CREDIT RISK PREMIA AND QUADRATIC BSDEs WITH A SINGLE JUMP

Abstract

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is devoted to fill that gap. By using a simple constructive algorithm, and known results on continuous quadratic BSDEs, we provide sufficient conditions for the existence and uniqueness of quadratic BSDEs with discontinuities at random times.

Lp solutions of backward stochastic differential equations with jumps

“…Since x− y, D(y) = 0, it holds for any α ∈ R that x − αD(y) 2 Lemma A.5. Let E be a vector space with norm · only.…”

“…In this subsection, we will generalize such kind of stochastic integral for random fields in ∪ p∈ [1,2) U p loc in spirit of [29,VIII.75].…”

“…As to BSDEs driven by other discontinuous random sources, Xia [72] and Bandini [6] studied BSDEs driven by a random measure; Confortola et al [25,26] considered BSDEs driven by a marked point process; [61,5,66,36] analyzed BSDEs driven by Lévy processes; [2,68,46] discussed BSDEs driven by a process with a finite number of marked jumps.…”

Existence and uniqueness results for BSDE with jumps: the whole nine yards

“…Some attention has also been given to the so-called stochastic Lipschitz case, where the generator is Lipschitz continuous in (y, z) but with constants which are actually random processes themselves. There are few papers going in this direction, among which we can Blanchet-Scalliet and Eyraud-Loisel [1], Lim and Quenez [109], Jenablanc, Matoussi and Ngoupeyou [89], Kharroubi, Quenez and Sulem [127], Lim and Ngoupeyou [96], Kharroubi and Lim [95], Laeven and Stadje [105], Richter [130], Jeanblanc, Mastrolia, Possamaï and Réveillac [88], Kazi-Tani, Possamaï and Zhou [93,94], Fujii and Takahashi [71], Dumitrescu, Quenez and Sulem [58], and El Karoui, Matoussi and Ngoupeyou [61], while the specific case of Lévy processes was treated by Nualart and Schoutens [123] and later Bahlali, Eddahbi and Essaky [5]. A general presentation has been proposed recently by Kruse and Popier [102,103], to which we refer for more references (see also the recent paper of Yao [140]).…”

“…Moreover, all authors gratefully acknowledge the financial support from the PROCOPE project "Financial markets in transition: mathematical models and challenges". 1 The authors are indebted to Saïd Hamadène for pointing out this reference. The published version of [51] states that the article was received on October 27, 1971.…”