Backward stochastic differential equations with Young drift

We prove via a direct fixpoint argument the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite $p$-variation with $p \in [1,2)$. An application to the Feynman-Kac representation of semilinear rough partial differential equations is given.


Introduction
Stochastic differential equations (SDEs) driven by Brownian motion W and an additional deterministic path η of low regularity (so called "mixed SDEs") have been well-studied.In [GN08] the wellposedness of such SDEs is established if η has finite q-variation with q ∈ [1, 2). 1 The integral with respect to the latter is handled via fractional calculus.Independently, in [Die12] the same problem is studied using Young integration for the integral with respect to η.Interestingly, both approaches need to establish (unique) existence of solutions via a Yamada-Watanabe theorem.A direct proof using a contraction argument is not obvious to implement.
For paths of q-variation with q ∈ (2, 3) integration has to be dealt with via the theory of rough paths.Motivated by a problem in stochastic filtering, [CDF13] give a formal meaning to the mixed SDE by using a flow decomposition which seperates the stochastic integration from the deterministic rough path integration.It is not shown that the resulting object actually satisfies any integral equation.
In [DOR15] well-posedness of the corresponding mixed SDE is established by first constructing a joint rough path "above" W and η.The determinstic theory of rough paths then allows to solve the mixed SDE.The main difficulty in that work is the proof of exponential integrability of the resulting process, which is needed for applications.In [DFS14] these results have been used to study linear "rough" partial differential equations via Feyman-Kac formulae.
Backward stochastic differential equations (BSDEs) were introduced by Bismut in 1973.In [Bis73] he applied linear BSDEs to stochastic optimal control.In 1990 Pardoux and Peng [PP90] then considered non-linear equations.A solution to a BSDE with driver f and random variable ξ ∈ L 2 (F T ) is an adapted pair of processes (Y, Z) in suitable spaces, satisfying Key words and phrases.Rough paths theory, Young integration, BSDE, rough PDE.This research was partially supported in part by the DAAD P.R.I.M.E.program and NSF grant DMS 1413717.Part of this work was carried out while the first author was visiting the University of Souther California, and he would like to thank Jin Ma and Jianfeng Zhang for their hospitality.
1 See Section 4 for background on the variation norm and Young integration.
Under appropriate conditions on f and ξ they showed the existence of a unique solution to such an equation.One important use for BSDEs is their application to semilinear partial differential equations.This "nonlinear Feynman-Kac" formula is for example studied in [PP92].
In this work we are interested in showing wellposedness of the following equation Here W is a multidimensional Brownian motion, η is a multidimensional (determinstic) path of finite q-variation, q ∈ [1, 2) and ξ is a bounded random variable, measurable at time T .Such equations have previously been studied in [DF12].In that work η is even allowed to be a rough path, i.e. every q ≥ 1 is feasible.The drawback of that approach is that no intrinsic meaning is given to the equation, that is a solution to (1) is only defined as the limit of smooth approximations.In the current work we solve (1) directly via a fixed point argument.The resulting object solves the integral equation, where the integral with respect to η is a pathwise Young integral.
In Section 2 we state and prove our main result.In Section 3 we give an application to partial differential equations.In Section 4 we recall the notions of p-variation and Young integration.

Main result
We shall need the following spaces.
Definition 1.For p > 2 define B p to be the space of adapted process

Denote by BMO the space of all progressively measurable
where the dη integral is a well-defined (pathwise) Young integral. and is locally uniformly continuous.(iv) Fixing f, g there exists for every Remark 3. The refined continuity statement in (iv) will be imporant for our application to rough PDEs in Section 3.
Proof.For R > 0 define This is well-defined as usual in the BSDE literature (see for example [PP90]), by setting and letting Z be the integrand in the Itô representation of the martingale In what follows A B means there exists a constant C > 0 that is independent of η, ξ such that

Unique existence on small interval
We first show that for T small enough, Φ leaves a ball invariant, i.e. for T small enough, R large enough Using the Young estimate (Theorem 7 in the Appendix) we estimate The Burkholder-Davis-Gundy inequality for p-variation ([FV10, Theorem 14.12]) gives Now the dη integral satisfies the usual product rule, so together with Itō's formula we get By Lemma 9 (again in Appendix below) Taking conditional expectation we get We estimate trivially which we can bound, using (3), ( 4) and ( 5), by a constant times Combining with (6), we get Using |a| ≤ 1 + |a| 2 and picking T > 0 such that T + T 2 ≤ 1/2 we get, with F (T ) → 0, as T → 0 (here we use that ||η|| q-var;[0,T ] → 0 for T → 0). Then which can be made smaller than R/2 by picking first R large and then T small.So indeed the ball stays invariant.
We now show that for T small enough, Φ is a contraction on Using the Young estimate (Theorem 7) and Lemma 8 (in Appendix below) we have for some constant c that can change from line to line where On the other hand . We estimate the Lebesgue integral as So, after taking conditional expectation, Picking λ small, then T small, we get Define the modified norm We hence have a contraction and thereby existence of a unique solution on small enough time intervals.

Continuity on small time interval
This follows from virtually the same argument as the contraction mapping argument.

Comparison on small time interval
Let C B > 0 be given, and pick T = T (C B ) so small that the BSDE is well-posed for any f, g Let ξ 1 , ξ 2 ∈ F T be given with ||ξ 1 || ∞ < C B and η ∈ C q-var with ||η|| q-var;[0,T ] < C B .Let η n be a sequence of smooth paths approximating η in q-variation norm, with ||η n || q-var;[0,T ] < C B for all n ≥ 1.

By assumption
Consider the following Young ODEs: Note that (Y , 0) and (Y , 0) solve the following BSDEs respectively: Choose δ such that the BSDE (2) is wellposed on a time interval of length δ whenever the terminal condition is bounded by

Continuity
Using the previous step we can use the continuity result on small intervals to get continuity of the solution map on arbitrary intervals.
We finish by showing the second continuity statement.Since the dη-term is more difficult then the dt-term we will assume f ≡ 0 for ease of presentation.First note that since the ||ξ|| ∞ , ||ξ ′ || ∞ < M , the local uniform continuity of the solution map in Theorem 2 we get Note that So that By Ito's formula, together with the classical product rule for the dη-term, we get so that if the latter is an honest martingale we get Let us calculate the conditional moments of Γ t := Further, by the product rule, Iterating, we get that for some In particular, for every So there is ε > 0 such that In particular for every c ∈ R E[exp(c| So the statement follows with is an honest martingale.But this follows from Here we used

Application to rough PDEs
It is well-known that BSDEs provide a stochastic representation for solutions to semi-linear parabolic partial differential equations (PDEs), in what is sometimes called the "nonlinear Feynman-Kac formula" [PP92].In this section we show how to use BSDEs with Young drift for the stochastic representation for PDEs of the form Here η has finite q-variation, with q ∈ [1, 2) and the last term is hence not well-defined.There are several approaches to make sense of such a "rough" PDE.Here we shall define the solution as the limit of solutions to smooth approximations, see Theorem 5 below.x ∈ R m let X s,x be the solution to the SDE Then u(t, x) := Y t,x t is the unique viscosity solution in BUC([0, T ] × R m ) to the PDE The following theorem extends this representation property to BSDEs with Young drift.
Theorem 5. Let η ∈ C 0,q-var , q ∈ [1, 2) and let η n smooth be given such that η n → η in C 0,q-var .Let f (t, y, z) Then there exists u ∈ BC([0, T ] × R m ) such that u n → u locally uniformly and the limit does not depend on the approximating sequence.Formally, u solves the PDE Moreover u(t, x) = Y t,x t , where X s,x is the solution to the SDE dX s,x t = σ(X s,x t )dW t + b(X s,x t )dt X s,x s = x and Y s,x the solution to the BSDE with Young drift Proof.By Theorem 4 we can write u n (t, x) = Y n,t,x t where dX s,x t = σ(X s,x t )dW t + b(X s,x t )dt, X s,x s = x and Y n,s,x is the solution to the BSDE By Theorem 2 we have that for fixed s, x, Y n,s,x → Y s,x in B p , where Y s,x solves the corresponding BSDE with Young drift.In particular Y n,s,x s → Y s,x s , and hence we get pointwise convergence of u n .We now show that u n is locally uniformly continuous in (t, x) uniformly in n.By Theorem 2 (iv), uniformly in n, where we used Lipschitzness of the map R where we used the uniform boundedness of Y n p,2 in the last step (as in the proof of Theorem 2).
It follows that u n is locally uniformly continuous in (t, x) uniformly in n.Hence u n converges to u locally uniformly.
Remark 6.In the vain of [DFS14] one can also, under appropriate assumptions on the coefficients, verify that u solves an integral equation.

Appendix -Young integration
For p ≥ 1, V some Banach space, we denote by C p-var = C p-var ([0, T ], V ) the space of V -valued continuous paths X with finite p-variation Here the supremum runs over all partitions of the interval [0, T ] and X u,v := X v − X u .
We shall also need the space C 0,p-var = C 0,p-var ([0, T ], V ), defined as the The proof of the following result goes back to [You36].A short modern proof can be found in [FH14,Chapter 4].In this statement and in what follows a b, means that there exists a constant c > 0, not depending on the paths under considerations, such that a ≤ cb.The constant c can depend on the vector fields under considerations, the dimension and the time horizion T , but is bounded for T bounded.