Defaultable Game Options in a Hazard Process Model

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≥𝜏𝑑}} 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 ∫𝐻𝑑𝑋, as opposed to a more restricted notion of the component-wise stochastic integral, which is defined as the sum ∑^𝑑𝑖=1∫𝐻^𝑖𝑑𝑋^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 ∫𝐻𝑑𝑋 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 ∫^𝑡0 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_𝑢𝑑𝒟_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 𝛽𝑋 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 ∫𝐻𝑑𝑌 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_{{𝜏𝑑≤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^𝑡𝑇 to denote the set of all 𝔾-stopping times with values in [𝑡,𝑇] and we let 𝒢^𝑡𝑇 stand for the set {𝜏∈𝒢^𝑡𝑇;𝜏⋀𝜏_𝑑≥𝜏⋀𝜏_𝑑}, where the lifting time of a call protection 𝜏 belongs to 𝒢^0𝑇.

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^0𝑇 is a game option with the ex-dividend cumulative discounted cash flows 𝛽_𝑡𝜋(𝑡;𝜏_𝑝,𝜏_𝑐) given by the formula, for any 𝑡∈[0,𝑇] and (𝜏_𝑝,𝜏_𝑐)∈𝒢^𝑡𝑇×𝒢^𝑡𝑇, 𝛽_𝑡𝜋𝑡;𝜏_𝑝,𝜏_𝑐=^𝜏𝑡𝛽_𝑢𝑑𝐷_𝑢+1_{{𝜏<𝜏𝑑}}𝛽_𝜏1^{𝜏=𝜏_𝑝^<𝑇}𝐿_𝜏𝑝+1_{{𝜏<𝜏𝑝}}𝑈_𝜏𝑐+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𝑑𝐶_𝑢+_[]0,𝑡𝑅_𝑢𝑑𝐻_𝑢=𝐶_𝜏−1_{{𝑡≥𝜏}}+𝐶_𝑡1_{𝑡<𝜏}+𝑅_𝜏1_{{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^𝑡𝑇×𝒢^𝑡𝑇, the random variable 𝜋(𝑡;𝜏_𝑝,𝜏_𝑐) is 𝒢_{𝜏⋀𝜏𝑑}-measurable.
(ii) For any (𝜏_𝑝,𝜏_𝑐)∈𝒢^0𝑇×𝒢^0𝑇, 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_𝑡ℒ_𝑡∶=_[0,𝑡]𝛽_𝑢𝑑𝐷_𝑢+1_{{𝑡<𝜏𝑑}}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_𝑡𝒰_𝑡∶=_[0,𝑡]𝛽_𝑢𝑑𝐷_𝑢+1_{{𝑡<𝜏𝑑}}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 esssu=p_{𝜏𝑝∈𝒢𝑡𝑇}Πes_𝑡sinf_{𝜏𝑐∈𝒢𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡=essinf_{𝜏𝑐∈𝒢𝑡𝑇}esssup_{𝜏𝑝∈𝒢𝑡𝑇}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 esssup_ℚ∈ℳ𝔼_ℚsup_{𝑡∈[0,𝑇]}_[0,𝑡]𝛽_𝑢𝑑𝐷_𝑢+1_{{𝑡<𝜏𝑑}}𝛽_𝑡1_{𝑡<𝑇}𝐿_𝑡+1_{𝑡=𝑇}𝜉∣𝒢₀<∞,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 𝛽ℒ (cf. (2.4) attains its maximum at 𝑇, that is, 𝛽_𝑡ℒ_𝑡≤𝛽_𝑇ℒ_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_𝑡𝜙(𝑡)=^𝑇𝑡𝛽_𝑢𝑑𝐷_𝑢+1_{{𝜏𝑑>𝑇}}𝛽_𝑇𝜉,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 (𝑉₀,𝜁) such that (i)𝑉₀ is a 𝒢₀-measurable real-valued random variable representing the initial wealth,(ii)𝜁 is an ℝ^1⊗𝑑-valued, 𝛽𝑋-integrable process representing holdings in primary risky assets.

Remark 2.8. The reason why we do not assume that 𝒢₀ 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 (𝑉₀,𝜁) is given by the formula, for 𝑡∈[0,𝑇], 𝛽_𝑡𝑉_𝑡=𝛽₀𝑉₀+^𝑡0𝜁_𝑢𝑑𝛽_𝑢𝑋_𝑢.(2.9)

Given the wealth process 𝑉 of a primary strategy (𝑉₀,𝜁), we uniquely specify a 𝔾-optional process 𝜁⁰ by setting 𝑉_𝑡=𝜁^0𝑡𝛽^𝑡−1+𝜁_𝑡𝑋_𝑡,𝑡∈[0,𝑇].(2.10) The process 𝜁⁰ represents the number of units held in the savings account at time 𝑡, when we start from the initial wealth 𝑉₀ 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 𝛽𝜋 given by (2.2).
(i)An issuer hedge with cost process 𝜌 is represented by a quadruplet (𝑉₀,𝜁,𝜌,𝜏_𝑐) such that(a)(𝑉₀,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 𝜌₀=0,(c)a (fixed) call time 𝜏_𝑐 belongs to 𝒢^0𝑇,(d)the following inequality is valid, for every put time 𝜏_𝑝∈𝒢^0𝑇, 𝛽_𝜏𝑉_𝜏+^𝜏0𝛽_𝑢𝑑𝜌_𝑢≥𝛽₀𝜋0;𝜏_𝑝,𝜏_𝑐,a.s.(2.11)(ii)A holder hedge with cost process 𝜌 is a quadruplet (𝑉₀,𝜁,𝜌,𝜏_𝑝) such that(a)(𝑉₀,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 𝜌₀=0,(c)a (fixed) put time 𝜏_𝑝 belongs to 𝒢^0𝑇,(d)the following inequality is valid, for every call time 𝜏_𝑐∈𝒢^0𝑇, 𝛽_𝜏𝑉_𝜏+^𝜏0𝛽_𝑢𝑑𝜌_𝑢≥−𝛽₀𝜋0;𝜏_𝑝,𝜏_𝑐,a.s.(2.12)

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^𝜏0𝛽_𝑢𝑑𝜌_𝑢≥𝛽^𝜏−1^𝜏0𝛽_𝑢𝑑𝐷_𝑢+1_{𝜏<𝜏𝑑}1^𝜏=𝜏_𝑝^<𝑇𝐿_𝜏𝑝+1_{𝜏<𝜏𝑝}𝑈_𝜏𝑐+1_{𝜏𝑝=𝜏𝑐=𝑇}𝜉,a.s.(2.13) 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 𝑑𝜌_𝑡≥0) or withdrawn from (if 𝑑𝜌_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 (𝑉₀,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 (𝑉₀,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^𝑐0 (resp., 𝒱^𝑝0) as the set of initial values 𝑉₀ for which there exists an issuer (resp., holder) hedge of the game option with the initial value 𝑉₀ (resp., −𝑉₀) 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, essinf∅=∞) essinf_{𝜏𝑐∈𝒢0𝑇}esssup_{𝜏𝑝∈𝒢0𝑇}𝔼_ℚ𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀≤essinf_{𝑉0∈𝒱𝑐0}𝑉₀,a.s.(2.14)
(ii) If inequality (2.5) is valid then esssup_{𝜏𝑝∈𝒢0𝑇}essinf_{𝜏𝑐∈𝒢0𝑇}𝔼_ℚ𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀≥−essinf_{𝑉0∈𝒱𝑝0}𝑉₀,a.s.(2.15)

Proof. (i) Assume that for some stopping time 𝜏_𝑐∈𝒢^0𝑇 the quadruplet (𝑉₀,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^0𝑇, 𝛽₀𝑉₀=≥𝛽𝛽_{𝜏𝑝∧𝜏𝑐0}𝑉_{𝜏𝑝∧𝜏𝑐}−^{𝜏𝑝∧𝜏𝑐}₀𝜁_𝑢𝑑𝛽_𝑢𝑋_𝑢𝜋0;𝜏_𝑝,𝜏_𝑐−^{𝜏𝑝∧𝜏𝑐}₀𝜁^𝑢𝑑𝛽_𝑢𝑋_𝑢+𝛽^𝑢𝑑𝜌_𝑢.(2.16) In particular, by taking 𝜏_𝑝=𝑡, we obtain that, for any 𝑡∈[0,𝑇], 𝛽₀𝑉₀=≥𝛽𝛽^𝑡∧𝜏_𝑐0𝑉_{𝑡∧𝜏𝑐}−^{𝑡∧𝜏𝑐}₀𝜁_𝑢𝑑𝛽_𝑢𝑋_𝑢𝜋0;𝑡,𝜏_𝑐−^{𝑡∧𝜏𝑐}₀𝜁^𝑢𝑑𝛽_𝑢𝑋_𝑢+𝛽^𝑢𝑑𝜌_𝑢.(2.17) The stochastic integral ∫^𝑡0𝜁_𝑢𝑑(𝛽_𝑢𝑋_𝑢) with respect to a 𝔾-sigma martingale 𝛽𝑋 is a 𝔾-sigma martingale. Hence the stopped process ∫^{𝑡⋀𝜏𝑐}₀𝜁_𝑢𝑑(𝛽_𝑢𝑋_𝑢), as well as the process ^{𝑡∧𝜏𝑐}₀𝜁^𝑢𝑑𝛽_𝑢𝑋_𝑢+𝛽^𝑢𝑑𝜌_𝑢(2.18) 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^0𝑇 (recall that 𝜏_𝑐 is fixed) 𝛽₀𝑉₀≥𝔼_ℚ𝛽₀𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀,∀𝜏_𝑝∈𝒢^0𝑇,(2.19) and thus, by the assumed positivity of the process 𝛽,𝑉₀≥essinf_{𝜏𝑐∈𝒢0𝑇}esssup_{𝜏𝑝∈𝒢0𝑇}𝔼_ℚ𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀,a.s.(2.20) The required inequality (2.14) is an immediate consequence of the last formula.
(ii) Let (𝑉₀,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^0𝑇. Then (2.9) and (2.12) imply that, for any 𝑡∈[𝜏,𝑇], 𝛽₀𝑉₀≥=𝛽−𝛽^𝑡∧𝜏_𝑝0𝑉_{𝑡∧𝜏𝑝}−^{𝑡∧𝜏𝑝}₀𝜁_𝑢𝑑𝛽_𝑢𝑋_𝑢𝜋0;𝜏_𝑝,𝑡−^{𝑡∧𝜏𝑝}₀𝜁^𝑢𝑑𝛽_𝑢𝑋_𝑢+𝛽^𝑢𝑑𝜌_𝑢.(2.21) 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^0𝑇, we obtain 𝛽₀𝑉₀≥𝔼_ℚ−𝛽₀𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀,a.s.,∀𝜏_𝑐∈𝒢^0𝑇,(2.22) so that 𝑉₀≥−esssup_{𝜏𝑝∈𝒢0𝑇}essinf_{𝜏𝑐∈𝒢0𝑇}𝔼_ℚ𝜋0;𝜏_𝑝,𝜏_𝑐∣𝒢₀,a.s.,(2.23) 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_{{𝜏𝑑≤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=−ln(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 𝜏̃, 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 eesssssinuf_{𝜏𝑐∈𝒢𝑡𝑇}p_{𝜏𝑝∈𝒢𝑡𝑇}essinf_{𝜏𝑐∈𝒢𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡=esssup_{𝜏𝑝∈ℱ𝑡𝑇}essinf_{𝜏𝑐∈ℱ𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡,esssup_{𝜏𝑝∈𝒢𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡=essinf_{𝜏𝑐∈ℱ𝑡𝑇}esssup_{𝜏𝑝∈ℱ𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡.(3.1)

Proof. For (𝜏_𝑝,𝜏_𝑐)∈𝒢^𝑡𝑇×𝒢^𝑡𝑇, one has that 𝜋𝑡;𝜏_𝑝,𝜏_𝑐=𝜋𝑡;𝜏_𝑝∧𝜏_𝑑,𝜏_𝑐∧𝜏_𝑑=𝜋𝑡;𝜏̃_𝑝∧𝜏_𝑑,𝜏̃_𝑐∧𝜏_𝑑=𝜋𝑡;𝜏̃_𝑝,𝜏̃_𝑐(3.2) for some stopping times (𝜏̃_𝑝,𝜏̃_𝑐)∈ℱ^𝑡𝑇×ℱ^𝑡𝑇, where the middle equality follows from Lemma A.4, and the other two from the definition of 𝜋. Since, clearly, ℱ^𝑡𝑇⊆𝒢^𝑡𝑇 and ℱ^𝑡𝑇⊆𝒢^𝑡𝑇, 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_𝑡exp(−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^𝑡𝑇 and 𝜏_𝑐∈ℱ^𝑡𝑇 one has that 𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣𝒢_𝑡=1_{{𝑡<𝜏𝑑}}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_𝑡𝜋𝑡;𝜏_𝑝,𝜏_𝑐=^𝜏𝑡𝛼_𝑢𝑑𝐶_𝑢+𝑅_𝑢𝑑Γ_𝑢+𝛼_𝜏1^{𝜏=𝜏_𝑝^<𝑇}𝐿_𝜏𝑝+1_{{𝜏<𝜏𝑝}}𝑈_𝜏𝑐+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⋀𝜏𝑑}-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<𝜏𝑑}}Π_𝑡,(3.5) where Π_𝑡 satisfies esssup_{𝜏𝑝∈ℱ𝑡𝑇}essinf_{𝜏𝑐∈ℱ𝑡𝑇}𝔼_ℚ𝜋𝑡;𝜏_𝑝,𝜏_𝑐∣ℱ_𝑡=Π_𝑡=essinf_{𝜏𝑐∈ℱ𝑡𝑇}esssup_{𝜏𝑝∈ℱ𝑡𝑇}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^𝑡𝑇×ℱ^𝑡𝑇 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^𝑡𝑇×ℱ^𝑡𝑇, 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^𝑡𝑇×𝒢^𝑡𝑇, 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_𝑡Θ_𝑡=𝛼_𝑇𝜉+𝛼_𝑇𝐹_𝑇𝐿−𝛼_𝑡𝐹_𝑡+^𝑇𝑡𝛼_𝑢𝑑𝐾_𝑢−^𝑇𝑡𝛼_𝑢𝑑𝑀_𝑢,𝑡∈[0,𝑇_𝑡],≤Θ_𝑡≤𝑈_𝑡,𝑡∈[0,𝑇_𝑇0],Θ_𝑢−𝐿_𝑢𝑑𝐾^+𝑢=^𝑇0𝑈_𝑢−Θ_𝑢𝑑𝐾^−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)∫^⋅0𝛼𝑑𝑀 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⁻ denote the Jordan components of 𝐾, and where the convention that 0×±∞=0 is made in the third line.

By the Jordan decomposition, we mean the decomposition 𝐾=𝐾⁺−𝐾⁻, where the nondecreasing continuous processes 𝐾⁺ and 𝐾⁻ vanish at time 0 and define mutually singular measures.

The state process Θ in a solution to (3.7) is clearly an 𝔽-semimartingale. So there are obvious (though rather artificial) cases in which (3.7) does not admit a solution: it suffices to take 𝜏=0 and 𝐿=𝑈, assumed not to be an 𝔽-semimartingale. It is also clear that a solution would not necessarily be unique if we did not impose the condition of a mutual singularity of the nonnegative measures defined by 𝐾⁺ and 𝐾⁻ (see, e.g., [31, Remark  4.1]).

Remark 3.10. In applications (see [4, 9, 32, 33]), the input process 𝐹 is typically given in the form of the Lebesgue integral 𝛼𝐹=∫𝛼𝑓𝑑𝑢 and the component 𝑀 of a solution to (3.7) is usually searched for in the form 𝑀=∫𝑍𝑑𝑁+𝑛 for some ℝ^𝑞-valued and real-valued square-integrable 𝔽-martingales 𝑁 and 𝑛 (see also Assumption 4.7 in Section 4.3). For more explicit (in particular, Markovian) specifications of the present setup and sufficient conditions for the existence and uniqueness of a solution to (3.7), the interested reader is referred to, for example, [4, 30–33].

Basically, in any model endowed with the martingale representation property, the existence (and uniqueness) of a solution to (3.7) (supplemented by suitable integrability conditions on the data and the solution) is equivalent to the so-called Mokobodski condition, namely, the existence of a quasimartingale 𝑍 such that 𝐿≤𝑍≤𝑈 on [0,𝑇] (see, in particular, Crépey and Matoussi [33], Hamadène and Hassani [31, Theorem  4.1], and previous works in this direction, starting with Cvitanić and Karatzas [30]). It is thus satisfied when one of the barriers is a quasimartingale and, in particular, when one of the barriers is given as 𝑆∨ℓ, where 𝑆 is an Itô-Lévy process 𝑆 with square-integrable special semimartingale decomposition components (see [33]) and ℓ is a constant in ℝ∪{−∞}. This framework covers, for instance, the payoff at call of a convertible bond examined in [3, 4].

Remark 3.11. (i) Since 𝐾, and thus 𝐾⁺ and 𝐾⁻, are continuous, the minimality conditions (third line) in (3.7) are equivalent to ^𝑇0Θ_𝑢−−𝐿_𝑢−𝑑𝐾^+𝑢=^𝑇0𝑈_𝑢−−Θ_𝑢−𝑑𝐾^−𝑢=0.(3.9)
Indeed the related integrands here and in the third line of (3.7) differ on an at most countable set whereas the integrators define atomless measures on [0,𝑇]; see, for example, [33]. In the preprint version [34] of this work, we defined more general notions of 𝜀-hedges that were pertaining in the case where there may be jumps in the process 𝐾. Since in all existing works on doubly reflected BSDEs the process 𝐾 is actually found to be a continuous process (see [4, 30, 31, 33]), we decided to impose here the continuity of 𝐾 in Definition 3.9 and we only consider hedges, as opposed to 𝜀-hedges. Note, however, that essentially all the results of this paper can be extended to possible jumps in 𝐾, using the generalized notion of 𝜀-hedge defined in [34], and with the minimality conditions stated as (3.9) instead of the third line in condition (3.7) of Definition 3.9.
(ii) Since 𝐹 is a given process, the BSDE (3.7) can be rewritten as 𝛼_𝑡Θ_𝑡=𝛼_𝑇𝜉̂𝐿+^𝑇𝑡𝛼_𝑢𝑑𝐾_𝑢−^𝑇𝑡𝛼_𝑢𝑑𝑀_𝑢,𝑡∈[0,𝑇_𝑡],≤Θ_𝑡≤𝑈_𝑡,𝑡∈[0,𝑇_𝑇0],Θ^𝑢−𝐿_𝑢𝑑𝐾^+𝑢=^𝑇0𝑈^𝑢−Θ_𝑢𝑑𝐾^−𝑢=0,(3.10) where Θ_𝑡=Θ_𝑡+𝐹_𝑡,𝜉̂=𝜉+𝐹_𝑇,𝐿_𝑡=𝐿_𝑡+𝐹_𝑡, and 𝑈_𝑡=𝑈_𝑡+𝐹_𝑡. This shows that the problem of solving (3.7) can be formally reduced to the case of 𝐹=0 with suitably modified reflecting barriers 𝐿,𝑈 and terminal condition 𝜉̂. However, the freedom to choose the driver of a related BSDE associated with a game option is important from the point of view of applications (this is apparent in the followup papers [4, 9]; see also [34]).
(iii) In the special case where all 𝔽-martingales are continuous and where the 𝔽-semimartingale 𝐹 and the barriers 𝐿 and 𝑈 are continuous (see [4, 30, 35]), it is natural to look for a continuous solution of (3.7), that is, a solution of (3.7) given by a triplet of continuous processes (Θ,𝑀,𝐾).
(iv) In the context of a Markovian setup, the probabilistic BSDE approach may be complemented by a related analytic variational inequality approach; this issue is dealt with in the followup papers [4, 9]. Note, however, that the variational inequality approach strongly relies on the BSDE approach. Moreover, a simulation method based on the BSDE is the only efficient way of numerically solving the pricing problem whenever the problem dimension (number of model factors) is greater than three or four. Indeed, in that case the computational cost of deterministic numerical schemes based on the variational inequality approach becomes prohibitive.

In order to establish a relationship between a solution to the related doubly reflected BSDE and the arbitrage ex-dividend ℚ-price of the defaultable game option, we first recall the general relationship between doubly reflected BSDEs and Dynkin games with purely terminal cost, before applying this result to dividend-paying game options in the fictitious default-free market in Proposition 3.12.

Observe that if (Θ,𝑀,𝐾) solves (3.7) then one has, for any stopping time 𝜏∈ℱ^𝑡𝑇, 𝛼_𝑡Θ_𝑡=𝛼_𝜏Θ_𝜏+𝛼_𝜏𝐹_𝜏−𝛼_𝑡𝐹_𝑡+^𝜏𝑡𝛼_𝑢𝑑𝐾_𝑢−^𝜏𝑡𝛼_𝑢𝑑𝑀_𝑢.(3.11)

Proposition 3.12 (Verification principle for a Dynkin game). Let (Θ,𝑀,𝐾) be a solution to (3.7) with 𝐹=0. Then Θ_𝑡 is the value of the Dynkin game with cost criterion 𝔼_ℚ(𝜃(𝑡;𝜏_𝑝,𝜏_𝑐)∣ℱ_𝑡) on ℱ^𝑡𝑇×ℱ^𝑡𝑇, where 𝜃(𝑡;𝜏_𝑝,𝜏_𝑐) is the ℱ_𝜏-measurable random variable defined by 𝛼_𝑡𝜃𝑡;𝜏_𝑝,𝜏_𝑐=𝛼_𝜏1^{𝜏=𝜏_𝑝^<𝑇}𝐿_𝜏𝑝+1_{{𝜏=𝜏𝑐<𝜏𝑝}}𝑈_𝜏𝑐+1_{𝜏=𝑇}𝜉,(3.12) where 𝜏=𝜏_𝑝⋀𝜏_𝑐. Moreover, for any 𝑡∈[0,𝑇], the pair of stopping times (𝜏^∗𝑝,𝜏^∗𝑐)∈ℱ^𝑡𝑇×ℱ^𝑡𝑇 given by 𝜏^∗𝑝=inf𝑢∈[𝑡,𝑇];Θ_𝑢≤𝐿_𝑢∧𝑇,𝜏^∗𝑐=inf𝑢∈𝜏∨𝑡,𝑇;Θ_𝑢≥𝑈_𝑢∧𝑇,(3.13) is a saddle-point of this Dynkin game, in the sense that one has, for any (𝜏_𝑝,𝜏_𝑐)∈ℱ^𝑡𝑇×ℱ^𝑡𝑇, 𝔼_ℚ𝜃𝑡;𝜏_𝑝,𝜏^∗𝑐∣ℱ_𝑡≤Θ_𝑡≤𝔼_ℚ𝜃𝑡;𝜏^∗𝑝,𝜏_𝑐∣ℱ_𝑡.(3.14)

Proof. Except for the presence of 𝜏, the result is standard (see, e.g., Lepeltier and Maingueneau [19]). Let us first check that the right-hand side inequality in (3.14) is valid for any 𝜏_𝑐∈ℱ^𝑡𝑇. Let 𝜏^∗denote 𝜏^∗𝑝⋀𝜏_𝑐. By the definition of 𝜏^∗𝑝 and continuity of 𝐾⁺, we see that 𝐾⁺ equals 0 on [𝑡,𝜏^∗]. Since 𝐾⁻ is nondecreasing, (3.11) is applied to yield 𝛼_𝑡Θ_𝑡≤𝛼_𝜏∗Θ_𝜏∗−^𝜏∗_𝑡𝛼_𝑢𝑑𝑀_𝑢.(3.15) Taking conditional expectations (recall that ∫^⋅𝑡𝛼_𝑢𝑑𝑀_𝑢 is an 𝔽-martingale), and using also the facts that Θ_𝜏^∗𝑝≤𝐿_𝜏^∗𝑝 if 𝜏^∗𝑝<𝑇,Θ_𝜏^∗𝑝=𝜉 if 𝜏^∗𝑝=𝑇 and Θ_𝜏𝑐≤𝑈_𝜏𝑐 (recall that 𝜏_𝑐∈ℱ^𝑡𝑇, so that 𝜏_𝑐≥𝜏 and 𝑈_𝜏𝑐=𝑈_𝜏𝑐), we obtain 𝛼_𝑡Θ_𝑡≤≤𝔼𝔼_ℚℚ𝛼_𝜏^∗Θ_𝜏∗∣ℱ_𝑡𝛼^𝜏∗1^{𝜏∗=𝜏∗𝑝<𝑇}𝐿_𝜏^∗𝑝+1_{𝜏∗=𝜏𝑐<𝜏}^∗𝑝_𝑈_𝜏𝑐+1_{𝜏∗=𝑇}𝜉∣ℱ_𝑡.(3.16) We conclude that Θ_𝑡≤𝔼_ℚ(𝜃(𝑡;𝜏^∗𝑝,𝜏_𝑐)∣ℱ_𝑡) for any 𝜏_𝑐∈ℱ^𝑡𝑇. This completes the proof of the right-hand side inequality in (3.14). The left-hand side inequality can be shown similarly. It is in fact standard, since it does not involve 𝜏, and thus the details are left to the reader.

Let us now apply Proposition 3.12 to a defaultable game option. To this end, we first rewrite (3.4) as follows 𝛼_𝑡𝜋𝑡;𝜏_𝑝,𝜏_𝑐=𝛼_𝜏𝐹_𝜏−𝛼_𝑡𝐹_𝑡+𝛼_𝜏1^{𝜏=𝜏_𝑝^<𝑇}𝐿_𝜏𝑝+1_{{𝜏<𝜏𝑝}}𝑈_𝜏𝑐+1_{𝜏=𝑇}𝜉,(3.17) where 𝐹_𝑡∶=𝛼^𝑡−1_[0,𝑡]𝛼_𝑢𝑑𝐷_𝑢with𝐷_𝑡∶=_[0,𝑡]𝑑𝐶_𝑢+𝑅_𝑢𝑑Γ_𝑢.(3.18) Let us denote by (ℰ) equation (3.10) with 𝐹_𝑡=𝐹_𝑡, that is, 𝛼_𝑡Θ_𝑡=𝛼_𝑇𝜉̂𝐿+^𝑇𝑡𝛼_𝑢𝑑𝐾_𝑢−^𝑇𝑡𝛼_𝑢𝑑𝑀_𝑢,𝑡∈[0,𝑇_𝑡],≤Θ_𝑡≤𝑈_𝑡,𝑡∈[0,𝑇_𝑇0],Θ_𝑢−𝐿_𝑢𝑑𝐾^+𝑢=^𝑇0𝑈^𝑢−Θ_𝑢𝑑𝐾^−𝑢=0,(ℰ) with 𝜉̂=𝜉+𝐹_𝑇,𝐿_𝑡=𝐿_𝑡+𝐹_𝑡, and 𝑈_𝑡=𝑈_𝑡+𝐹_𝑡.

Assumption 3.13. The doubly reflected BSDE (ℰ) admits a solution (Θ,𝑀,𝐾).

Let us stress that Assumption 3.13, heroic as it may seem in the general hazard process setup, is in fact a plausible assumption in any reasonable application one may think of (cf. the comments following Definition 3.9).

We denote, for 𝑡∈[0,𝑇], Π_𝑡=Θ_𝑡−𝐹_𝑡,Π_𝑡=1_{{𝑡<𝜏𝑑}}Π_𝑡,Π_𝑡=Π_𝑡+𝛽^𝑡−1_[0,𝑡]𝑚𝛽_𝑢𝑑𝐷_𝑢,(3.19)_𝑡=𝛽_𝑡Π_𝑡+^{𝑡∧𝜏𝑑}₀𝛽_𝑢𝑑𝐾_𝑢.(3.20) The following lemma is crucial in what follows (Lemma 3.14(i) is actually the key of the proof of Proposition 4.1 below).

Lemma 3.14. (i) The process 𝑚 given by (3.20) is 𝔾-martingale stopped at 𝜏_𝑑.
(ii) The process Π is a 𝔾-semimartingale.
(iii) The process 𝛽Π is a special 𝔾-semimartingale.

Proof. (i) The triplet (Π,𝑀,𝐾) satisfies (3.7) with 𝐹 given by 𝐹 in (3.18). Therefore, for every 𝑡∈[0,𝑇], 𝛼_𝑡Π_𝑡=𝛼_𝑇𝜉+^𝑇𝑡𝛼_𝑢𝑑𝐷_𝑢+^𝑇𝑡𝛼_𝑢𝑑𝐾_𝑢−^𝑇𝑡𝛼_𝑢𝑑𝑀_𝑢(3.21) and thus ^𝑡0𝛼_𝑢𝑑𝑀_𝑢=𝛼_𝑡Π_𝑡−𝛼₀Π₀+^𝑡0𝛼_𝑢𝑑𝐾_𝑢+^𝑡0𝛼_𝑢𝑑𝐷_𝑢.(3.22) Using Lemma A.5, it is easy to check that one has, for any 0≤𝑡≤𝑢≤𝑇, 1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ^𝑢𝑡𝛼_𝑣𝑑𝑀_𝑣∣ℱ_𝑡=𝔼^ℚ𝑚_𝑢−𝑚_𝑡∣𝒢_𝑡.(3.23) Since the integral ∫^⋅𝑡𝛼_𝑣𝑑𝑀_𝑣 is an 𝔽-martingale, the process 𝑚 is a 𝔾-martingale. It is also clear that it is stopped at 𝜏_𝑑.
(ii) In view of (3.19), (3.20) and part (i), the process Π is clearly a 𝔾-semimartingale.
(iii) By (3.20), one has that 𝛽_𝑡Π_𝑡=𝑚_𝑡−^{𝑡∧𝜏𝑑}₀𝛽_𝑢𝑑𝐾_𝑢,(3.24) where 𝑚 is a 𝔾-martingale, by (i), and where the second term in the right-hand side is a 𝔾-adapted and continuous (hence 𝔾-predictable) processes of finite variation.

Remark 3.15. In view of (3.24) and since 𝐾 is continuous, the process 𝑚 given by (3.20) can equivalently be redefined as the canonical 𝔾-local martingale component of the discounted cumulative ℚ-value process 𝛽Π. The processes 𝑚 and 𝛽Π are easily seen to coincide on the random interval [0,𝜏^∗𝑐⋀𝜏^∗𝑝⋀𝜏_𝑑⋀𝑇]. Therefore, both 𝑚 and 𝛽Π can be interpreted on this interval as the discounted cumulative ℚ-value of a defaultable game option.

The following result establishes a useful connection between (Θ,𝑀,𝐾) and the arbitrage ex-dividend ℚ-price of the defaultable game option.

Proposition 3.16 (Verification principle for a defaultable game option). The process Π is the arbitrage ex-dividend ℚ-price for the game option. Moreover, for any 𝑡∈[0,𝑇], the saddle-point (𝜏^∗𝑝,𝜏^∗𝑐)∈ℱ^𝑡𝑇×ℱ^𝑡𝑇 for the related Dynkin game (2.6) on 𝒢^𝑡𝑇×𝒢^𝑡𝑇 is given by 𝜏^∗𝑝=inf𝑢∈[𝑡,𝑇];Π_𝑢≤𝐿_𝑢∧𝑇,𝜏^∗𝑐=inf𝑢∈𝜏∨𝑡,𝑇;Π_𝑢≥𝑈_𝑢∧𝑇.(3.25)

Proof. In view of (3.4), the present assumptions imply that Π_𝑡 is the value of the Dynkin game (3.6), by Proposition 3.12, with saddle-point (𝜏^∗𝑝,𝜏^∗𝑐). Therefore, by Lemmas 3.5 and 3.6, Π_𝑡 is the value of the Dynkin game associated with the game option on 𝒢^𝑡𝑇×𝒢^𝑡𝑇, with saddle-point (𝜏^∗𝑝,𝜏^∗𝑐). Moreover, Π is a 𝔾-semimartingale, by Lemma 3.14(ii). To conclude the proof, it suffices to make use of the last statement in Theorem 3.8.

4. Hedging in a Hazard Process Setup

In the remaining part of this work, we examine in some detail the existence and basic properties of hedging strategies for defaultable game options in a hazard process setup.

4.1. Cost Process of a Hedging Strategy

From now on, we will work under Assumption 3.13. Let thus (Θ,𝑀,𝐾) denote a solution to (ℰ) and let Π and Π be defined by (3.19). In particular, Π is the arbitrage ℚ-price for the game option (by Proposition 3.16) and the left-hand sides in (2.14) and (2.15) are equal to Π₀. Finally, recall that the 𝔾-martingale 𝑚 is defined by (3.20).

Let us stress that some of the key arguments underlying the following result are classical, and they are already contained in Lepeltier and Maingueneau [19] (see, in particular, Theorem  11 therein). Proposition 4.1 can thus be seen as a natural extension of their results to the defaultable case, in which two filtrations are involved. It is notable that our assumptions are made relative to the filtration 𝔽, whereas conclusions are drawn relative to the filtration 𝔾.

Proposition 4.1 (Hedging with a local martingale cost). Let 𝜁 be an arbitrary ℝ^1⊗𝑑- valued and 𝛽𝑋-integrable process. Then the following statements are valid.
(i) Let the process 𝜌(𝜁) be given by 𝜌₀(𝜁)=0 and 𝛽_𝑡𝑑𝜌_𝑡(𝜁)=𝑑𝑚_𝑡−𝜁_𝑡𝑑𝛽_𝑡𝑋_𝑡.(4.1) Then (Π₀,𝜁,𝜌(𝜁),𝜏^∗𝑐) is an issuer hedge with 𝔾-sigma (local, in case 𝛽𝑋 and 𝜁 are locally bounded) martingale cost.
(ii) Let the process 𝜌(𝜁) be given by 𝜌₀(𝜁)=0 and 𝛽_𝑡𝑑𝜌_𝑡(𝜁)=−𝑑𝑚_𝑡−𝜁_𝑡𝑑𝛽_𝑡𝑋_𝑡.(4.2) Then (−Π₀,𝜁,𝜌(𝜁),𝜏^∗𝑝) is a holder hedge with a 𝔾-sigma martingale (local martingale, when 𝛽𝑋 and 𝜁 are locally bounded) cost process.

Recall that, according to our convention (see Section 1.2), the 𝛽𝑋-integrability of an ℝ^1⊗𝑑-valued stochastic process 𝜁 implies its 𝔾-predictability. Note also that the equality 𝜌(−𝜁)=−𝜌(𝜁) is valid for any process 𝜁, since −𝛽_𝑡𝑑𝜌_𝑡(𝜁)=−𝑑𝑚_𝑡+𝜁_𝑡𝑑𝛽_𝑡𝑋_𝑡=−𝑑𝑚^𝑡−−𝜁_𝑡𝑑𝛽_𝑡𝑋_𝑡.(4.3)

Proof of Proposition 4.1. The arguments for a holder are essentially symmetrical to those for an issuer; we thus only prove part (i). By Lemma 3.14(i), the process 𝜌(𝜁) is a 𝔾-sigma martingale, and a 𝔾-local martingale if 𝛽𝑋 and 𝜁 are locally bounded processes. For the ease of notation, we write 𝜌=𝜌(𝜁). Let 𝑉 denote the wealth process of the primary strategy (Π₀,𝜁). By combining (2.9) with (4.1), we obtain 𝑉₀=Π₀ and, for every 𝑡∈[0,𝑇], 𝑑𝛽_𝑡𝑉_𝑡=𝜁_𝑡𝑑𝛽_𝑡𝑋_𝑡=𝑑𝑚^𝑡−𝛽_𝑡𝑑𝜌_𝑡(4.4) and thus 𝛽_𝑡𝑉_𝑡+^𝑡0𝛽_𝑢𝑑𝜌_𝑢=𝑚_𝑡+𝛽₀Π⁰−Π₀=𝛽^𝑡Π_𝑡+^{𝑡∧𝜏𝑑}₀𝛽_𝑢𝑑𝐾_𝑢+𝛽₀Π⁰−Π₀,(4.5) where the second equality follows from (3.20). Recall that the stopping time 𝜏^∗𝑐∈ℱ^0𝑇 is given by (see Proposition 3.16) 𝜏^∗𝑐=inf𝑡∈𝜏,𝑇;Π_𝑡≥𝑈_𝑡∧𝑇.(4.6) In order to prove that the quadruplet (Π₀,𝜁,𝜌,𝜏^∗𝑐) is an issuer hedge for the game option, it is enough to show that one has for any 𝜏_𝑝∈ℱ^0𝑇, with 𝜏=𝜏_𝑝⋀𝜏^∗𝑐 (cf. (2.13), 𝛽_𝜏𝑉_𝜏+^𝜏0𝛽_𝑢𝑑𝜌_𝑢−𝑑𝐷_𝑢≥1_{𝜏<𝜏𝑑}𝛽_𝜏1^𝜏=𝜏_𝑝^<𝑇𝐿_𝜏𝑝+1_{𝜏<𝜏𝑝}𝑈_𝜏^∗𝑐+1_{𝜏𝑝=𝜏}^∗𝑐_=𝑇𝜉.(4.7) From the definition of 𝜏^∗𝑐, the minimality conditions in (ℰ) and the continuity of 𝐾⁻ it follows that 𝐾⁻=0 and thus 𝐾≥0 on [0,𝜏^∗𝑐]. Since 𝜏≤𝜏^∗𝑐, (4.5) thus yields 𝛽_𝜏𝑉_𝜏+^𝜏0𝛽_𝑢𝑑𝜌_𝑢−𝑑𝐷_𝑢=𝛽_𝜏Π_𝜏+_{[0,𝜏∧𝜏𝑑]}𝛽_𝑢𝑑𝐾_𝑢≥𝛽_𝜏Π_𝜏=1_{{𝜏<𝜏𝑑}}𝛽_𝜏Π_𝜏,(4.8) where, by (ℰ), one has that Π_𝜏≥1_{𝜏<𝑇}𝐿_𝜏+1_{𝜏=𝑇}𝜉.(4.9) In addition, by the definition of 𝜏^∗𝑐, one has that Π_𝜏∗𝑐≥𝑈_𝜏^∗𝑐 on the event {𝜏^∗𝑐<𝑇}. It is now easy to see that (4.7) is satisfied and thus (𝑉₀,𝜁,𝜌,𝜏^∗𝑐) is indeed an issuer hedge.

Remark 4.2. (i) The situation where 𝜌 can be made equal to zero by the choice of a suitable strategy 𝜁 in Proposition 4.1 corresponds to a particular form of hedgeability of a game option in which an issuer and a holder are able to hedge all risks embedded in a defaultable game option. The case where 𝜌≠0 corresponds either to nonhedgeability of a game option or to the situation in which an issuer (or a holder) is able to hedge, but she prefers not to hedge all risks associated with a game option, for instance, she may be willing to take some directional bets regarding specific risks. For this reason, we decided not to postulate a priori that 𝜌 should be minimized in some sense as, for instance, in Schweizer [24].
(ii) It is possible to introduce the issuer trivial hedge (Π₀,0,𝜌⁰,𝜏^∗𝑐) (resp., the holder trivial hedge (−Π₀,0,−𝜌⁰,𝜏^∗𝑝)) with the 𝔾-local martingale cost 𝜌^0𝑡=^𝑡0𝛽^𝑢−1𝑑𝑚_𝑢,𝑡∈[0,𝑇].(4.10) Obviously, this hedge is of no practical interest, since it implicitly assumes that one is not interested in hedging any risks. The trivial hedge or, more precisely, the existence of any hedge is used in the proof of Proposition 4.3, however.

Let us now draw some conclusions from Lemma 2.13 and Proposition 4.1. In the context of specific (Cox-Ross-Rubinstein or Black-Scholes, say) models, analogous results can be found in Kifer [8]. Our main contribution here is an extension of these results to the present setup involving a reduction of filtration, as well as to a fairly general class of semimartingale models. We use here the notation ess min (instead of a more common symbol essinf) in order to emphasize that the respective bounds are in fact attained.

Proposition 4.3. Under the assumptions of Proposition 4.1, the following statements are valid. (i)The equality Π₀=essmin𝒱^𝑐0 holds, so that Π₀ is the minimum of initial wealths of an issuer hedge with a 𝔾-sigma martingale cost.(ii)One has that −Π₀∈𝒱^𝑝0. If, in addition, (2.5) holds then Π₀=−essmin𝒱^𝑝0 and −Π₀ is the minimum of initial wealths of a holder hedge with a 𝔾-sigma martingale cost.(iii)The above statements are also valid with local martingale instead of sigma martingale therein.

Proof. (i) By applying Proposition 4.1 to the trivial hedge of Remark 4.2(ii), we get, in particular, that Π₀∈𝒱^𝑐0, where Π₀ is also equal to the ℚ-value of the related Dynkin game, by Proposition 3.16. Therefore, the infimum is attained and one has equality, rather than inequality, in Lemma 2.13(i).
(ii) In view of (2.5) and Lemma 2.13(ii), the second claim can be proven in the same way as part (i).
(iii) This follows immediately from parts (i) and (ii), since the cost 𝜌⁰ of the trivial hedge is a 𝔾-local martingale.

Given our definition of hedging with a cost and the definition of Π₀, the fact that there exists a hedge with an initial wealth Π₀ and a 𝔾-sigma martingale cost (or a local martingale cost, in suitable cases) is by no means surprising. The minimality statement establishes a connection between arbitrage prices and hedging in a general incomplete market. Let us conclude this section by mentioning that one could state analogous definitions and results regarding hedging strategies for a defaultable game option starting at any date 𝑡∈[0,𝑇].

4.2. Risk Factors of a Defaultable Game Option

Let 𝑁^𝑑=𝐻−Γ_{⋅⋀𝜏𝑑} stand for the compensated default process. Under our standing assumption that the 𝔽-hazard process Γ of 𝜏_𝑑 is a continuous and nondecreasing process (cf. Remark 3.2(ii), the process 𝑁^𝑑 is known to be a 𝔾-martingale. Recall also that the avoidance property holds, in the sense that ℚ(𝜏_𝑑=𝜏)=0 for any 𝔽-stopping time 𝜏 (cf. Remark 3.2(i).

An analysis of hedging strategies in the next section hinges on the following lemma, which yields the risk decomposition of the discounted cumulative value process of a defaultable game option. More formally, the martingale component 𝑚 (cf. Remark 3.15) is represented in terms of the pure jump martingale 𝑁^𝑑 and a real-valued 𝔽-martingale 𝑀, which arise as the second component of a solution to the doubly reflected BSDE (3.7). Intuitively, the process 𝑀 models the pre-default risk associated with a defaultable game option, as opposed to the event risk, which is due to an unexpected occurrence of the default event, and which is modeled through the jump martingale 𝑁^𝑑.

Lemma 4.4. The 𝔾-martingale 𝑚 defined by (3.20) satisfies 𝑑𝑚_𝑡=1_{𝑡≤𝜏𝑑}𝛽_𝑡𝑑𝑀_𝑡+𝑌_𝑡𝑑𝑁^𝑑𝑡,(4.11) where the 𝔽-predictable process 𝑌 equals 𝑌_𝑡=𝑅_𝑡−Π_𝑡−.

Proof. Let us introduce the Doléans-Dade martingale (see, e.g., [29]) ℰ_𝑡=1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡=1−^𝑡0ℰ_𝑢−𝑑𝑁^𝑑𝑢,(4.12) so that 𝛼_𝑡ℰ_𝑡=𝛽_𝑡1_{{𝑡<𝜏𝑑}} and 𝛼_𝑡ℰ_𝑡−=𝛽_𝑡1_{{𝑡≤𝜏𝑑}}. Then (cf. (3.19) and (3.20) 𝑑𝑚_𝑡=𝑑𝛽_𝑡Π_𝑡+1^{𝑡≤𝜏_𝑑^}𝛽_𝑡𝑑𝐾_𝑡=𝑑ℰ_𝑡𝛼_𝑡Π_𝑡+1^{𝑡≤𝜏_𝑑^}𝛽_𝑡𝑑𝐾_𝑡+𝛽_𝑡𝑑𝐷_𝑡.(4.13) It may happen that the 𝔽-semimartingale 𝛼Π fails to be also a 𝔾-semimartingale, so a direct application of the (𝔾-)integration by parts formula to ℰ𝛼Π is not possible. However, by Lemma A.1(iv), the process 𝛼Πstopped at 𝜏_𝑑 is a 𝔾-semimartingale. It is also clear that ℰ𝛼Π=ℰ𝛼_{⋅⋀𝜏𝑑}Π_{⋅⋀𝜏𝑑}. Hence by applying the integration by parts formula to ℰ𝛼_{⋅⋀𝜏𝑑}Π_{⋅⋀𝜏𝑑}, we obtain 𝑑ℰ_𝑡𝛼_{𝑡∧𝜏𝑑}Π_{𝑡∧𝜏𝑑}=ℰ^𝑡−𝑑𝛼^𝑡∧𝜏_𝑑Π_{𝑡∧𝜏𝑑}−𝛼^𝑡Π_𝑡−𝑑𝑁^𝑑𝑡+𝑑ℰ,𝛼^⋅∧𝜏_𝑑Π_{⋅∧𝜏𝑑}_𝑡,(4.14) where, in addition, one has that [ℰ,𝛼_{⋅⋀𝜏𝑑}Π_{⋅⋀𝜏𝑑}]_𝑡=−𝑒^Γ𝜏𝑑𝛼_𝜏𝑑ΔΠ_𝜏𝑑𝐻_𝑡. Using the avoidance property of Remark 3.2(i), formula (3.22), and the assumptions that the coupon process 𝐶 is 𝔽-predictable and the hazard process Γ is continuous (so that Δ𝐶_𝜏𝑑=ΔΓ_𝜏𝑑=0), we obtain the equality ΔΠ_𝜏𝑑=0. Using (3.22), we next deduce from (4.13) that 𝑑𝑚_𝑡=ℰ_𝑡−𝑑𝛼^𝑡∧𝜏_𝑑Π_{𝑡∧𝜏𝑑}−𝛼^𝑡Π_𝑡−𝑑𝑁^𝑑𝑡+1^{𝑡≤𝜏_𝑑^}𝛽_𝑡𝑑𝐾_𝑡+𝛽_𝑡𝑑𝐷_𝑡=1_{{𝑡≤𝜏𝑑}}𝛽_𝑡−𝑑𝐾^𝑡−𝑑𝐶_𝑡−𝑅_𝑡𝑑Γ_𝑡+𝑑𝑀_𝑡−Π_𝑡−𝑑𝑁^𝑑𝑡+1^{𝑡≤𝜏_𝑑^}𝛽_𝑡𝑑𝐾_𝑡+𝛽_𝑡𝑑𝐷_𝑡=1_{𝑡≤𝜏𝑑}𝛽_𝑡−𝑑𝐶^𝑡−𝑅_𝑡𝑑Γ_𝑡+𝑑𝑀_𝑡−Π_𝑡−𝑑𝑁^𝑑𝑡+𝛽^𝑡𝑑𝐷_𝑡.(4.15) Using (2.3) and the equality Δ𝐶_𝜏𝑑=0, we finally arrive at the formula 𝑑𝑚_𝑡=1_{{𝑡≤𝜏𝑑}}𝛽_𝑡𝑑𝑀^𝑡+𝑅_𝑡−Π_𝑡−𝑑𝑁^𝑑𝑡,(4.16) which is the required result.

4.3. Hedging of Risk Factors

In order to study nontrivial cases of hedging strategies for a defaultable game option in the general setup of this paper, we need to impose more assumptions on prices of primary traded assets. Since we are working in a fairly general framework, we will be able to provide only general results concerning hedging strategies. The interested reader is referred to the followup papers [4, 9] for a more detailed analysis of assumptions made in this section and particular examples.

First, we recall that the ex-dividend price 𝑋 of primary risky assets satisfies 𝑋_𝑡=(1−𝐻_𝑡)𝑋_𝑡, for every 𝑡∈[0,𝑇], where the ℝ^𝑑-valued, 𝔽-adapted process 𝑋 formally represents the pre-default value of 𝑋. We thus assume, by convention, that any residual value of the primary asset at 𝜏_𝑑 is embedded in the recovery part of the dividend process for 𝑋. We denote by ℛ an ℝ^𝑑-valued and 𝔽-predictable process, which is aimed to represent the recovery processes of primary risky assets. Inspired by decomposition (4.11) of Lemma 4.4, we make also the following natural postulate regarding the behavior of the cumulative price process 𝑋 stopped at 𝜏_𝑑⋀𝑇.

Assumption 4.5. The dynamics under ℚ of the cumulative price process 𝑋 of primary risky assets are, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝛽_𝑡𝑋_𝑡=𝛽^𝑡𝑑𝑀^𝑡+𝑌_𝑡𝑑𝑁^𝑑𝑡(4.17) for some ℝ^𝑑-valued 𝔽-martingale 𝑀, where the ℝ^𝑑-valued, 𝔽-predictable process 𝑌 is given by the equality 𝑌_𝑡=ℛ_𝑡−𝑋_𝑡− for every 𝑡∈[0,𝑇].

By inserting (4.11) and (4.17) into (4.1), we obtain, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝜌_𝑡(𝜁)=𝑑𝑀_𝑡−𝜁_𝑡𝑑𝑀_𝑡+𝑌_𝑡−𝜁_𝑡𝑌_𝑡𝑑𝑁^𝑑𝑡.(4.18) At this stage, we were only able to separate the two principal components of the cost process that correspond to pre-default and default event risks, respectively, where the pre-default risk is now modeled by the 𝔽-martingales 𝑀 and 𝑀 associated with a game option and primary traded assets, respectively.

Remark 4.6. In what follows, we will only be interested in hedging on the random interval [0,𝜏_𝑑⋀𝑇]. Therefore, without loss of generality, we may and do assume that 𝜁 is 𝔽-predictable (see Lemma A.2(ii). This means that the reduction of filtration method can also be applied to hedging of a defaultable game option, and not only to its valuation as was already shown in Section 3.2.

Within the present framework, the event risk factor is common for all traded primary and derivative assets. Therefore, in the next step, we are going to get a closer look on pre-default risks of traded and derivative assets. To this end, we make a further standing assumption, in which the concept of the systematic risk factor (also known as the market risk factor) is introduced.

Assumption 4.7. We are given an ℝ^𝑞-valued 𝔽-martingale, denoted by 𝑁, which is aimed to represent the systematic risk factor for the underlying market model. We postulate that the 𝔽-martingales 𝑀 and 𝑀 of (4.11) and (4.17) satisfy the following decompositions, for every 𝑡∈[0,𝑇], 𝑑𝑀_𝑡=𝑍_𝑡𝑑𝑁_𝑡+𝑑𝑛_𝑡,𝑑𝑀_𝑡=𝑍_𝑡𝑑𝑁_𝑡+𝑑𝑛_𝑡̂,(4.19) where 𝑍 (resp., 𝑍) is some 𝔽-adapted, ℝ^1⊗𝑞-valued (resp., ℝ^𝑑⊗𝑞-valued), 𝑁-integrable processes and 𝑛 (resp., 𝑛̂) is a real-valued (resp., ℝ^𝑑-valued) 𝔽-martingale.

It is natural to refer to 𝔽-martingales 𝑛 and 𝑛̂ appearing in Assumption 4.7 as idiosyncratic risk factors associated with a defaultable game option and primary traded assets, respectively. In this context, we find it convenient to refer to 𝑁^𝑑 as the event risk factor.

Remark 4.8. A specification of the systematic risk factor 𝑁 depends on a particular market model and on a problem at hand, so that it is not possible to make it more explicit in the abstract setup considered here. As it will become apparent in the sequel, the idiosyncratic risk factors are expected to be in some sense orthogonal to the systematic risk factor. For this reason, one cannot simply make 𝑍 and 𝑍 to vanish in (4.19). Once again, for more information on particular models, we refer to [4, 9] (see also Remark 3.10).

Let us denote 𝒩=[^𝑁𝑁_𝑑] and let [𝑍,𝑌] stand for the concatenation of 𝑍 and 𝑌. The next lemma is an immediate consequence of (4.18) and (4.19). The idea behind formula (4.20) is the separation of risk factors 𝑁,𝑁^𝑑,𝑛, and 𝑛̂ in the dynamics of the cost process of a trading strategy.

Lemma 4.9. For any ℝ^1⊗𝑑-valued, 𝛽𝑋-integrable process 𝜁, the cost process satisfies, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝜌_𝑡(𝜁)=𝑍_𝑡,𝑌_𝑡−𝜁_𝑡𝑍^𝑡,𝑌_𝑡𝑑𝒩^𝑡+𝑑𝑛_𝑡−𝜁_𝑡𝑑𝑛_𝑡̂.(4.20)

Example 4.10. To provide some intuition underpinning the present setup, let us first consider a situation where the perfect hedgeability of risks can be achieved, at least in principle. Let us set 𝑞=1 and we take 𝑑𝑀_𝑡=𝑍_𝑡𝑑𝑁_𝑡, so that 𝑛 vanishes. For 𝑑=2, we further postulate that 𝑛̂=0 and 𝑑𝑀_𝑡=𝑑𝑀^1𝑡𝑀^2𝑡=𝑍𝑍^1𝑡2𝑡𝑑𝑀^𝑡(4.21) or, equivalently, that for 𝑖=1,2𝑑𝛽_𝑡𝑋^𝑖𝑡=𝛽^𝑡𝑍^𝑖𝑡𝑑𝑀_𝑡+𝑌^𝑖𝑡𝑑𝑁^𝑑𝑡.(4.22) Assume that there exists a 𝛽𝑋-integrable process 𝜁 solving the equation 𝜁_𝑡𝑍^𝑡,𝑌_𝑡=𝜁^1𝑡,𝜁^2𝑡𝑍^1𝑡𝑌^1𝑡𝑍^2𝑡𝑌^2𝑡=𝑍^𝑡,𝑌_𝑡.(4.23) Then it follows from (4.20) (or (4.18) that the cost process 𝜌(𝜁) vanishes and thus the strategy 𝜁 (resp., −𝜁) is an issuer's (resp., holder's) superhedge for a defaultable game option, in the sense of Definition 2.10. Note that the first (resp., the second) equation in formula (4.23) is used to eliminate the pre-default risk (resp., the event risk). As was expected, the strategy 𝜁 obtained by solving (4.23) is 𝔽-predictable (cf. Remark 4.6).

Remark 4.11. In [4], we further specify the setup of Example 4.10, by examining the exact replication of a convertible bond with the equity and the credit default swap on the underlying credit name in an equity-to-credit intensity-based model, in which the systematic risk factor is modeled by the Brownian motion driving the equity value and all processes appearing in (4.23) can be computed explicitly.

In the foregoing result, we examine two typical situations regarding the partial hedgeability of risk factors when superhedging is either not possible or not desirable. The case considered in part (i) refers to elimination of event and systematic risks. In contrast, part (ii) deals with hedging of the systematic risk only. Of course, it is also possible to hedge the event risk only, but we do not formulate here the corresponding result. Since the proof of the lemma follows easily from (4.20), it is omitted.

Lemma 4.12 (Hedging of risk factors). (i) Assume that the equation [𝑍,𝑌]=𝜁[𝑍,𝑌] admits a 𝛽𝑋-integrable solution 𝜁̂ on [0,𝑇⋀𝜏_𝑑]. Then the cost process 𝜌̂=𝜌(𝜁̂) satisfies, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝜌̂_𝑡=𝑑𝑛_𝑡−𝜁̂_𝑡𝑑𝑛_𝑡̂.(4.24)
(ii) Assume that the equation 𝑍=𝜁𝑍 admits a 𝛽𝑋-integrable solution 𝜁̌ on [0,𝑇⋀𝜏_𝑑]. Then the dynamics of the cost process 𝜌̌=𝜌(𝜁̌) are, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝜌_𝑡=𝑌_𝑡−𝜁̌_𝑡𝑌_𝑡𝑑𝑁^𝑑𝑡+𝑑𝑛_𝑡−𝜁̌_𝑡𝑑𝑛_𝑡̂.(4.25)

Part (i) in Lemma 4.12 corresponds to the case where the common risks (systematic and event) can be completely eliminated. In contrast, part (ii) refers either to the case of unhedgeable event risk (e.g., when 𝑌=0 in dynamics (4.17) or to the situation when the issuer (or holder) is not willing to hedge that risk.

As was already mentioned, practically useful decompositions of 𝑀 and 𝑀 will depend on a particular model for the primary market, as well as on the game option under study. In an abstract setup considered here, they may be formally deduced from martingale representation theorems with orthogonal components.

Let thus ℋ² stand for the class of real-valued 𝔽-martingales with integrable quadratic variation over [0,𝑇] or, by a slight abuse of notation, the class of vector-valued processes with mutually strongly orthogonal components in ℋ². It is worth recalling here that an 𝔽-martingale stopped at 𝜏_𝑑 is also a 𝔾-local martingale, by virtue of Lemma A.1(iii).

The Galtchouk-Kunita-Watanabe (GKW) decomposition of 𝑀 and 𝑀 with respect to 𝑁 and the filtration 𝔽 (see, e.g., Protter [16, Section IV.3, Corollary  1]) thus yields the decompositions (4.19) of 𝑀 and 𝑀 with 𝑛 and 𝑛̂ strongly orthogonal to 𝑁 in ℋ². Since 𝑁 is meant to represent the systematic risk factor, we may and do assume, without loss of generality, that the idiosyncratic risk factors 𝑛 and 𝑛̂ are also mutually strongly orthogonal.

The following proposition justifies the informal statement that the strategy 𝜁̂ (resp., 𝜁̌) hedges the risk factor 𝒩 (resp., 𝑁). We use hereafter the standard symbol [⋅,⋅] to denote the square bracket between 𝔾-semimartingales.

Proposition 4.13 (Orthogonality of risk factors). Assume that the processes 𝑁,𝑛, and 𝑛̂ in decompositions (4.19) of 𝑀 and 𝑀 are mutually strongly orthogonal in ℋ².
(i) Under assumptions of Lemma 4.12(i), the processes 𝜌̂ and 𝒩_{⋅⋀𝜏𝑑} are orthogonal in 𝔾, in the sense that [𝜌̂,𝒩_{⋅⋀𝜏𝑑}] is a 𝔾-sigma martingale (and a 𝔾-local martingale if 𝜁̂ is locally bounded).
(ii) Under assumptions of Lemma 4.12(ii), the processes 𝜌̌ and 𝑁_{⋅⋀𝜏𝑑} are orthogonal in 𝔾, in the sense that [𝜌̌,𝑁_{⋅⋀𝜏𝑑}] is a 𝔾-sigma martingale (and a 𝔾-local martingale if 𝜁̌,𝑅, and ℛ are locally bounded processes).

Proof. We first note that 𝑛_{⋅⋀𝜏𝑑} and 𝑁_{⋅⋀𝜏𝑑} are 𝔾-local martingales, by Lemma A.1(iii). Since 𝑛 is strongly orthogonal to 𝑁 in ℋ², the process [𝑛_{⋅⋀𝜏𝑑},𝑁_{⋅⋀𝜏𝑑}] is a 𝔾-local martingale, as an 𝔽-local martingale stopped at 𝜏_𝑑 (cf. Lemma A.1(iii). Furthermore, by Lemma A.6, [𝑛_{⋅⋀𝜏𝑑},𝑁^𝑑] is a 𝔾-local martingale. We conclude that [𝑛_{⋅⋀𝜏𝑑},𝒩_{⋅⋀𝜏𝑑}] is a 𝔾-local martingale. So are also [𝑛,̂𝑁_{⋅⋀𝜏𝑑}] and [𝑛,̂𝒩_{⋅⋀𝜏𝑑}], since the integral ∫^⋅0𝛽_𝑡𝑑𝑛_𝑡̂ is strongly orthogonal to 𝒩_{⋅⋀𝜏𝑑}. Furthermore, by Lemma A.6, [𝑁_{⋅⋀𝜏𝑑},𝑁^𝑑] is a 𝔾-local martingale.
Using (4.24), we conclude for part (i) that [𝜌̂,𝒩_{⋅⋀𝜏𝑑}] is a 𝔾-sigma martingale and thus it follows a 𝔾-local martingale if 𝜁̂ is a locally bounded process.
For part (ii), we conclude in view of (4.25) that [𝜌̌,𝑁_{⋅⋀𝜏𝑑}] is a 𝔾-sigma martingale and thus a 𝔾-local martingale if 𝜁̌, 𝑅 and ℛ are locally bounded processes.

4.4. Hedging with Orthogonal Cost

Before concluding this work, let us examine briefly an alternative approach to hedging a defaultable game option, which is formally defined as the problem of finding a strategy 𝜁 that makes the cost process 𝔾-orthogonal under a given risk-neutral probability measure ℚ to a predetermined ℝ^𝑞-valued 𝔾-local martingale 𝑁 where, without loss of generality, the process 𝑁 is assumed to be stopped at 𝜏_𝑑. In reference to Proposition 4.13, by the 𝔾-orthogonality, we mean here that [𝜌,𝑁] is a 𝔾-local martingale under ℚ.

Remark 4.14. In the financial interpretation, the process 𝑁 may represent the wealth processes of some preexisting portfolios, rather than risk factors as in Sections 4.2 and 4.3. Admittedly, we consider here a reduced concept of hedging, at least from the theoretical perspective. It is possible to argue, however, that this approach may be of practical relevance, since some kind of a relative hedging (as opposed to replication or superhedging) is a common market practice.

For the purpose of this section, the process 𝑚 arising in (4.1) may be defined either by (3.20), in reference to a solution of a related doubly reflected BSDE with respect to the filtration 𝔽 or, more generally (cf. Remark 3.15), as the 𝔾-local martingale component of the discounted cumulative ℚ-value process 𝛽Π of a game option, provided that 𝛽Π is a 𝔾-special semimartingale.

In the following proposition we denote (whenever well-defined) ℂov_𝑡𝑑𝑋_𝑡,𝑑𝑌_𝑡=lim_ℎ→0ℎ⁻¹ℂov𝑋_𝑡+ℎ−𝑋_𝑡,𝑌_𝑡+ℎ−𝑌_𝑡∣𝒢_𝑡(4.26) and 𝕍ar_𝑡(𝑑𝑋_𝑡)=ℂov_𝑡(𝑑𝑋_𝑡,𝑑𝑋_𝑡).

Proposition 4.15 (Hedging with orthogonal cost). Assume that 𝑋 admits the decomposition, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝑑𝛽_𝑡𝑋_𝑡=𝛽^𝑡𝑍_𝑡𝑑𝑁_𝑡+𝛽_𝑡𝑑𝑛_𝑡(4.27) with 𝑛 and 𝑁 orthogonal in 𝔾 and an ℝ^𝑑⊗𝑞-valued, 𝑁-integrable process 𝑍, which is left-invertible on [0,𝑇⋀𝜏_𝑑]. Let us set, for every 𝑡∈[0,𝑇⋀𝜏_𝑑], 𝜁_𝑡=ℂov_𝑡𝑑𝑚^𝑡,𝛽_𝑡𝑑𝑁_𝑡𝕍ar^𝑡𝛽^𝑡𝑑𝑁_𝑡⁻¹Λ_𝑡,(4.28) where Λ is the left inverse of the transpose of 𝑍 on [0,𝑇⋀𝜏_𝑑]. Then the cost process 𝜌(𝜁) is orthogonal to 𝑁 in 𝔾.

Proof. By combining (4.1) with (4.27), we obtain 𝛽_𝑡𝑑𝜌_𝑡(𝜁)=𝑑𝑚_𝑡−𝜁_𝑡𝛽_𝑡𝑍_𝑡𝑑𝑁_𝑡−𝛽_𝑡𝜁_𝑡𝑑𝑛_𝑡.(4.29) Therefore, in order to have the cost 𝜌 orthogonal to 𝑁 in 𝔾, it suffices to select a strategy 𝜁 for which 𝑚−∫^⋅0𝛽_𝑡𝜁_𝑡𝑍_𝑡𝑑𝑁_𝑡 is 𝔾-orthogonal to 𝑁. Relying on the multilinear regression formula, this can be achieved by setting 𝜁 as in (4.28).

The problem of hedging a defaultable game option with respect to 𝑁 can thus be solved, at least formally, provided that one can find a decomposition (4.27) with the required properties. Such a decomposition can be obtained as the GKW decomposition of 𝛽𝑋 in 𝔾 with respect to 𝑁, provided that the related matrix 𝑍 is left-invertible on [0,𝑇⋀𝜏_𝑑]. It is then natural to conjecture that the strategy given by (4.28) for 𝑁=𝒩_{⋅⋀𝜏𝑑} (resp., 𝑁=𝑁_{⋅⋀𝜏𝑑}) will coincide with the strategy 𝜁̂ (resp., 𝜁̌) of Proposition 4.13.

The following result examines the special case when 𝑛=0 in (4.27). As can be seen from formula (4.31) below, this corresponds to the assumption that the process 𝑁 represents the wealth processes of some portfolios of primary traded assets.

Corollary 4.16 (Min-variance hedging). Assume that 𝑛=0 in (4.27) and 𝑍 is left-invertible on [0,𝑇⋀𝜏_𝑑]. Then the strategy 𝜁 of Proposition 4.15 becomes 𝜁_𝑡=ℂov_𝑡𝑑𝑚^𝑡,𝑑𝛽_𝑡𝑋_𝑡𝕍ar^𝑡𝑑𝛽^𝑡𝑋_𝑡⁻¹.(4.30)

Proof. Under the present assumptions, we obtain from (4.27) 𝛽_𝑡𝑑𝑁_𝑡=Λ_𝑡𝑑𝛽_𝑡𝑋_𝑡,𝑡∈0,𝑇∧𝜏^𝑑.(4.31) Consequently, formula (4.28) reduces to 𝜁_𝑡=ℂov_𝑡𝑑𝑚^𝑡,Λ_𝑡𝑑𝛽_𝑡𝑋_𝑡𝕍ar^𝑡Λ^𝑡𝑑𝛽_𝑡𝑋_𝑡⁻¹Λ_𝑡,(4.32) and this in turn yields (4.31).

We recognize here a strategy, which is known to arise in the context of the min-variance hedging in incomplete markets. In the present setup, it was not derived by suitable optimization arguments, however, but obtained by simply postulating that the cost process should be orthogonal to prices of primary assets under the preselected risk-neutral probability measure ℚ. It should be noted that in the context of a game (or even American) option, the min-variance hedging approach should also incorporate optimization with respect to exercise times. This would lead to the optimization problem of the form (from the issuer's perspective, cf. (2.11) essinf_{𝜏𝑐∈𝒢0𝑇,𝜁}esssup_{𝜏𝑝∈𝒢0𝑇}𝔼_ℙ𝛽^𝜏𝑉_𝜏−𝛽₀𝜋0;𝜏_𝑝,𝜏_𝑐²∣𝒢₀(4.33) for a given level 𝑉₀ of the initial wealth, where 𝜏=𝜏_𝑐⋀𝜏_𝑝⋀𝜏_𝑑 and where the expectation is taken under the statistical probability measure ℙ. In a discrete-time setup, this problem was recently proposed and solved by Dolinsky and Kifer [36], who have also observed that the corresponding problem in a continuous-time framework is very difficult to deal with. Whether such an approach in continuous time is amenable to mathematical and practical solution is indeed far from trivial, so it is left as an open challenging problem.

Appendix

A. Auxiliary Lemmas

Recall that an 𝔽-pseudo-stopping time 𝜏 is a random time such that 𝔼_ℚ𝑀_𝜏=𝔼_ℚ𝑀₀ for every bounded 𝔽-martingale 𝑀 (see Nikeghbali and Yor [28, Remark  1]).

We work throughout under the standing Assumption 3.1.

Lemma A.1. (i)𝐺 is a nonincreasing process.
(ii) The 𝔾-stopping time 𝜏_𝑑 is an 𝔽-pseudo-stopping time.
(iii) Any 𝔽-local martingale stopped at 𝜏_𝑑 is a 𝔾-local martingale.
(iv) Any 𝔽-semimartingale stopped at 𝜏_𝑑 is a 𝔾-semimartingale.
(v) The integral process of a continuous integrand with respect to an 𝔽-martingale stopped at 𝜏_𝑑 is a 𝔾-local martingale.

Proof. Since 𝐺 is a continuous supermartingale, it admits the Doob-Meyer decomposition 𝐺=𝑀−𝐴 with a continuous martingale component 𝑀 [15, page 44, Lemma  4.24]. Hence 𝑀 is in fact constant, as a continuous martingale with finite variation, and thus (i) holds. By [28, Theorem  4.5], (i) implies (ii) (note that the continuity of the filtration 𝔽 is only used for the converse in [28, Theorem  4.5]). By [28, Theorem  4.4], (ii) implies (iii), which also immediately yields (iv). For (v), we note that an 𝔽-martingale stopped at 𝜏_𝑑 is a 𝔾-local martingale, by virtue of (iii). The integral process of a continuous (hence predictable and locally bounded) integrand, with respect to an 𝔽-martingale stopped at 𝜏_𝑑, is thus a 𝔾-local martingale (cf. Remark 2.2).

We recall the following well-known results. We refer the interested reader to Bielecki and Rutkowski [29, Lemma  5.1.2(ii) and Corollary  5.1.2] for (i) and Dellacherie et al. [37, page 186, Section  75] for (ii) (see also Proposition  9.12 of Nikeghbali [38]).

Lemma A.2. (i) Let 𝜒 be a 𝒢_∞-measurable random variable. For any 𝑡∈ℝ₊ such that one of the members of the following equality is well defined in ℝ (e.g., 𝜒 bounded from one side), the other one is well defined as well and one has that 1_{𝑡<𝜏𝑑}𝔼_ℚ𝜒∣𝒢_𝑡=1_{𝑡<𝜏𝑑}𝑒^Γ𝑡𝔼_ℚ1^𝑡<𝜏_𝑑^𝜒∣ℱ_𝑡.(A.1) In particular, if 𝜒 is 𝒢_𝑡-measurable then 1_{{𝑡<𝜏𝑑}}𝜒=1_{{𝑡<𝜏𝑑}}𝜒 where 𝜒=𝑒^Γ𝑡𝔼_ℚ(1_{{𝑡<𝜏𝑑}}𝜒∣ℱ_𝑡) is an ℱ_𝑡-measurable random variable. So for any 𝔾-adapted process 𝑌 over [0,𝑇], there exists an 𝔽-adapted process 𝑌 over [0,𝑇] such that 1_{{𝑡<𝜏𝑑}}𝑌_𝑡=1_{{𝑡<𝜏𝑑}}𝑌_𝑡,𝑡∈[0,𝑇].(A.2)
(ii) For any 𝔾-predictable process 𝑌 over [0,𝑇], there exists an 𝔽-predictable process 𝑌 over [0,𝑇] such that 1_{{𝑡≤𝜏𝑑}}𝑌_𝑡=1_{{𝑡≤𝜏𝑑}}𝑌_𝑡,𝑡∈[0,𝑇].(A.3)

Remark A.3. In the 𝔾-predictable case, the process 𝑌 satisfying (A.3) is uniquely defined under Assumption 3.1, by [37, page 186].

For any 𝑡∈[0,𝑇], we denote by ℱ^𝑡𝑇 the set of all 𝔽-stopping times with values in [𝑡,𝑇]. Also, given a stopping time 𝜏∈ℱ^0𝑇 let ℱ^𝑡𝑇 stand for the class {𝜏∈ℱ^𝑡𝑇;𝜏≥𝜏}. The following result examines the relevant properties of these classes of stopping times.

Lemma A.4. (i) If 𝜏∈𝒢^𝑡𝑇 for some 𝑡∈[0,𝑇] then there exists 𝜏̃∈ℱ^𝑡𝑇 such that 𝜏⋀𝜏_𝑑=𝜏̃⋀𝜏_𝑑. Moreover, if 𝜏∈𝒢^0𝑇 and if 𝜏∈𝒢^𝑡𝑇 for some 𝑡∈[0,𝑇] then one has 𝜏̃⋀𝜏_𝑑≥𝜏⋀𝜏_𝑑.
(ii) If 𝜏∈ℱ^0𝑇 and 𝜏∈𝒢^𝑡𝑇 for some 𝑡∈[0,𝑇] then there exists 𝜏̃∈ℱ^𝑡𝑇 such that 𝜏⋀𝜏_𝑑=𝜏̃⋀𝜏_𝑑.

Proof. Since 𝜏 is a 𝔾-stopping time, by [37, page 186, Section  75] there exists an 𝔽-stopping time 𝜏̂ such that 𝜏⋀𝜏_𝑑=𝜏̂⋀𝜏_𝑑. Moreover, since 𝜏∈𝒢^𝑡𝑇, one has 𝜏∧𝜏_𝑑=(𝜏∨𝑡)∧𝜏_𝑑=𝜏∧𝜏_𝑑∨𝑡∧𝜏_𝑑=𝜏̂∧𝜏_𝑑∨𝑡∧𝜏_𝑑=(𝜏̂∨𝑡)∧𝜏_𝑑,(A.4) so that we may take 𝜏̃=𝜏̂∨𝑡∈ℱ^𝑡𝑇. Moreover, if 𝜏⋀𝜏_𝑑≥𝜏⋀𝜏_𝑑 for some stopping time 𝜏∈𝒢^0𝑇, then we also have that 𝜏̃⋀𝜏_𝑑=𝜏⋀𝜏_𝑑≥𝜏⋀𝜏_𝑑, which proves (i).
For (ii), let 𝜏̆∈ℱ^𝑡𝑇 be such that 𝜏⋀𝜏_𝑑=𝜏̆⋀𝜏_𝑑, by (i). Assuming that 𝜏∈ℱ^0𝑇, one has that 𝜏̃=𝜏̆∨𝜏∈ℱ^𝑡𝑇. So 𝜏̃∧𝜏_𝑑=𝜏̆∨𝜏∧𝜏_𝑑=𝜏̆∧𝜏_𝑑∨𝜏∧𝜏_𝑑=𝜏̆∧𝜏_𝑑=𝜏∧𝜏_𝑑,(A.5) where the third equality holds, since 𝜏∈𝒢^𝑡𝑇 implies that 𝜏̆⋀𝜏_𝑑≥𝜏⋀𝜏_𝑑, by (i).

Let us recall that for any 𝔽-stopping time 𝜏 one has that (formula (A.6) can be found, e.g., in Dellacherie [39, T47]) ℚ𝜏_𝑑>𝜏∣ℱ_𝜏=𝑒^−Γ𝜏.(A.6) The following lemma is of independent interest.

Lemma A.5. Assume that 𝜏∈ℱ^𝑡𝑇 for some 𝑡∈[0,𝑇]. Then one has the following.
(i) For any ℱ_𝜏-measurable random variable 𝜒 such that at least one side of the following equality is well defined in ℝ (e.g., 𝜒 bounded from one side), the other one is also well defined and one has 𝔼_ℚ1^𝜏<𝜏_𝑑^𝜒∣𝒢_𝑡=1^𝑡<𝜏_𝑑^𝑒^Γ𝑡𝔼_ℚ𝑒^−Γ𝜏𝜒∣ℱ_𝑡.(A.7)
(ii) For any 𝔽-predictable process 𝑍 such that at least one side of the following equality is well defined in ℝ (e.g., 𝑍 is bounded from one side), the other one is also well defined and one has 𝔼_ℚ1_{{𝑡<𝜏𝑑≤𝜏}}𝑍_𝜏𝑑∣𝒢_𝑡=1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ^𝜏𝑡𝑍_𝑢𝑒^−Γ𝑢𝑑Γ_𝑢∣ℱ_𝑡.(A.8)
(iii) For any finite variation 𝔽-predictable process 𝐴 such that at least one side of the following equality is well defined in ℝ (e.g., the variation of 𝐴 over [0,𝑇] is bounded from one side), the other one is also well defined and one has 𝔼_ℚ^{𝜏∧𝜏𝑑}_{𝑡∧𝜏𝑑}𝑑𝐴_𝑢∣𝒢_𝑡=1^{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ^𝜏𝑡𝑒^−Γ𝑢𝑑𝐴_𝑢∣ℱ_𝑡.(A.9)

Proof. (i) Since 𝜏∈ℱ^𝑇𝑡, one has ℱ_𝑡⊂ℱ_𝜏⊂ℱ_𝑇, hence by Lemma A.2𝔼_ℚ1^𝜏<𝜏_𝑑^𝜒∣𝒢_𝑡==11^𝑡<𝜏_𝑑^_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ1^𝜏<𝜏_𝑑^𝜒∣ℱ_𝑡=1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ𝜒ℚ𝜏<𝜏_𝑑∣ℱ_𝜏∣ℱ_𝑡𝑒^Γ𝑡𝔼_ℚ𝜒𝑒^−Γ𝜏∣ℱ_𝑡,(A.10) where in the last equality one has used (A.6).
(ii) If suffices to prove the formula for an elementary predictable process of the form 𝑍_𝑡=1_]𝑢,𝑣](𝑡)𝐵_𝑢 for an arbitrary event 𝐵_𝑢∈ℱ_𝑢. For such a process, the formula follows easily from part (i).
(iii) One has that ^{𝜏∧𝜏𝑑}_{𝑡∧𝜏𝑑}𝑑𝐴_𝑢=1_{{𝑡<𝜏𝑑}}^{𝜏∧𝜏𝑑}_{𝑡∧𝜏𝑑}𝑑𝐴_𝑢=1_{{𝜏<𝜏𝑑}}^𝜏𝑡𝑑𝐴_𝑢+1_{{𝑡<𝜏𝑑≤𝜏}}^𝜏𝑑_𝑡𝑑𝐴_𝑢,(A.11) where 𝐴 is 𝔽-predictable. Using parts (i) and (ii), we obtain 𝔼_ℚ1^{𝜏<𝜏𝑑}^𝜏𝑡𝑑𝐴_𝑢∣𝒢_𝑡=1^{𝑡<𝜏𝑑}𝔼_ℚ𝑒^{Γ𝑡−Γ𝜏}^𝜏𝑡𝑑𝐴_𝑢∣ℱ_𝑡𝔼_ℚ,1^{{𝑡<𝜏𝑑≤𝜏}}^𝜏𝑑_𝑡𝑑𝐴_𝑢∣𝒢_𝑡=1^{{𝑡<𝜏𝑑}}𝔼_ℚ^𝜏𝑡^𝑠𝑡𝑑𝐴_𝑢𝑒^{Γ𝑡−Γ𝑠}𝑑Γ_𝑠∣ℱ_𝑡,(A.12) where, by Fubini's theorem, ^𝜏𝑡^𝑠𝑡𝑑𝐴_𝑢^{Γ𝑡−Γ𝑠}𝑑Γ_𝑠=^𝜏𝑡^𝑠𝑡𝑑𝐴_𝑢𝑒^{Γ𝑡−Γ𝑠}𝑑Γ_𝑠=^𝜏𝑡𝑒^{Γ𝑡−Γ𝑢}𝑑𝐴_𝑢−𝑒^{Γ𝑡−Γ𝜏}^𝜏𝑡𝑑𝐴_𝑢.(A.13) Hence 𝔼_ℚ^{𝜏∧𝜏𝑑}_{𝑡∧𝜏𝑑}𝑑𝐴_𝑢∣𝒢_𝑡=1^{{𝑡<𝜏𝑑}}𝔼_ℚ^𝜏𝑡𝑒^{Γ𝑡−Γ𝑢}𝑑𝐴_𝑢∣ℱ_𝑡=1^{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ^𝜏𝑡𝑒^−Γ𝑢𝑑𝐴_𝑢∣ℱ_𝑡,(A.14) as was expected.

In the next result, [𝑀_{⋅⋀𝜏𝑑},𝑁^𝑑] refers to the square bracket of 𝑀_{⋅⋀𝜏𝑑} and 𝑁^𝑑 with respect to the filtration 𝔾, where 𝑁^𝑑 denotes, as usual, the compensated jump-to-default process. This bracket is well defined, since 𝑁^𝑑 is a 𝔾-martingale and 𝑀_{⋅⋀𝜏𝑑} is a 𝔾-local martingale, by Lemma A.1(iii).

Lemma A.6. For any 𝔽-martingale 𝑀, the process [𝑀_{⋅⋀𝜏𝑑},𝑁^𝑑] is a 𝔾-local martingale.

Proof. Let us write 𝐻^𝑑=(1−𝐻)𝑒^Γ. Since Γ is continuous and nondecreasing, one has that 𝑑𝐻^𝑑𝑡=−𝐻^𝑑𝑡−𝑑𝑁^𝑑𝑡 (see [29]). By an application of Lemma A.5(i) with 𝜏=𝑇 and 𝜒=𝑒^Γ𝑇𝑀_𝑇, we obtain, for every 𝑡∈[0,𝑇], 𝐻^𝑑𝑡𝑀_{𝑡∧𝜏𝑑}=1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝑀_𝑡=1_{{𝑡<𝜏𝑑}}𝑒^Γ𝑡𝔼_ℚ𝑀_𝑇∣ℱ_𝑡=𝔼_ℚ1_{{𝑇<𝜏𝑑}}𝑒^Γ𝑇𝑀_𝑇∣𝒢_𝑡,(A.15) so 𝑀_{⋀𝜏𝑡⋅𝑑}𝐻^𝑑𝑡,𝑡∈[0,𝑇], is a 𝔾-uniformly integrable martingale, hence [𝑀_{⋅⋀𝜏𝑑},𝐻^𝑑] is a 𝔾-local martingale (since 𝑀_{⋅⋀𝜏𝑑} and 𝐻^𝑑 are 𝔾-local martingales). Now one has that 𝑀_{⋅∧𝜏𝑑},𝑁^𝑑_𝑡=Δ𝑀_𝜏𝑑𝐻_𝑡=−𝑒^−Γ𝑡𝑀_{𝑡∧𝜏𝑑},𝐻^𝑑_𝑡,(A.16) and thus the conclusion follows.

Acknowledgments

The research of T. R. Bielecki was supported by NSF Grant 0202851 and Moody’s Corporation grant 5-55411. The research of S. Crépey was supported by Ito33 and the Europlace Institute of Finance. The research of M. Jeanblanc was supported by Ito33, FIRN, and Moody’s Corporation grant 5-55411 and Fédération Bancaire Française. The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460.