Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 695798, 33 pages
doi:10.1155/2009/695798
Research Article

Defaultable Game Options in a Hazard Process Model

Tomasz R. Bielecki,1 Stéphane Crépey,2 Monique Jeanblanc,2,3 and Marek Rutkowski4,5

1Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
2Département de Mathématiques, Université d'Évry Val d'Essonne, 91025 Évry Cedex, France
3Europlace Institute of Finance, Palais Brongniart-28 Place de la Bourse, 75002 Paris, France
4School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
5Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland

Received 22 October 2008; Accepted 4 April 2009

Academic Editor: Salah-Eldin Mohammed

Copyright © 2009 Tomasz R. Bielecki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The valuation and hedging of defaultable game options is studied in a hazard process model of credit risk. A convenient pricing formula with respect to a reference filteration is derived. A connection of arbitrage prices with a suitable notion of hedging is obtained. The main result shows that the arbitrage prices are the minimal superhedging prices with sigma martingale cost under a risk neutral measure.

1. Introduction

The goal of this work is to analyze valuation and hedging of defaultable contracts with game option features within a hazard process model of credit risk. Our motivation for considering American or game clauses together with defaultable features of an option is not that much a quest for generality, but rather the fact that the combination of early exercise features and defaultability is an intrinsic feature of some actively traded assets. It suffices to mention here the important class of convertible bonds, which were studied by, among others, Andersen and Buffum [1], Ayache et al. [2], Bielecki et al. [3, 4], Davis and Lischka [5], Kallsen and Kühn [6], and Kwok and Lau [7].

In Bielecki et al. [3], we formally defined a defaultable game option, that is, a financial contract that can be seen as an intermediate case between a general mathematical concept of a game option and much more specific convertible bond with credit risk. We concentrated there on developing a fairly general framework for valuing such contracts. In particular, building on results of Kifer [8] and Kallsen and Kühn [6], we showed that the study of an arbitrage price of a defaultable game option can be reduced to the study of the value process of the related Dynkin game under some risk-neutral measure ℚ for the primary market model. In this stochastic game, the issuer of a game option plays the role of the minimizer and the holder of the maximizer. In [3], we dealt with a general market model, which was assumed to be arbitrage-free, but not necessarily complete, so that the uniqueness of a risk-neutral (or martingale) measure was not postulated. In addition, although the default time was introduced, it was left largely unspecified. An explicit specification of the default time will be an important component of the model considered in this work.

As is well known, there are two main approaches to modeling of default risk: the structural approach and the reduced-form approach. In the latter approach, also known as the hazard process approach, the default time is modeled as an exogenous random variable with no reference to any particular economic background. One may object to reduced-form models for their lack of clear reference to economic fundamentals, such as the firm's asset-to-debt ratio. However, the possibility of choosing various parameterizations for the coefficients and calibrating these parameters to any set of CDS spreads and/or implied volatilities makes them very versatile modeling tools, well suited to price and hedge derivatives consistently with plain-vanilla instruments. It should be acknowledged that structural models, with their sound economic background, are better suited for inference of reliable debt information, such as: risk-neutral default probabilities or the present value of the firm's debt, from the equities, which are the most liquid among all financial instruments. The structure of these models, as rich as it may be (and which can include a list of factors such as stock, spreads, default status, and credit events) never rich enough to yield consistent prices for a full set of CDS spreads and/or implied volatilities of related options. As we ultimately aim to specify models for pricing and hedging contracts with optional features (such as convertible bonds), we favor the reduced-form approach in the sequel.

1.1. Outline of the Paper

From the mathematical perspective, the goal of the present paper is twofold. First, we wish to specialize our previous valuation results to the hazard process setup, that is, to a version of the reduced-form approach, which is slightly more general than the intensity-based setup. Hence we postulate that filtration 𝔾 modeling the information flow for the primary market admits the representation 𝔾 = ℍ ∨ 𝔽 , where the filtration ℍ is generated by the default indicator process 𝐻 𝑡 = 1 { 𝑡 ≥ 𝜏 𝑑 } and 𝔽 is some reference filtration. The main tool employed in this section is the effective reduction of the information flow from the full filtration 𝔾 to the reference filtration 𝔽 . The main results in this part are Theorems 3.7 and 3.8, which give convenient pricing formulae with respect to the reference filtration 𝔽 .

The second goal is to study the issue of hedging of a defaultable game option in the hazard process setup. Some previous attempts to analyze hedging strategies for defaultable convertible bonds were done by Andersen and Buffum [1] and Ayache et al. [2], who worked directly with suitable variational inequalities within the Markovian intensity-based setup.

Our preliminary results for hedging strategies in a hazard process setup, Propositions 4.1 and 4.3, can be informally stated as follows: under the assumption that a related doubly reflected BSDE admits a solution ( Θ , 𝑀 , 𝐾 ) under some risk-neutral measure ℚ , for which various sets of sufficient conditions are given in literature, the state-process Θ of the solution is the minimal (pre-default) super-hedging price up to a ( 𝔾 , ℚ ) -sigma (or local) martingale cost process. More specific properties of hedging strategies are subsequently analyzed in Propositions 4.13 and 4.15, in which we resort to suitable Galtchouk-Kunita-Watanabe decompositions of a solution to the related doubly reflected BSDE and discounted prices of primary assets with respect to various risk factors corresponding to systematic, idiosyncratic and event risks. It is noteworthy that these decompositions, though seemingly rather abstract in a general setup considered here, are by no means artificial. On the contrary, they arise naturally in the context of particular Markovian models that are studied in the followup paper by Bielecki et al. [4, 9]. We conclude the paper by briefly commenting on some alternative approaches to hedging of defaultable game options.

1.2. Conventions and Standing Notation

Throughout this paper, we use the concept of the vector stochastic integral, denoted as ∫ 𝐻 𝑑 𝑋 , as opposed to a more restricted notion of the component-wise stochastic integral, which is defined as the sum ∑ 𝑑 𝑖 = 1 ∫ 𝐻 𝑖 𝑑 𝑋 𝑖 of integrals with respect to one-dimensional integrators 𝑋 𝑖 . For a detailed exposition of the vector stochastic integration, we refer to Shiryaev and Cherny [10] (see also Chatelain and Stricker [11] and Jacod [12]). Given a stochastic basis satisfying the usual conditions, an ℝ 𝑑 -valued semimartingale integrator 𝑋 and an ℝ 1 ⊗ 𝑑 -valued (row vector) predictable integrand 𝐻 , the notion of the vector stochastic integral ∫ 𝐻 𝑑 𝑋 allows one to take into account possible “interferences’’ of local martingale and finite variation components of a (scalar) integrator process, or of different components of a multidimensional integrator process. Well-defined vector stochastic integrals include, in particular, all integrals with a predictable and locally bounded integrand (e.g., any integrand of the form 𝐻 = 𝑌 − where 𝑌 is an adapted càdlàg process, see He et al. [13, Theorem  7.7]). The usual properties of stochastic integral, such as: linearity, associativity, invariance with respect to equivalent changes of measures and with respect to inclusive changes of filtrations, are known to hold for the vector stochastic integral. Moreover, unlike other kinds of stochastic integrals, vector stochastic integrals form a closed space in a suitable topology. This feature makes them well adapted to many problems arising in the mathematical finance, such as Fundamental Theorems of Asset Pricing (see, e.g., Delbaen and Schachermayer [14] or Shiryaev and Cherny [10]).

By default, we denote by ∫ 𝑡 0 the integrals over ( 0 , 𝑡 ] . Otherwise, we explicitly specify the domain of integration as a subscript of ∫ . Note also that, depending on the context, 𝜏 will stand either for a generic stopping time or it will be given as 𝜏 = 𝜏 𝑝 ⋀ 𝜏 𝑐 for some specific stopping times 𝜏 𝑐 and 𝜏 𝑝 . Finally, we consider the right-continuous and completed versions of all filtrations, so that they satisfy the so-called “usual conditions.’’

2. Semimartingale Setup

After recalling some fundamental valuation results from [3], we will examine basic features of hedging strategies for defaultable game options that are valid in a general semimartingale setup. The important special case of a hazard process framework is studied in the next section.

We assume throughout that the evolution of the underlying primary market is modeled in terms of stochastic processes defined on a filtered probability space ( Ω , 𝔾 , ℙ ) , where ℙ denotes the statistical probability measure.

Specifically, we consider a primary market composed of the savings account and of 𝑑 risky assets, such that, given a finite horizon date 𝑇 > 0 :

(i)the discount factor process 𝛽 , that is, the inverse of the savings account, is a 𝔾 -adapted, finite variation, positive, continuous and bounded process,(ii) the risky assets are 𝔾 -semimartingales with càdlàg sample paths.

The primary risky assets, with ℝ 𝑑 -valued price process 𝑋 , pay dividends, whose cumulative value process, denoted by 𝒟 , is assumed to be a 𝔾 -adapted, càdlàg and ℝ 𝑑 -valued process of finite variation. Given the price process 𝑋 , we define the cumulative price 𝑋 of primary risky assets as 𝑋 𝑡 = 𝑋 𝑡 + 𝛽 𝑡 − 1 [ 0 , 𝑡 ] 𝛽 𝑢 𝑑 𝒟 𝑢 . ( 2 . 1 ) In the financial interpretation, the last term in (2.1) represents the current value at time 𝑡 of all dividend payments from the assets over the period [ 0 , 𝑡 ] , under the assumption that all dividends are immediately reinvested in the savings account. We assume that the primary market model is free of arbitrage opportunities, though presumably incomplete. In view of the First Fundamental Theorem of Asset Pricing (cf. [10, 14]), and accounting in particular for the dividends, this means that there exists a risk-neutral measure ℚ ∈ ℳ , where ℳ denotes the set of probability measures ℚ ∼ ℙ for which 𝛽 𝑋 is a sigma martingale with respect to 𝔾 under ℚ (for the definition of a sigma martingale, see [10, Definition  1.9]). The following well-known properties of sigma martingales will be used in the sequel.

Proposition 2.1 (see [10, 15, 16]). (i) The class of sigma martingales is a vector space containing all local martingales. It is stable with respect to (vector) stochastic integration, that is, if 𝑌 is a sigma martingale and 𝐻 is a (predictable) 𝑌 -integrable process then the (vector) stochastic integral ∫ 𝐻 𝑑 𝑌 is a sigma martingale.
(ii) Any locally bounded sigma martingale is a local martingale, and any bounded from below sigma martingale is a supermartingale.

Remark 2.2. In the same vein, we recall that stochastic integration of predictable, locally bounded integrands preserves local martingales (see, e.g., Protter [16]).

We now introduce the concept of a dividend paying game option (see also Kifer [8]). In broad terms, a dividend paying game option, with the inception date 0 and the maturity date 𝑇 , is a contract with the following cash flows that are paid by the issuer of the contract and received by its holder:

(i)a dividend stream with the cumulative dividend at time 𝑡 denoted by 𝐷 𝑡 , (ii)a terminal put payment 𝐿 𝑡 made at time 𝑡 = 𝜏 𝑝 if 𝜏 𝑝 ≤ 𝜏 𝑐 and 𝜏 𝑝 < 𝑇 ; time 𝜏 𝑝 is called the put time and is chosen by the holder, (iii)a terminal call payment 𝑈 𝑡 made at time 𝑡 = 𝜏 𝑐 provided that 𝜏 𝑐 < 𝜏 𝑝 ⋀ 𝑇 ; time 𝜏 𝑐 , known as the call time, is chosen by the issuer and may be subject to the constraint that 𝜏 𝑐 ≥ 𝜏 , where 𝜏 is the lifting time of the call protection, (iv)a terminal payment at maturity 𝜉 made at maturity date 𝑇 provided that 𝑇 ≤ 𝜏 𝑝 ⋀ 𝜏 𝑐 .

The (possibly random) time 𝜏 in (iii) is used to model the restriction that the issuer of a game option may be prevented from making a call on some random time interval [ 0 , 𝜏 ) .

Of course, there is also the initial cash flow, namely, the purchasing price of the contract, which is paid at the initiation time by the holder and received by the issuer.

Let us now be given an [ 0 , + ∞ ] -valued 𝔾 -stopping time 𝜏 𝑑 representing the default time of a reference entity, with default indicator process 𝐻 𝑡 = 1 { 𝜏 𝑑 ≤ 𝑡 } . A defaultable dividend paying game option is a dividend paying game option such that the contract is terminated at 𝜏 𝑑 , if it has not been put or called and has not matured before. In particular, there are no more cash flows related to this contract after the default time. In this setting, the dividend stream 𝐷 is assumed to include a possible recovery payment made at the default time.

We are interested in the problem of the time evolution of an arbitrage price of the game option. Therefore, we formulate the problem in a dynamic way by pricing the game option at any time 𝑡 ∈ [ 0 , 𝑇 ] . We write 𝒢 𝑡 𝑇 to denote the set of all 𝔾 -stopping times with values in [ 𝑡 , 𝑇 ] and we let 𝒢 𝑡 𝑇 stand for the set { 𝜏 ∈ 𝒢 𝑡 𝑇 ⋀ 𝜏 ; 𝜏 𝑑 ≥ 𝜏 ⋀ 𝜏 𝑑 } , where the lifting time of a call protection 𝜏 belongs to 𝒢 0 𝑇 .

We are now in the position to state the formal definition of a defaultable game option.

Definition 2.3. A defaultable game option with lifting time of the call protection 𝜏 ∈ 𝒢 0 𝑇 is a game option with the ex-dividend cumulative discounted cash flows 𝛽 𝑡 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) given by the formula, for any 𝑡 ∈ [ 0 , 𝑇 ] and ( 𝜏 𝑝 , 𝜏 𝑐 ) ∈ 𝒢 𝑡 𝑇 × 𝒢 𝑡 𝑇 , 𝛽 𝑡 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 = 𝜏 𝑡 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 { 𝜏 < 𝜏 𝑑 } 𝛽 𝜏 1 { 𝜏 = 𝜏 𝑝 < 𝑇 } 𝐿 𝜏 𝑝 + 1 { 𝜏 < 𝜏 𝑝 } 𝑈 𝜏 𝑐 + 1 { 𝜏 = 𝑇 } 𝜉 , ( 2 . 2 ) where 𝜏 = 𝜏 𝑝 ⋀ 𝜏 𝑐 and (i)the dividend process 𝐷 = ( 𝐷 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] equals 𝐷 𝑡 = [ ] 0 , 𝑡 1 − 𝐻 𝑢 𝑑 𝐶 𝑢 + [ ] 0 , 𝑡 𝑅 𝑢 𝑑 𝐻 𝑢 = 𝐶 𝜏 − 1 { 𝑡 ≥ 𝜏 } + 𝐶 𝑡 1 { 𝑡 < 𝜏 } + 𝑅 𝜏 1 { 𝑡 ≥ 𝜏 } , ( 2 . 3 ) for some coupon process 𝐶 = ( 𝐶 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] , which is a 𝔾 -predictable, real-valued, càdlàg process with bounded variation, and some real-valued, 𝔾 -predictable recovery process 𝑅 = ( 𝑅 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] , (ii)the put payment 𝐿 = ( 𝐿 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] and the call payment 𝑈 = ( 𝑈 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] are 𝔾 -adapted, real-valued, càdlàg processes, (iii)the inequality 𝐿 𝑡 ≤ 𝑈 𝑡 holds for every 𝑡 ∈ [ 𝜏 𝑑 ⋀ 𝜏 , 𝜏 𝑑 ⋀ 𝑇 ) , (iv)the payment at maturity 𝜉 is a 𝒢 𝑇 -measurable, real-valued random variable.

The following result easily follows from Definition 2.3.

Lemma 2.4. (i) For any 𝑡 and ( 𝜏 𝑝 , 𝜏 𝑐 ) ∈ 𝒢 𝑡 𝑇 × 𝒢 𝑡 𝑇 , the random variable 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) is 𝒢 𝜏 ⋀ 𝜏 𝑑 -measurable.
(ii) For any ( 𝜏 𝑝 , 𝜏 𝑐 ) ∈ 𝒢 0 𝑇 × 𝒢 0 𝑇 , the processes 𝜋 ( 0 ; ⋅ , 𝜏 𝑐 ) and 𝜋 ( 0 ; 𝜏 𝑝 , ⋅ ) are 𝔾 -adapted.

We further assume that 𝑅 , 𝐿 , and 𝜉 are bounded from below, so that there exists a constant 𝑐 such that, for every 𝑡 ∈ [ 0 , 𝑇 ] , 𝛽 𝑡 ℒ 𝑡 ∶ = [ 0 , 𝑡 ] 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 { 𝑡 < 𝜏 𝑑 } 𝛽 𝑡 1 { 𝑡 < 𝑇 } 𝐿 𝑡 + 1 { 𝑡 = 𝑇 } 𝜉 ≥ − 𝑐 . ( 2 . 4 )

Symmetrically, we should sometimes additionally assume that 𝑅 , 𝑈 , and 𝜉 are bounded (from below and from above), or that (2.4) is supplemented by the inequality, for every 𝑡 ∈ [ 0 , 𝑇 ] , 𝛽 𝑡 𝒰 𝑡 ∶ = [ 0 , 𝑡 ] 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 { 𝑡 < 𝜏 𝑑 } 𝛽 𝑡 1 { 𝑡 < 𝑇 } 𝑈 𝑡 + 1 { 𝑡 = 𝑇 } 𝜉 ≤ 𝑐 . ( 2 . 5 )

2.1. Valuation of a Defaultable Game Option

We will state the following fundamental pricing result without proof, referring the interested reader to [3, Proposition  3.1 and Theorem  4.1] for more details. The goal is to characterize the set of arbitrage ex-dividend prices of a game option in terms of values of related Dynkin games (for the general theory of Dynkin games, see, e.g., Dynkin [17], Kifer [18], and Lepeltier and Maingueneau [19]). The notion of an arbitrage price of a game option referred to in Theorem 2.5 is the dynamic notion of arbitrage price for game options, as defined in Kallsen and Kühn [6], and extended in [3] to the case of dividend-paying primary assets and dividend-paying game options by resorting to the transformation of prices into cumulative prices. Note that in the sequel, the statement “ ( Π 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] is an arbitrage price for the game option’’ is in fact to be understood as “ ( 𝑋 𝑡 , Π 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ]  is an arbitrage price for the extended market consisting of the primary market and the game option.’’

Theorem 2.5 (Arbitrage price of a defaultable game option). Assume that a process Π is a 𝔾 -semimartingale and there exists ℚ ∈ ℳ such that Π is the value of the Dynkin game related to a game option, meaning that e s s s u p 𝜏 𝑝 ∈ 𝒢 𝑡 𝑇 e s s i n f 𝜏 𝑐 ∈ 𝒢 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 = Π 𝑡 = e s s i n f 𝜏 𝑐 ∈ 𝒢 𝑡 𝑇 e s s s u p 𝜏 𝑝 ∈ 𝒢 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 [ ] . , 𝑡 ∈ 0 , 𝑇 ( 2 . 6 ) Then Π is an arbitrage ex-dividend price of the game option, called the ℚ -price of the game option. The converse holds true (thus any arbitrage price is a ℚ -price for some ℚ ∈ ℳ ) under the following integrability assumption e s s s u p ℚ ∈ ℳ 𝔼 ℚ s u p 𝑡 ∈ [ 0 , 𝑇 ] [ 0 , 𝑡 ] 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 { 𝑡 < 𝜏 𝑑 } 𝛽 𝑡 1 { 𝑡 < 𝑇 } 𝐿 𝑡 + 1 { 𝑡 = 𝑇 } 𝜉 ∣ 𝒢 0 < ∞ , a . s . ( 2 . 7 )

Note that defaultable American (or European) options can be seen as special cases of defaultable game options.

Definition 2.6. A defaultable American option is a defaultable game option with 𝜏 = 𝑇 . A defaultable European option is a defaultable American option such that the process 𝛽 ℒ (cf. (2.4) attains its maximum at 𝑇 , that is, 𝛽 𝑡 ℒ 𝑡 ≤ 𝛽 𝑇 ℒ 𝑇 for every 𝑡 ∈ [ 0 , 𝑇 ] .

In view of Theorem 2.5, the cash flows 𝜙 ( 𝑡 ) of a defaultable European option can be redefined by 𝛽 𝑡 𝜙 ( 𝑡 ) = 𝑇 𝑡 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 { 𝜏 𝑑 > 𝑇 } 𝛽 𝑇 [ ] 𝜉 , 𝑡 ∈ 0 , 𝑇 . ( 2 . 8 )

2.2. Hedging of a Defaultable Game Option

We adopt the definition of hedging game options stemming from successive developments, starting from the hedging of American options examined by Karatzas [20], and subsequently followed by El Karoui and Quenez [21], Kifer [8], Ma and Cvitanić [22], and Hamadène [23]. One of our goals is to show that this definition is consistent with the concept of arbitrage valuation of a defaultable game option in the sense of Kallsen and Kühn [6].

Recall that 𝑋 (resp., 𝑋 ) is the price process (resp., cumulative price process) of primary traded assets, as given by (2.1). The following definitions are standard, accounting for the dividends on the primary market.

Definition 2.7. By a (self-financing) primary trading strategy we mean any pair ( 𝑉 0 , 𝜁 ) such that (i) 𝑉 0 is a 𝒢 0 -measurable real-valued random variable representing the initial wealth,(ii) 𝜁 is an ℝ 1 ⊗ 𝑑 -valued, 𝛽 𝑋 -integrable process representing holdings in primary risky assets.

Remark 2.8. The reason why we do not assume that 𝒢 0 is trivial (which would, of course, simplify several statements) is that we apply our results in the subsequent work [4] to situations, where this assumption fails to hold (e.g., when studying convertible bonds with a positive call notice period).

Definition 2.9. The wealth process 𝑉 of a primary trading strategy ( 𝑉 0 , 𝜁 ) is given by the formula, for 𝑡 ∈ [ 0 , 𝑇 ] , 𝛽 𝑡 𝑉 𝑡 = 𝛽 0 𝑉 0 + 𝑡 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 . ( 2 . 9 )

Given the wealth process 𝑉 of a primary strategy ( 𝑉 0 , 𝜁 ) , we uniquely specify a 𝔾 -optional process 𝜁 0 by setting 𝑉 𝑡 = 𝜁 0 𝑡 𝛽 𝑡 − 1 + 𝜁 𝑡 𝑋 𝑡 [ ] , 𝑡 ∈ 0 , 𝑇 . ( 2 . 1 0 ) The process 𝜁 0 represents the number of units held in the savings account at time 𝑡 , when we start from the initial wealth 𝑉 0 and we use the strategy 𝜁 in the primary risky assets. Recall that we denote 𝜏 = 𝜏 𝑝 ⋀ 𝜏 𝑐 .

Definition 2.10. Consider the game option with the ex-dividend cumulative discounted cash flows 𝛽 𝜋 given by (2.2).
(i)An issuer hedge with cost process 𝜌 is represented by a quadruplet ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑐 ) such that(a) ( 𝑉 0 , 𝜁 ) is a primary strategy with the wealth process 𝑉 given by (2.9),(b)a cost process 𝜌 is a real-valued, càdlàg 𝔾 -semimartingale with 𝜌 0 = 0 ,(c)a (fixed) call time 𝜏 𝑐 belongs to 𝒢 0 𝑇 ,(d)the following inequality is valid, for every put time 𝜏 𝑝 ∈ 𝒢 0 𝑇 , 𝛽 𝜏 𝑉 𝜏 + 𝜏 0 𝛽 𝑢 𝑑 𝜌 𝑢 ≥ 𝛽 0 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 , a . s . ( 2 . 1 1 ) (ii)A holder hedge with cost process 𝜌 is a quadruplet ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑝 ) such that(a) ( 𝑉 0 , 𝜁 ) is a primary strategy with the wealth process 𝑉 given by (2.9),(b)a cost process 𝜌 is a real-valued, càdlàg 𝔾 -semimartingale with 𝜌 0 = 0 ,(c)a (fixed) put time 𝜏 𝑝 belongs to 𝒢 0 𝑇 ,(d)the following inequality is valid, for every call time 𝜏 𝑐 ∈ 𝒢 0 𝑇 , 𝛽 𝜏 𝑉 𝜏 + 𝜏 0 𝛽 𝑢 𝑑 𝜌 𝑢 ≥ − 𝛽 0 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 , a . s . ( 2 . 1 2 )

Issuer or holder hedges at no cost (i.e., with 𝜌 = 0 ) are thus in effect issuer or holder superhedges. A more explicit form of condition (2.11) reads (for (2.12), we need to insert the minus sign in the right-hand side of (2.13) 𝑉 𝜏 + 𝛽 𝜏 − 1 𝜏 0 𝛽 𝑢 𝑑 𝜌 𝑢 ≥ 𝛽 𝜏 − 1 𝜏 0 𝛽 𝑢 𝑑 𝐷 𝑢 + 1 𝜏 < 𝜏 𝑑 1 𝜏 = 𝜏 𝑝 < 𝑇 𝐿 𝜏 𝑝 + 1 𝜏 < 𝜏 𝑝 𝑈 𝜏 𝑐 + 1 𝜏 𝑝 = 𝜏 𝑐 = 𝑇 𝜉 , a . s . ( 2 . 1 3 ) The left-hand side in (2.13) is the value at time 𝜏 of a strategy with a cost process 𝜌 , when the players adopt their respective exercise policies 𝜏 𝑝 and 𝜏 𝑐 , whereas the right-hand side represents the payoff to be done by the issuer, including past dividends and the recovery at default.

Remark 2.11. (i) The process 𝜌 is to be interpreted as the (running) financing cost, that is, the amount of cash added to (if 𝑑 𝜌 𝑡 ≥ 0 ) or withdrawn from (if 𝑑 𝜌 𝑡 ≤ 0 ) the hedging portfolio in order to get a perfect, but no longer self-financing, hedge. In the special case where 𝜌 is a 𝔾 -martingale under ℚ we thus recover the notion of mean self-financing hedge, in the sense of Schweizer [24].
(ii) Regarding the admissibility of hedging strategies (see, e.g., Delbaen and Schachermayer [14]), note that the left-hand side in formula (2.11) (discounted wealth process inclusive of financing costs) is bounded from below for any issuer hedge with a cost ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑐 ) . Likewise, in the case of a bounded payoff 𝜋 (i.e., assuming (2.5), the left-hand side in formula (2.12) is bounded from below for any holder hedge with a cost ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑝 ) .

Obviously, the class of all hedges with semimartingale cost processes is too large for any practical purposes. Therefore, we will restrict our attention to hedges with a 𝔾 -sigma martingale cost 𝜌 under a particular risk-neutral measure ℚ .

Assumption 2.12. In the sequel, we work under a fixed, but arbitrary, risk-neutral measure ℚ ∈ ℳ .

All the measure-dependent notions like (local) martingale and compensator, implicitly refer to the probability measure ℚ . In particular, we define 𝒱 𝑐 0 (resp., 𝒱 𝑝 0 ) as the set of initial values 𝑉 0 for which there exists an issuer (resp., holder) hedge of the game option with the initial value 𝑉 0 (resp., − 𝑉 0 ) and with a 𝔾 -sigma martingale cost under ℚ .

The following result gives some preliminary conclusions regarding the initial cost of a hedging strategy for the game option under the present, rather weak, assumptions. In Proposition 4.3, we will see that, under stronger assumptions, the infima are attained and thus we obtain equalities, rather than merely inequalities, in (2.14) and (2.15).

Lemma 2.13. (i) One has (by convention, e s s i n f ∅ = ∞ ) e s s i n f 𝜏 𝑐 ∈ 𝒢 0 𝑇 e s s s u p 𝜏 𝑝 ∈ 𝒢 0 𝑇 𝔼 ℚ 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 ≤ e s s i n f 𝑉 0 ∈ 𝒱 𝑐 0 𝑉 0 , a . s . ( 2 . 1 4 )
(ii) If inequality (2.5) is valid then e s s s u p 𝜏 𝑝 ∈ 𝒢 0 𝑇 e s s i n f 𝜏 𝑐 ∈ 𝒢 0 𝑇 𝔼 ℚ 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 ≥ − e s s i n f 𝑉 0 ∈ 𝒱 𝑝 0 𝑉 0 , a . s . ( 2 . 1 5 )

Proof. (i) Assume that for some stopping time 𝜏 𝑐 ∈ 𝒢 0 𝑇 the quadruplet ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑐 ) is an issuer hedge with a 𝔾 -sigma martingale cost 𝜌 for the game option. It is easily seen from (2.9) and (2.11) that, for any stopping time 𝜏 𝑝 ∈ 𝒢 0 𝑇 , 𝛽 0 𝑉 0 = 𝛽 𝜏 𝑝 ∧ 𝜏 𝑐 𝑉 𝜏 𝑝 ∧ 𝜏 𝑐 − 𝜏 𝑝 ∧ 𝜏 𝑐 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 ≥ 𝛽 0 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 − 𝜏 𝑝 ∧ 𝜏 𝑐 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 + 𝛽 𝑢 𝑑 𝜌 𝑢 . ( 2 . 1 6 ) In particular, by taking 𝜏 𝑝 = 𝑡 , we obtain that, for any 𝑡 ∈ [ 0 , 𝑇 ] , 𝛽 0 𝑉 0 = 𝛽 𝑡 ∧ 𝜏 𝑐 𝑉 𝑡 ∧ 𝜏 𝑐 − 𝑡 ∧ 𝜏 𝑐 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 ≥ 𝛽 0 𝜋 0 ; 𝑡 , 𝜏 𝑐 − 𝑡 ∧ 𝜏 𝑐 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 + 𝛽 𝑢 𝑑 𝜌 𝑢 . ( 2 . 1 7 ) The stochastic integral ∫ 𝑡 0 𝜁 𝑢 𝑑 ( 𝛽 𝑢 𝑋 𝑢 ) with respect to a 𝔾 -sigma martingale 𝛽 𝑋 is a 𝔾 -sigma martingale. Hence the stopped process ∫ 𝑡 ⋀ 𝜏 𝑐 0 𝜁 𝑢 𝑑 ( 𝛽 𝑢 𝑋 𝑢 ) , as well as the process 𝑡 ∧ 𝜏 𝑐 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 + 𝛽 𝑢 𝑑 𝜌 𝑢 ( 2 . 1 8 ) are 𝔾 -sigma martingales. The latter process is bounded from below (this follows from (2.2)–(2.4) and (2.17), so that it is a bounded from below local martingale [15, page 216] and thus also a supermartingale. By taking conditional expectations in (2.16), we thus obtain for any stopping time 𝜏 𝑝 ∈ 𝒢 0 𝑇 (recall that 𝜏 𝑐 is fixed) 𝛽 0 𝑉 0 ≥ 𝔼 ℚ 𝛽 0 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 , ∀ 𝜏 𝑝 ∈ 𝒢 0 𝑇 , ( 2 . 1 9 ) and thus, by the assumed positivity of the process 𝛽 , 𝑉 0 ≥ e s s i n f 𝜏 𝑐 ∈ 𝒢 0 𝑇 e s s s u p 𝜏 𝑝 ∈ 𝒢 0 𝑇 𝔼 ℚ 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 , a . s . ( 2 . 2 0 ) The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let ( 𝑉 0 , 𝜁 , 𝜌 , 𝜏 𝑝 ) be a holder hedge with a 𝔾 -sigma martingale cost 𝜌 for the game option for some stopping time 𝜏 𝑝 ∈ 𝒢 0 𝑇 . Then (2.9) and (2.12) imply that, for any 𝑡 ∈ [ 𝜏 , 𝑇 ] , 𝛽 0 𝑉 0 = 𝛽 𝑡 ∧ 𝜏 𝑝 𝑉 𝑡 ∧ 𝜏 𝑝 − 𝑡 ∧ 𝜏 𝑝 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 ≥ − 𝛽 0 𝜋 0 ; 𝜏 𝑝 − , 𝑡 𝑡 ∧ 𝜏 𝑝 0 𝜁 𝑢 𝑑 𝛽 𝑢 𝑋 𝑢 + 𝛽 𝑢 𝑑 𝜌 𝑢 . ( 2 . 2 1 ) Under condition (2.5), the stochastic integral in the last formula is bounded from below and thus we conclude, by the same arguments as in part (i) that it is a supermartingale. Consequently, for a fixed stopping time 𝜏 𝑝 ∈ 𝒢 0 𝑇 , we obtain 𝛽 0 𝑉 0 ≥ 𝔼 ℚ − 𝛽 0 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 , a . s . , ∀ 𝜏 𝑐 ∈ 𝒢 0 𝑇 , ( 2 . 2 2 ) so that 𝑉 0 ≥ − e s s s u p 𝜏 𝑝 ∈ 𝒢 0 𝑇 e s s i n f 𝜏 𝑐 ∈ 𝒢 0 𝑇 𝔼 ℚ 𝜋 0 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 0 , a . s . , ( 2 . 2 3 ) and this in turn implies (2.15).

3. Valuation in a Hazard Process Setup

In order to get more explicit pricing and hedging results for defaultable game options, we will now study the so-called hazard process setup.

3.1. Standing Assumptions

Given an [ 0 , + ∞ ] -valued 𝔾 -stopping time 𝜏 𝑑 , we assume that 𝔾 = ℍ ∨ 𝔽 , where the filtration ℍ is generated by the process 𝐻 𝑡 = 1 { 𝜏 𝑑 ≤ 𝑡 } and 𝔽 is some reference filtration. As expected, our approach will consist in effectively reducing the information flow from the full filtration 𝔾 to the reference filtration 𝔽 .

Let 𝐺 stand for the process 𝐺 𝑡 = ℚ ( 𝜏 𝑑 > 𝑡 ∣ ℱ 𝑡 ) for 𝑡 ∈ ℝ + . The process 𝐺 is a (bounded) 𝔽 -supermartingale, as the optional projection on the filtration 𝔽 of the nonincreasing process 1 − 𝐻 (see Jeulin [25]).

In the sequel, we will work under the following standing assumption.

Assumption 3.1. We assume that the process 𝐺 is (strictly) positive and continuous with finite variation, so that the 𝔽 -hazard process Γ 𝑡 = − l n ( 𝐺 𝑡 ) , 𝑡 ∈ ℝ + , is well defined and continuous with finite variation.

Remark 3.2. (i) The assumption that 𝐺 is continuous implies that 𝜏 𝑑 is a totally inaccessible 𝔾 -stopping time (see, e.g., [26]). Moreover, 𝜏 𝑑 avoids 𝔽 -stopping times, in the sense that ℚ ( 𝜏 𝑑 = 𝜏 ) = 0 for any 𝔽 -stopping time 𝜏 (see Coculescu and Nikeghbali [27]).
(ii) If 𝐺 is continuous, the additional assumption that 𝐺 has a finite variation implies in fact that 𝐺 is nonincreasing (see Lemma A.1(i). This lies somewhere between assuming further the (stronger) ( ℋ ) Hypothesis and assuming further that 𝜏 𝑑 is an 𝔽 -pseudo-stopping time (see Nikeghbali and Yor [28]). Recall that the ( ℋ ) Hypothesis means that all local 𝔽 -martingales are local 𝔾 -martingales (see, e.g., [29]), whereas 𝜏 𝑑 is an 𝔽 -pseudo-stopping time whenever all 𝔽 -local martingales stopped at 𝜏 𝑑 are 𝔾 -local martingales (see Nikeghbali and Yor [28] and the appendix).

Some consequences of Assumption 3.1 useful for this work are summarized in Lemma A.1. The next definition refers to some auxiliary results, which are relegated to the appendix.

Definition 3.3. The 𝔽 -stopping time ̃ 𝜏 , the ℱ 𝑡 -measurable random variable 𝜒 and the 𝔽 -adapted or 𝔽 -predictable process 𝑌 introduced in Lemmas A.2 and A.4 are called the 𝔽 -representatives of 𝜏 , 𝜒 and 𝑌 , respectively. In the context of credit risk, where 𝜏 𝑑 represents the default time of a reference entity, they are also known as the pre-default values of 𝜏 , 𝜒 and 𝑌 .

To simplify the presentation, we find it convenient to make additional assumptions. Strictly speaking, these assumptions are superfluous, in the sense that all the results below are true without Assumption 3.4. Indeed, by making use of Lemmas A.2 and A.4 and Definition 3.3, it is always possible to reduce the original problem to the case described in Assumption 3.4. Since this would make the notation heavier, without adding much value, we prefer to work under this standing assumption.

Assumption 3.4. (i) The discount factor process 𝛽 is 𝔽 -adapted.
(ii) The coupon process 𝐶 is 𝔽 -predictable.
(iii) The recovery process 𝑅 is 𝔽 -predictable.
(iv) The payoff processes 𝐿 , 𝑈 are 𝔽 -adapted and the random variable 𝜉 is ℱ 𝑇 -measurable.
(v) The call protection 𝜏 is an 𝔽 -stopping time.

3.2. Reduction of a Filtration

The next lemma shows that the computation of the lower and upper value of the Dynkin games (2.6) with respect to 𝔾 -stopping times can be reduced to the computation of the lower and upper value with respect to 𝔽 -stopping times.

Lemma 3.5. One has that e s s s u p 𝜏 𝑝 ∈ 𝒢 𝑡 𝑇 e s s i n f 𝜏 𝑐 ∈ 𝒢 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 = e s s s u p 𝜏 𝑝 ∈ ℱ 𝑡 𝑇 e s s i n f 𝜏 𝑐 ∈ ℱ 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 , e s s i n f 𝜏 𝑐 ∈ 𝒢 𝑡 𝑇 e s s s u p 𝜏 𝑝 ∈ 𝒢 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 = e s s i n f 𝜏 𝑐 ∈ ℱ 𝑡 𝑇 e s s s u p 𝜏 𝑝 ∈ ℱ 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 . ( 3 . 1 )

Proof. For ( 𝜏 𝑝 , 𝜏 𝑐 ) ∈ 𝒢 𝑡 𝑇 × 𝒢 𝑡 𝑇 , one has that 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 = 𝜋 𝑡 ; 𝜏 𝑝 ∧ 𝜏 𝑑 , 𝜏 𝑐 ∧ 𝜏 𝑑 = 𝜋 𝑡 ; ̃ 𝜏 𝑝 ∧ 𝜏 𝑑 , ̃ 𝜏 𝑐 ∧ 𝜏 𝑑 = 𝜋 𝑡 ; ̃ 𝜏 𝑝 , ̃ 𝜏 𝑐 ( 3 . 2 ) for some stopping times ( ̃ 𝜏 𝑝 , ̃ 𝜏 𝑐 ) ∈ ℱ 𝑡 𝑇 × ℱ 𝑡 𝑇 , where the middle equality follows from Lemma A.4, and the other two from the definition of 𝜋 . Since, clearly, ℱ 𝑡 𝑇 ⊆ 𝒢 𝑡 𝑇 and ℱ 𝑡 𝑇 ⊆ 𝒢 𝑡 𝑇 , we conclude that the lemma is valid.

Under our assumptions, the computation of conditional expectations of cash flows 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) with respect to 𝒢 𝑡 can be reduced to the computation of conditional expectations of 𝔽 -equivalent cash flows 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) with respect to ℱ 𝑡 . Let 𝛼 𝑡 ∶ = 𝛽 𝑡 e x p ( − Γ 𝑡 ) stand for the credit-risk adjusted discount factor. Note that, similarly to 𝛽 , the process 𝛼 is bounded.

Lemma 3.6. For any stopping times 𝜏 𝑝 ∈ ℱ 𝑡 𝑇 and 𝜏 𝑐 ∈ ℱ 𝑡 𝑇 one has that 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ 𝒢 𝑡 = 1 { 𝑡 < 𝜏 𝑑 } 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ ℱ 𝑡 , ( 3 . 3 ) where 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) is given by, with 𝜏 = 𝜏 𝑝 ⋀ 𝜏 𝑐 , 𝛼 𝑡 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 = 𝜏 𝑡 𝛼 𝑢 𝑑 𝐶 𝑢 + 𝑅 𝑢 𝑑 Γ 𝑢 + 𝛼 𝜏 1 { 𝜏 = 𝜏 𝑝 < 𝑇 } 𝐿 𝜏 𝑝 + 1 { 𝜏 < 𝜏 𝑝 } 𝑈 𝜏 𝑐 + 1 { 𝜏 = 𝑇 } 𝜉 . ( 3 . 4 )

Proof. Formula (3.3) is an immediate consequence of formula (2.2) and Lemma A.5.

Note that 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) is an ℱ 𝜏 -measurable random variable. A comparison of formulae (2.2) and (3.4) shows that we have effectively moved our considerations from the original market subject to the default risk, in which cash flows are discounted according to the discount factor 𝛽 , to the fictitious default-free market, in which cash flows are discounted according to the credit risk adjusted discount factor 𝛼 . Recall that the original cash flows 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) are given as 𝒢 𝜏 ⋀ 𝜏 𝑑 -measurable random variables, whereas the 𝔽 -equivalent cash flows 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) are manifestly ℱ 𝜏 -measurable and they depend on the default time 𝜏 𝑑 only via the hazard process Γ . For the purpose of computation of the ex-dividend price of a defaultable game option these two market models are in fact equivalent. This follows from the next result, which is obtained by combining Theorem 2.5 with Lemmas 3.5 and 3.6.

Theorem 3.7 (Pre-default price of a defaultable game option). Assuming condition (2.7), let Π be the arbitrage ex-dividend ℚ -price for a game option. Then one has, for any 𝑡 ∈ [ 0 , 𝑇 ] , Π 𝑡 = 1 { 𝑡 < 𝜏 𝑑 } Π 𝑡 , ( 3 . 5 ) where Π 𝑡 satisfies e s s s u p 𝜏 𝑝 ∈ ℱ 𝑡 𝑇 e s s i n f 𝜏 𝑐 ∈ ℱ 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ ℱ 𝑡 = Π 𝑡 = e s s i n f 𝜏 𝑐 ∈ ℱ 𝑡 𝑇 e s s s u p 𝜏 𝑝 ∈ ℱ 𝑡 𝑇 𝔼 ℚ 𝜋 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ∣ ℱ 𝑡 . ( 3 . 6 ) Hence the Dynkin game with cost criterion 𝔼 ℚ ( 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) ∣ ℱ 𝑡 ) on ℱ 𝑡 𝑇 × ℱ 𝑡 𝑇 admits the value Π 𝑡 , which coincides with the pre-default ex-dividend price at time 𝑡 of the game option under the risk-neutral measure ℚ .

The following result is the converse of Theorem 3.7. It is an immediate consequence of Lemmas 3.5 and 3.6 and the “if’’ part of Theorem 2.5 (noting also that Π defined by (3.5) is obviously a 𝔾 -semimartingale if Π is a 𝔾 -semimartingale).

Theorem 3.8. Let Π 𝑡 be the value of the Dynkin game with the cost criterion 𝔼 ℚ ( 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) ∣ ℱ 𝑡 ) on ℱ 𝑡 𝑇 × ℱ 𝑡 𝑇 , for any 𝑡 ∈ [ 0 , 𝑇 ] . Then Π 𝑡 defined by (3.5) is the value of the Dynkin game with the cost criterion 𝔼 ℚ ( 𝜋 ( 𝑡 ; 𝜏 𝑝 , 𝜏 𝑐 ) ∣ 𝒢 𝑡 ) on 𝒢 𝑡 𝑇 × 𝒢 𝑡 𝑇 , for any 𝑡 ∈ [ 0 , 𝑇 ] . If, in addition, Π is a 𝔾 -semimartingale then Π is the arbitrage ex-dividend ℚ -price for the game option.

Theorems 3.7 and 3.8 thus allow us to reduce the study of a game option to the study of Dynkin games (3.6) with respect to the reference filtration 𝔽 .

3.3. Valuation via Doubly Reflected BSDEs

In this section, we will characterize the arbitrage ex-dividend ℚ -price of a game option as a solution to an associated doubly reflected BSDE. To this end, we first recall some auxiliary results concerning the relationship between Dynkin games and doubly reflected BSDEs.

Given an additional 𝔽 -adapted process 𝐹 of finite variation, we consider the following doubly reflected BSDE with the data 𝛼 , 𝐹 , 𝜉 , 𝐿 , 𝑈 , 𝜏 (see Cvitanić and Karatzas [30], Hamadène and Hassani [31], Crépey [32], Crépey and Matoussi [33], Bielecki et al. [4, 9]): 𝛼 𝑡 Θ 𝑡 = 𝛼 𝑇 𝜉 + 𝛼 𝑇 𝐹 𝑇 − 𝛼 𝑡 𝐹 𝑡 + 𝑇 𝑡 𝛼 𝑢 𝑑 𝐾 𝑢 − 𝑇 𝑡 𝛼 𝑢 𝑑 𝑀 𝑢 [ ] , 𝐿 , 𝑡 ∈ 0 , 𝑇 𝑡 ≤ Θ 𝑡 ≤ 𝑈 𝑡 [ ] , , 𝑡 ∈ 0 , 𝑇 𝑇 0 Θ 𝑢 − 𝐿 𝑢 𝑑 𝐾 + 𝑢 = 𝑇 0 𝑈 𝑢 − Θ 𝑢 𝑑 𝐾 − 𝑢 = 0 , ( 3 . 7 ) where the process 𝑈 = ( 𝑈 𝑡 ) 𝑡 ∈ [ 0 , 𝑇 ] equals, for 𝑡 ∈ [ 0 , 𝑇 ] , 𝑈 𝑡 = 1 { 𝑡 < 𝜏 } ∞ + 1 { 𝑡 ≥ 𝜏 } 𝑈 𝑡 . ( 3 . 8 )

Definition 3.9. By a ( ℚ -) solution to the doubly reflected BSDE (3.7), we mean a triplet ( Θ , 𝑀 , 𝐾 ) such that (i)the state process Θ is a real-valued, 𝔽 -adapted, càdlàg process,(ii) ∫ ⋅ 0 𝛼 𝑑 𝑀 is a real-valued 𝔽 -martingale vanishing at time 0 , (iii) 𝐾 is an 𝔽 -adapted, continuous, finite variation process vanishing at time 0 , (iv)all conditions in (3.7) are satisfied, where in the third line 𝐾 + and 𝐾