Existence, uniqueness and comparisons for BSDEs in general spaces
Abstract
We present a theory of backward stochastic differential equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the left continuity of the filtration, of the predictable quadratic variations of martingales or of the measure integrating the driver. We present conditions for existence and uniqueness of squareintegrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem and hence construct time consistent nonlinear expectations in these general spaces.
10.1214/11AOP679 \volume40 \issue5 2012 \firstpage2264 \lastpage2297 \newproclaimdefinitionDefinition[section] \newproclaimremarkRemark[section]
BSDEs in general spaces
A]\fnmsSamuel N. \snmCohen\corref\thanksreft1label=e1]samuel.cohen@maths.ox.ac.uklabel=u1,url]http://people.maths.ox.ac.uk/cohens/ and B]\fnmsRobert J. \snmElliottlabel=e2]relliott@ucalgary.calabel=u2,url]http://people.ucalgary.ca/~relliott/
t1Samuel Cohen is now at the University of Oxford; this work was completed at the University of Adelaide.
class=AMS] \kwd[Primary ]60H20 \kwd[; secondary ]60H10 \kwd91B16. BSDE \kwdcomparison theorem \kwdgeneral filtration \kwdseparable probability space \kwdGrönwall inequality \kwdnonlinear expectation.
60H20, 60H10, 91B16, BSDE, comparison theorem, general filtration, separable probability space, Gronwall inequality, nonlinear expectation
1 Introduction
The theory of backward stochastic differential equations (BSDEs) has been extensively studied. Typically, results have been obtained only in the context of a filtration generated by a Brownian motion, possibly with the addition of Poisson jumps. Specifically, attention has been given to equations of the form
where is the martingale generating the filtration (typically Brownian motion), is a fixed finite terminal time, is a stochastic terminal value, is a progressively measurable function, denotes matrix/vector transposition (and hence denotes the inner product of and ) and the solution is a square integrable pair of processes , where is adapted and is predictable.
A notable exception to this is the work of El Karoui and Huang El1997a (), where a general probability space is considered. In the case considered in El1997a (), the martingale is specified a priori, and the equation considered is
(1) 
where each term is as above, the filtration is quasileft continuous, is a continuous process such that is absolutely continuous with respect to and is a martingale strongly orthogonal to , that is, , where denotes the predictable quadratic covariation process.
These equations depend heavily on the continuity of and, therefore, are unable to deal with any situation where martingales may jump at a point with positive probability. However, these situations may arise in various applications. For example, when using BSDEs in modeling dividend paying assets, the martingales involved may jump at the time of the dividend announcement. Similarly, if we consider embedding a discrete time process in continuous time, we obtain processes which jump with positive probability at every integer.
A significant use of these equations is to generate “nonlinear expectations” or “nonlinear evaluations,” in the sense of Peng2004 (). These are operators
satisfying certain basic properties. They have important applications in mathematical finance and stochastic control. Given the results of Coquet2002 () and Hu2008 (), it is known that in the Brownian setting, under certain conditions, these operators are completely described by BSDEs. Furthermore, it is clear, given the comparison theorem in Cohen2009 (), BSDEs of the form of (1) in arbitrary spaces, under some conditions, also describe nonlinear expectations. However, it is not known how large a class of nonlinear expectations in a general space is given by a BSDE.
To establish such a result for BSDEs of the form of (1), one faces a significant problem. If is given as the solution to (1) for some not dependent on , once is fixed, for any martingale orthogonal to with , we have the property
This property is clearly not true for most nonlinear expectations, whenever there are nontrivial examples of such processes , which is not the case in the Brownian setting (as a martingale representation theorem holds). It follows that these equations cannot describe any nonlinear expectations which do not possess this property.
Furthermore, the fact that the martingale must be specified a priori is arguably unsatisfying. Conceptually, it may be preferable if, in some sense, the probability space itself dictated what martingales are needed for the BSDE. In this case, one could proceed either by specifying the probability space using a collection of martingales (which, given a representation theorem holds, will then describe all martingales in the space), or vice versa.
In this paper we establish such a general result. We show that there is a sense in which the original BSDE can be interpreted in a general space, using only a separability assumption on . We establish conditions on the existence and uniqueness of BSDEs in this setting, where the driver is integrated with respect to an arbitrary deterministic Stieltjes measure (Theorem 6.1). We also prove a comparison theorem for these solutions, which shows under which conditions they do indeed describe nonlinear expectations and evaluations.
A similar approach is taken in Hassani2002 (), where a form of BSDE is considered using generic maps from a space of semimartingales to the spaces of squareintegrable martingales and of finitevariation processes integrable with respect to a given continuous increasing process. Using Browder’s theorem, they demonstrate the existence of solutions to these equations on an infinite horizon. Our approach differs from theirs by considering a classical form of BSDE on a finite horizon and deriving an existence result using a contraction mapping technique. Because of this, our conditions for existence are a more straightforward extension of those in the classical case. More significantly, our approach does not require the driver of the BSDE to be integrated with respect to a continuous measure, which allows a unification of the discrete and continuous time theory of BSDEs.
2 Martingale representations
The key result used in the construction of BSDEs is the Martingale representation theorem. In the Brownian setting, this result is well known (see, e.g., Revuz1999 (), Chapter V.3, or Elliott1982 (), Theorem 12.33). In other cases, for example, when dealing with martingales generated by Markov chains, a similar result is available (see Cohen2008 ()); however it is also known that there exist probability spaces in which no finitedimensional martingale representation theorem exists.
Consider a probability space with a filtration , satisfying the usual conditions of completeness and right continuity. The timeinterval is given the Borel field . {definition} For any nondecreasing process of finite variation , we define the measure induced by to be the measure over given by
Here , and the integral is taken pathwise in a Stieltjes sense. {remark} If is a deterministic process, then this definition gives the product measure . We can also consider these as measures on the space , where is the predictable algebra.
Under the assumption that the Hilbert space is separable, a paper of Davis and Varaiya Davis1974 () gives the following result (see also Malamud Malamud2007 ()).
Theorem 2.1 ((Martingale representation theorem; Davis1974 ()))
Suppose that is a separable Hilbert space, with an inner product . Then there exists a finite or countable sequence of squareintegrable martingales such that every square integrable martingale has a representation
for some sequence of predictable processes . This sequence satisfies
(2) 
These martingales are orthogonal (i.e., for all ), and the predictable quadratic variation processes satisfy
where denotes absolute continuity of the induced measures (Definition 2). Furthermore, these martingales are unique, in that if is another such sequence, then , where denotes equivalence of the induced measures.
Corollary 2.1.1
For any predictable processes satisfying (2), the process is well defined and is a squareintegrable martingale.
When a finitedimensional martingale representation theorem holds, as when the space is generated by a Brownian motion, then all but finitely many of the martingales given by Theorem 2.1 will be zero. We shall not, in general, assume that this is the case, but acknowledge that, in this situation, significant simplification of the equations considered is possible.
We shall use this result to construct a form of BSDE on this general space.
We denote by the space of infinite valued sequences. We note that the predictable processes in Theorem 2.1 can be written as a vector process , which takes values in .
3 BSDEs in general spaces: A definition
We seek to construct BSDEs, assuming only the usual properties of the filtration and that is a separable Hilbert space. For simplicity, we shall also assume that is trivial, which, by right continuity, ensures that, almost surely, no martingale has a jump at . {definition} Let be a deterministic signed Stieltjes measure. For , a BSDE is an equation of the form
(3) 
where is the (countably infinite) vector with entries . For a terminal value , a predictable driver function , a solution is a pair of processes taking values in , where is predictable, and is adapted. We shall restrict our attention to the case when is square integrable, and satisfies (2). {remark} We note that this type of equation encompasses most previously studied forms of BSDEs. When the filtration is Brownian, we can take to be the th component of the generating Brownian motion, , and the equation is standard. When the filtration is generated by a Poisson random measure over a separable space and a Brownian motion, as in Barles1997 (), Royer2006 () and others, or by a Markov chain, as in Cohen2008 (), Cohen2008b (), we have a similar reduction. When we consider the analogous equations in discrete time, we can form the discretetime filtration embedded in this continuous time context (see Jacod2003 (), Chapter 1f) and hence obtain the backward stochastic difference equations considered in Cohen2008c () and Cohen2009a ().
Comparing with the work of El1997a (), we see that if depends only on the projection of into a finitedimensional subspace of , then it is possible to reduce the equation to a form similar to (1).
We shall present a result (Theorem 6.1) demonstrating conditions under which there exists a unique solution to such an equation.
4 Inequalities for Stieltjes integrals
To give conditions under which solutions to a BSDE exist, we must first establish the following results regarding integrals with respect to Stieltjes measures. These results are standard whenever the measures are continuous.
4.1 Stieltjes exponentials
{definition}For any càdlàg function of finite variation , we write
and call this the Stieltjes exponential of . Note that this is also a càdlàg function.
Note that should be more properly written as , as it is a function of not just of . We use the former notation purely for compactness, whenever this does not lead to confusion. We note the following useful bound.
Lemma 4.1
If is a càdlàg function, then , where is the classical exponential of .
As , it is clear that for all . The result follows.
Lemma 4.2
For any càdlàg function of finite variation, the Stieltjes exponential is well defined. Furthermore, if , then . If , then , and is well defined. In this case, the process is the solution to the Lebesgue–Stieltjes integral equation,
As the process is càdlàg and of finite variation, it is a (deterministic) semimartingale. is then the standard Doléans–Dade exponential of this process, and so its existence and basic properties can be seen in Elliott1982 (), Theorem 13.5 ff. This guarantees the convergence of the infinite products considered and solves the desired integral equation. The nonnegativity result is clear by inspection.
For the positivity result, we need only show that . By continuity of the logarithm, this is equivalent to showing that
We then note that we can consider three cases. First, if , then , and hence
Second, we note that is finite, as is of finite variation, and hence there are only finitely many such that . Therefore
Finally, we know that for . Hence, we have
Combining these three sums gives the desired constraint on the logarithm, and hence the strict positivity of the desired product.
Lemma 4.3
For a càdlàg function of finite variation with , we have the stronger result
By the same argument as in Lemma 4.2, we have
It follows that
for all . Hence
Let be a càdlàg function of finite variation with for all . Then the leftjump inversion of is defined by
Similarly if for all , the rightjump inversion is defined by
Lemma 4.4
For a function as in Definition 4.1, the left and rightjump inversions are finite (whenever they are defined), and satisfy
and
Consider first the leftjumpinversion. We know that and . Hence it follows that has only finitely many values in any neighborhood not containing zero and hence is bounded away from . That is, there exists some such that for all . To show finiteness, write
and
Combining these sums gives the desired finiteness result.
We now note that, algebraically,
Hence
The proof for the rightjump inversion follows in the same way, where finiteness is because
and satisfies the requirements given above for the leftjump inversion. The algebraic result is then that
and the result is as given.
Lemma 4.5
For a càdlàg function of bounded variation with , the rightjump inversion of the leftjump inversion of is the original function, that is,
Similarly, if , then .
For simplicity, we decompose into a discontinuous part and a continuous part . Clearly, taking either the left or rightjump inversion will not alter the continuous part , and so it is sufficient to show that the discontinuous parts are equal, that is, for all , whenever these terms are well defined. From Definition 4.1 we have
and hence
and similarly , as desired.
4.2 Integrating factors
It is useful to have some results relating to the solutions of equations of the form These are similar the the classical results on the use of integrating factors and Grönwall’s inequality in the study of ordinary differential equations. {definition} Let be two measures on a algebra . We write if, for any , . {remark} When is a nonnegative measure, and is absolutely continuous with respect to , this definition is equivalent to requiring that the Radon–Nikodym derivative satisfies , a.e.
Lemma 4.6
Let , and be signed Stieltjes measures on , such that for all , and
then
where is the rightjump inversion of .
Applying the product rule for Stieltjes integrals we have
As and , this gives
Lemma 4.7 ((Backward Grönwall inequality))
Let be a process such that, for a nonnegative Stieltjes measure with and a integrable process, is integrable and
then
If is constant, this simplifies to
First note that and that . Then let
From the product rule for stochastic integrals, as is of finite variation,
Note that and are both nonnegative. Therefore, by integration,
Substitution yields
and the desired inequalities follow from . If , then this simplifies to
Lemma 4.8 ((Forward Grönwall inequality))
Let be a function such that, for a nonnegative Stieltjes measure and a integrable process, is integrable and
then
If is constant, this simplifies to
This result follows in an almost identical fashion to Lemma 4.7, and the proof is therefore omitted.
5 Existence of BSDE solutions: Fundamental results
In this section we shall establish the existence of solutions to BSDEs when the process satisfies particular properties. {definition} Let be a deterministic nondecreasing rightcontinuous function . The measure will serve in the place of the Lebesgue measure in our BSDE.
As is of finite variation, its discontinuities are bounded. We assume that assigns positive measure to any nonempty open interval in .
Unless otherwise indicated, all (in)equalities should be read as “up to evanescence.” {definition} We denote by the standard Euclidean norm on , and note that , where denotes vector transposition. {definition} For a given and fixed , we define the stochastic seminorm on as follows. For each , consider as a measure on the predictable algebra; cf. Remark 2. Let have the Lebesgue decomposition
where is absolutely continuous with respect to , and is orthogonal to . As they represent bounded measures on the predictable algebra, both and will be nondecreasing predictable processes.
We define, for ,
where is the th element in , considered as a series of values in .
We note that, for any predictable, progressively measurable process taking values in , and, in particular, for processes satisfying (2) in each of their components, we have the inequality
for any predictable set . (Note the latter equalities are simply the standard isometry used in the construction of the stochastic integral, by the orthogonality of the .) {definition} We define the following spaces of equivalence classes:
where two elements of are deemed equivalent if
two elements of are deemed equivalent if they are indistinguishable and two elements of are equivalent if they are equal a.s. Note that is here taken as fixed. {remark} We note that is itself a complete metric space, with norm given by ; similarly for . Note also that the martingale representations constructed in Theorem 2.1 are unique in .
A key assumption in the study of BSDEs is the continuity of the driver function . When the measure is continuous, we shall show that it is sufficient that is uniformly Lipschitz continuous for the BSDE (3) to have a solution. On the other hand, as is clear in discrete time (cf. Cohen2009a ()), when is not continuous, a stronger condition is needed on . We shall call this a firm Lipschitz bound on , as is defined in the following theorem.
Theorem 5.1
For as in Definition 5, assume . Let be a predictable, progressively measurable function such that:

;

there exists a linear firm Lipschitz bound on , that is, a measurable deterministic function uniformly bounded by some , such that, for any , ,
and
Note that the variable bound need only apply to the behavior of with respect to .
A function satisfying these conditions will be called standard. Then for any , the BSDE (3) with driver has a unique solution . ( and are defined in Definition 5.)
To prove this theorem, we first establish the following results.
Lemma 5.1
If assigns positive measure to every nonempty open interval, then two càdlàg processes in are indistinguishable if and only if they are equivalent in . Similarly, two càdlàg processes are equivalent in if and only if their left limits are equivalent in .
Clearly indistinguishability implies equivalence of the processes, and their left limits, in . By right continuity (resp., left continuity), if on some nonnull set , two processes (resp., their left limits) differ at any point, they must differ on some nonempty open interval. As assigns positive measure to such an interval, it follows that the processes will not be equivalent in .
Lemma 5.2
Let be the solution to a BSDE with data . If is standard, and , then if and only if the left limit process .
Clearly, if , then as is càdlàg and adapted, and hence progressive, . For the converse, write
and by the assumptions of the lemma, as , and so is a square integrable martingale, by Doob’s inequality Jacod2003 (), Theorem 1.43, this quantity is finite in expectation.
The following lemma provides the key bounds on BSDE solutions, which we shall use to prove existence and uniqueness of solutions.
Lemma 5.3
Let and be the solutions to two BSDEs with standard parameters and . Define
(5)  
where and are the Lipschitz constants of , and , are any nonnegative measurable functions such that and for all , and the integrands defining and are uniformly bounded.
Then
(6)  
and
(7)  
Let . By application of the differentiation rule for stochastic integrals, we have