝑃 𝑃 processes satisfying 𝐸 [∣𝐾𝑇 ∣] < ∞ such that (𝑃∫) 𝑇

𝑌𝑡 = 𝑋 −

𝑍𝑠 𝑑𝐵𝑠 + 𝐾𝑇𝑃 − 𝐾𝑡𝑃 ,

0 ≤ 𝑡 ≤ 𝑇,

𝑃

𝔽

solution

of the

-adapted increasing

𝑃 -a.s.

for all

𝑃 ∈𝒫

𝑡 (4.19) and such that the following minimality condition holds for all

ess inf 𝑃

𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

] ′[ ′ ′ 𝐸 𝑃 𝐾𝑇𝑃 − 𝐾𝑡𝑃 ℱ𝑡 = 0 𝑃 -a.s.

We note that (4.20) is essentially the

for all

0 ≤ 𝑡 ≤ 𝑇:

𝑃 ∈ 𝒫.

ℰ -martingale condition (4.4):

(4.20)

if the

𝑃 can be aggregated into a single process 𝐾 and 𝐾 ∈ ℋ, then processes 𝐾 𝑇 −𝐾 = ℰ(−𝐾𝑇 ). Regarding the aggregation of (𝐾 𝑃 ), see also Remark 4.17. A second notion is needed to state the main result.

𝑌

is said to be

under

𝑃

for all

of class (D,𝒫 )

𝑃 ∈ 𝒫,

where

if the family

𝜎

{𝑌𝜎 }𝜎

runs through all

A càdlàg process

is uniformly integrable

F-stopping times.

example, we have seen in Corollary 4.12 that all symmetric

As an

ℰ -martingales

are of class (D,𝒫 ). (Of course, it is important here that we work with a nite

𝑇 .) For 𝑝 ∈ [1, ∞), we 𝑝 ℋ := {𝑋 ∈ ℋ : ∣𝑋∣𝑝 ∈ ℋ}.

time horizon well as

dene

∥𝑋∥𝐿𝑝 =: sup𝑃 ∈𝒫 𝐸[∣𝑋∣𝑝 ]1/𝑝 𝒫

as

Lemma 4.15. If 𝑋 ∈ ℋ𝑝 for some 𝑝 ∈ (1, ∞), then ℰ(𝑋) is of class (D,𝒫 ). Proof.

Let

𝑃 ∈ 𝒫.

If

𝜎

is an

F-stopping time, Jensen's inequality and (4.12)

yield that ′

∣ℰ𝜎 (𝑋)∣𝑝 ≤ ess sup𝑃 𝐸 𝑃 [∣𝑋∣𝑝 ∣ℱ𝜎 ] = ℰ𝜎 (∣𝑋∣𝑝 ) 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ𝜎 ,𝑃 )

In particular,

∥ℰ𝜎 (𝑋)∥𝑝𝐿𝑝 (𝑃 ) ≤ 𝐸 𝑃 [ℰ𝜎 (∣𝑋∣𝑝 )]

and thus Lemma 4.4 yields ′

∥ℰ𝜎 (𝑋)∥𝑝𝐿𝑝 (𝑃 ) ≤ 𝐸 𝑃 [ℰ𝜎 (∣𝑋∣𝑝 )∣ℱ0 ] ≤ ess sup𝑃 𝐸 𝑃 [ℰ𝜎 (∣𝑋∣𝑝 )∣ℱ0 ] 𝑃 -a.s. 𝑃 ′ ∈𝒫(ℱ0 ,𝑃 )

The right hand side

𝑃 -a.s. equals ℰ0 (∣𝑋∣𝑝 ) by (4.11), so we conclude with (4.7)

that ′

∥ℰ𝜎 (𝑋)∥𝑝𝐿𝑝 (𝑃 ) ≤ ℰ0 (∣𝑋∣𝑝 ) ≤ sup 𝐸 𝑃 [∣𝑋∣𝑝 ] = ∥𝑋∥𝑝𝐿𝑝 < ∞ 𝑃 -a.s. 𝑃 ′ ∈𝒫

{ℰ𝜎 (𝑋)}𝜎 is bounded in 𝐿𝑝 (𝑃 ) under 𝑃 . This holds for all 𝑃 ∈ 𝒫 .

Therefore, the family formly integrable

We can now state the main result of this section.

21

𝒫

and in particular uni-

Theorem 4.16. Let 𝑋 ∈ ℋ. (i) The pair (ℰ(𝑋), 𝑍 𝑋 ) is the minimal solution of the 2BSDE (4.19); i.e., if (𝑌, 𝑍) is another solution, then ℰ(𝑋) ≤ 𝑌 𝒫 -q.s. (ii) If (𝑌, 𝑍) is a solution of (4.19) such that 𝑌 is of class (D,𝒫 ), then (𝑌, 𝑍) = (ℰ(𝑋), 𝑍 𝑋 ). In particular, if 𝑋 ∈ ℋ𝑝 for some 𝑝 > 1, then (ℰ(𝑋), 𝑍 𝑋 ) is the unique solution of (4.19) in the class (D,𝒫 ). Proof. (i) Let 𝑃 ∈ 𝒫 . To show that (ℰ(𝑋), 𝑍 𝑋 ) is a solution, we only have to show that

𝐾𝑃

from the decomposition (4.17) satises the minimality con-

dition (4.20). We denote this decomposition by

ℰ(𝑋)

It follows from Proposition 4.5(i) that As

𝐾 𝑃 ≥ 0,

is

ℰ(𝑋) = ℰ0 (𝑋) + 𝑀 𝑃 − 𝐾 𝑃 . 𝑃 an (𝔽 , 𝑃 )-supermartingale.

we deduce that

𝑃

ℰ0 (𝑋) + 𝑀 𝑃 ≥ ℰ(𝑋) ≥ 𝐸 𝑃 [𝑋∣𝔽 ] 𝑃 -a.s., 𝑃

𝑃

𝐸 𝑃 [𝑋∣𝔽 ] denotes the càdlàg (𝔽 , 𝑃 )-martingale with terminal value 𝑋 . Hence 𝑀 𝑃 is a local 𝑃 -martingale bounded from below by a 𝑃 -martingale 𝑃 is an (F, 𝑃 )-supermartingale by a standard argument using and thus 𝑀 Fatou's lemma. This holds for all 𝑃 ∈ 𝒫 . Therefore, (4.4) yields where

′

0 = ℰ𝑡 (𝑋) − ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡 ] 𝑃 ′ ∈𝒫(ℱ

= ess inf

𝑃

𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

𝐸

𝑃′

[

𝑡 ,𝑃 )

] ℰ𝑡 (𝑋) − ℰ𝑇 (𝑋) ℱ𝑡

] ′ ′ ′ ′[ ′ = ess inf 𝑃 𝐸 𝑃 𝑀𝑡𝑃 − 𝑀𝑇𝑃 + 𝐾𝑇𝑃 − 𝐾𝑡𝑃 ℱ𝑡 ′ 𝑃 ∈𝒫(ℱ𝑡 ,𝑃 ) ] ′[ ′ ′ ≥ ess inf 𝑃 𝐸 𝑃 𝐾𝑇𝑃 − 𝐾𝑡𝑃 ℱ𝑡 𝑃 -a.s. for all 𝑃 ∈ 𝒫. 𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

Since

𝐾𝑃

′

is nondecreasing, the last expression is also nonnegative and (4.20)

follows. Thus

(ℰ(𝑋), 𝑍 𝑋 )

is a solution.

(𝑌, 𝑍) be another solution of (4.19). It (F, 𝑃 )-supermartingale for all 𝑃 ∈ 𝒫 . ∫ (𝑃 ) As above, the integrability of 𝑋 implies that 𝑌0 + 𝑍 𝑑𝐵 is bounded below by a 𝑃 -martingale. Noting also that 𝑌0 is 𝑃 -a.s. equal to a constant ∫ (𝑃 ) 𝑍 𝑑𝐵 and 𝑌 are (F, 𝑃 )-supermartingales. by Lemma 4.4, we deduce that Since 𝑌 is càdlàg and 𝑌𝑇 = 𝑋 , the minimality property in Proposition 4.5(i) shows that 𝑌 ≥ ℰ(𝑋) 𝒫 -q.s. ∫ (𝑃 ) 𝑍 𝑑𝐵 is a true 𝑃 -martingale (ii) If in addition 𝑌 is of class (D,𝒫 ), then To prove the minimality, let

follows from (4.19) that

𝑌

is a local

by the Doob-Meyer theorem and we have

] ′[ ′ ′ 0 = ess inf 𝑃 𝐸 𝑃 𝐾𝑇𝑃 − 𝐾𝑡𝑃 ℱ𝑡 𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

′

= 𝑌𝑡 − ess sup𝑃 𝐸 𝑃 [𝑋∣ℱ𝑡 ] 𝑃 ′ ∈𝒫(ℱ𝑡 ,𝑃 )

= 𝑌𝑡 − ℰ𝑡 (𝑋) 𝑃 -a.s. 22

for all

𝑃 ∈ 𝒫.

The last statement in the theorem follows from Lemma 4.15.

Remark 4.17.

If we use axioms of set theory stronger than the usual ZFC,

such as the Continuum Hypothesis, then the integrals

∫ {(𝑃 ) 𝑍 𝑑𝐵}𝑃 ∈𝒫

can

be aggregated into a single (universally measurable) continuous process, denoted by

∫

𝑍 𝑑𝐵 ,

for any

𝑍

which is

𝐵 -integrable

under all

𝑃 ∈ 𝒫.

This

follows from a recent result on pathwise stochastic integration, cf. [25]. In

(𝐾 𝑃 )𝑃 ∈𝒫 of increasing ∫ 𝐾 := ℰ0 (𝑋) − ℰ(𝑋) + 𝑍 𝑋 𝑑𝐵 .

Proposition 4.11, we can then aggregate the family processes into a single process

𝐾

by setting

Moreover, we can strengthen Theorem 4.16 by asking for a universal process

𝐾

the Denition 4.14 of the 2BSDE.

4.4

Application to Superhedging and Replication

We now turn to the interpretation of the previous results for the super-

𝑑 problem. Let 𝐻 be an ℝ -valued 𝔽-predictable process satisfying ∫hedging 𝑇 2 ∣ 𝑑⟨𝐵⟩𝑠 < ∞ 𝒫 -q.s. Then 𝐻 is called an admissible trading strategy 0 ∣𝐻 ∫𝑠 (𝑃 ) if 𝐻 𝑑𝐵 is a 𝑃 -supermartingale for all 𝑃 ∈ 𝒫 . (We do not insist that the integral be dened without reference to 𝑃 , since this is not necessary economically.

But see also Remark 4.17.)

As usual in continuous-time -

nance, this denition excludes doubling strategies. proof of Theorem 4.16 that

𝑍𝑋

is admissible for

We have seen in the

𝑋 ∈ ℋ.

The minimality

property in Proposition 4.5(i) and the existence of the decomposition (4.17) yield the following conclusion:

ℰ0 (𝑋) is the minimal ℱ0 -measurable initial 𝑋 ; i.e., ℰ0 (𝑋) is the 𝒫 -q.s. minimal ℱ0 -

capital which allows to superhedge measurable random variable

𝐻

𝜉0

such that there exists an admissible strategy

satisfying

(𝑃∫) 𝑇

𝐻𝑠 𝑑𝐵𝑠 ≥ 𝑋

𝜉0 +

𝑃 -a.s.

for all

𝑃 ∈ 𝒫.

0 Moreover, the overshoot

𝐾𝑃

for the strategy

𝑍𝑋

satises the minimality

condition (4.20). As seen in Example 4.6, the

ℱ0 -superhedging

price

ℰ0 (𝑋)

need not be

a constant, and therefore it is debatable whether it is a good choice for a conservative price, in particular if the raw ltration

𝔽∘

is seen as the initial

information structure for the model. Indeed, the following illustration shows that knowledge of

ℱ0

can be quite signicant. Consider a collection

positive constants and

𝒫 = {𝑃 𝛼 : 𝛼 ≡ 𝑎𝑖

for some

𝑖}.

(Such a set

(𝑎𝑖 ) of 𝒫 can

indeed satisfy the assumptions of this section.) In this model, knowledge of

ℱ0

ℱ0 contains { } ∘ 𝐴𝑖 := lim sup 𝑡−1 ⟨𝐵⟩𝑡 = lim inf 𝑡−1 ⟨𝐵⟩𝑡 = 𝑎𝑖 ∈ ℱ0+

completely removes the volatility uncertainty since

𝑡→0

𝑡→0

23

the sets

which form a

𝒫 -q.s.

partition of

Ω.

Hence, one may want to use the more

conservative choice

𝑥 = ℰ0∘ (𝑋) = sup 𝐸 𝑃 [𝑋] = inf{𝑦 ∈ ℝ : 𝑦 ≥ ℰ0 (𝑋)} 𝑃 ∈𝒫

ℰ -martingale as follows. ℱ0− be the smallest 𝜎 -eld containing the 𝒫 -polar sets, then ℱ0− is trivial 𝒫 -q.s. If we adjoin ℱ0− as a new initial state to the ltration 𝔽, we can extend ℰ(𝑋) by setting as the price. This value can be embedded into the

Let

ℰ0− (𝑋) := sup 𝐸 𝑃 [𝑋],

𝑋 ∈ ℋ.

𝑃 ∈𝒫 The resulting process

{ℰ𝑡 (𝑋)}𝑡∈[−0,𝑇 ]

satises the properties from Proposi-

𝑥 = ℰ0− (𝑋) ℱ0− -superhedging price of 𝑋 . (Of course, all this becomes superuous ∘ the case where ℰ(𝑋) is a 𝒫 -modication of {ℰ𝑡 (𝑋)}.)

tion 4.5 in the extended ltration and in particular the constant is the in

In the remainder of the section, we discuss replicable claims and adopt the previously mentioned conservative choice.

Denition 4.18. a constant

𝒫 -q.s.

𝑥∈ℝ

A random variable and an

𝑋 ∈ ℋ is called replicable if there exist ∫𝑇 2 𝑑⟨𝐵⟩ < ∞ ∣𝐻 ∣ 𝑠 𝑠 0

F-predictable process 𝐻 with

such that

(𝑃∫) 𝑇

𝑋 =𝑥+

𝐻𝑡 𝑑𝐵𝑡

𝑃 -a.s.

for all

𝑃 ∈𝒫

(4.21)

0 and such that

(𝑃 )

∫

𝐻 𝑑𝐵

is an

(F, 𝑃 )-martingale

for all

𝑃 ∈ 𝒫.

The martingale assumption is needed to avoid strategies which throw away money. Moreover, as in Corollary 4.12, the stochastic integral can necessarily be dened without reference to

𝑃,

by setting

∫

𝐻 𝑑𝐵 := ℰ(𝑋) − 𝑥.

The following result is an analogue of the standard characterization of replicable claims in incomplete markets (e.g., [8, p. 182]).

Proposition 4.19. Let 𝑋 ∈ ℋ be such that −𝑋 ∈ ℋ. The following are equivalent: (i) ℰ(𝑋) is a symmetric ℰ -martingale and ℰ0 (𝑋) is constant 𝒫 -q.s. (ii) 𝑋 is replicable. (iii) There exists 𝑥 ∈ ℝ such that 𝐸 𝑃 [𝑋] = 𝑥 for all 𝑃 ∈ 𝒫 . Proof.

The equivalence (i)⇔(ii) is immediate from Corollary 4.12 and the

implication (ii)⇒(iii) follows by taking expectations in (4.21). prove (iii)⇒(ii).

By (4.7) we have

Hence we

ℰ0 (−𝑋) ≤ sup𝑃 ∈𝒫 𝐸 𝑃 [−𝑋] = −𝑥

24

and

ℰ(𝑋) ≤ 𝑥. Thus, ℰ(𝑋) show that (𝑃∫) 𝑇 𝑍 −𝑋 𝑑𝐵 −𝑋 ≤ −𝑥 +

similarly

ℰ(−𝑋)

𝑃 ∈ 𝒫,

given

the decompositions (4.17) of

and

(𝑃∫) 𝑇

𝑋 ≤𝑥+

and

𝑍 𝑋 𝑑𝐵

𝑃 -a.s.

(4.22)

0

0

∫ (𝑃 ) 𝑇 (𝑍 −𝑋 0

+ 𝑍 𝑋 ) 𝑑𝐵 𝑃 -a.s. As we 𝑋 and 𝑍 −𝑋 know from the proof of Theorem 4.16 that the integrals of 𝑍 ∫ ∫ 𝑇 𝑇 (𝑃 ) −𝑋 𝑑𝐵 = −(𝑃 ) 𝑋 are supermartingales, it follows that 0 𝑍 0 𝑍 𝑑𝐵 𝑃 -a.s. ∫ 𝑇 (𝑃 ) 𝑋 Now (4.22) yields that 𝑋 = 𝑥 + 0 𝑍 𝑑𝐵 . In view of (iii), this integral

Adding the inequalities yields

0 ≤

is a supermartingale with constant expectation, hence a martingale.

5

Uniqueness of Time-Consistent Extensions

In the introduction, we have claimed that

{ℰ𝑡∘ (𝑋)}

as in (1.2) is the natural

dynamic extension of the static sublinear expectation

𝑋 7→ sup𝑃 ∈𝒫 𝐸 𝑃 [𝑋].

In this section, we add some substance to this claim by showing that the extension is unique under suitable assumptions. (We note that by Proposition 3.6, the question of existence is essentially reduced to the technical problem of aggregation.) The setup is as follows. We x a nonempty set on

(Ω, ℱ𝑇∘ ); it is not important whether 𝒫

𝒫

of probability measures

consists of martingale laws. On the

other hand, we impose additional structure on the set of random variables. In this section, we consider a chain of vector spaces

ℝ = ℋ0 ⊆ ℋ𝑠 ⊆ ℋ𝑡 ⊆ ℋ𝑇 =: ℋ ⊆ 𝐿1𝒫 , We assume that addition

𝑌

𝑋, 𝑌 ∈ ℋ𝑡

implies

is bounded. As before,

(ℋ𝑡 )0≤𝑡≤𝑇

satisfying

0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇.

𝑋 ∧ 𝑌, 𝑋 ∨ 𝑌 ∈ ℋ𝑡 , and 𝑋𝑌 ∈ ℋ𝑡 if in ℋ should be seen as the set of nancial

ℋ𝑡 will serve as test functions; the main example ℋ𝑡 = ℋ ∩ 𝐿1𝒫 (ℱ𝑡∘ ). We consider a family (𝔼𝑡 )0≤𝑡≤𝑇 of

claims. The elements of to have in mind is mappings

𝔼𝑡 : ℋ → 𝐿1𝒫 (ℱ𝑡∘ ) (𝔼𝑡 ) as a dynamic extension of 𝔼0 . Our aim is to nd conditions under which 𝔼0 already determines the whole family (𝔼𝑡 ), or more precisely, determines 𝔼𝑡 (𝑋) up to a 𝒫 -polar set for all 𝑋 ∈ ℋ and 0 ≤ 𝑡 ≤ 𝑇 .

and think of

Denition 5.1. if for all

(𝔼𝑡 )0≤𝑡≤𝑇 𝑋 ∈ ℋ,

The family

𝑡 ∈ [0, 𝑇 ]

and

𝔼𝑡 (𝑋𝜑) = 𝔼𝑡 (𝑋)𝜑 𝒫 -q.s.

is called

(ℋ𝑡 )-positively

for all bounded nonnegative

Note that this property excludes trivial extensions of

𝔼0 ,

we can always dene the (time-consistent) extension

{ 𝔼0 (𝑋), 0 ≤ 𝑡 < 𝑇, 𝔼𝑡 (𝑋) := 𝑋, 𝑡 = 𝑇, 25

𝔼0 .

homogeneous 𝜑 ∈ ℋ𝑡 . Indeed, given

but this family choices of

(𝔼𝑡 )

is not

(ℋ𝑡 )-positively

homogeneous for nondegenerate

(ℋ𝑡 ).

To motivate the next denition, we rst recall that in the classical setup under a reference measure

𝑃∗ ,

strict monotonicity of

tion for uniqueness of extensions; i.e., should imply that

𝔼0 (𝑋) > 𝔼0 (𝑌 ).

𝔼0 is the crucial condi𝑋 ≥ 𝑌 𝑃∗ -a.s. and 𝑃∗ {𝑋 > 𝑌 } > 0

In our setup with singular measures,

𝔼0 (⋅) = sup𝑃 ∈𝒫 𝐸 𝑃 [ ⋅ ], it is completely reasonable to have random variables 𝑋 ≥ 𝑌 satisfying 𝔼0 (𝑋) = 𝔼0 (𝑌 ) and 𝑃1 {𝑋 > 𝑌 } > 0 for some 𝑃1 ∈ 𝒫 , since the suprema can be attained at some 𝑃2 ∈ 𝒫 whose support is disjoint from {𝑋 > 𝑌 }. the corresponding condition is too strong. E.g., for

In the following denition, we allow for an additional localization by a test function.

Denition 5.2.

(ℋ𝑡 )-locally strictly monotone if for every 𝑡 ∈ [0, 𝑇 ] and any 𝑋, 𝑌 ∈ ℋ𝑡 satisfying 𝑋 ≥ 𝑌 𝒫 -q.s. and 𝑃 (𝑋 > 𝑌 ) > 0 for some 𝑃 ∈ 𝒫 , there exists 𝑓 ∈ ℋ𝑡 such that 0 ≤ 𝑓 ≤ 1 and We say that

𝔼0

is

𝔼0 (𝑋𝑓 ) > 𝔼0 (𝑌 𝑓 ). Here the delicate point is the regularity required for tempted to try

𝑓.

Indeed, one is

𝑓 := 1{𝑋>𝑌 +𝛿} (for some constant 𝛿 > 0), but in applications ℋ𝑡 may exclude this choice and require a more rened

the denition of

construction. We defer this task to Proposition 5.5 and rst show how local strict monotonicity yields uniqueness.

Proposition 5.3. Let 𝔼0 be (ℋ𝑡 )-locally strictly monotone. Then there exists at most one extension of 𝔼0 to a family (𝔼𝑡 )0≤𝑡≤𝑇 which is (ℋ𝑡 )-positively homogeneous and satises 𝔼𝑡 (ℋ) ⊆ ℋ𝑡 and 𝔼0 ∘ 𝔼𝑡 = 𝔼0 on ℋ. ˜ 𝑡 ) be two such extensions and suppose for contradicProof. Let (𝔼𝑡 ) and (𝔼 ˜ 𝑡 (𝑋) for some 𝑋 ∈ ℋ; i.e., there exists 𝑃 ∈ 𝒫 such that 𝔼𝑡 (𝑋) ∕= 𝔼 ˜ 𝑡 (𝑋)} > 0 or 𝑃 {𝔼𝑡 (𝑋) < 𝔼 ˜ 𝑡 (𝑋)} > 0. Without loss of 𝑃 {𝔼𝑡 (𝑋) > 𝔼

tion that either

generality, we focus on the rst case. Dene

𝜑 := Then over,

([ ] ) ˜ 𝑡 (𝑋) ∨ 0 ∧ 1. 𝔼𝑡 (𝑋) − 𝔼

𝜑 ∈ ℋ𝑡 , since ℋ𝑡 is a lattice containing the constant functions; more˜ 𝑡 (𝑋)}. Setting 𝑋 ′ := 𝑋𝜑 and 0 ≤ 𝜑 ≤ 1 and {𝜑 = 0} = {𝔼𝑡 (𝑋) ≤ 𝔼

using the positive homogeneity, we arrive at

˜ 𝑡 (𝑋 ′ ) 𝔼𝑡 (𝑋 ′ ) ≥ 𝔼

and

{ } ˜ 𝑡 (𝑋 ′ ) > 0. 𝑃 𝔼𝑡 (𝑋 ′ ) > 𝔼

𝑓 ∈ ℋ𝑡 such that 0 ≤ 𝑓 ≤ 1 ( ) ( ) ˜ 𝑡 (𝑋 ′ )𝑓 . Now 𝔼0 = 𝔼0 ∘ 𝔼𝑡 yields that 𝔼0 𝔼𝑡 (𝑋 ′ )𝑓 > 𝔼0 𝔼 ( ) ( ) ˜ 𝑡 (𝑋 ′ )𝑓 = 𝔼 ˜ 0 (𝑋 ′ 𝑓 ), 𝔼0 (𝑋 ′ 𝑓 ) = 𝔼0 𝔼𝑡 (𝑋 ′ )𝑓 > 𝔼0 𝔼 By local strict monotonicity there exists

which contradicts

˜0. 𝔼0 = 𝔼 26

and

We can extend the previous result by applying it on dense subspaces. This relaxes the assumption that

𝔼𝑡 (ℋ) ⊆ ℋ𝑡

and simplies the verication

of local strict monotonicity since one can choose convenient spaces of test

ˆ 𝑡 )0≤𝑡≤𝑇 satisfying the same assump(ℋ ˆ 𝑇 is a ∥ ⋅ ∥ 1 -dense subspace of ℋ. We ℋ 𝐿

functions. Consider a chain of spaces

(ℋ𝑡 )0≤𝑡≤𝑇 that (𝔼𝑡 )0≤𝑡≤𝑇

tions as say

and such that

𝒫

𝐿1𝒫 -continuous if ( ) ( ) 𝔼𝑡 : ℋ, ∥ ⋅ ∥𝐿1 → 𝐿1𝒫 (ℱ𝑡∘ ), ∥ ⋅ ∥𝐿1 is

𝒫

𝒫

is continuous for every 𝑡. We remark that the motivating example

(ℰ𝑡∘ ) from

Assumption 4.1 satises this property (it is even Lipschitz continuous).

Corollary 5.4. Let 𝔼0 be (ℋˆ 𝑡 )-locally strictly monotone. Then there exists at most one extension of 𝔼0 to an 𝐿1𝒫 -continuous family (𝔼𝑡 )0≤𝑡≤𝑇 on ˆ 𝑡 )-positively homogeneous and satises 𝔼𝑡 (ℋ ˆ𝑇 ) ⊆ ℋ ˆ 𝑡 and ℋ which is (ℋ ˆ𝑇 . 𝔼0 ∘ 𝔼𝑡 = 𝔼0 on ℋ Proof.

ˆ𝑇 . 𝔼𝑡 (𝑋) is uniquely determined for 𝑋 ∈ ℋ ˆ 𝑇 ⊆ ℋ is dense and 𝔼𝑡 is continuous, 𝔼𝑡 is also determined on ℋ. Since ℋ Proposition 5.3 shows that

𝔼0 (⋅) = sup𝑃 ∈𝒫 𝐸 𝑃 [ ⋅ ]

In our last result, we show that

is

(ℋ𝑡 )-locally

strictly monotone in certain cases. The idea here is that we already have an

(𝔼𝑡 ) (as in Assumption 4.1), whose uniqueness we try to establish. 𝐶𝑏 (Ω) the set of bounded continuous functions on Ω and by ∘ 𝐶𝑏 (Ω𝑡 ) the ℱ𝑡 -measurable functions in 𝐶𝑏 (Ω), or equivalently the bounded functions which are continuous with respect to ∥𝜔∥𝑡 := sup0≤𝑠≤𝑡 ∣𝜔𝑠 ∣. Similarly, UC𝑏 (Ω) and UC𝑏 (Ω𝑡 ) denote the sets of bounded uniformly continuous 1 1 ∞ functions. We also dene 𝕃𝑐,𝒫 to be the closure of 𝐶𝑏 (Ω) in 𝐿𝒫 , while 𝕃𝑐,𝒫 1 ∞ ∘ denotes the 𝒫 -q.s. bounded elements of 𝕃𝑐,𝒫 . Finally, 𝕃𝑐,𝒫 (ℱ𝑡 ) is obtained ∞ ∘ similarly from 𝐶𝑏 (Ω𝑡 ), while 𝕃𝑢𝑐,𝒫 (ℱ𝑡 ) is the space obtained when starting from UC𝑏 (Ω𝑡 ) instead of 𝐶𝑏 (Ω𝑡 ). extension

We denote by

Proposition 5.5. Let 𝔼0 (⋅) = sup𝑃 ∈𝒫 𝐸 𝑃 [ ⋅ ]. Then strictly monotone for each of the cases (i) ℋ𝑡 = 𝐶𝑏 (Ω𝑡 ), (ii) ℋ𝑡 = UC𝑏 (Ω𝑡 ), ∘ (iii) ℋ𝑡 = 𝕃∞ 𝑐,𝒫 (ℱ𝑡 ), ∘ (iv) ℋ𝑡 = 𝕃∞ 𝑢𝑐,𝒫 (ℱ𝑡 ).

𝔼0

is (ℋ𝑡 )-locally

Together with Corollary 5.4, this yields a uniqueness result for extensions. Before giving the proof, we indicate some examples covered by this result;

(𝔼𝑡 ) is ℋ = 𝕃1𝑢𝑐,𝒫 in both cases. (This 1 1 that 𝕃𝑢𝑐,𝒫 = 𝕃𝑐,𝒫 when 𝒫 is tight; cf. the

see also Example 2.1. The domain of statement implicitly uses the fact proof of [23, Proposition 5.2].)

27

(a) Let

(𝔼𝑡 )

be the

lary 5.4 applies: if

ˆ𝑇 ) ⊆ ℋ ˆ𝑡 𝔼𝑡 (ℋ

ˆ𝑡 ℋ

𝐺-expectation

as introduced in [27, 28]. Then Corol-

is any of the spaces in (i)(iv), the invariance property

ˆ𝑇 ℋ

is satised and

is dense in

ℋ.

(b) Using the construction given in [23], the

𝐺-expectation

can be ex-

tended to the case when there is no nite upper bound for the volatility. This

𝐺 (and then 𝒫 need not be tight). ˆ 𝑡 = UC𝑏 (Ω𝑡 ) since 𝔼𝑡 (ℋ ˆ𝑇 ) ⊆ ℋ ˆ 𝑡 is satised ℋ ∞ ∘ [23, Corollary 3.6], or also with ℋ𝑡 = 𝕃𝑢𝑐,𝒫 (ℱ𝑡 ).

corresponds to a possibly innite function Here Corollary 5.4 applies with by the remark stated after

Proof of Proposition 5.5.

𝑡 ∈ [0, 𝑇 ]. All topological notions in this proof ′ ′ are expressed with respect to 𝑑(𝜔, 𝜔 ) := ∥𝜔 − 𝜔 ∥𝑡 . Let 𝑋, 𝑌 ∈ ℋ𝑡 be such that 𝑋 ≥ 𝑌 𝒫 -q.s. and 𝑃∗ (𝑋 > 𝑌 ) > 0 for some 𝑃∗ ∈ 𝒫 . By translating and multiplying with positive constants, we may assume that 1 ≥ 𝑋 ≥ 𝑌 ≥ 0. Fix

We prove the cases (i)(iv) separately. (i) Choose

𝛿>0

small enough so that

𝐴1 := {𝑋 ≥ 𝑌 + 2𝛿}, Then

𝐴1

and

𝐴2

𝐴1 .

0≤𝑓 ≤1

(5.1)

as well as

𝑓 =0

on

𝐴2

and

It remains to check that

𝔼0 (𝑋𝑓 ) > 𝔼0 (𝑌 𝑓 ), If

𝐴2 := {𝑋 ≤ 𝑌 + 𝛿}.

𝑑(𝜔, 𝐴2 ) 𝑑(𝜔, 𝐴1 ) + 𝑑(𝜔, 𝐴2 )

is a continuous function satisfying on

and let

are disjoint closed sets and

𝑓 (𝜔) :=

𝑓 =1

𝑃∗ {𝑋 ≥ 𝑌 + 2𝛿} > 0

𝔼0 (𝑌 𝑓 ) = 0,

sup 𝐸 𝑃 [𝑋𝑓 ] > sup 𝐸 𝑃 [𝑌 𝑓 ].

i.e.,

𝑃 ∈𝒫

the observation that

𝑃 ∈𝒫

𝔼0 (𝑋𝑓 ) ≥ 𝐸 𝑃∗ [𝑋𝑓 ] ≥ 2𝛿𝑃∗ (𝐴1 ) > 0

already yields the proof.

𝔼0 (𝑌 𝑓 ) > 0. For 𝜀 > 0, let 𝑃𝜀 ∈ 𝒫 be 𝑓 ] ≥ 𝔼0 (𝑌 𝑓 ) − 𝜀. Since 𝑋 > 𝑌 + 𝛿 on {𝑓 > 0} and since have 𝑋𝑓 ≥ (𝑌 + 𝛿)𝑓 ≥ (𝑌 + 𝛿𝑌 )𝑓 and therefore

Hence, we may assume that

𝑃 such that 𝐸 𝜀 [𝑌

0 ≤ 𝑌 ≤ 1,

we

𝔼0 (𝑋𝑓 ) ≥ lim sup 𝐸 𝑃𝜀 [(𝑌 + 𝛿𝑌 )𝑓 ] 𝜀→0

= lim sup (1 + 𝛿)𝐸 𝑃𝜀 [𝑌 𝑓 ] 𝜀→0

= (1 + 𝛿) 𝔼0 (𝑌 𝑓 ). As

𝛿>0

and

𝔼0 (𝑌 𝑓 ) > 0,

this ends the proof of (i).

(ii) The proof for this case is the same; we merely have to check that the function

𝑓

𝑍 := 𝑋 − 𝑌 is 𝑌 are. Thus there exists 𝜀 > 0 such that 𝑑(𝜔, 𝜔 ′ ) ≤ 𝜀. We observe that 𝑑(𝐴1 , 𝐴2 ) ≥ 𝜀

dened in (5.1) is uniformly continuous. Indeed,

uniformly continuous since

∣𝑍(𝜔) − 𝑍(𝜔 ′ )∣ < 𝛿

𝑋

whenever

and

and hence that the denominator in (5.1) is bounded away from zero. One then checks by direct calculation that

𝑓

28

is Lipschitz continuous.

(iii) We recall that

∘ 𝕃∞ 𝒫 (ℱ𝑡 )

coincides with the set of bounded

𝒫 -quasi

continuous functions (up to modication); cf. [10, Theorem 25]. That is, a bounded

ℱ𝑡∘ -measurable

function

ℎ

∘ 𝕃∞ 𝒫 (ℱ𝑡 ) if and only if for all 𝜀 > 0 that 𝑃 (Λ) > 1 − 𝜀 for all 𝑃 ∈ 𝒫 and

is in

Λ ⊆ Ω such ℎ∣Λ is continuous. For 𝛿 > 0 small enough, we have 𝑃∗ ({𝑋 ≥ 𝑌 + 2𝛿}) > 0. Then, we nd a closed set Λ ⊆ Ω such that 𝑋 and 𝑌 are continuous on Λ and there exists a closed set

such that the restriction

(1 + 𝛿) 𝔼0 (1Λ𝑐 ) < 𝛿 2 𝔼0 (1{𝑋≥𝑌 +2𝛿}∩Λ ).

can

(5.2)

Dene the disjoint closed sets

𝐴1 := {𝑋 ≥ 𝑌 + 2𝛿} ∩ Λ, and let

𝑓

𝐴2 := {𝑋 ≤ 𝑌 + 𝛿} ∩ Λ,

be the continuous function (5.1). We distinguish two cases. Sup-

pose rst that

𝛿𝔼0 (𝑌 𝑓 ) ≤ (1 + 𝛿) 𝔼0 (1Λ𝑐 );

then, using (5.2),

𝔼0 (𝑋𝑓 ) ≥ 2𝛿𝔼0 (1𝐴1 ) > (1 + 𝛿)𝛿 −1 𝔼0 (1Λ𝑐 ) ≥ 𝔼0 (𝑌 𝑓 ) and we are done. Otherwise, we have

𝛿𝔼0 (𝑌 𝑓 ) > (1 + 𝛿) 𝔼0 (1Λ𝑐 ).

Moreover,

𝔼0 (𝑋𝑓 1Λ ) ≥ (1 + 𝛿)𝔼0 (𝑌 𝑓 1Λ ) can be shown as in (i); we simply replace 𝑓 𝑓 1Λ in that argument. Using the subadditivity of 𝔼0 , we deduce that

by

𝔼0 (𝑋𝑓 ) + (1 + 𝛿) 𝔼0 (𝑌 𝑓 1Λ𝑐 ) ≥ 𝔼0 (𝑋𝑓 1Λ ) + (1 + 𝛿) 𝔼0 (𝑌 𝑓 1Λ𝑐 ) ≥ (1 + 𝛿) 𝔼0 (𝑌 𝑓 1Λ ) + (1 + 𝛿) 𝔼0 (𝑌 𝑓 1Λ𝑐 ) ≥ (1 + 𝛿) 𝔼0 (𝑌 𝑓 ) and hence

𝔼0 (𝑋𝑓 )−𝔼0 (𝑌 𝑓 ) ≥ 𝛿𝔼0 (𝑌 𝑓 )−(1+𝛿) 𝔼0 (𝑌 𝑓 1Λ𝑐 ) ≥ 𝛿𝔼0 (𝑌 𝑓 )−(1+𝛿) 𝔼0 (1Λ𝑐 ). The right hand side is strictly positive by assumption. (iv) The proof is similar to the one for (iii): we use [23, Proposition 5.2] instead of [10, Theorem 25] to nd

Λ,

and then the observation made in the

proof of (ii) shows that the resulting function

𝑓

is uniformly continuous.

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