On a class of reflected backward stochastic Volterra integral equations and related time-inconsistent optimal stopping problems

We introduce a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.


Introduction
In recent years backward stochastic Volterra integral equations (BSVIEs) have attracted a lot of interest due to their modelling potential in problems related to time-preferences of decision makers, asset allocation and risk management in mathematical finance among other fields, for which the related optimal control problems are time-inconsistent meaning that the associate value-function does not satisfy the dynamic programming principle.
Lin [1] was first to introduce and study a class of BSVIEs driven by Brownian motion.They can be described as follows: For a given Lipschitz driver f and a square integrable terminal condition ξ solving a BSVIE consists in finding an adapted process (Y , Z ) satisfying the equation In a series of papers, Yong [2][3][4] systematically studied a general class, including the M-solution, of BSVIEs of the form: ✩ Acknowledgements: Many thanks to the editor and the anonymous refer- ees for their insightful and valuable comments that helped improve the content of the paper.The authors gratefully acknowledge the financial support provided by the Swedish Research Council grants (2020-04697) and (2016-04086), respectively.* Corresponding author.E-mail addresses: nacira.agram@lnu.se(N.Agram), boualem@kth.se(B.Djehiche).Y (t) = ξ (t) + ∫ T t f (t, s, Y (s), Z (t, s), Z (s, t))ds − ∫ T t Z (t, s)dW (s).
In problems related to mathematical finance, the BSVIE is satisfied by the value function of time-inconsistent optimal control and equilibrium problems related to stochastic differential utility such as in Di Persio [14], dynamic risk measures such as in Yong [3], Wang and Yong [8] and Agram [15] and dynamic capital allocations such as in Kromer and Overbeck [16].The BSVIE is also satisfied by the adjoint process of a related stochastic maximum principle related to equilibrium recursive utility and equilibrium dynamic risk measures, such as in Djehiche and Huang [17], Wang, Sun and Yong [11] among many more papers (see [13] for further references).
The purpose of the paper is to introduce and study a version of the above BSVIE whose first component of the solution Y is constrained to be greater than or equal to a given obstacle process L. This is achieved by including as a part of the solution a process K (t, •) (parametrized by t), which should be adapted to the filtration generated by the one-dimensional Brownian motion W and increasing for each t, in the sense that the equation considered is of the form • ξ (t), which is a random variable (parametrized by t), is Moreover, it is also required the so-called Skorohod flatness condition under which K (t, •) is 'flat' whenever Y (t) > L(t) holds.
Our main results are: • to introduce the reflected BSVIE, • to establish existence and uniqueness of the solution, • to prove a related comparison result, • to show how the solution to the reflected BSVIE solves a time-inconsistent stopping problem.
We may remark that for simplicity, the driver in the above reflected BSVIE is only allowed to depend on Y and Z (t, s) but the results can be extended to driver depending also on Z (s, t).To the best of our knowledge, this class of reflected BSVIE and its connection to time-inconsistent optimal stopping problems seem new.
The content of our paper is as follows.After some preliminaries in Section 2, we give a formulation of the class of continuous reflected BSVIEs with lower obstacle.In Section 3 we derive existence, uniqueness and comparison results.Finally, in Section 4, we give an application to time-inconsistent optimal stopping problems.

Notation and preliminaries
Let (Ω, F , P) be a complete probability space on which is defined a standard one-dimensional Brownian motion W = (W (t)) 0≤t≤T .We denote by F := (F t ) 0≤t≤T its natural filtration augmented by all the P-null sets in F .
Let B(G) be the Borel σ -field of the metric space G.In the sequel, C > 0 represents a generic constant which can be different from line to line.
In this paper we only consider reflected BSVIEs driven by a one-dimensional Brownian motion.Extension to higher dimensions being straightforward.
We define the following spaces for the solution.
• H 2 is the space of progressively measurable processes • K 2 is the space of processes K which satisfy ) is an F-adapted and increasing process with K (t, 0) = 0;

Formulation of the problem
We investigate existence of a unique triple (Y , Z , K ) of processes taking values in R × R × R + which satisfy the following reflected BSVIE with one obstacle associated with (f , ξ , L): (where K (t, ds) is the Lebesgue-Stieltjes measure induced by the (c) The process K enjoys the following properties: (c2) The Skorohod flatness condition holds: for each 0 ≤ α < β ≤ T , ×K 2 satisfying (1) for every t ∈ [0, T ] P-a.s., and satisfying the obstacle condition (b) and the Skorohod flatness condition (c2).

Remark 2.
(1) The Skorohod flatness condition (c2) implies in fact that This form is more natural for the reflected BSVIE (1), since only (K (t, s), s ≥ t) is involved in the definition of Y (t).
(2) In the case of a standard reflected BSDEs, the condition (c2) is analogous to where the integral is to be interpreted in the Lebesgue-Stieltjes sense.It would be interesting to have a similar characterization for the Volterra type reflected equations.
We make the following assumptions on (f , ξ , L).
Assumption (iii) yields the continuity of Y and the bicontinuity of K (•, •) which in turn guarantees the Skorohod flatness condition (c2).

Existence and uniqueness of solutions to reflected BSVIEs
In this section we derive existence, uniqueness and a comparison result for continuous reflected BSVIEs.We have Theorem 4.
Remark 5. We note that the representation (5) does not imply that Y is a supermartingale, as it can easily be checked.
Inspired by the approach suggested in [1,10] and [3] to solve the ordinary BSVIE by identifying an accompanying true martingale, we derive a unique solution to the reflected BSVIE (1) associated with (f , ξ , L) by applying the notion of Snell envelope along with a contraction argument to an accompanying supermartingale Ỹ (t, •), parametrized by t, defined below from which we obtain Y by setting Y (t) := Ỹ (t, t).To this end, we first consider the case where the driver f does not depend on (Y , Z ).Then, we consider the general case where f depends on (Y , Z ).

Driver independent of Y and Z
Consider the following reflected BSVIE (6) where the driver f does not depend on (Y , Z ).We have Proposition 6.Under Assumptions (A1), (A2) and (A3), there exists a unique solution (Y , Z , K ) to the reflected BSVIE (6) associated with (f , ξ , L) which satisfies the properties (a), (b) and (c).Moreover, it admits the representation (5).
Proof.The proof is based on existence and uniqueness of the process ( Ỹ , Z , K ) which satisfies the following reflected BSDE, parametrized by t, associated with (f , ξ , L): P-a.s., for every t ∈ [0, T ], where •) is continuous and increasing, K (t, 0) = 0 and satisfies the following version of the Skorohod flatness condition: for which is equivalent to the property • Existence of a solution.For a fixed t ∈ [0, T ], set In view of (i) and the continuity of the obstacle process L, the map u ↦ → Γ (t, u) is continuous on [0, T ).Moreover, by (A1) and (A3), it holds that, for each t ∈ [0, T ], Consider the process ( Ỹ (t, u), u ∈ [0, T ]) defined by where the essential supremum is taken over i.e. it is the Snell envelope of the processes (Γ (t, u)) 0≤u≤T which is the smallest continuous supermartingale, parametrized by t, which dominates the continuous process Γ (t, •).Furthermore, by Doob's inequality it holds that In particular, the processes Γ (t, •) and Ỹ (t, •) are uniformly integrable and the processes Γ (•, u) and Ỹ (•, u)) are squareintegrable.Furthermore, the supermartingale ( By the well known techniques related the Snell envelope which use the Doob-Meyer decomposition along with the martingale representation theorem (see e.g.[18], Proposition 5.1), there exists a unique adapted increasing continuous process K (t, •) such that K (t, 0) = 0 and K (t, T ) ∈ L 2 (P) and an F-adapted process Z (t, •) ∈ H 2 such that, P-a.s., for every t ∈ [0, T ], for which the properties (d), (e) and (f) are satisfied.Moreover, the process Hence, since Y (t) = Ỹ (t, t), it satisfies P-a.s. the representation (5) and the following estimate for (Y , Z , K ) holds: which is finite by the assumptions (A1), (i) and (A3).Thus, Before we establish bi-continuity, we derive the following estimates.
Lemma 7.For any β > 1, there exists a positive constant C β depending only on β, c 1 and T such that Proof.First we note that, given t, t ′ ∈ [0, T ], denoting by ∆ξ := 2 and the Skorohod flatness condition (f), we have, for any 0 Taking expectation, we obtain Moreover, by Burkholder-Davis-Gundy's inequality, (iii) and Young's inequality it follows that, for any β > By Doob's maximal inequality and (iii) Hence, where .
• Continuity of the map (t, u) ↦ → K (t, u).This follows from the fact that and the bi-continuity of each of the terms on the r.h.s.

The general case. Proof of Theorem 4 Let
Consider the map Φ from E to itself for which (Y , Z ) = Φ(U, V ), where Proposition 8. Assume (A1), (A2) and (A3).Then there exists δ > 0 depending only on the Lipschitz constant of f such that Φ is a contraction mapping on the space E([T − δ, T ]).
Proof.Let X := (U, V ) and From Proposition 6, we have P-a.s. Thus, Noting that by the Cauchy-Schwarz inequality and (ii) we have Therefore, On the other hand, by Itô's formula applied to | Ỹ (t, u)|

2
, taking expectation and using Young's inequality, we obtain ] .
By repeating the same reasoning on each time interval [T − (m + 1)δ, T − mδ], m = 1, 2, . . ., n (where n is arbitrary) with a similar dynamics but terminal condition Ỹ m−1 (t, T − mδ) at time Pasting these processes, we obtain a unique solution (Y , Z , K ) of the reflected BSVIE (1) on the full time interval [0, T ]. □ A closer look at the way the estimate (10) is derived in e.g.[18] or [20], we obtain the following proposition as a direct consequence of the estimate leading to it.Proposition 9.If instead of (A1) and (A2), we assume that ] < ∞.

A comparison result for reflected BSVIEs
In this section we derive a comparison theorem, similar to that of [18], Theorem 4.1 for standard reflected BSDEs, which extends [8], Theorem 3.4., for non-reflected BSVIEs.We follow the method of the proof in [8] and first derive a comparison when the driver f does not depend of y, extending [8], Proposition 3.3, since in this case the proof is based on the comparison principle for reflected BSDEs, and then proceed to the proof of the general case using an approximation scheme.Some of the imposed conditions on the coefficients can be relaxed at the expense of heavy technical details that we omit to make the content easy to follow.Proposition 10.Let (f , ξ , L) and (f ′ , ξ ′ , L ′ ) be two sets of processes which satisfy the assumptions (A1), (A2) and (A3) and suppose further that Proof.For each fixed t ∈ [0, T ], let Ỹ (t, •) and Ỹ ′ (t, •) be the standard BSDEs accompanying Y and Y ′ respectively, constructed in a similar way to the one described in the proof of Proposition 6: For each fixed t ∈ [0, T ], A similar form for Ỹ ′ holds.We may apply the comparison the- orem ( [18], Theorem 4.1) for standard reflected BSDEs to obtain that, for each fixed t ∈ [0, T ], Let (Y , Z , K ) be a solution of the reflected BSVIE associated with (f , ξ , L) and (Y ′ , Z ′ , K ′ ) be a solution of the reflected BSVIE associated with (f ′ , ξ ′ , L ′ ).Then Y (t) ≤ Y ′ (t), 0 ≤ t ≤ T , P-a.s.
Proof.The proof follows the same steps of the proof of Theorem 3.2 in [8].We sketch it and leave some technical details related to Peng's monotone convergence theorem [21] which are by now standard in the literature related to reflected BSDEs.
Assume the map y ↦ → f (t, s, y, z) is nondecreasing.Set Y 0 (•) = Y ′ (•) and for n ≥ 1, consider, the following sequence of reflected BSVIEs with the same lower obstacle L In view of the assumption (H2) and the monotonicity of f in y, we may apply Proposition 10, to obtain that, for every n ≥ 1,  To show convergence, let Ỹn (t, •) be the solution to the stan- dard reflected BSDEs, parametrized by t ∈ [0, T ], accompanying Y n (i.e.Ỹn (t, t) = Y n (t)), constructed in a similar way to the one described in the proof of Proposition 6, defined, for each fixed t ∈ [0, T ], by For n, m ≥ 1, set δ Ỹ (t, s) := Ỹn (t, s) − Ỹm (t, s), δZ(t, u) := For any θ > ] Therefore, with Y n (t) = Ỹn (t, t), we have ] .
Moreover, in view of the estimate in Proposition 3.6 of [18], there exists a constant C independent of n and m such that Therefore, ] .

Application to time-inconsistent optimal stopping problems
Let (X (t), t ∈ [0, T ]) be an F-adapted process.In many financial applications X may model the price of a commodity.Suppose that (f , L, ξ ) satisfies the assumptions (A1), (A2) and (A3) of Section 2. We propose to solve the following optimal stopping problem associated with (f , ξ , L): for where the supremum is taken over F-stopping times τ taking values in [t, T ].More precisely, we would like to find an F-stopping time τ * t , indexed by t, such that be the value-function associated to the optimal stopping problem (20).
In financial applications, the functional J represents the yield of an investment in a commodity with price process X , where f is the utility rate per unit time, L is the utility function at the stopping time τ and ξ is the utility at the final time T .
The dependence on t in Eq. ( 21) implies that the optimal stopping problem is in general time-inconsistent which is similar to time-inconsistency in optimal stochastic control (see e.g.[17,22]).
In time-consistent optimal stopping problems (i.e. when f and ξ do not depend of t), the process Y would be the value function of the stopping problem (20).In particular Y (0) = sup τ ≥0 J(0, τ ).This is not the case here.However, we have sup Now, if we can find an F-stopping time τ * t such that In view of Proposition 6, there exists a unique process (Z , K ) ∈ provided that (f , ξ , L) satisfies (A1), (A2) and (A3).

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

2 × L 2 ×
yields that Φ is a contraction mapping on E([T − δ, T ]) and thus admits a unique fixed point which yields the unique solution of the reflected BSVIE (1) associated with (f , ξ , L) over [T − δ, T ]. □ We are now ready to give a proof of Theorem 4. Proof.By repeatedly applying the fixed point argument of Proposition 8 over adjacent time intervals of fixed length δ, satisfying δ 2 + δ < 1/8c f , and pasting the solutions that we describe below, we finally obtain existence of a unique solution in H K 2 to the reflected BSVIE (1) over the whole time interval [0, T ].