Second order backward SDE with random terminal time ††thanks: This work benefits from the financial support of the ERC Advanced Grant 321111 and the Chairs Financial Risk and Finance and Sustainable Development.
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).
MSC2010. 60H10, 60H30
Keywords. Backward SDE, second order backward SDE, quasi-sure stochastic analysis, random horizon
Let be a filtered probability space, supporting a dimensional Brownian motion . The martingale representation theorem states that any integrable measurable random variable , for some stopping time , can be represented as , for some square integrable predictable process , and some martingale with and . In particular when is the (augmented) canonical filtration of the Brownian motion, . This result can be seen as the path-dependent counterpart of the heat equation. Indeed, a standard density argument reduces to the case for an arbitrary partition of , where the representation follows from a backward resolution of the heat equation on each time interval , , and the 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 [PP90] introduced the theory of backward stochastic differential equations in finite horizon, extended further to the random horizon setting by Darling & Pardoux [DP97]. In words, this theory provides a representation of an measurable random variable with appropriate integrability as with , , where is a given random field. In the Markov setting where and , , it turns out that for some deterministic function , which is easily seen to correspond to the semilinear heat equation , by the fact that the process again identifies the space gradient of .
As our interest in this paper is on the random horizon setting, we refer the interested reader to the related works by Briand & Hu [BH98], Briand and Carmona [BC00], Royer [Roy04], Bahlali, Elouaflin & N’zi [BEN04], Popier [Pop07], Briand and Confortola [BC08]. We also mention the related work of Hamadène, Lepeltier & Wu [HLW99] which considers the 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 [STZ12], and further developed by Possamaï, Tan & Zhou [PTZ17], see also the first attempt by Cheridito, Soner, Touzi & Victoir [CSTV07], and the closely connected BSDEs in a nonlinear expectation framework of Hu, Ji, Peng & Song [HJPS14a, HJPS14b] (called GBSDEs). This extension is performed on the canonical space of continuous paths with canonical process denoted by . The key idea is to reduce the fully nonlinear representation to a semilinear representation which is required to hold simultaneously under an appropriate family of singular semimartingale measures on the canonical space. Namely, an random variable with appropriate integrability is represented as
Here, , and is a supermartingale with , , a.s. for all satisfying the minimality condition . Loosely speaking, in the Markov setting where for some deterministic function , the last representation implies that is a supersolution of a semilinear parabolic PDE parameterized by the diffusion coefficient , and the minimality condition induces the fully nonlinear parabolic PDE .
Our main contribution is to extend the finite horizon fully nonlinear representation of [STZ12] and [PTZ17] to the context of a random horizon defined by a finite 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 well-posedness 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 and the corresponding “space gradient” . 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.
2.1 Canonical space
Fix , , . Let
be the space of continuous paths starting from the origin equipped with the distance defined by Define the canonical process by
Let be the collection of all probability measures on , equipped with the topology of weak convergence. Denote by the raw filtration generated by the canonical process . Denote by the right limit of . For each , we denote by the augmented filtration of under . The filtration is the coarsest filtration satisfying the usual conditions. We denote by and the (right-continuous) universal completed filtration defined by
Clearly, is right-continuous. Simialrly, for , we introduce and , where
For any family , we say that a property holds quasi-surely, abbreviated as q.s., if it holds a.s. for all .
We denote by the collection of probability measures such that for each ,
is a continuous -local martingale whose quadratic variation is absolutely continuous in with respect to the Lebesgue measure;
is an -dimensional -Browinian motion such that is absolutely continuous in with respect to the Lebesgue measure.
Due to the continuity of , that is an -local martingale under implies that is an -local martingale. Similarly, is an -Brownian motion under .
As in [Kar95], we can define a pathwise version of a -matrix-valued process . The constructed process is -progressively measurable and coincides with the quadratic variation of under all . In particular, the -matrix-valued and -matrix-valued processes and are defined pathwisely, and we may introduce the corresponding -progressively measurable density processes
so that and , a.s., for all .
For later use, we observe that, as , the set of nonnegative-definite symmetric matrices, we may define a measurable555Any matrix has a decomposition for some orthogonal matrix , and a diagonal matrix , with Borel-measurable maps and , as this decomposition can be obtained by e.g. the Rayleigh quotient iteration. This implies the Borel measurability of the generalized inverse map , where is the diagonal element defined by , . generalized inverse .
Throughout this paper, we shall work with the the following subset of :
, and we have , -a.s. for all .
The measurability of follows from Nutz & von Handel [NvH13, Lemma 4.5]. We consider the extended space , where equipped with the filtration generated by the canonical process. Denote by the Wiener measure on . Set , and . Extend , , and from to in the obvious way, and denote these extensions by , , and . Note that
By [SV97, Theorem 4.5.2], there is a -dimensional Brownian motion on , such that
Obviously, we have . Then, , -a.s. which implies the desired result. ∎
2.2 Spaces and norms
Let and .
(i) One-measure integrability classes: for a probability measure , let be an -stopping time. We denote:
is the space of -valued and -measurable random variables , such that
is the space of -valued, -adapted processes with càdlàg paths, such that
is the space of -valued, -progressively measurable processes such that
is the space of -valued, -adapted martingales such that
is the set of scalar -predictable processes with càdlàg nondecreasing paths, s.t.
is the set of càdlàg -supermartingales , with Doob-Meyer decomposition into the difference of a martingale and a predictable non-decreasing process, such that
(ii) Integrability classes under dominated nonlinear expectation: For , denote by the set of all probability measures such that
for some -progressively measurable process uniformly bounded by , which is a fixed Lipschitz constant throughout this paper, see Assumption 3.1. By Girsanov’s Theorem,
for all . For , we denote
and we introduce the subspace of r.v. such that
We define similarly the subspaces , , , and the subsets , .
(iii) Integrability classes under non-dominated nonlinear expectation: Let be a subset of probability measures, and denote
Let be a filtration with for all , so that is also a -stopping time. We define the subspace as the collection of all -measurable -valued random variables , such that
We define similarly the subspaces and by replacing with .
3 Main results
3.1 Random horizon backward SDE
For a probability measure , a finite -stopping time , an -measurable r.v. , and a generator , -measurable 666By we denote the -algebra generated by progressively measurable processes. Consequently, for every fixed , the process is progressively measurable., we set
and we consider the following backward stochastic differential equation (BSDE):
Here, is a càdlàg adapted scalar process, is a predictable -valued process, and a càdlàg -valued martingale with orthogonal to , i.e., . We recall from Lemma 2.2 that , a.s.
By freezing the pair to , we set .
The generator satisfies the following conditions.
(i) Lipschitz: there is a constant , such that for all , , ,
(ii) Monotone: there is a constant , such that for all , , ,
is a finite stopping time, is measurable, and
Theorem 3.3 (Existence and uniqueness).
We emphasize that Darling & Pardoux [DP97] requires a similar integrability condition as Assumption 3.2 with instead of and instead of . The following example illustrates the relevance of our assumption in the simple case of a linear generator.
Let , be the Wiener measure on , so that is a Brownian motion. Let , where , , and for some constants . Notice that , and directly verification:
We next show that Darling & Pardoux’s condition is not satisfied. To see this, observe that the event set satisfies , and therefore
We also have the following comparison and stability results, which are direct consequences of Theorem 3.8 below, obtained by setting the obstacle to therein, together with the estimate (3.2) in Theorem 3.3.
Let , be two sets of parameters satisfying the conditions of Theorem 3.3 with some stopping time , and the corresponding solutions , .
Stability. Denoting , , , and , we have for all and :
Comparison. Assume , -a.s., and for all , a.e. Then, , -a.s. for all stopping time , a.s.
Following [EPQ97] we say that is a supersolution (resp. subsolution) of the BSDE with parameters if the martingale 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 is a subsolution of BSDE, and is a supersolution of BSDE.
3.2 Random horizon reflected backward SDE
We now consider an obstacle defined by , and we search for a representation similar to (3.1) with the additional requirement that . This is achieved at the price of pushing up the solution by substracting a supermartingale with minimal action. We then consider the following reflected backward stochastic differential equation (RBSDE):
where is a càdlàg supermartingale, for all , starting from , orthogonal to , i.e. . The last minimality requirement is the so-called Skorokhod condition.888This condition indeed coincides the standard Skorokhod condition in the literature. Indeed, by using the corresponding Doob-Meyer decomposition into a martingale and a nondecreasing process , and recalling that , it follows that is equivalent to , a.s. by the arbitrariness of .
Theorem 3.7 (Existence and uniqueness).
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.
Let and be two sets of parameters satisfying the conditions of Theorem 3.7, with corresponding solutions and .
Comparison. Assume , -a.s., for all , and , -a.e. Then, , -a.s., for all stopping time , -a.s.
Stability. Let , and denote , , , and . Then, for all and , we have:
where and .
Notice that the stability result is incomplete as the differences , and are controlled by the norms of and . However, in contrast with the estimate (3.2) in the backward SDE context, we have unfortunately failed to derive a similar control of by the ingredients and in the present context of random horizon reflected backward SDE .
3.3 Random horizon second order backward SDE
Following Soner, Touzi & Zhang [STZ12], 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 :
where we recall that . We also define for all finite stopping times :