Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison

Abstract

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance. . This reprint differs from the original in pagination and typographic detail. 1 2 S. CRÉPEY AND A. MATOUSSI bonds in finance (see Section 6 and [5,6,8]). In this case, the state process (first component) Y of a solution to a related R2BSDE may be interpreted in terms of an arbitrage price process for the bond. As demonstrated in [7], the mere existence of a solution to the related R2BSDE is a result with important theoretical consequences in terms of pricing and hedging the bond. Yet, in order to further develop these results in Markovian set-ups, we also need the R2BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and we also need a suitable comparison theorem. Now, as opposed to the situation prevailing for RBSDEs (see, e.g., El Karoui et al. [17]), universal a priori estimates cannot be obtained for R2BSDEs. In order to get estimates for R2BSDEs, one needs to specialize the problem somewhat. Likewise, universal comparison theorems do not hold in models with jumps (see [2] for a counterexample in the simple case of a BSDE without barriers).Section 2 presents an abstract set-up in which our results are derived, as well as the BSDEs under consideration (Section 2.1). In Sections 3 and 4, we establish the a priori bound and error estimates (Theorem 3.2) and our comparison theorem (Theorem 4.2). The a priori error estimates immediately imply uniqueness of a solution to our problems (Section 5.1). Assuming an additional martingale representation property and the quasi-left-continuity of the barriers, we then give existence results (Section 5.2). In Section 6, we show that all of the required assumptions are satisfied in the case of the convertible-bonds-related R2BSDEs, in a rather generic Markovian specification of our abstract set-up. These R2BSDEs thus admit (unique) solutions.These results can be used to develop a related variational inequality approach in the Markovian case (see [10,11,12]).2. Set-up. Throughout the paper we work with a finite time horizon T > 0, a probability space (Ω, F, P) and a filtration F = (F t ) t∈[0,T ] , with F T = F, satisfying the usual conditions of right-conti...

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“…• the state process Θ is a real-valued, F-adapted, càdlàg process, For various specifications of the present set-up and sets of technical assumptions ensuring the existence and uniqueness of a solution to (E), we refer the reader to [10,16,5,11]. Basically, for the data in suitable spaces of square-integrable processes and random variables, and allowing for jumps of L and U (at totally inaccessible F -stopping times, see Recall that a quasimartingale Y can equivalently be defined as a difference of two non-negative supermartingales, or in terms of a bound on some conditionally expected variations of Y on arbitrary partitions of [0, T ], or as a special semimartingale with predictable finite variation component of integrable variation (Protter [24, Chapter III, section 4]).…”

“…Basically, for the data in suitable spaces of square-integrable processes and random variables, and allowing for jumps of L and U (at totally inaccessible F -stopping times, see Recall that a quasimartingale Y can equivalently be defined as a difference of two non-negative supermartingales, or in terms of a bound on some conditionally expected variations of Y on arbitrary partitions of [0, T ], or as a special semimartingale with predictable finite variation component of integrable variation (Protter [24, Chapter III, section 4]). In particular, any square integrable Itô-Lévy process S (Itô-Lévy process with square integrable special semimartingale decomposition components), or S ∨ for any such process S and constant , is a quasimartingale (see Crépey and Matoussi [10]). Hence the Mokobodski condition is satisfied, and the existence of a solution for (E) holds, whenever L and/or U is given by S or S ∨ for such an Itô-Lévy process S, as it is the case in many practical applications (see [10,5]).…”

“…In particular, any square integrable Itô-Lévy process S (Itô-Lévy process with square integrable special semimartingale decomposition components), or S ∨ for any such process S and constant , is a quasimartingale (see Crépey and Matoussi [10]). Hence the Mokobodski condition is satisfied, and the existence of a solution for (E) holds, whenever L and/or U is given by S or S ∨ for such an Itô-Lévy process S, as it is the case in many practical applications (see [10,5]). …”