Second order backward SDE with random terminal time

Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).


Introduction
Let (Ω, F, {F t } t≥0 , P) be a filtered probability space, supporting a d−dimensional Brownian motion W . The martingale representation theorem states that any integrable F τ −measurable random variable ξ, for some F−stopping time τ , can be represented as ξ = Eξ + (Z · W ) τ + N τ , for some square integrable F−predictable process Z, and some martingale N with N 0 = 0 and [N, W ] = 0. In particular when F is the (augmented) canonical filtration of the Brownian motion, N = 0. This result can be seen as the path-dependent counterpart of the heat equation. Indeed, a standard density argument reduces to the case ξ = g(W t0 , . . . , W tn ) for an arbitrary partition 0 = t 0 < . . . < t n = T of [0, T ], where the representation follows from a backward resolution of the heat equation ∂ t v + 1 2 ∆v = 0 on each time interval [t i−1 , t i ], i = 1, . . . , n, and the Z process is identified to the space gradient of the solution.
As a first extension of the martingale representation theorem, the seminal work of Pardoux & Peng [31] introduced the theory of backward stochastic differential equations in finite horizon. In words, this theory provides a representation of an F T −measurable random variable ξ with appropriate integrability as ξ = Y T with where f is a given random field. In the Markov setting where ξ = g(W T ) and f t (ω, y, z) = f t, W t (ω), y, z , t ≥ 0, it turns out that Y t (ω) = v(t, W t (ω)) for some deterministic function v : R + × R d −→ R, which is easily seen to correspond to the semilinear heat by the fact that the Z process again identifies the space gradient of v.
It was extended further to the random horizon setting by [32], Darling & Pardoux [10]. On one hand, these results provide a representation for an F τ −measurable random variable ξ with appropriate integrability as ξ = Y τ with where f is a given random field. On the other hand, they give probabilistic interpretation to solutions of semilinear elliptic PDEs. As our interest in this paper is on the random horizon setting, we refer the interested reader to the related works by El Karoui & Huang [13], Briand & Hu [7], Briand & Carmona [5], Bender & Kohlmann [2], Royer [38], Bahlali, Elouaflin & N'zi [1], Hu and Tessitore [20], Popier [33], Briand and Confortola [6], Wang, Ran and Chen [43], Papapantoleon, Possamaï and Saplaouras [30]. We also mention the related works of Hamadène, Lepeltier & Wu [16], Chen & Wang [8] and Hu and Schweizer [19], which study BSDEs with infinite horizon. Our main interest in this paper is on the extension to the fully nonlinear second order parabolic equations, as initiated in the finite horizon setting by Soner, Touzi & Zhang [40], and further developed by Possamaï, Tan & Zhou [34], see also the first attempt by Cheridito, Soner, Touzi & Victoir [9], and the closely connected BSDEs in a nonlinear expectation framework of Hu, Ji, Peng & Song [17,18] (called GBSDEs). This extension is performed on the canonical space of continuous paths with canonical process denoted by X. The key idea is to reduce the fully nonlinear representation to a semilinear representation which is required to hold simultaneously under an appropriate family P of singular semimartingale measures on the canonical space. Namely, an F T − random variable ξ with appropriate integrability is represented as s. for all P ∈ P.
Here, σ 2 s ds = d X s , and U P is a supermartingale with U P 0 = 0, [U P , X] = 0, P−a.s. for all P ∈ P satisfying the minimality condition sup P∈P E P [U P T ] = 0. Loosely speaking, in the Markov setting where Y t (ω) = v t, X t (ω) for some deterministic function v, the last representation implies that v is a supersolution of a semilinear parabolic PDE parameterized by the diffusion coefficient Tr σσ D 2 v − F (t, x, v, Dv, σ) ≥ 0, Tr σσ D 2 v + F (t, x, v, Dv, σ) = 0.
Our main contribution is to extend the finite horizon fully nonlinear representation of [40] and [34] to the context of a random horizon defined by a finite F−stopping time.
In view of the formulation of second order backward SDEs as backward SDEs holding simultaneously under a non-dominated family of singular measures, we review -and in fact complement-the corresponding theory of backward SDEs, and we develop the theory of reflected backward SDEs, which is missing in the literature, and which plays a crucial role in the wellposedness of second order backward SDEs. Finally, we emphasize that backward SDEs and their second order extension provide a Sobolev-type of wellposedness as uniqueness holds within an appropriate integrability class of the solution Y and the corresponding "space gradient" Z. Also, our extension to the random horizon setting allows in particular to cover the elliptic fully nonlinear second order PDEs with convex dependence on the Hessian component.
The paper is organized as follows. Section 2 sets the notations used throughout the paper. Our main results are contained in Section 3, with proofs reported in the remaining sections. Namely, Section 4 contains the proofs related to backward SDEs and the corresponding reflected version, while Sections 5 and 6 focus on the uniqueness and the existence, respectively, for the second order backward SDEs.

Canonical space
Fix d ∈ N, and let Ω = ω ∈ C [0, ∞); R d : ω 0 = 0 be the space of continuous paths starting from the origin equipped with the distance defined by ω − ω ∞ := n≥0 2 −n sup 0≤t≤n ω t − ω t ∧ 1 . Denote by X the canonical process. Let M 1 be the collection of all probability measures on (Ω, F), equipped with the topology of weak convergence. Denote by F := (F t ) t≥0 the raw filtration generated by the canonical process X. Denote by F + := (F + t ) t≥0 the right limit of (F t ) t≥0 . For each P ∈ M 1 , we denote by F +,P the augmented filtration of F + under P. The filtration F +,P is the coarsest filtration satisfying the usual conditions. We denote by F U := F U t t≥0 and For any family P ⊆ M 1 , we say that a property holds P−quasi-surely, abbreviated as P−q.s., if it holds P-a.s. for all P ∈ P. Define P loc the subset of M 1 such that, for each P ∈ P loc , X is P-local martingale whose quadratic variation X is absolutely continuous in t with respect to the Lebesgue measure. Note that the d × d-matrix-valued processes X can be defined pathwisely, and we may introduce the corresponding F-progressively measurable density processes Second order backward SDE with random terminal time so that X t = t 0 a s ds, t ≥ 0, P-a.s., for all P ∈ P loc .
For later use, we observe that, as a t ∈ S d + , the set of d × d nonnegative-definite symmetric matrices, we may define a measurable 1 generalized inverse a −1 t , and a measurable square root a 1 2 t =: σ t .

Spaces and norms
Let p > 1 and α ∈ R.
(i) One-measure integrability classes: for a probability measure P ∈ M 1 , let τ be an F +,P -stopping time. We denote: • L p α,τ (P) is the space of R-valued and F +,P τ -measurable random variables ξ, such • I p α,τ (P) is the set of scalar F +,P -predictable processes K with càdlàg nondecreasing paths, s.t.
• U p α,τ (P) is the set of càdlàg F-supermartingales U , with Doob-Meyer decomposition U = N − K into the difference of a martingale and a predictable non-decreasing process, such that 1 Any matrix S ∈ S d + has a decomposition S = Q S Λ S Q S for some orthogonal matrix Q S , and a diagonal matrix Λ S , with Borel-measurable maps S → Q S and S → Λ S , as this decomposition can be obtained by e.g. the Rayleigh quotient iteration. This implies the Borel measurability of the generalized inverse map 2 If the stopping time τ is finite, the norm is indeed Y p D p α,τ (P) := E P sup 0≤t≤τ e αt Yt p < ∞.
(ii) Integrability classes under dominated nonlinear expectation: Let us enlarge the canonical space to Ω := Ω × Ω and denote by (X, W ) the coordinate process in Ω. Denote by F the filtration generated by (X, W ). For each P ∈ P loc , we may construct a probability measure P on Ω such that P • X −1 = P, W is a P-Brownian motion and dX t = σ t dW t , P-a.s. From now on, we abuse the notation, and keep using P to represent P on Ω. Denote by Q L (P) the set of all probability measures Q λ such that for some F +,P -progressively measurable process λ = (λ) t≥0 uniformly bounded by L. By Girsanov's theorem, W λ := W − · 0 λ s ds is a Q λ -Brownian motion on any finite horizon, and thus X λ := X − · 0 σ t λ t dt is a Q λ -martingale on any finite horizon. For P ∈ P loc , we denote E P [·] := sup We define similarly the subspaces D p α,τ (P), H p α,τ (P), N p α,τ (P), and the subsets I p α,τ (P), U p α,τ (P). Let G := {G t } t≥0 be a filtration with G t ⊇ F t for all t ≥ 0, so that τ is also a G-stopping time. We define the subspace L p α,τ (P, G) as the collection of all G τ -measurable R-valued random variables ξ, such that ξ p L p α,τ (P) := E P e ατ ξ p < ∞.
We define similarly the subspaces D p α,τ (P, G) and H p α,τ (P, G) by replacing F +,P with G.

Random horizon backward SDE
For a probability measure P ∈ P loc , an F-stopping time τ , which may be infinite, an F +,P τ -measurable random variable ξ, and a generator F : and we consider the following backward stochastic differential equation (BSDE): for t, (3.1) 3 By Prog we denote the σ-algebra generated by progressively measurable processes. Consequently, for every fixed (y, z) ∈ R × R d , the process Ft(y, z, σt) t≥0 is progressively measurable.
Here, Y is a càdlàg adapted scalar process, Z is a predictable R d -valued process, and N a càdlàg R-valued martingale with N 0 = 0 orthogonal to X, i.e., [X, N ] = 0. We recall that dX s = σ s dW s , P−a.s.

Remark 3.3.
In the context of a bounded stopping time τ ≤ T , the monotonicity assumption can be deduced from the Lipschitz assumption by the following standard argument.
Clearly, F inherits the Lipschitz property of F , and satisfies the monotonicity condition for sufficiently large λ. Finally,ξ is in the same integrability class as ξ for bounded τ . We emphasize that the above mentioned technique applies throughout this paper, and thus when pulling back to the context of finite horizon, the monotonicity assumption could be removed.
However, if one applies the previous argument in the case as τ is not bounded, thenξ would fit different integrability condition from ξ. Therefore, the monotonicity condition is necessary.
Except for the estimate (3.2), whose proof is reported in Section 4.5, the wellposedness part of the last result is a special case of Theorem 3.9 below, with obstacle S ≡ −∞.
Remark 3.5. The norm, with which we propose the integrability condition on the coefficients (Assumption 3.2) and the solution space in Theorem 3.4, is novel. It is mainly motivated by the following reasons.
• In the initial investigation on the random horizon backward SDE by Peng [32] and Darling & Pardoux [10], it requires a similar integrability condition as Assumption 3.2 withρ := ρ + L 2 /2 instead of ρ and E P instead of E P . The following Example 3.6 illustrates the relevance of our assumption in the simple case of a linear generator.
In the works generalizing the result in [10], see e.g. [7,38], to our knowledge, it is always assumed that µ > 0, i.e., the generator is strictly monotone, and the coefficients ξ, f 0 are bounded, which is a special case of our Assumption 3.2. For µ = 0, i.e., the generator f is monotone, Royer [38] provided the existence and uniqueness under assumptions that the generator f depending only on z is bounded and ξ is bounded. • The backward SDE can be viewed as a nonlinear representation of a random variable by an Itô process with a particular generator function. For the sake of applications, we would like that the representation is a 'one-to-one mapping' between the random variable space and the solution space of backward SDE. Here, on the one hand, according to Theorem 3.4, given ξ1 {τ <∞} ∈ q>1,ρ>−µ L q ρ,τ , we may find the solution in q>1,ρ>−µ D q ρ,τ (P) × H q ρ,τ (P) × N q ρ,τ (P). On the other hand, given Y 0 ∈ R, (Z, U ) ∈ q>1,ρ>−µ H q ρ,τ (P) × N q ρ,τ (P), we may construct an Itô process (by solving an ODE) such that Y τ 1 {τ <∞} ∈ q>1,ρ>−µ L q ρ,τ . This builds up the desired one-to-one correspondence.
Again, we remind that, unlike in [37], the application of the new norm in Assumption 3.2 is not to pursue a weaker integrability condition for the wellposedness of backward SDE. Example 3.6. Let P := P 0 , be the Wiener measure on Ω, so that X is a P 0 −Brownian motion. Let τ := H 1 , where H x := inf{t > 0 : X t ≥ x}, ξ := |X 1∧τ |, and f t (ω, y, z) := −µy + Lz for some constants 0 < µ < 1 ≤ L. Notice that f 0 = 0, and ξ ∈ L 2 0,τ (P 0 ) by direct verification: We next show that Darling & Pardoux's condition is not satisfied. To see this, observe that the event set A := ω ∈ Ω : sup 0≤t≤1 X t < 1, We also have the following comparison and stability results, which are direct consequences of Theorem 3.10 below, obtained by setting the obstacle to −∞ therein, together with the estimate (3.2) in Theorem 3.4.  δf = f − f , we have for all 1 < p < p < q and −µ < η < η < ρ: s. for all finite stopping time τ 0 ≤ τ , P-a.s. Remark 3.8. Following [15] we say that (Y, Z) is a supersolution (resp. subsolution) of the BSDE with parameters (f, ξ) if the martingale N in (3.1) is replaced by a supermartingale (resp. submartingale). A direct examination of the proof of the last comparison result reveals that the conclusion is unchanged if (Y, Z) is a subsolution of BSDE(f, ξ), and (Y , Z ) is a supersolution of BSDE(f , ξ ).

Random horizon reflected backward SDE
We now consider an obstacle defined by (S t ) t≥0 , and we search for a representation similar to (3.1) with the additional requirement that Y ≥ S. This is achieved at the price of pushing up the solution Y by substracting a supermartingale U with minimal action.
The existence part of this result is proved in Section 4.4. The uniqueness is a consequence of claim (i) of the following stability and comparison results. (i) Comparison. Assume ξ ≤ ξ , P-a.s. on {τ < ∞}, f (y, z) ≤ f (y, z) for all (y, z) ∈ R × R d , and S ≤ S , dt ⊗ P-a.e. Then, Y τ0 ≤ Y τ0 , P-a.s., for all finite stopping time τ 0 ≤ τ , P-a.s. 5 This condition coincides the standard Skorokhod condition in the literature. Indeed, by using the corresponding Doob-Meyer decomposition U = N − K into a martingale N and a nondecreasing process K, and recalling that Y ≥ S , it follows that 0 = E P τ ∧t is equivalent to τ 0 (Y r− − S r− )dKr = 0, P−a.s. by the arbitrariness of t ≥ 0.
Notice that the stability result is incomplete as the differences δY , δZ and δU are controlled by the norms of Y and Y . However, in contrast with the estimate (3.2) in the backward SDE context, we have unfortunately failed to derive a similar control of (Y, Z, U ) by the ingredients ξ, f 0 and S in the present context of random horizon reflected backward SDE due to the presence of the orthogonal martingale N in the general filtration, see also [4].

Random horizon second order backward SDE
Following Soner, Touzi & Zhang [40], we introduce second order backward SDE as a family of backward SDEs defined on the supports of a convenient family of singular probability measures. For this reason, we introduce the subset of P loc : P 0 = P ∈ P loc : f 0 t (ω) < ∞, for Leb⊗P-a.e. (t, ω) ∈ R + × Ω , (3.4) where we recall that f 0 t (ω) = F t ω, 0, 0, σ t (ω) . Note that in the context of stochastic control, which is the major application of second order backward SDE, the set P 0 defined above is the set of all admissible controls of volatility. We also define for all finite stopping times τ 0 : P P (τ 0 ) := P ∈ P 0 : P = P on F τ0 , and P + P (τ 0 ) := h>0 P P τ 0 + h .
We remark that the definition of P + P (τ 0 ) differs slightly from the one in [40,41], in which the authors studied second order backward SDEs under the extra uniform continuity condition.
For a finite F-stopping time τ , the second order backward SDE (2BSDE, hereafter) is defined by Second order backward SDE with random terminal time Definition 3.11. Let p > 1 and η ∈ R. A process (Y, Z) ∈ D p η,τ P 0 , F +,P0 ×H p η,τ P 0 , F P0 is said to be a solution of the 2BSDE (3.5), if for all P ∈ P 0 , the process is a càdlàg P-local supermartingale starting from U P 0 = 0, orthogonal to X, i.e. [X, U P ] = 0, P−a.s. and satisfying the minimality condition Remark 3.12. Notice that the last definition relaxes slightly (3.5) by allowing for a dependence of U on the underlying probability measure. This dependence is due to the fact that the stochastic integral Z • X := · 0 Z s · dX s is defined P−a.s. under all P ∈ P 0 , and should rather be denoted by (Z • X) P in order to emphasize the P−dependence.
By Theorem 2.2 in Nutz [27], the family {(Z • X) P } P∈P0 can be aggregated as a medial limit (Z • X) under the acceptance of Zermelo-Fraenkel set theory with axiom of choice together with the continuum hypothesis into our framework. In this case, (Z • X) can be chosen as an F +,P0 -adapted process, and the family {U P } P∈P0 can be aggregated into the resulting medial limit U , i.e., U = U P , P−a.s. for all P ∈ P 0 .
Under the additional Assumption 3.13 (ii), such a solution (Y, Z) for the 2BSDE (3.5) exists.
If P 0 is saturated 6 , then U P is a P-a.s. non-increasing process for all P ∈ P 0 .
Similar to Soner, Touzi & Zhang [40], the following comparison result for second order backward SDEs is a by-product of our construction; the proof is provided in Proposition 5.2.

Wellposedness of random horizon reflected BSDEs
Throughout this section, we fix a probability measure P ∈ P loc , and we omit the dependence on P in all of our notations. We also observe that Q L : For all Q λ ∈ Q L , it follows from Girsanov's Theorem that martingale, and we may rewrite the RBSDE as satisfies the Assumption 3.1 with Lipschitz coefficient 2L.
• U remains a Q λ -supermartingale, with the same Doob-Meyer decomposition as under P.

Auxiliary inequalities
We first state a Doob-type inequality. For simplicity, we write E[·] := E P [·]. Then, From the definition of concatenation and the optional sampling theorem, we obtain for all Q ∈ Q L : 6 We say that the family P 0 is saturated if, for all P ∈ P 0 , we have Q ∈ P 0 for every probability measure Q ∼ P on (Ω, F ) such that X is Q−local martingale. The assertion follows by the same argument as in [ and we deduce that The required inequality follows from the arbitrariness of Q ∈ Q L .
The following result is well-known, we report its proof for completeness as we could not find a reference for it. We shall denote sgn(x) : Proof. Consider a decreasing sequence of C 2 , symmetric convex functions ϕ n on R, such that ϕ n (x) = |x| on (− 1 n 2 , 1 n 2 ) c , and ϕ n (x) increases to 1 for x > 0 and ϕ n (x) decreases to −1 for x < 0, i.e., ϕ n (x) converges to sgn(x). By Itô's formula and convexity of ϕ n , we obtain that By convexity of ϕ n , this implies that ϕ n (X t ) − ϕ n (X 0 ) ≥ t 0 ϕ n (X s− )dX s . The required inequality follows by sending n → ∞ in the above inequality and by applying the dominated convergence theorem for stochastic integrals (see, e.g., [35, Section IV, Theorem 32]).

A priori estimates
Proof. Let U = N − K be the Doob-Meyer decomposition of the supermartingale U .

1.
We first prove that and Second order backward SDE with random terminal time We only prove (4.1), the second claim follows by similar arguments.
We continue by estimating the right hand side term: where we used the estimate It follows from Assumption 3.1 and Young's inequality that for an arbitrary α ∈ (α, ρ). Let t → ∞. It follows that Second order backward SDE with random terminal time by the BDG inequality. Since K is non-decreasing, we applying Young's inequality with an arbitrary ε > 0 to deduce where the last inequality follows from (4.1). Plugging this estimate into (4.3), and using (4.2) together with the fact that 2α − α < α, we obtain 3. We shall prove in Step 4 below that for δ < δ < ρ: Plugging this inequality with δ := 2α − α and δ := α in (4.4), and using the left hand side inequality of (4.2), we see that we may choose ε > 0 conveniently such that for some constant C Z p,α,α ,L > 0. Plugging this inequality into (4.5) with (δ, δ ) := (α, α ) induces the estimate for some constant C K p,α,α ,L . Combining with (4.4), and recalling that 2α − α < α, in turn, this implies an estimate for U λ p N p α,τ (Q λ ) which can be plugged into (4.1) to provide: Since the constants in (4.6), (4.7) and (4.8) do not depend on Q ∈ Q L , the proof of this proposition is completed by taking supremum over the family of measures Q ∈ Q L .

Stability of reflected backward SDEs
Proof of Theorem 3.10 (ii). Clearly, the process (δY, δZ, δU ) satisfies the following equa- 1. In this step, we prove that, for some constant C p,p , (4.14) It follows from Proposition 4.2 that As f and f satisfy Assumption 3.1, we obtain that Considering the Doob-Meyer decomposition U = N − K and U = N − K , and denoting δN and δK the corresponding differences, it follows from the Skorokhod condition that Then, denoting it follows from inequality (4.15) and −µ < η that As δZ ∈ H p η,τ (P) and δN ∈ N p η,τ (P), we deduce from the BDG inequality that the last two terms are Q λ −uniformly integrable martingales. Then, with τ n := n ∧ τ and n ≥ t: From the dominated convergence theorem and monotone convergence theorem and the fact that e η t Y t and e η t Y t are uniformly integrable.
By Lemma 4.1, we deduce that for any p ∈ (p, q): which induces the required inequality (4.14).

Let
and therefore, together with (4.16) and the fact that η + µ ≥ 0, we obtain that for an arbitrary ε > 0. Therefore, by choosing an ε > 0 conveniently, we obtain for some constant C p,η,η > 0. By the BDG inequality, Young's inequality and the Cauchy-Schwarz inequality, we obtain Together with (4.14) from Step 1, and Proposition 4.3, this induces the first estimate in Theorem 3.10 (ii). By the BDG inequality and the Cauchy-Schwarz inequality, we obtain for β ∈ (α, ρ):

It remains to verify the announced estimate on
for some constant C p,α,β,L .

Wellposedness of reflected backward SDEs
We start from the so-called Snell envelope defined by the dynamic optimal stopping problem 7 : where T t,τ denotes the set of all F +,P −stopping times θ with t ∧ τ ≤ θ ≤ τ . Following the proof of [14, Proposition 5.1] and the theory of optimal stopping, see e.g., [12], we deduce that there exists an X−integrable process z, such that: where u is local supermartingale, starting from u 0 = 0, orthogonal to X, i.e., [X, u] = 0. In other words, ( y, z, u) is a solution of the RBSDE with generator f ≡ 0 and obstacle S. Then, it follows by Itô's formula that the triple (y, z, u), defined by y t := e µt y t , z t := e µt z t , u t := t 0 e µs d u s , t ≥ 0, is a solution of the following RBSDE, for t, t ∈ R + , t ≤ t , where u is local supermartingale, starting from u 0 = 0, orthogonal to X, i.e., [X, u] = 0.
Proof. By the definition of y, we have for all 1 < p < p . By our assumption on ξ and S + , we see that we need to restrict to p < q in order to ensure that the last bound is finite. Moreover, by Proposition 4.3, we have for some α > α, Second order backward SDE with random terminal time By our assumption on ξ and S + , we see that we need to restrict α to the interval [−µ, ρ) in order to ensure that the last bound is finite. Now, we construct a sequence of approximating solutions to the RBSDE, using the finite horizon RBSDE result in [4] and on the optimal stopping problem above. Let τ n := τ ∧ n, and (Y n , Z n , U n ) be the solution to the following RBSDE We extend the definition of Y n for t ∧ τ ≥ τ n by Y n t∧τ = y t∧τ , Z n t∧τ = z t∧τ , U n t∧τ = u t∧τ , so that (Y n , Z n , U n ) is a solution of the RBSDE with parameters (f n , ξ, S): for t, t ∈ R + , The following result justifies the existence statement in Theorem 3.9.
Proof. 1. We first show that {(Y n , Z n , U n )} n∈N is a Cauchy sequence in D p η,τ ×H p η,τ ×U p η,τ , which induces the convergence of (Y n , Z n , U n ) towards some (Y, Z, U ) in D p η,τ × H p η,τ × U p η,τ .
By the stability result of Theorem 3.10 (ii), we have the following estimate for the differences (δY, δZ, δU ) : where, by the Lipschitz property of f in Assumption 3.1, Similarly, for η < η < η < ρ, we obtain that E τn τm e η s σ s z s ds , and E τn τm e η s |y s |ds The last three estimates show that ∆ f −→ 0 as m, n → ∞, so that the required Cauchy property would follow from (4.19) once we establish that Y n 2. We next prove that the limit process U is a càdlàg supermartingale with [U, X] = 0. Then, there exists a limit process U ∈ D p 0,τ (P). As U n is a càdlàg Q−uniformly integrable supermartingale for all Q ∈ Q L , we may deduce that its limit U is also a càdlàg Q−uniformly integrable supermartingale for all Q ∈ Q L . Define U t := t 0 e −ηs dU s , t ≥ 0. Clearly, U ∈ D p η,τ (P). As the integrand e −ηs is positive, the process U is a supermartingale. By Kunita-Watanabe inequality for semimartingales, we obtain Second order backward SDE with random terminal time Theorem 3.10 (ii) also states that the right-hand side converges a.s. to 0, at least along a subsequence, which implies that [U, X] = 0.

3.
Clearly, Y ≥ S, P−a.s. In this step, we prove that the limit supermartingale U satisfies the Skorokhod condition. To do this, denote ϕ n := 1 ∧ (Y n − S), ϕ := 1 ∧ (Y − S), and let us show that the convergence of (Y n , U n ) to (Y, U ) implies that Since ϕ ε is piecewise constant, bounded by 1, and U n → U in D p η,τ , we get For the second term, we have By (4.7) and |f n,0 | ≤ |f 0 | we obtain that Hence, we may control the second term by choosing ε arbitrarily small. For the first term, we have Again we may show that E (K n τ ∧t ) p is bounded by a constant, independent of n ∈ N. We choose α < η. Then, it is easily seen that e α(t∧τ ) Y n t∧τ −→ e α(t∧τ ) Y t∧τ , for all t ≥ 0, and so that e ατ Y n τ −→ e ατ ξ, on {τ < ∞}.
From a similar argument, we also have t ∧τ t∧τn e αs f s (Y n s , Z n s ) + µY n s ds −→ 0, in L p , and by Lipschitz continuity of f we see that Therefore, we have proved that We now prove the comparison result. In particular, this justifies the uniqueness statement in Theorem 3.9.
Proof of Theorem 3.10 (i). Denote by Y n , Z n , U n n∈N and Y n , Z n , U n n∈N the approximating sequence of (Y, Z, U ) and (Y , Z , U ), using the triples (y, z, u) and (y , z , u ), respectively, as in the last proof. Since ξ ≤ ξ and S ≤ S , we have y τn ≤ y τn .
By standard comparison argument of BSDEs, see e.g. [39,Proposition 3.2], this in turns implies that Y n τ0 ≤ Y n τ0 for all stopping time τ 0 ≤ τ . The required result follows by sending n → ∞.

Special case: backward SDE
Proof of Theorem 3.4. By setting S = −∞, the existence and uniqueness results follow from Theorem 3.9. In particular, the Skorokhod condition implies in the present setting that U = N is a P−martingale orthogonal to X. It remains to verify the estimates (3.2).

Second order backward SDE: representation and uniqueness
We shall use the additional notation: We also note that E P,+ t 1 {τ ≥n} is a P-supermartingale. Then, by Doob's martingale inequality, we have where ξ 0 is an F τ0 −measurable random variable for some stopping time τ 0 ≤ τ . Under our conditions on (F, ξ), the wellposedness of these BSDEs for ξ 0 ∈ L p η,τ0 (P) follows from Theorem 3.4. Remark that in the sequel we always consider the version of Y P such that Y P t∧τ ∈ F + t∧τ by the result of Lemma 6.3. The following statement provides a representation for the 2BSDE, and justifies the comparison (and uniqueness) result of Proposition 3.15.
Proof. The uniqueness of Y is an immediate consequence of (5.3), and implies the uniqueness of Z, a t dt ⊗ P 0 -q.s. by the fact that This representation also implies the comparison result as an immediate consequence of the corresponding comparison result of the BSDEs Y P [ξ, τ ].

1.
We first prove (5.2). Fix some arbitrary P ∈ P 0 and P ∈ P + P (t 1 ∧ τ ). By Definition 3.11 of the solution of the 2BSDE (3.5), we see that Y is a supersolution of the BSDE on t 1 ∧τ, t 2 ∧τ under P with terminal condition Y t2∧τ at time t 2 ∧τ . By the comparison result of Theorem 3.7 (ii), see also Remerk 3.8 t1∧τ -measurable, the inequality also holds P-a.s., by definition of P + P (t 1 ) and the fact that measures extend uniquely to the completed σ-algebras. Then, by arbitrariness of P , We next prove the reverse inequality.
As a P u , b P u are uniformly bounded by L, it follows from the Doob maximal inequality that where C p is a constant independent of P . Then, it follows from Hölder's inequality that Second order backward SDE with random terminal time As it follows from the minimality condition in Definition 3.11 that and C P,p,α t1 < ∞ (see (6.18)), we obtain that thus providing the required equality.

3.
We finally verify the estimate (3.6). By the representation (5.3) proved in the previous step, and following the proof of Proposition 6.8, we may show that As, for each P ∈ P 0 , Y, Z, U P is a solution of the RBSDE (6.17), the required estimate for the Z component follows from Proposition 4.3.

Second order backward SDE: existence
In view of the representation (5.3) in Proposition 5.2, we follow the methodology of Soner, Touzi & Zhang [40,41] by defining the dynamic version of this representation (which requires the additional notations of the next section), and proving that the induced process defines a solution of the 2BSDE. In order to bypass the strong regularity conditions of [40,41], we adapt the approach of Possamaï, Tan & Zhou [34] to ensure measurability of the process of interest.
By a standard monotone class argument, we see that ξ t,ω is F s −measurable whenever ξ is F t+s -measurable. In particular, for an F-stopping time τ , t ≤ τ , then τ t,ω := τ t,ω − t is still an F-stopping time. Similarly, for any F-progressively measurable process Y , the shifted process Y t,ω s (ω ) := Y t+s (ω ⊗ t ω ), s ≥ 0, is also F-progressively measurable. The above notations can naturally be extended to (τ, ω)− shifting for any finite F-stopping time τ .

Backward SDEs on the shifted spaces
For all P ∈ P(t, ω), we introduce a family of random horizon BSDEs In this section, we will prove the following measurability result, which is important for the discussion of the dynamic programming.
We will first review in Section 6.2.1 the finite horizon argument of [34], and we next adapt it to our random horizon setting in Section 6.2.2.

Measurability -finite horizon
Let τ = T , where T is a finite deterministic time. For the convenience of the reader we repeat the argument in [34] in order to prove the finite horizon version of Proposition 6.2. For each P ∈ P loc , we consider the following shifted BSDE Y t,ω,P s = ξ t,ω + T −t s F t,ω r Y t,ω,P r , Z t,ω,P r , σ r dr − Z t,ω,P r · dX r − dN t,ω,P r , P-a.s. (6.3) for s ∈ [0, T − t]. Lemma 6.3. Let τ = T be a deterministic time. Then, there exists a version of Y t,ω,P such that the mapping (t, ω, s, ω , P) Proof. We shall exploit the construction of the solution of the BSDE (6.3) by the Picard iteration, thus proving that for each step of the iteration, the induced process Y n,t,ω,P satisfies the required measurability.

1.
We start from the first step of the Picard iteration. Take the initial value Y 0,t,ω,P ≡ 0 and Z 0,t,ω,P ≡ 0. Define for all t ≤ T Y 1,t,ω,P s We extend the definition so that Y 1,t,ω,P s := ξ t,ω on {s > T − t} ∩ {t ≤ T } and Y 1,t,ω,P s ≡ ξ(ω T ∧· ) for t > T . By Lemma 6.1, the mapping ξ ·,· (·) : It follows from Lemma 3.1 in [25] that there exists a version, still noted by Y 1,t,ω,P , such that the mapping (t, ω, ω , P) → Y The function Y 1,t,ω,P s we just constructed is not necessarily P-a.s. càdlàg in s. We next construct a version Y 1,t,ω,P (i.e., Y 1,t,ω,P s = Y 1,t,ω,P s , P-a.s. for all s) which is measurable and P-a.s. càdlàg in s. Let t n i := i2 −n (T − t), and set for s ≥ 0: and so is (t, ω, s, ω , P) → Y 1,t,ω,P s (ω ). Since the filtration F +,P satisfies the usual conditions and the conditional expectation in (6.4) is an F +,P -martingale, one can prove by a standard argument (see e.g. [23,Proposition I 3.14]) that Y 1,t,ω,P is a P-a.s. càdlàg version of Y 1,t,ω,P .
As before, we extend the definition so that Y t,ω,P s := ξ t,ω on {s > T − t} ∩ {t ≤ T } and Y t,ω,P s ≡ ξ(ω T ∧· ) for t > T . Then it follows from [25,Lemma 3.2] that there exists an increasing sequence {n P k } k∈N ⊆ N such that P −→ n P k is measurable for each k and lim k→∞ sup 0≤s≤T −t Y n P k ,t,ω,P s − Y t,ω,P s = 0, P-a.s.

Dynamic programming principle
The goal of this section is to prove that the dynamic value process V satisfies the dynamic programming principle. We first focus on the underlying BSDEs for which the dynamic programming principle reduces to the following tower property, where we denote by Y[ξ 0 , τ 0 ] the Y component of the solution of the BSDE with the terminal time τ 0 and value ξ 0 . Lemma 6.5. Let Assumptions 3.1 and 3.2 hold true. Then, for all stopping time τ 0 ≤ τ , and P ∈ P loc : The proof is omitted as (i) is a direct consequence of the uniqueness of the solution to BSDE, and (ii) is similar to [34,Lemma 2.7]. In order to apply the classic measurable selection results, we need the following properties of the probability families {P(t, ω)} (t,ω)∈ 0,τ . Lemma 6.6. The graph P := {(t, ω, P) : P ∈ P(t, ω)}, is Borel-measurable in R + × Ω × M 1 . Moreover for all (t, ω) ∈ 0, τ and all stopping time τ 0 valued in [t, τ ], denoting τ t,ω 0 := τ t,ω 0 − t, we have: (i) P(t, ω) = P(t, ω ·∧t ), and for all P ∈ P(t, ω), the r.c.p.d. P τ t,ω 0 ,ω ∈ P(τ 0 , ω ⊗ t ω ), for P-a.e. ω ∈ Ω.
F τ0 (ω), P-a.s., for all P ∈ P 0 . Therefore, for Q ∈ Q L (P), we have Then, by the estimate (3.2), we obtain which induces the required estimate by sending ε → 0.
Indeed, for all P ∈ P 0 , and Q ∈ Q L (P): by (6.8) in Theorem 6.7, implying that E P0 e ητ V + τ p ≤ v p . Then δ n := e ητ V + τ − e ητ n Vτn satisfies for an arbitrary m ≥ 1: which implies the required convergence.

2.
We now prove that V + τ0 ≥ Y P τ0 V + τ1 , τ 1 , P−a.s. for all P ∈ P + P (τ 0 ), where the right hand is well defined by the integrability of V + obtained in step 1. Recall from Theorem 6.7 that , P-a.s.
Since for each m ∈ N, P P (τ m 0 ) ⊆ P + P (τ 0 ) = h>0 P P (τ 0 + h), we have for any P ∈ P + P (τ 0 ) and for m large enough that where τ m 0 and τ n 1 are defined from τ 0 and τ 1 as in the previous step. By the stability result of BSDEs in Proposition 4.6, and the result of Step 1 of the present proof, we have Then, , P-a.s., and therefore where the last equality is due to Y P V + τ1 , τ 1 ∈ D p η,τ1 (P ).

3.
We next prove the reverse inequality. By the comparison result together with the last step of the present proof, we have P ess sup So it remains to prove that (6.14) In the remainder of Step 3, we omit the parameter [ξ, τ ] without causing confusion. For any η ∈ [−µ, ρ), we obtain by the dominated convergence theorem together with the estimate (6.8) of Theorem 6.7 that . (6.15) By the Lipschitz property of F in Assumption 3.1, we estimate that , which provides (6.14) in view of (6.15).

4.
It remains to prove that V + inherits the integrability property of V . By which induces the required result by Remark 5.1.
Notice that τ ε ≤ τ , as the two processes are equal to ξ at time τ . From the Skorokhod condition, it follows that U P is a martingale on [0, τ ε ], thus reducing the RBSDE to a BSDE on this time interval. Denoting as usual by Y P V + τε , τ ε , we obtain by standard BSDE techniques that, for some probability measure Q ∈ Q L (P), where the last inequality follows from the crucial dynamic programming principle of Proposition 6.8. By the definition of ε, the last inequality cannot happen. Consequently Y P = V + . In particular, V + is a càdlàg semimartingale which would satisfy (6.16) once we prove that the family {Z P } P∈P0 may be aggregated. By Karandikar [22], the quadratic covariation process V + , X may be defined on R + × Ω. Moreover, V + , X is P 0 -q.s. continuous and hence is F +,P0 -predictable, or equivalently F P0 -predictable. Similar to the proof of [28,Theorem 2.4], we can define a universal F P0predictable process Z by Z t dt := a −1 t d V + , X t , and by comparing to the corresponding covariation under each P ∈ P 0 , we see that Z = Z P , P-a.s. for all P ∈ P 0 . This completes the proof of (6.16).

2.
It remains to prove that the family of supermartingales U P P∈P0 satisfies the minimality condition. Let 0 ≤ s ≤ t, P ∈ P 0 , P ∈ P + P (s∧τ ), and denote by Y P , Z P , N P the solution of the BSDE with parameters (F, ξ). Define δY := V + − Y P , δZ := Z − Z P and δU := U P − N P . By Itô's formula, we have for α ∈ [−µ, ρ), e α(s∧τ ) δY s∧τ = τ s∧τ e α(r∧τ ) F r V + r , Z r , σ r − F r Y P r , Z P r , σ r − αδY r dr − δZ r · dX r − dδU r = τ s∧τ e α(r∧τ ) a P r δY r + b P r · σ r δZ r dr − σ r δZ r · dW r + dδU r , for some bounded processes a P and b P , by Assumption 3.1. This provides that Γ P t∧τ e α(t∧τ ) δY t∧τ − Γ P s∧τ e α(s∧τ ) δY s∧τ = t∧τ s∧τ Γ P r e αr δY r b P r + σ r δZ r · dW r + dδU r , where Γ P r := exp r s∧τ a P u − Recall that δY ≥ 0, and U P is a P −supermartingale with decomposition U P = N P − K P , for some P −martingale N P and nondecreasing process K P . Now, the minimality condition in Definition 3.11 follows immediately from Proposition 6.8, provided that C P,p s,t < ∞, P−a.s. which we now prove.
The family E P t∧τ s∧τ dK P s p F + s∧τ , P ∈ P + P (t ∧ τ ) is directed upward. 9 Then, it follows from [26, Proposition V-1-1] that the ess sup in (6.18) is attained as an increasing limit along some sequence {P n } n∈N ⊆ P + P (s ∧ τ ). By the monotone convergence theorem, we see that η,τ (P0) < ∞ due to the wellposedness of the RBSDE. Hence, C P,p s,t < ∞, P-a.s.

Connection to a fully nonlinear elliptic path-dependent PDE
In this section, we present an example of pricing under volatility uncertainty from the so-called robust finance. The canonical process X represents the price process a financial asset. The objective is the hedging of the derivative security defined by the payoff ξ(X) at some maturity H Q defined as the exiting time from some set Q.
In contrast with the standard approach, we assume that the volatility is uncertain.
The probability space (Ω, F) is endowed with a family of probability measures P UVM , P UVM := P : X is a continuous P-martingale and d X t dt ∈ σ 2 , σ 2 , P-a.s. .
We assume that σ > 0. The superhedging problem under volatility uncertainty was initially formulated by Denis and Martini [11] and Neufeld and Nutz [24]. Their superhedging result expresses the cost of robust superhedging as where r > 0 is the discount rate, Q is a bounded open convex subset of R d containing 0, and H Q := inf{t ≥ 0 : ω t / ∈ Q}.
Since the family P UVM is saturated, we consider the following saturated 2BSDE dK s , P UVM -q.s. Y P 0 , P-a.s. for all P ∈ P UVM , where for all P ∈ P UVM is the solution of the following BSDE under P Z P s · dX s , P-a.s. 9 This follows from the same argument as in [40,Theorem 4.3]. For P 1 , P 2 ∈ P + P (s ∧ τ ), denote κ P i := E P i s∧τ t∧τ s∧τ dK P r p , and A := {κ P 1 > κ P 2 }, and define E ∈ F −→ P 3 (E) := P 1 (A ∩ E) + P 2 (A c ∩ E); clearly, P 3 ∈ P + P (t ∧ τ ), and κ P 3 = κ P 1 ∨ κ P 2 .
where the last equality follows from the fact, which is easy to show, that the family E P e −rH Q ξ F + 0 , P ∈ P + P (0) is upward directed.
We refer the interested reader to [36] and the references therein for more details about the theory of path-dependent PDE.

Remark 7.2.
Here we connect the random horizon 2BSDE to the elliptic path-dependent PDE via the value function of the stochastic control problem (7.1). In order to verify that the value function is a viscosity solution to the path-dependent PDE, one need first prove it is uniformly continuous (according to the definition in [36]). This regularity requirement is closely related to the generator and the boundary of the equation. In the example above, we address the most simple case in which the generator is uniformly elliptic and the boundary is convex. In such setting, one may prove the desired uniform continuity, using an elementary argument of which the key ingredient is to verify the uniform continuity of x → H x Q := inf{t ≥ 0 : x + ω t / ∈ Q} under a nonlinear expectation. For more details, see [36,Proposition 8.1].