Minimal supersolutions of BSDEs with lower semicontinuous generators

www.imstat.org/aihp 2014, Vol. 50, No. 2, 524–538

DOI:10.1214/12-AIHP523

© Association des Publications de l’Institut Henri Poincaré, 2014

Minimal supersolutions of BSDEs with lower semicontinuous generators

Gregor Heyne, Michael Kupper and Christoph Mainberger

Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.

E-mail:heyne@math.hu-berlin.de;kupper@math.hu-berlin.de;mainberg@math.hu-berlin.de Received 4 January 2012; revised 27 August 2012; accepted 29 August 2012

Abstract. We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.

Résumé. Nous étudions des sur-solutions minimales d’équations stochastiques rétrogrades. Nous montrons l’existence et l’unicité de telles sur-solutions minimales lorsque le générateur est conjointement semi-continu inférieurement, minoré par une fonction affine de la variable de contrôle et satisfait une condition spécifique de normalisation. Le résultat principal est obtenu en utilisant une convergence de semi-martingales.

MSC:60H20; 60H30

Keywords:Supersolutions of backward stochastic differential equations; Semimartingale convergence

1. Introduction

On a filtered probability space, the filtration of which is generated by ad-dimensional Brownian motion, we give conditions ensuring that the setA(ξ, g), consisting of all supersolutions(Y, Z)of a backward stochastic differential equation with terminal conditionξ and generatorg, has a minimal element. Recall that a supersolution can be seen as, compare for instance [7,8,13], a càdlàg value processY and a control processZ, such that, for all 0≤s≤t≤T,

Y_s− t

s

g_u(Y_u, Z_u)du+ t

s

Z_udWu≥Y_t and Y_T ≥ξ is satisfied.

Our ansatz to find the minimal supersolution is partially inspired by the methods and the setting introduced in Drapeau et al. [7]. More precisely, we start by considering the processEˆ^g(ξ ), defined by

Eˆt^g(ξ )=ess inf

Y_t∈L⁰(Ft): (Y, Z)∈A(ξ, g)

, t∈ [0, T].

It was shown in [7] that under a positivity assumption on the generator – this can be relaxed to a linear bound from below – the processEˆ^g(ξ )is in fact a supermartingale. Moreover, under such an assumption, given an adequate space of control processes, it follows that every value process of a supersolution is also a supermartingale. This is one of the key features of the approach in [7], and we will also adhere to the concept of supermartingale supersolutions. It allows us to consider the processE^g(ξ )=lims↓·,s∈QEˆs^g(ξ )as a candidate for the value process of the minimal supersolution.

Now, given this candidate value process, one needs to find a candidate control processZˆ such that(E^g(ξ ),Z)ˆ ∈ A(ξ, g). In [7] this was done by constructing a monotone decreasing sequence of supersolutions converging toE^g(ξ ) and by drawing on compactness results for sequences of martingales given in Delbaen and Schachermayer [4]. Owing to this approach, it was possible to characterize the candidate control process as the limit of a sequence of convex combinations of control processes. Therefore, in order to verify that the pair(E^g(ξ ),Z)ˆ is a supersolution, it was crucial that the generator is convex with respect to the control variable. The principal aim of the current paper is to drop this convexity assumption.

In order to obtain the existence of a minimal supersolution without taking convex combinations, we proceed as follows. Our first idea is to use results on semimartingale convergence given in Barlow and Protter [2]. Loosely speaking, given a sequence of special semimartingales that converges uniformly, in some sense to be made precise, to some limit process, their result guarantees that the limit process is also a special semimartingale and that the locale martingale parts converge inH¹to the local martingale in the decomposition of the limit process. Interpreted in our setting, this implies that, if we can construct a sequence((Yⁿ, Zⁿ))of supersolutions such that(Yⁿ)converges in the R^∞-norm toE^g(ξ ), then we obtain the existence of a candidate control processZˆ as the limit of the sequence(Zⁿ).

Now, our second main idea shows how to construct a sequence converging in the sense of [2]. To that end, we prove that, forε >0, there exists(Y^ε, Z^ε)∈A(ξ, g)such thatY^ε−E^g(ξ )_R^∞≤ε. Note that it is not possible to infer the existence of such a supersolution from the approach taken in [7], where the approximating sequence was decreasing, but only uniform on a finite set of rationals. Therefore, we have to develop a new method. The central idea is to define a suitable preorder on the set of supersolutions and to use Zorn’s lemma to show the existence of a maximal element. To set up our preorder, we associate with each supersolution(Y, Z)the stopping timeτ, at which Y first leaves theε-neighborhood ofE^g(ξ ). With this at hand, we say(Y¹, Z¹)dominates(Y², Z²), if and only if τ¹≥τ²and the processes coincide up toτ². Given this preorder, we have to show that each totally ordered chain has an upper bound. In order to achieve this, we assume a mild normalization condition on the generator. In its simplest form it states thatgequals zero as soon as the control variable is zero. This assumption is well known especially in the context ofg-expectations, see for example Peng [12] and [7]. More generally, we ask for a certain very simple SDE to have a solution on some short time interval. Combining this assumption with the supermartingale structure of our setting, in particular with arguments based on supermartingale convergence, yields the existence of an upper bound. Moreover, we can show that the stopping time associated with the maximal element provided by Zorn’s lemma equalsT.

The previous arguments show that we obtain indeed a pair of candidate processes(E^g(ξ ),Z). It remains to verifyˆ that the candidate pair is an element ofA(ξ, g). However, this is straightforward by assuming that the generator is jointly lower semicontinuous and can be done by similar arguments as in [7].

Let us briefly discuss the existing literature on related problems, a broader discussion of which can be found in [7]. Nonlinear BSDEs were first introduced in Pardoux and Peng [11]. In this seminal work existence and uniqueness results were given for the case of Lipschitz generators and square integrable terminal conditions. Kobylanski [10]

studies BSDEs with quadratic generators, whereas Delbaen et al. [5] consider superquadratic BSDEs with positive generators that are convex inzand independent ofy. BSDEs with generators that are not locally Lipschitz are studied in Bahlali et al. [1]. Among the first introducing supersolutions of BSDEs were El Karoui et al. [8], Section 2.3.

Further references can also be found in Peng [13], who studies the existence and uniqueness of constrained minimal supersolutions under the assumption of a Lipschitz generator and square integrable terminal conditions. For a link between minimal supersolutions of BSDEs and solutions of reflected BSDEs see Peng and Xu [14]. Most recently, Cheridito and Stadje [3] have analyzed existence and stability of supersolutions of BSDEs. They consider terminal conditions which are functionals of the underlying Brownian motion and generators that are convex inzand Lipschitz in y, and they work with discrete time approximations of BSDEs. Furthermore, the concept of supersolutions is closely related to Peng’s g-expectations, see for instance [7,12], since the mapping ξ →E₀^g(ξ ) can be seen as a nonlinear expectation.

The remainder of this paper is organized as follows. Setting and notations are specified in Section2. A precise definition of minimal supersolutions and important structural properties of Eˆ^g(ξ ), along with the main existence theorem, can then be found in Sections3.1and3.2. Possible relaxations on the assumptions imposed on the generator are discussed in Section3.3. Finally, we conclude the paper with a generalization of our results to the case of arbitrary continuous local martingales in Section3.4.

2. Setting and notations

We consider a filtered probability space(Ω,F, (Ft)t≥0, P ), where the filtration(Ft)is generated by ad-dimensional Brownian motionWand is assumed to satisfy the usual conditions. For some fixed time horizonT >0 and for allt∈ [0, T], the sets ofFt-measurable random variables are denoted byL⁰(Ft), where random variables are identified in the P-almost sure sense. Let furthermore denoteL^p(Ft)the set of random variables inL⁰(Ft)with finitep-norm, forp∈ [1,+∞]. Inequalities and strict inequalities between any two random variables or processesX¹, X²are understood in theP-almost sure or in theP ⊗dt-almost everywhere sense, respectively. We denote byT the set of stopping times with values in[0, T]and hereby call an increasing sequence of stopping times(τⁿ)such thatP[

n{τⁿ=T}] =1 a localizing sequence of stopping times. ByS:=S(R)we denote the set of càdlàg progressively measurable processes Y with values inR. For p∈ [1,+∞[, we further denote by H^p the set of càdlàg local martingales M with finite H^p-norm on[0, T], that isM_H^p:=E[M, M^p/2_T ]^1/p<∞. ByL^p:=L^p(W )we denote the set ofR^1×d-valued, progressively measurable processesZ such that

ZdW ∈H^p, that is, Z_L^p:=E[(T

0 |Zs|²ds)^p/2]^1/p is finite.

ForZ∈L^p, the stochastic integral

ZdW is well defined, see Protter [15], and is by means of the Burkholder–

Davis–Gundy inequality [15], Theorem 48, a continuous martingale. We further denote by L:=L(W ) the set of R^1×d-valued, progressively measurable processesZ such that there exists a localizing sequence of stopping times (τⁿ)withZ1_[0,τⁿ]∈L¹, for alln∈N. ForZ∈L, the stochastic integral

ZdW is well defined and is a continuous local martingale. Furthermore, for a processX, letX^∗denote the following expressionX^∗:=sup_t∈[0,T]|X_t|, by which we define the normX_R^∞:= X^∗L^∞.

We call a càdlàg semimartingaleXa special semimartingale, if it can be decomposed intoX=X₀+M+A, where Mis a local martingale andAa predictable process of finite variation such thatM₀=A₀=0. Such a decomposition is then unique, compare for instance [15], Chapter III, Theorem 30, and is called the canonical decomposition ofX.

3. Minimal supersolutions of BSDEs

3.1. First definitions and structural properties

Throughout this paper, a generator is a jointly measurable functiongfromΩ× [0, T] ×R×R^1×d toR∪ {+∞}

whereΩ× [0, T]is endowed with the progressiveσ-field. Given a generatorgand a terminal conditionξ∈L⁰(FT), a pair(Y, Z)∈S×Lis a supersolution of a BSDE, if, for 0≤s≤t≤T, holds

Y_s− _t

s

g_u(Y_u, Z_u)du+ _t

s

Z_udWu≥Y_t and Y_T ≥ξ. (3.1)

For a supersolution(Y, Z), we callY the value process andZits corresponding control process. Note that the formu- lation in (3.1) is equivalent to the existence of a càdlàg increasing processK, withK0=0, such that

Y_t=ξ+ _T

t

g_u(Y_u, Z_u)du+(K_T −K_t)− _T

t

Z_udWu, t∈ [0, T]. (3.2) Although the notation in (3.2) is standard in the literature concerning supersolutions of BSDEs, see for example [8]

and [13], we will work with (3.1), since the proof of our main result exploits this structure. A control processZis said to be admissible, if the continuous local martingale

ZdW is a supermartingale. Throughout this paper a generator gis said to be:

(LSC) If(y, z)→g(ω, t, y, z)is lower semicontinuous, for all(ω, t)∈Ω× [0, T]. (POS) Positive, ifg(y, z)≥0, for all(y, z)∈R×R^1×d.

(NOR) Normalized, ifgt(y,0)=0, for all(t, y)∈ [0, T] ×R.

We are now interested in the set A(ξ, g):=

(Y, Z)∈S×L: Zis admissible and (3.1) holds

(3.3)

and the process Eˆt^g(ξ ):=ess inf

Y_t∈L⁰(Ft): (Y, Z)∈A(ξ, g)

, t∈ [0, T]. (3.4)

A pair (Y, Z) is called minimal supersolution, if (Y, Z)∈A(ξ, g), and if for any other supersolution (Y, Z)∈ A(ξ, g), holdsYt≤Y_t, for allt∈ [0, T].

For the proof of our main existence theorem we will need some auxiliary results concerning structural properties ofEˆ^g(ξ )and supersolutions(Y, Z)inA(ξ, g).

Lemma 3.1. Let g be a generator satisfying (POS). Assume further that A(ξ, g)=∅ and that for the terminal conditionξ holds ξ⁻∈L¹(FT).Thenξ ∈L¹(FT)and,for any(Y, Z)∈A(ξ, g),the controlZ is unique and the value processY is a supermartingale such thatY_t≥E[ξ|Ft].Moreover,the unique canonical decomposition ofY is given by

Y =Y0+M−A, (3.5)

whereM=

ZdW andAis an increasing,predictable,càdlàg process withA0=0.

The proof of Lemma3.1can be found in [7], Lemma 3.2.

Proposition 3.2. Suppose thatA(ξ, g)=∅and letξ∈L⁰(FT)be a terminal condition such thatξ⁻∈L¹(FT).Ifg satisfies(POS),then the processEˆ^g(ξ )is a supermartingale.In particular,

Et^g(ξ ):= lim

s↓t,s∈QEˆs^g(ξ ) for allt∈ [0, T ) and E_T^g(ξ ):=ξ is a càdlàg supermartingale such that

Eˆt^g(ξ )≥Et^g(ξ ) for allt∈ [0, T].

Furthermore,the following two pasting properties hold true.

1. Let (Zⁿ)⊂Lbe admissible,σ ∈T,and(B_n)⊂Fσ be a partition ofΩ.Then the pasted processZ¯=Z¹1_[0,σ]+

n≥1Zⁿ1Bn1_]σ,T]is admissible.

2. Let((Yⁿ, Zⁿ))⊂A(ξ, g),σ∈T and(Bn)⊂Fσ be as before.IfY_σ¹₋1B_n≥Y_σⁿ1B_n holds true for alln∈N,then (Y ,¯ Z)¯ ∈A(ξ, g),where

Y¯=Y¹1_[0,σ[+

n≥1

Yⁿ1Bn1_[σ,T] and Z¯=Z¹1_[0,σ]+

n≥1

Zⁿ1Bn1_]σ,T].

Proof. The proof of the part concerning the processE^g(ξ )can be found in [7], Proposition 3.4.Z¯ is admissible by [7], Lemma 3.1.1. We can approximateσ from below by some foretelling sequence of stopping times(η_m),¹and then show, analogously to [7], Lemma 3.1.2, that the pair(Y ,¯ Z)¯ satisfies Inequality (3.1) and is thus an element of

A(ξ, g).

Remark 3.3. Whenever a stopping timeσ takes values in a countable subsetSof[0, T],the adapted processEˆ^g(ξ ) evaluated atσ is defined by

Eˆσ^g(ξ ):=

s∈S

1AsEˆs^g(ξ ) withA_s:= {σ=s}.

1Such a sequence satisfyingηm≤ηm+1< σ, for allm∈N, and limmηm=σ, always exists, since in a Brownian filtration every stopping time is predictable, compare Revuz and Yor [16], Corollary V.3.3.

It is straigthforward to show thatEˆσ^g(ξ )isFσ-measurable and consistent with(3.4),in the sense that Eˆ_σ^g(ξ )=ess inf

Y_σ: (Y, Z)∈A(ξ, g) .

Proposition3.2yields that the set{Yσ: (Y, Z)∈A(ξ, g)}is directed downwards,see[7],Proposition3.3.1,and as a consequence we can find,for anyε >0,some(Y^ε, Z^ε)∈A(ξ, g)such that

Y_σ^ε≤ ˆEσ^g(ξ )+ε.

Proposition 3.4. Let0=τ₀≤τ₁≤τ₂≤ · · ·be a sequence of stopping times converging to the finite stopping time τ^∗=limn→∞τ_n.Further,let (Yⁿ)be a sequence of càdlàg supermartingales such that Y_τⁿ

n−≥Y_τⁿ⁺¹

n ,and which satisfiesYⁿ1_[τn−1,τn[≥M1_[τn−1,τn[,whereMis a uniformly integrable martingale.Then,for any sequence of stopping timesσ_n∈ [τ_n−1, τ_n[,the limitY^∞:=limn→∞Y_σⁿ

nexists and the process Y¯:=

n≥1

Yⁿ1_[τ_n−1,τn[+Y^∞1_[τ^∗,∞[

is a càdlàg supermartingale.Moreover,the limitY^∞is independent of the approximating sequence(Y_σⁿ_n)and,if all Yⁿare continuous andY_τⁿ

n=Y_τⁿ⁺¹

n ,for alln∈N,thenY¯ is continuous.

Proof. Note that(Y_σⁿ

n)is a(Fσn)-supermartingale. Indeed, if(η˜_m)↑τ_n is a foretelling sequence of stopping times, then, withη_m:= ˜η_m∨τ_n−1, the family((Y_ηⁿ

m)⁻)_m∈Nis uniformly integrable and we obtain E Y_σⁿ⁺¹

n+1|Fσn

=E E Y_σⁿ⁺¹

n+1|FτnFσn

≤E Y_τⁿ⁺¹

n |Fσn

≤E Y_τⁿ

n−|Fσn

≤lim inf

m E Y_ηⁿ

m|Fσn

≤lim inf

m Y_ηⁿ

m∧σn=Y_σⁿ

n. Moreover,((Y_σⁿ

n)⁻)is uniformly integrable. Hence, the sequence(Y_σⁿ

n)converges by the supermartingale convergence theorem, see Dellacherie and Meyer [6], Theorems V.28, 29, to some random variable Y^∞,P-almost surely, and thusY¯ is well-defined. Furthermore, the limitY^∞is independent of the approximating sequence(Y_σⁿ

n). Indeed, for any other sequence(σ˜n)withσ˜n∈ [τn−1, τn[, the limit limnY_σ˜ⁿ

n exists by the same argumentation. Now limnY_σⁿ_n= limnY_σⁿ_˜

n=Y^∞holds, since the sequence(σˆn)defined by ˆ

σ_n:=

σn/2∨ ˜σn/2 forneven, σ(n+1)/2∧ ˜σ(n+1)/2 fornodd satisfiesσˆ_n∈ [τ_n−1, τ_n[and limnY_ˆⁿ

σnexists. Thus, all limits must coincide. Next, we show thatY¯^σnis a supermartin- gale, for alln∈N. To this end first observe that, for all 0≤s≤t,

E Y¯_t^σn− ¯Y_s^σn|Fs

=

n−2

k=0

E E Y¯_(τ^σn

k+1∨s)∧t− ¯Y_(τ^σn

k∨s)∧t|F(τ_k∨s)∧tFs

+E E Y¯_(σ^σn

n∨s)∧t− ¯Y_(τ^σn

n−1∨s)∧t|F(τ_n−1∨s)∧tFs

+E E Y¯_t^σn− ¯Y_(σ^σn

n∨s)∧t|F(σn∨s)∧tFs

.

Note further that, for eachn∈N, the processY¯^σnis càdlàg and can only jump downwards, that is,Y¯_t^σ₋ⁿ≥ ¯Y_t^σn, for all t∈R. Observe to this end that, on the one hand,Y¯_τ^σn

k−=Y_τ^k

k−≥Y_τ^k+1

k = ¯Y_τ^σ_kⁿ, for all 0≤k≤n−1, by assumption, where we assumedτ_k−1< τ_k, without loss of generality. On the other hand,Y^k can only jump downwards. Indeed, as càdlàg supermartingales, allY^k can be decomposed intoY^k=Y₀^k+M^k−A^k, by the Doob–Meyer decomposition theorem [15], Chapter III, Theorem 13, whereM^k is a local martingale andA^k a predictable, increasing process with A^k₀=0. Since in a Brownian filtration every local martingale is continuous, the claim follows.

Thus, for all 0≤k≤n−2, and(η˜_m)↑τ_k+1a foretelling sequence of stopping times, it holds withη_m:= ˜η_m∨τ_k, E Y¯_(τ^σn

k+1∨s)∧t− ¯Y_(τ^σn

k∨s)∧t|F(τk∨s)∧t

≤E Y¯_((τ^σn

k+1−)∨s)∧t− ¯Y_(τ^σn

k∨s)∧t|F(τk∨s)∧t

=E

lim inf

m Y¯_(η^σn

m∨s)∧t− ¯Y_(τ^σn

k∨s)∧tF(τ_k∨s)∧t

≤E

lim inf

m Y_(η^k+1

m∨s)∧tF(τk∨s)∧t

−Y_(τ^k+1

k∨s)∧t

≤lim inf

m E Y_(η^k+1

m∨s)∧t|F(τk∨s)∧t

−Y_(τ^k+1

k∨s)∧t≤0.

Moreover,E[ ¯Y_t^σn− ¯Y_(σ^σn

n∨s)∧t|F(σn∨s)∧t] =0, as well as E Y¯_(σ^σn

n∨s)∧t− ¯Y_(τ^σn

n−1∨s)∧t|F(τ_n−1∨s)∧t

≤E Y_(σⁿ

n∨s)∧t−Y_(τⁿ

n−1∨s)∧t|F(τ_n−1∨s)∧t

≤0.

Combining this we obtain that E[ ¯Y_t^σn|Fs] ≤ ¯Y_s^σn. Furthermore, limnY¯_t^σn = ¯Y_t, for all t∈R. Indeed, let us write limnY¯_t^σn=limnY¯_t^σn1_{t <τ^∗}+limnY¯_t^σn1_{t≥τ^∗}. Then, limnY¯_t^σn1_{t≥τ^∗}=limnY_σⁿ

n1_{t≥τ^∗}=Y^∞1_{t≥τ^∗}= ¯Y_t1_{t≥τ^∗}and limnY¯_σn∧t1_{t <τ^∗}= ¯Y_t1_{t <τ^∗}. Hence, the claim follows. As a consequence of Fatou’s lemma it now holds that

E[ ¯Y_t|Fs] ≤lim inf

n→∞ E Y¯_t^σn|Fs

≤lim inf

n→∞ Y¯_s^σn= ¯Y_s,

since the family ((Y¯_t^σn)⁻)is uniformly integrable. Hence,Y¯ is a supermartingale, which by construction has right- continuous paths and Karatzas and Shreve [9], Theorem 1.3.8, then yields thatY¯ is even càdlàg. Finally, whenever all Yⁿare continuous andY_τⁿ

n=Y_τⁿ⁺¹

n holds, for alln∈N, the processY¯ is continuous per construction.

3.2. Existence and uniqueness of minimal supersolutions

We are now ready to state our main existence result. Possible relaxations of the assumptions (POS) and (NOR) imposed on the generator are discussed in Section3.3. Note that it is not our focus to investigate conditions assuring the crucial assumption thatA(ξ, g)=∅. See [7] and the references therein for further details.

Theorem 3.5. Letgbe a generator satisfying(LSC), (POS)and(NOR)andξ∈L⁰(FT)be a terminal condition such thatξ⁻∈L¹(FT).IfA(ξ, g)=∅,thenE^g(ξ )is the value process of the unique minimal supersolution,that is,there exists a unique control processZˆ ∈Lsuch that(E^g(ξ ),Z)ˆ ∈A(ξ, g).

Observe that Theorem3.5and Proposition3.2imply thatE^g(ξ )is a modification ofEˆ^g(ξ ).

Proof. Step 1: Uniform limit and verification. SinceA(ξ, g)=∅, there exist(Y^b, Z^b)∈A(ξ, g). From now on we restrict our focus to supersolutions (Y ,¯ Z)¯ in A(ξ, g) satisfyingY¯0≤Y₀^b. Indeed, since we are only interested in minimal supersolutions, we can paste any value process of(Y, Z)∈A(ξ, g)atτ:=inf{t >0: Y_t^b> Y_t} ∧T such that Y¯:=Y^b1_[0,τ[+Y1_[τ,T]satisfiesY¯₀≤Y₀^b, where the corresponding controlZ¯ is obtained as in Proposition3.2.

Assume for the beginning that we can find a sequence((Yⁿ, Zⁿ))withinA(ξ, g)such that

nlim→∞Yⁿ−E^g(ξ )

R^∞=0. (3.6)

Since allYⁿare càdlàg supermartingales, they are, by the Doob–Meyer decomposition theorem, special semimartin- gales with canonical decompositionYⁿ=Y₀ⁿ+Mⁿ−Aⁿas in (3.5). The supermartingale property of all

ZⁿdW andξ <∞, compare Lemma3.1, imply thatE[Aⁿ_T] ≤Y₀^b−E[ξ] ∈L¹(FT). Hence, since eachAⁿ is increasing,

sup_nE[_T

0 |dAⁿ_s|]<∞. As (3.6) implies in particular that limn→∞E[(Yⁿ−E^g(ξ ))^∗] =0, it follows from [2], Theo- rem 1 and Corollary 2, thatE^g(ξ )is a special semimartingale with canonical decompositionE^g(ξ )=E₀^g(ξ )+M−A and that

nlim→∞Mⁿ−M

H¹=0, lim

n→∞E Aⁿ−A_∗

=0.

The local martingaleMis continuous and allows a representation of the formM=ZˆdW, whereZˆ ∈L, compare [15], Chapter IV, Theorem 43. Since

E _T

0

Zⁿ_u− ˆZ_u2

du 1/2

n→+∞−→ 0,

we have that, up to a subsequence,(Zⁿ)convergesP⊗dt-almost everywhere toZˆ and limn→∞t

0ZⁿdW=t 0ZˆdW, for allt∈ [0, T],P-almost surely, due to the Burkholder–Davis–Gundy inequality. In particular, limn→∞Zⁿ(ω)= Z(ω), dtˆ -almost everywhere, for almost allω∈Ω.

In order to verify that(E^g(ξ ),Z)ˆ ∈A(ξ, g), we will use the convergence obtained above. More precisely, for all 0≤s≤t≤T, Fatou’s lemma together with (3.6) and the lower semicontinuity of the generator yields

Es^g(ξ )− _t

s

g_u

Eu^g(ξ ),Zˆ_u du+

_t

s

Zˆ_udWu

≥lim sup

n

Y_sⁿ−

_t

s

g_u Y_uⁿ, Z_uⁿ

du+ _t

s

Z_uⁿdWu

≥lim sup

n

Y_tⁿ=Et^g(ξ ).

The above, the positivity ofgandE^g(ξ )≥E[ξ|F·]imply thatZˆdW≥E[ξ|F·] −E₀^g(ξ ). Hence, being bounded from below by a martingale, the continuous local martingaleZˆdW is a supermartingale. Thus, Zˆ is admissible and(E^g(ξ ),Z)ˆ ∈A(ξ, g)and therefore, by Lemma3.1,Zˆ is unique. Since we know by Proposition3.2thatEˆt^g(ξ )≥ Et^g(ξ ), for allt∈ [0, T], we deduce thatEˆt^g(ξ )=Et^g(ξ ), for allt∈ [0, T], by the definition ofEˆ^g(ξ ). Hence,(E^g(ξ ),Z)ˆ is the unique minimal supersolution.

Step 2: A preorder onA(ξ, g). As to the existence of((Yⁿ, Zⁿ))satisfying (3.6), it is sufficient to show that, for arbitraryε >0, we can find a supersolution(Y^ε, Z^ε)satisfying

Y^ε−E^g(ξ )

R^∞≤ε. (3.7)

We define the following preorder²onA(ξ, g) Y¹, Z¹

Y², Z²

⇔ τ1≤τ2 and Y¹, Z¹

1_[0,τ₁[= Y², Z²

1_[0,τ₁[, (3.8)

where, fori=1,2, τ_i=inf

t≥0:Y_tⁱ >Et^g(ξ )+ε

∧T . (3.9)

For any totally ordered chain((Yⁱ, Zⁱ))_i∈IwithinA(ξ, g)with corresponding stopping timesτ_i, we want to construct an upper bound. If we consider

τ^∗=ess sup

i∈I

τ_i,

2Note that, in order to apply Zorn’s lemma, we need a partial order instead of just a preorder. To this end we consider equivalence classes of processes. Two supersolutions(Y¹, Z¹), (Y², Z²)∈A(ξ, g)are said to be equivalent, if(Y¹, Z¹)(Y², Z²)and(Y², Z²)(Y¹, Z¹). This means that they are equal up to their corresponding stopping timeτ1=τ2as in (3.9). This induces a partial order on the set of equivalence classes and hence the use of Zorn’s lemma is justified.

we know by the monotonicity of the stopping times that we can find a monotone subsequence(τ_m)of(τ_i)_i∈I such thatτ^∗=limm→∞τ_m. In particular,τ^∗is a stopping time. Furthermore, the structure of the preorder (3.8) yields that the value processes of the supersolutions((Y^m, Z^m))corresponding to the stopping times(τm)satisfy

Y_τ^m+1

m ≤Y_τ^m+1

m− =Y_τ^m

m− for allm∈N, (3.10)

where the inequality follows from the fact that allY^mare càdlàg supermartingales, see the proof of Proposition3.4.

Step 3: A candidate upper bound(Y ,¯ Z)¯ for the chain((Yⁱ, Zⁱ))_i∈I. We construct a candidate upper bound(Y ,¯ Z)¯ for((Yⁱ, Zⁱ))i∈IsatisfyingP[τ (Y ) > τ¯ ^∗|τ^∗< T] =1, withτ (Y )¯ as in (3.9).

To this end, let(σ¯n)be a decreasing sequence of stopping times taking values in the rationals and converging towardsτ^∗from the right.³Then the stopping timesσˆ_n:= ¯σ_n∧T satisfyσˆ_n> τ^∗andσˆ_n∈Q, on{τ^∗< T}, for alln big enough. Let us furthermore define the following stopping time

¯ τ :=inf

t > τ^∗: E_τ^g∗(ξ )−Et^g(ξ )>ε 2

∧T . (3.11)

Due to the right-continuity ofE^g(ξ )inτ^∗, it follows thatτ > τ¯ ^∗on{τ^∗< T}. We now set

σ_n:= ˆσ_n∧ ¯τ for alln∈N. (3.12)

The above stopping times still satisfy limn→∞σ_n=τ^∗andσ_n> τ^∗on{τ^∗< T}, for alln∈N. We further define the following sets

A_n:=E_τ^g∗(ξ )− ˆEσ^g_m(ξ )<ε

8,for allm≥n

∩

σ_n∈Q∪ {T}

. (3.13)

They satisfyA_n⊂A_n+1 and

nA_n=Ω, by definition of the sequence(σ_m).⁴ Note further thatA_n∈Fσn, since Eˆσ^g_m(ξ )isFσ_m-measurable, for allm≥n, see Remark3.3. Since the range of eachσnis countable on the setAn, we deduce by Remark3.3that, for eachn∈N, there exists(Y˜ⁿ,Z˜ⁿ)∈A(ξ, g)such that

Y˜_σⁿ

n≤ ˆE_σ^g_n(ξ )+ε

8 on the setA_n. (3.14)

Next we partitionΩintoB_n:=A_n\A_n−1, where we setA₀:=∅andτ₀:=0, and define the candidate upper bound as

Y¯ =

m≥1

Y^m1_[τ_m−1,τ_m[+1_{τ^∗<T}

n≥1

1B_n

E_τ^g∗(ξ )+ε 2

1_[τ^∗,σ_n[

+1_{τ^∗<T}

n≥1

1B_nY˜ⁿ1_[σ_n,T[, Y¯_T =ξ, (3.15)

Z¯ =

m≥1

Z^m1_]τ_m−1,τm]+1_{τ^∗<T}

n≥1

Z˜ⁿ1Bn1_]σn,T]. (3.16)

Step 4: Verification of (Y ,¯ Z)¯ ∈A(ξ, g). By verifying that the pair(Y ,¯ Z)¯ is an element ofA(ξ, g), we identify (Y ,¯ Z)¯ as an upper bound for the chain((Yⁱ, Zⁱ))_i∈I. Even more,P[τ (Y ) > τ¯ ^∗|τ^∗< T] =1 holds true, since, on the setB_n, we haveY¯_t=E_τ^g∗(ξ )+^ε₂≤Et^g(ξ )+ε, for allt∈ [τ^∗, σ_n[, due to the definition ofτ¯in (3.11).

Step 4a: The value processY¯ is an element ofS. By construction, the only thing to show is thatY¯τ∗−, the left limit atτ^∗, exists. This follows from Proposition3.4, since, by means of((Y^m, Z^m))⊂A(ξ, g)andξ ∈L¹(FT), allY^m

3Compare [9], Problem 2.24.

4Since on{τ^∗< T},τ > τ¯ ^∗and limnσˆn=τ^∗withσˆn∈Q∪ {T}, it is ensured that there exists somen₀∈N, depending onω, such thatσntakes values in the rationals for alln≥n0. By definition ofE^g(ξ )as the right-hand side limit ofEˆ^g(ξ )on the rationals, the inequality in the definition of Anis satisfied for alln≥n0.