Dynkin games with Poisson random intervention times
aa r X i v : . [ m a t h . O C ] J u l DYNKIN GAMES WITH POISSON RANDOM INTERVENTIONTIMES ∗ GECHUN LIANG, HAODONG SUN † Abstract.
This paper introduces a new class of Dynkin games, where the two players areallowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The valuefunction and the associated optimal stopping strategy are characterized by the solution of a backwardstochastic differential equation. The paper further provides a replication strategy for the game, andapplies the model to study the optimal conversion and calling strategies of convertible bonds, andtheir asymptotics when the Poisson intensity goes to infinity.
Key words. constrained Dynkin game, penalized BSDE, optimal stopping strategy, replicationstrategy, convertible bond.
AMS subject classifications.
1. Introduction.
Dynkin games are the games on stopping times, where twoplayers determine their optimal stopping times as their strategies. The game wasfirst introduced by Dynkin [14], and later generalized by Neveu [28] in 1970s. In thisgame, two players observe two stochastic processes, say L and U , and their aims areto maximize/minimize the expected value of the payoff R ( σ, τ ) = L τ { τ ≤ σ } + U σ { σ<τ } over stopping times τ and σ , respectively. In a discrete-time setting, under the as-sumption that U ≥ L , Neveu proved the existence of the game value and its associatedoptimal strategy.Since then, there has been a considerable development of Dynkin games. Thecorresponding continuous time models were developed, among others, by Bismut [6],Alario-Nazaret et al [1], Lepeltier and Maingueneau [21] and Morimoto [27]. In orderto relax the condition U ≥ L , Yasuda [36] proposed to extend the class of strategiesto randomized stopping times, and proved that the game value exists under merelyan integrability condition. Rosemberg et al [30], Touzi and Vielle [34] and Laraki andSolan [19] further extended his work in this direction. If the two players in the gameare with asymmetric payoffs, then it gives arise to a nonzero-sum Dynkin game. See,for example, Hamadene and Zhang [16] and more recently De Angelis et al [12] withmore references therein. A robust version of Dynkin games can be found in Bayraktarand Yao [3] if the players are ambiguous about their probability model.The setups in all the aforementioned works are either in continuous time wherestopping times take any value in a certain time interval, or in discrete time wherestopping times only take values in a pre-specified time grid. In this paper, we considera hybrid of continuous and discrete times, and introduce a new type of Dynkin games,where both players are allowed to stop at a sequence of random times generated by anexogenous Poisson process serving as a signal process. We call such a Dynkin game a constrained Dynkin game .The underlying Poisson process can be regarded as an exogenous constraint onthe players’ abilities to stop, so it may represent the liquidity effect, i.e. the Poisson ∗ The work is partially supported by a start-up research fund from the University of Warwick andNSFC No. 11771158. † Department of Statistics, University of Warwick, Coventry, CV4 7AL, U.K. Email adress: [email protected]; [email protected] Gechun Liang and Haodong Sun process indicates the times at which the underlying stochastic processes are availableto stop. Moreover, the Poisson process can also be seen as an information constraint.The players are allowed to make their stopping decisions at all times, but they areonly able to observe the underlying stochastic processes at Poisson times.Our first main result is Theorem 2.3, which characterizes the value of the con-strained Dynkin game and its associated optimal stopping strategy in terms of thesolution of a penalized backward stochastic differential equation (BSDE). The latteris widely used to approximate the solution of a reflected BSDE with double obstaclesand the corresponding continuous time Dynkin game. The main idea to solve the con-strained Dynkin game is to introduce a family of auxiliary games (see (3.9)-(3.10)),for which standard dynamic programming principle holds. Furthermore, followingfrom the convergence of penalized BSDE to reflected BSDE (see, for example, [11]and [15]) and the penalized BSDE characterization (2.6) of the constrained Dynkingame, we also make a connection with standard Dynkin games in continuous time.That is, the value of the constrained Dynkin game will converge to the value of itscontinuous time counterpart when the Poisson intensity goes to infinity.Our second main result is about replication of the constrained Dynkin game (seeTheorem 5.1). This has an application to the hedging problems in finance. In theexisting literature of financial applications of optimal stopping with Poisson times,the vast majority of papers focus on the risk-neutral valuation without even men-tioning the issue of hedging (see [13] and [20] among others). This somewhat lacks afoundation since, as is well known, the major argument supporting the risk-neutralvaluation is the existence of hedging strategies. We address this issue by constructinga replication strategy for the constrained Dynkin game (which in particular covers theoptimal stopping case). For such a replication problem, a new element is the jump riskstemming from the Poisson process. To hedge this jump risk, we introduce a pricingprocess generated by the jump times of the Poisson process. We then construct thereplication strategies recursively for a sequence of constrained Dynkin games startingfrom different Poisson arrival times, and for each game, the replication strategy isconstructed via two linear BSDEs. The first BSDE is used to replicate the payoff ofthe game before the next jump time, and the second equation is used to replicate thepayoff after this jump time.With the above replication strategies behind the risk-neutral valuation, we thenapply the constrained Dynkin game to study convertible bonds. In a convertible bond,the bondholder decides whether to keep the bond to collect coupons or to convert it tothe firm’s stocks. She will choose a conversion strategy to maximize the bond value.On the other hand, the issuing firm has the right to call the bond, and presumablyacts to maximize the equity value of the firm by minimizing the bond value. Thiscreates a two-person, zero-sum Dynkin game.Traditionally, convertible bond models often assume that both the bond holderand the firm are allowed to stopped at any stopping time adapted to the firm’s fun-damental (such as its stock prices). In reality, there may exist some liquidation con-straint as an external shock, and both players only make their decisions when sucha shock arrives. We model such a liquidation shock as the arrival times of an exoge-nous Poisson process, and thus the convertible bond model falls into the frameworkof constrained Dynkin games. A similar idea has first appeared in the modeling ofdebt run problems (see [23]), which can be formulated as optimal stopping problemswith Poisson arrival times.Furthermore, in a Markovian setting, we derive explicitly the optimal stopping ynkin games with Poisson random intervention times c , dividend rate q , interest rate r and surrender price K . For the firm,its optimal stopping strategy is to either call the bond back as soon as possible (if c ≥ rK ) or postpone the calling time of the bond as late as possible (if c < rK ). Incontrast, the investor’s optimal stopping strategy depends on the relationship between c and qK . If c > qK , the investor will delay her conversion time as late as possible;if c ≤ qK , her conversion strategy is determined by an optimal conversion boundary,the latter of which is obtained by solving a free boundary problem.Turning to the literature, the optimal stopping problem with constraints on thestopping times was introduced by Dupuis and Wang [13], when they used it to modelperpetual American options exercised at exogenous Poisson arrival times. See alsoLempa [20] and Menaldi and Robin [25] for further extensions of this type of optimalstopping problems. On the other hand, Liang [22] made a connection between suchkind of optimal stopping problems with penalized BSDE. The corresponding optimalswitching (impulse control) problems were studied by Liang and Wei [24] and morerecently by Menaldi and Robin [26] with more general signal times and state spaces.The study of convertible bonds dated back to Brennan and Schwartz [7] andIngersoll [17]. However, it was Sirbu et al [31] who first analyzed the optimal strategyof perpetual convertible bonds (see also Sirbu and Shreve [32] for the finite horizoncounterpart). They reduced the problem from a Dynkin game to an optimal stoppingproblem, and discussed when call precedes conversion and vice versa. Several morerealistic features of convertible bonds have been taken into account since then. Forexample, Bielecki et al [4] considered the problem of the decomposition of a convertiblebond into bond component and option component. Crepey and Rahal [10] studiedthe convertible bond with call protection, which is typically path dependent. Chenet al [9] considered the tax benefit and bankruptcy cost for convertible bonds. Fora complete literature review, we refer to the aforementioned papers with referencestherein.The paper is organized as follows. Section 2 contains the problem formulation andmain result, with its proof provided in section 3. In section 4, we establish a connectionwith standard Dynkin games. Section 5 is about replication of the constrained Dynkingame. In section 6, we apply the constrained Dynkin game to study the convertiblebonds in a Markovian setting, and derive the explicit optimal stopping strategies andthe corresponding free boundaries under various situations. Section 7 carries out anasymptotic analysis of the game values and the free boundaries when the Poissonintensity goes to infinity.
2. Constrained Dynkin games.
Let ( W t ) t ≥ be a d -dimensional standardBrownian motion defined on a filtered probability space (Ω , F , F = {F t } t ≥ , P ) with F being the minimal augmented filtration of W . Let { T i } i ≥ be the arrival times ofan independent Poisson process with intensity λ and minimal augmented filtration H = {H t } t ≥ . Denote the smallest filtration generated by F and H as G = {G t } t ≥ ,i.e. G t = F t ∨H t . Without loss of generality, we also assume that T = 0 and T ∞ = ∞ .Let T be a finite F -stopping time representing the terminal time of the game,and ξ be an F T -measurable random variable representing the corresponding payoff.Define a random variable M : Ω → N such that T M is the next Poisson arrival timeafter T , i.e. M ( ω ) = P i ≥ i { T i − ( ω ) ≤ T ( ω ) For any integer i ≥ 0, define the control set R T i ( λ ) = { G -stopping time τ for τ ( ω ) = T N ( ω ) where i ≤ N ≤ M ( ω ) } . The subscript T i in R T i ( λ ) represents the smallest stopping time that is allowed tochoose, and λ represents the intensity of the underlying Poisson process.Consider the following constrained Dynkin game , where two players choose theirrespective stopping times σ, τ ∈ R T ( λ ) in order to minimize/maximize the expectedvalue of the discounted payoff(2.1) R ( σ, τ ) = Z σ ∧ τ ∧ T e − rs f s ds + e − rT ξ { σ ∧ τ ≥ T } + e − rτ L τ { τ 0. Note that the above BSDE (2.6) is often used to construct the solution ofa reflected BSDE with two reflecting barriers L and U (cf. (4.3)). Intuitively, when V λ falls below L (or goes above U ), there will be a penalty λ ( L − V λ ) (or λ ( V λ − U ))incurred, so BSDE (2.6) is also refereed to as the penalized equation. Assumption 2.1. For t ∈ [0 , T ] , L t ≤ U t , a.s. Moreover, (i) when T is anunbounded stopping time, the running payoff f and the terminal payoffs L , U and ξ are all bounded; (ii) when T is a bounded stopping time, f , L , U and ξ are square-integrable, i.e. E [sup ≤ t ≤ T | X t | ] < ∞ for X = f, L, U and ξ . The assumption L ≤ U is crucial to the existence of the game value. On the otherhand, the conditions (i) and (ii) are to guarantee the existence and uniqueness of thesolution to BSDE (2.6), which will in turn be used to construct the game value andits associated optimal stopping strategy. Proposition 2.2. Suppose that Assumption 2.1 holds. Then, there exists aunique solution ( V, Z ) to BSDE (2.6). Moreover, (i) when T is unbounded, V is abounded and continuous F -adapted process, and Z ∈ M loc (0 , T ; R d ) , where the latterdenotes the space of all F -progressively measurable processes Z such that || Z || loc := E "Z t ∧ T | Z s | ds < ∞ for t ≥ (ii) when T is bounded, then V is a continuous square-integrable F -adapted process,and Z ∈ M (0 , T ; R d ) . The proof essentially follows from Theorem 4.1 in [29] (for bounded T ) and Section5 in [8] (for unbounded T ), so we omit its proof and refer to [29] and [8] for the details.We are now in a position to state the main result of this paper. Theorem 2.3. Suppose that Assumption 2.1 holds. Let ( V λ , Z λ ) be the uniquesolution to BSDE (2.6). Then, the value of the constrained Dynkin game (2.2)-(2.3)exists and is given by v λ = v λ = v λ = V λ . The corresponding optimal stoppingstrategy is given by (2.7) (cid:26) σ ∗ T = inf { T N ≥ T : V λT N ≥ U T N } ∧ T M ; τ ∗ T = inf { T N ≥ T : V λT N ≤ L T N } ∧ T M . Moreover, the value of the Dynkin game (2.4)-(2.5) also exists and is given by ˆ v λ =min { U , max { v λ , L }} , with the associated optimal stopping strategy σ ∗ T and τ ∗ T . Theorem 2.3 solves a wide class of problems in a unified man-ner, covering from Markovian to non-Markovian situations and from finite to infinitehorizons. In the one-dimensional homogenous Markovian setting, there usually existsa threshold strategy. For this, we will discuss a specific convertible-bond example insection 6. In the rest of the section, we list several path-dependent examples, whichare difficult to dealt with under Markovian framework (at least it needs a case-by-casestudy) but covered by Theorem 2.3.(i) Path-dependent payoffs L and U . Let T be fixed so it is a constant stoppingtime and S be a one-dimensional positive diffusion process adapted to F . For δ > Gechun Liang and Haodong Sun consider an Israeli option written on S with maturity T , where the holder may exerciseto get a normal claim but the writer is punished by an amount δS for annulling thecontract early (see [18]). The payoffs L and U may take the form L t = max { m, S ∗ t } and U t = max { m, S ∗ t } + δS t for m > S and S ∗ t = sup ≤ u ≤ t S u . This is so calledIsraeli Russian option. For L t = R t S u du and U t = R t S u du + δS t , it is called Israeliintegral option (see [2]). Under mild integrability assumption on S as in Assumption2.1, Theorem 2.3 shows that the values of both Israeli options exist and the associatedoptimal strategies can be characterized via the solution to (2.6).(ii) Path-dependent stopping time T . Stopping times are widely used in insuranceas indicators of a variety of risks. Let S be a one-dimensional positive diffusion processadapted to F . We may consider the following stopping times as the terminal time ofthe game: drawdown stopping time T = inf { t ≥ S ∗ t − S t ≥ m } for m ≥ T = inf { t ≥ m : R t { S u ∈ A } du ≥ m } for A ⊂ R + . Notethat unlike the standard first-passage-time (see θ λ in section 6), both types of path-dependent stopping times need tailor-made analysis under Markovian framework, butcan be covered by Theorem 2.3 in a unified manner. 3. Proof of Theorem 2.3. We first give an equivalent formulation of the con-strained Dynkin game (2.2)-(2.3). Given the arrival time T i , define pre- T i σ -field G T i = A ∈ _ s ≥ G s : A ∩ { T i ≤ s } ∈ G s for s ≥ and ˜ G = {G T i } i ≥ . It is obvious that the upper and lower values of the constrainedDynkin game can be rewritten as(3.1) v λ = inf N σ ∈N ( λ ) sup N τ ∈N ( λ ) E [ R ( T N σ , T N τ )] , (3.2) v λ = sup N τ ∈N ( λ ) inf N σ ∈N ( λ ) E [ R ( T N σ , T N τ )] , where N n ( λ ) = n ˜ G -stopping time N for n ≤ N ( ω ) ≤ M ( ω ) o . The subscript n in N n ( λ ) represents the smallest stopping time that is allowed tochoose, and λ represents the intensity of the underlying filtration ˜ G . Both players areallowed to stop at a sequence of integers n, n + 1 , · · · , M .We also observe that a pair of processes (cid:0) V λ , Z λ (cid:1) solve (2.6), if and only if thecorresponding discounted processes ( Q λt , ˜ Z λt ) = ( e − rt V λt , e − rt Z λt ), for t ∈ [0 , T ], solvethe following BSDE(3.3) Q λt ∧ T = ˜ ξ + Z Tt ∧ T (cid:20) ˜ f s + λ (cid:16) ˜ L s − Q λs (cid:17) + − λ (cid:16) Q λs − ˜ U s (cid:17) + (cid:21) ds − Z Tt ∧ T ˜ Z λs dW s , where ˜ ξ = e − rT ξ and ˜ φ s = e − rs φ s for φ = f, L, U .Thus, to prove Theorem 2.3, it is equivalent to show that Q λ = q λ = q λ , where(3.4) q λ := inf N σ ∈N ( λ ) sup N τ ∈N ( λ ) E h ˜ R ( T N σ , T N τ ) i , ynkin games with Poisson random intervention times q λ := sup N τ ∈N ( λ ) inf N σ ∈N ( λ ) E h ˜ R ( T N σ , T N τ ) i , with ˜ R ( σ, τ ) = Z σ ∧ τ ∧ T ˜ f s ds + ˜ ξ { σ ∧ τ ≥ T } + ˜ L τ { τ Lemma 3.1. Suppose that Assumption 2.1 holds. Then, for any ≤ n ≤ M , thesolution of BSDE (3.3) at time T n − is the unique solution of the recursive equation Q λT n − = E "Z T n ∧ TT n − ˜ f s ds + ˜ ξ { T n >T } (3.7)+ (cid:16) { Q λTn ≥ ˜ U Tn } ˜ U T n + { Q λTn ≤ ˜ L Tn } ˜ L T n + { ˜ L Tn Suppose that Assumption 2.1 holds. Then, for any ≤ n ≤ M , thevalue of the auxiliary constrained Dynkin game (3.9)-(3.10) exists. Its value, denotedby ˆ q λT n − , satisfies the recursive equation (3.8), namely, ˆ q λT n − = min ( ˜ U T n − , max ( E " Z T n ∧ TT n − ˜ f s ds + ˜ ξ { T n >T } + ˆ q λT n { T n ≤ T } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G T n − , ˜ L T n − )) . Hence, ˆ q λT n − = ˆ Q λT n − a.s. The optimal stopping strategy of (3.9)-(3.10) is given by (3.11) (cid:26) ˆ N σ, ∗ n − = inf { N ≥ n − q λT N = ˜ U T N } ∧ M ;ˆ N τ, ∗ n − = inf { N ≥ n − q λT N = ˜ L T N } ∧ M. Proof . Without loss of generality, we may assume ˜ f s = 0. Step 1. Since T M − ≤ T < T M , the upper value of the auxiliary game (3.9) isequivalent toˆ q λT n − = ess inf N σ ∈N n − ( λ ) ess sup N τ ∈N n − ( λ ) E h ˜ ξ { N σ = N τ = M } + ˜ L T Nτ { n − ≤ N τ ≤ M − ,N τ ≤ N σ } + ˜ U T Nσ { n − ≤ N σ ≤ M − ,N σ Next, we show (3.12)-(3.13). Indeed, for i = M − q λT M − = ess inf N σ ∈N M − ( λ ) ess sup N τ ∈N M − ( λ ) E h ˜ ξ { N σ = N τ = M } + ˜ L T M − { M − N τ ≤ N σ } + ˜ U T M − { M − N σ In general, for n − ≤ i ≤ M − 2, we haveˆ q λT i = ess inf N σ ∈N i ( λ ) ess sup N τ ∈N i ( λ ) E h ˜ ξ { N σ = N τ = M } + ˜ L T Nτ { i ≤ N τ ≤ M − ,N τ ≤ N σ } + ˜ U T Nσ { i ≤ N σ ≤ M − ,N σ In this step, we show the operations ess inf N σ ∈N i +1 ( λ ) ess sup N τ ∈N i +1 ( λ ) and E [ ·|G T i ] are interchangeable, i.e. (3.16) below holds. To this end, for fixed i and N σ ∈ N i ( λ ), we note that the family(3.14) (cid:16) E h ˜ R i ( T N σ , T N τ ) |G T i i , N τ ∈ N i ( λ ) (cid:17) is an increasing directed set. Indeed, if we choose arbitrary N τ , N τ ∈ N i ( λ ) and let X j = E h ˜ R i ( T N σ , T N τj ) |G T i i , for j = 1 , 2. Then, defining the stopping time N τ as N τ = N τ { X ≥ X } + N τ { X It remains to prove that (cid:16) ˆ N σ, ∗ n − , ˆ N τ, ∗ n − (cid:17) in (3.11) are indeed the optimalstopping times for the auxiliary Dynkin game (3.9)-(3.10), i.e. for every ( N σ , N τ ) ∈ ynkin games with Poisson random intervention times N n − ( λ ) × N n − ( λ ), E h ˜ R n − (cid:16) T ˆ N σ, ∗ n − , T N τ (cid:17) |G T n − i ≤ E h ˜ R n − (cid:16) T ˆ N σ, ∗ n − , T ˆ N τ, ∗ n − (cid:17) |G T n − i ≤ E h ˜ R n − (cid:16) T N σ , T ˆ N τ, ∗ n − (cid:17) |G T n − i . To this end, it suffices to prove that(i) (cid:18) ˆ q λT m ∧ ˆ Nσ, ∗ n − ∧ ˆ Nτ, ∗ n − (cid:19) m ≥ n − is a ˜ G -martingale;(ii) (cid:18) ˆ q λT m ∧ ˆ Nσ, ∗ n − ∧ Nτ (cid:19) m ≥ n − is a ˜ G -supermartingale for any N τ ∈ N n − ( λ );(iii) (cid:18) ˆ q λT m ∧ Nσ ∧ ˆ Nτ, ∗ n − (cid:19) m ≥ n − is a ˜ G -submartingale for any N σ ∈ N n − ( λ ).Indeed, we have E (cid:20) ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − ∧ ˆ Nτ, ∗ n − (cid:12)(cid:12)(cid:12)(cid:12) G T m (cid:21) = E m X j = n − { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − = j } + { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − ≥ m +1 } ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − ∧ ˆ Nτ, ∗ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G T m = m X j = n − { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − = j } ˆ q λT j + { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − ≥ m +1 } E h ˆ q λT m +1 (cid:12)(cid:12)(cid:12) G T m i = m X j = n − { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − = j } ˆ q λT j + { ˆ N σ, ∗ n − ∧ ˆ N τ, ∗ n − ≥ m +1 } ˆ q λT m = ˆ q λT m ∧ ˆ Nσ, ∗ n − ∧ ˆ Nτ, ∗ n − , where the second last equality follows from the definition of (cid:16) ˆ N σ, ∗ n − , ˆ N τ, ∗ n − (cid:17) in (3.11),so the martingale property (i) has been proved.To prove the supermartingale property (ii), we note that E (cid:20) ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − ∧ Nτ (cid:12)(cid:12)(cid:12)(cid:12) G T m (cid:21) = E (cid:20) ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − { N τ ≥ m +1 } + ˆ q λT ˆ Nσ, ∗ n − ∧ Nτ { N τ ≤ m } (cid:12)(cid:12)(cid:12)(cid:12) G T m (cid:21) = E m X j = n − { ˆ N σ, ∗ n − = j } + { ˆ N σ, ∗ n − ≥ m +1 } ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − { N τ ≥ m +1 } + ˆ q λT ˆ Nσ, ∗ n − ∧ Nτ { N τ ≤ m } (cid:12)(cid:12)(cid:12)(cid:12) G T m (cid:21) = m X j = n − { ˆ N σ, ∗ n − = j } ˆ q λT j + { ˆ N σ, ∗ n − ≥ m +1 } E h ˆ q λT m +1 (cid:12)(cid:12)(cid:12) G T m i { N τ ≥ m +1 } + ˆ q λT ˆ Nσ, ∗ n − ∧ Nτ { N τ ≤ m } . Using the definition of ˆ N σ, ∗ n − in (3.11), we further have E h ˆ q λT m +1 (cid:12)(cid:12)(cid:12) G T m i ≤ max n E h ˆ q λT m +1 (cid:12)(cid:12)(cid:12) G T m i , ˜ L T m o = ˆ q λT m on { ˆ N σ, ∗ n − ≥ m + 1 } . Gechun Liang and Haodong Sun In turn, E (cid:20) ˆ q λT ( m +1) ∧ ˆ Nσ, ∗ n − ∧ Nτ (cid:12)(cid:12)(cid:12)(cid:12) G T m (cid:21) ≤ m X j = n − { ˆ N σ, ∗ n − = j } ˆ q λT j + { ˆ N σ, ∗ n − ≥ m +1 } ˆ q λT m { N τ ≥ m +1 } + ˆ q λT ˆ Nσ, ∗ n − ∧ Nτ { N τ ≤ m } = ˆ q λT m ∧ ˆ Nσ, ∗ n − { N τ ≥ m +1 } + ˆ q λT ˆ Nσ, ∗ n − ∧ Nτ { N τ ≤ m } = ˆ q λT m ∧ ˆ Nσ, ∗ n − ∧ Nτ , which proves the supermartingale property (ii). Likewise, the submartingale property(iii) can be proved in a similar way, and the proof of the lemma is completed.We are now in a position to prove Theorem 2.3. By Lemmas 3.1 and 3.2, we have Q λ = E "Z T ∧ T ˜ f s ds + ˜ ξ { T >T } + ˆ Q λT { T ≤ T } = E "Z T ∧ T ˜ f s ds + ˜ ξ { T >T } + ˆ q λT { T ≤ T } ≥ E (cid:20) Z T ∧ T ˜ f s ds + ˜ ξ { T >T } + E h ˜ R ( T ˆ N σ, ∗ , T N τ ) |G T i { T ≤ T } (cid:21) (3.17)for any N τ ∈ N ( λ ), where last inequality follows from the supermartingale property(ii). Moreover, recall that E h ˜ R ( T ˆ N σ, ∗ , T N τ ) |G T i = E "Z T ˆ Nσ, ∗ ∧ T Nτ ∧ TT ∧ T ˜ f s ds + ˜ ξ n T ˆ Nσ, ∗ ∧ T Nτ ≥ T o + ˜ L T Nτ n T Nτ 4. Connection with standard Dynkin games. We show that, when λ → ∞ ,the value v λ of the constrained Dynkin game converges to the value of a standardDynkin game. The setup is the same as in section 2 except that the control set isreplaced with R t , which is defined as R t = { F -stopping time τ for t ≤ τ ( ω ) ≤ T } . Define the corresponding upper and lower values of the standard Dynkin game as(4.1) v = inf σ ∈R sup τ ∈R E [ R ( σ, τ )] , (4.2) v = sup τ ∈R inf σ ∈R E [ R ( σ, τ )] . This game is said to have value v if v = v = v , and ( σ ∗ , τ ∗ ) ∈ R × R is calleda saddle point of the game if E [ R ( σ ∗ , τ )] ≤ E [ R ( σ ∗ , τ ∗ )] ≤ E [ R ( σ, τ ∗ )] for every( σ, τ ) ∈ R × R . Proposition 4.1. Suppose that Assumption 2.1 holds and, moreover, both L and U are continuous and satisfy L T ≤ ξ ≤ U T . Then, the value v of the Dynkin game(4.1)-(4.2) exists and, moreover, lim λ ↑∞ v λ = v. Proof . To solve the Dynkin game (4.1)-(4.2), we introduce the following reflectedBSDE defined on a random horizon [0 , T ]:(4.3) V t ∧ T = ξ + Z Tt ∧ T ( f s − rV s ) ds + Z Tt ∧ T dK + s − Z Tt ∧ T dK − s − Z Tt ∧ T Z s dW s for t ≥ 0, under the constraints (i) L t ≤ V t ≤ U t , for 0 ≤ t ≤ T ; (ii) R T ( V t − L t ) dK + t = R T ( U t − V t ) dK − t = 0. By a solution to the reflected BSDE (4.3), we mean a tripletof F -progressively measurable processes ( V, Z, K ), where K := K + − K − with K + and K − being increasing processes starting from K +0 = K − = 0.4 Gechun Liang and Haodong Sun It follows from Hamadene et al [15] that (4.3) is well-posed and admits a uniquesolution. Using arguments similar to the ones in Cvitanic and Karatzas [11], it isstandard to show that the value of the Dynkin game (4.1)-(4.2) exists and is given bythe solution of the reflected BSDE (4.3), i.e. v = v = v = V .To prove the second assertion, we note that BSDE (2.6) can be regarded as asequence of penalized BSDEs for (4.3), where the local time processes K + and K − are approximated by K λ, + t := Z t λ (cid:0) L s − V λs (cid:1) + ds ; K λ, − t := Z t λ (cid:0) V λs − U s (cid:1) + ds, with K λ := K λ, + − K λ, − . Since lim λ ↑∞ E [sup t ∈ [0 ,T ] | V λt − V t | ] = 0 (see, for example,[15] and [11]), the second assertion follows immediately. 5. Replication of constrained Dynkin games. In this section, we discussabout replication of the constrained Dynkin game. This provides a foundation for therisk-neutral valuation of convertible bonds introduced in the next section.We interpret P as a risk-neutral probability measure, and let ¯ N t := P n ≥ { T n ≤ t } − λt , t ≥ 0, be the compensated Poisson martingale. Suppose there exist ( d + 2) under-lying assets, whose pricing processes follow dS it = S it ( r − q i ) dt + S it σ i dW t , ≤ i ≤ d ;(5.1) dP t = P t − rdt + P t − ¯ σd ¯ N t ;(5.2) dB t = B t rdt, (5.3)where r > σ > P , and q i and σ i := ( σ ij ) ≤ j ≤ d represent, respectively, the dividend and volatility of S i . Assumethat the volatility matrix σ := ( σ ij ) ≤ i,j ≤ d is invertible. The risky assets ( S i ) ≤ i ≤ d are the underlying assets used to hedge the Brownian noise of the game. The riskyasset P is used to hedge the jump risk of the Poisson process. In practice, it could bethe cash flow of a credit default swap delivering payoffs at jump times ( T n ) n ≥ (see,for example, [5] for the single jump case). Finally, B represents the risk-free bankaccount.From section 2.3 (Lemmas 3.1 and 3.2 in particular), we know that the solution V λ of BSDE (2.6) provides the values of the constrained Dynkin game (2.2)-(2.3)starting at different Poisson arrival times T n − for 1 ≤ n ≤ M , and they satisfy therecursive equation e − rT n − V λT n − = E "Z T n ∧ TT n − e − rs f s ds + e − rT ξ { T n >T } (5.4) + e − rT n min { U T n , max { V λT n , L T n }} { T n ≤ T } |G T n − (cid:3) . Thus, the discounted payoff of the game starting at T n − is Z TT n − e − rs f s ds + e − rT ξ ! { T n >T } (5.5) + Z T n T n − e − rs f s ds + e − rT n min { U T n , max { V λT n , L T n }} ! { T n ≤ T } , ynkin games with Poisson random intervention times V λT n being the value of the game starting at T n . Compared to the original payoff(2.1), the above payoff (with n = 1) only involves the first Poisson arrival time T ,and the optimality of stopping strategies is encoded in V λT . Thus, the replication ofthe constrained Dynkin game (2.2)-(2.3) naturally depends on the replication of thesame game but starting at Poisson arrival time T , the later of which in turn dependson the replication of the game starting from T and so on and so forth. In particular,the discounted payoff of the game starting at T M − is R TT M − e − rs f s ds + e − rT ξ, since T M − ≤ T < T M by the definition of the random variable M .For 1 ≤ n ≤ M , consider the constrained Dynkin game starting at Poissonarrival time T n − . We aim to construct a replication portfolio ( π S,nt , π P,nt , π B,nt ), t ∈ [ T n − , T ], to replicate the discounted payoff (5.5), where π S,n = ( π S i ,n ) ≤ i ≤ d representthe amount of the money invested in ( S i ) ≤ i ≤ d , and π P,n and π B,n represent theamount of the money invested in P and B , respectively. Let X nt be the correspondingwealth of each player at time t . Then, X nt = P di =1 π S i ,nt + π P,nt + π B,nt , and theself-financing condition implies that X nt = X nT n − + Z tT n − d X i =1 π S i ,ns S is dS is + π P,ns P s − dP s + π B,ns B s dB s + d X i =1 q i π S i ,ns ds ! (5.6) = X nT n − + Z tT n − (cid:0) rX ns ds + π S,ns σdW s + π P,ns ¯ σd ¯ N s (cid:1) , for t ∈ [ T n − , T ]. The problem is to find a replication portfolio ( π S,n , π P,n , π B,n )such that the discounted wealth e − rT X nT replicates the discounted payoff (5.5), andto prove that X nT n − = V λT n − , i.e. the constrained Dynkin game starting from T n − is replicable and its value is indeed given by V λT n − . Theorem 5.1. Let ( Y ξ,θ , Z ξ,θ ) be the unique solution of the linear BSDE definedon the random horizon [ θ, T ] with a parameter θ ∈ [0 , T ] , i.e. (5.7) Y ξ,θt ∧ T = Z Tθ e r ( T − s ) f s ds + ξ ! − Z Tt ∧ T rY ξ,θs ds − Z Tt ∧ T Z ξ,θs dW s , for t ≥ θ . Then, for the constrained Dynkin game starting at T M − , its replicationwealth and the corresponding replication portfolio are given by X Mt = Y ξ,T M − t ;(5.8) ( π S,Mt , π P,Mt , π B,Mt ) = ( Z ξ,T M − t σ − , , X Mt − π S,Mt ) , t ∈ [ T M − , T ] , where σ − is the inverse of the volatility matrix ( σ ij ) ≤ i,j ≤ d . Moreover, the value ofthe game is given by X MT M − = V λT M − .In general, let ( Y θ , Z θ ) be the unique solution of the linear BSDE defined on [ θ, T ] with a parameter θ ∈ [0 , T ] , i.e. (5.9) Y θt ∧ T = Z Tθ e r ( T − s ) f s ds + ξ ! − Z Tt ∧ T (cid:2) rY θs − λ ( Y θ,ss − Y θs ) (cid:3) ds − Z Tt ∧ T Z θs dW s , for t ≥ θ , and ( Y ( θ, ¯ θ ) , Z ( θ, ¯ θ )) be the unique solution of the linear BSDE defined on Gechun Liang and Haodong Sun [¯ θ, T ] with parameters θ, ¯ θ satisfying ≤ θ < ¯ θ ≤ T , i.e. Y θ, ¯ θt ∧ T = Z ¯ θθ e r ( T − s ) f s ds + e r ( T − ¯ θ ) min { U ¯ θ , max { V λ ¯ θ , L ¯ θ }} ! (5.10) − Z Tt ∧ T rY θ, ¯ θs ds − Z Tt ∧ T Z θ, ¯ θs dW s , for t ≥ θ , where V λ is the unique solution to BSDE (2.6). Then, for the constrainedDynkin game starting at T n − for ≤ n ≤ M − , its replication wealth and thecorresponding replication portfolio are given by X nt = Y T n − t { t 0) satisfies the wealth equation (5.6) and, moreover,by applying Itˆo’s formula to e − rt Y ξ,θt , we further have e − r ( t ∧ T ) Y ξ,θt ∧ T = Z Tθ e − rs f s ds + e − rT ξ ! − Z Tt ∧ T e − rs Z ξ,θs dW s . Thus, e − rT Y ξ,T M − T replicates the discounted payoff (5.5) with n = M . Furthermore, X MT M − = Y ξ,T M − T M − = E " Z TT M − e − r ( s − T M − ) f s ds + e − r ( T − T M − ) ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G T M − = V λT M − . In general, we prove the assertion for 1 ≤ n ≤ M − T n and X n +1 T n = V λT n . Then, for the gamestarting at T n − , by the construction of X n in (5.11) and the terminal data for BSDEs(5.9) and (5.10), we have e − rT X nT = e − rT (cid:16) Y T n − T { T P,n ) given in (5.11) indeed satisfies the wealthequation (5.6). To this end, note that X nt = X nt ∧ T n − + ( X nt ∧ T n − X nt ∧ T n − ) + ( X nt − X nt ∧ T n ) , ynkin games with Poisson random intervention times t ∈ [ T n − , T ]. Since X nt ∧ T n − = Y T n − t ∧ T n − by the definition of X n , we have X nt ∧ T n − = Y T n − T n − + Z t ∧ T n − T n − (cid:2) rY T n − s − λ ( Y T n − ,ss − Y T n − s ) (cid:3) ds + Z t ∧ T n − T n − Z T n − s dW s = X nT n − + Z t ∧ T n − T n − rY T n − s ds + Z t ∧ T n − T n − Z T n − s dW s − Z tT n − ( Y T n − ,ss − Y T n − s ) { s ≤ T n } λds. Furthermore, at the Poisson arrival time T n , X n has a jump with size X nt ∧ T n − X nt ∧ T n − = (cid:16) Y T n − ,T n T n − Y T n − T n − (cid:17) { T n ≤ t } = Z t ( Y T n − ,ss − Y T n − s ) { s ≤ T n } ) dN s . On the other hand, since X nt = Y T n − ,T n t on the event { t ≥ T n } , we have X nt − X nt ∧ T n = Z tt ∧ T n rY T n − ,T n s ds + Z tt ∧ T n Z T n − ,T n s dW s . In turn, we deduce, using the constructions of X n , π S,n and π P,n in (5.11), that X nt = X nT n − + Z tT n − rX ns ds + Z tT n − π S,ns σdW s + Z tT n − π P,ns ¯ σd ¯ N s . Finally, applying Itˆo’s formula to e − rt X nt and using (5.12), we obtain that e − r ( t ∧ T ) X nt ∧ T = Z TT n − e − rs f s ds + e − rT ξ ! { T n >T } + Z T n T n − e − rs f s ds + e − rT n min { U T n , max { V λT n , L T n }} ! { T n ≤ T } − Z Tt ∧ T e − rs π S,ns σdW s − Z Tt ∧ T e − rs π P,ns ¯ σd ¯ N s . In turn, taking conditional expectation with respect to G T n − and using (5.4), weconclude that X nT n − = V λT n − . 6. Application to convertible bonds with random intervention times. In this section, using the constrained Dynkin game introduced in section 2, we studyconvertible bonds for which both players are only allowed to stop at a sequence ofrandom intervention times.Traditionally, convertible bond models often assume that both the bond holderand the issuing firm are allowed to stopped at any stopping time adapted to the firm’sfundamental (such as its stock prices). In reality, there may exist some liquidationconstraint as an external shock, and both players only make their decisions whensuch a shock arrives. We model such a liquidation shock as the arrival times of anexogenous Poisson process. A similar idea has first appeared in the modeling of debtrun problems (see [23]), which can be formulated as optimal stopping problems withPoisson arrival times. Assumption 6.1. Let d = 1 . The firm’s stock price S s , under the risk-neutralprobability measure P , follows (6.1) S st = s + Z t ( r − q ) S su du + Z t σS su dW u , Gechun Liang and Haodong Sun with S s = s > , where the constants r , q , σ represent the risk-free interest rate,the dividend rate and the volatility of the stock, satisfying the parameter assumption r > q . The firm issues convertible bonds as perpetuities with a constant coupon rate c .Consider an investor purchasing a share of this convertible bond at initial time t = 0.By holding the convertible bond, the investor will continuously receive the couponrate c from the firm until the contract is terminated. The investor has the right toconvert her bond to the firm’s stocks, while the firm has the right to call the bondand force the bondholder to surrender her bond to the firm at a sequence of Poissonarrival times { T n } n ≥ with a constant intensity λ > 0. Hence, there are two situationsthat the contract maybe terminated:(i) if the firm calls the bond at some G -stopping time σ firstly, the bondholderwill receive a pre-specified surrender price K at time σ ;(ii) if the investor chooses to convert her bond at some G -stopping time τ firstlyor both players choose to stop the contract simultaneously, the bondholder will obtain γS τ at time τ from converting her bond with a pre-specified conversion rate γ ∈ (0 , t = 0:(6.2) P ( s ; σ, τ ) = Z σ ∧ τ e − ru c du + e − rτ γS sτ { τ ≤ σ } + e − rσ K { σ<τ } , with σ, τ ∈ ˜ R T ( λ ), where˜ R T i ( λ ) = { G -stopping time τ for τ ( ω ) = T N ( ω ) where N ≥ i } . The investor will choose τ ∈ ˜ R T ( λ ) to maximize the bond value, while the firmwill choose σ ∈ ˜ R T ( λ ) to maximize the equity value of the firm by minimizing thebond value. This leads to a constrained Dynkin game as introduced in section 2. Theupper value and lower value of this constrained convertible bond are(6.3) v λ ( s ) = inf σ ∈ ˜ R T ( λ ) sup τ ∈ ˜ R T ( λ ) E [ P ( s ; σ, τ )] , (6.4) v λ ( s ) = sup τ ∈ ˜ R T ( λ ) inf σ ∈ ˜ R T ( λ ) E [ P ( s ; σ, τ )] . Note that the constrained Dynkin game in section 2 does not exactly cover theabove constrained convertible bond, since the model in section 2 has a random ter-minal time T , while the convertible bond is perpetual. However, in the followingproposition, we shall show that when s ≥ ¯ s λ := q + λr + λ Kγ , the optimal stopping strategy is trivial. In this region, it is always optimal for boththe investor and the firm to stop at the first Poisson arrival time. Intuitively, whenthe stock price is high, the stock is attractive enough to lead both the investor to The case r ≤ q can be treated in a similar way.ynkin games with Poisson random intervention times Proposition 6.2. Suppose that Assumption 6.1 holds. Then, the value of theconstrained convertible bond, denoted as v λ ( s ) , exists and satisfies L λ ( s ) ≤ v λ ( s ) ≤ U λ for s ∈ (0 , ∞ ) , where L λ ( s ) := cr + λ + λq + λ γs ; U λ := c + λKr + λ . Moreover, in the domain s ∈ (cid:2) ¯ s λ , ∞ (cid:1) , it holds that v λ ( s ) = L λ ( s ) , and the optimalstopping strategy is τ ∗ ,λ = σ ∗ ,λ = T .Proof . Choosing τ ≡ T in (6.4) yields a lower bound of the convertible bondprice: v λ ( s ) = sup τ ∈ ˜ R T ( λ ) inf σ ∈ ˜ R T ( λ ) E (cid:20)Z σ ∧ τ e − ru c du + e − rτ γS sτ { τ ≤ σ } + e − rσ K { σ<τ } (cid:21) ≥ inf σ ∈ ˜ R T ( λ ) E "Z T e − ru c du + e − rT γS sT = E (cid:20)Z ∞ λe − λm (cid:18)Z m e − ru c du + e − rm γS sm (cid:19) dm (cid:21) = Z ∞ λe − λm Z m e − ru c du dm + λγ E (cid:20)Z ∞ e − ( r + λ ) m S sm dm (cid:21) = cr + λ + λq + λ γs = L λ ( s ) , where we used the integration by parts in the last equality.On the other hand, by choosing σ ≡ T in (6.3), we get an upper bound of theconvertible bond price: v λ ( s ) = inf σ ∈ ˜ R T ( λ ) sup τ ∈ ˜ R T ( λ ) E (cid:20)Z σ ∧ τ e − ru c du + e − rτ γS sτ { τ ≤ σ } + e − rσ K { σ<τ } (cid:21) ≤ sup τ ∈ ˜ R T ( λ ) E "Z T e − ru c du + e − rT γS sT { τ = T } + e − rT K { τ>T } = cr + λ + max (cid:26) λq + λ γs, λKr + λ (cid:27) = max { L λ ( s ) , U λ } . In the domain s ∈ (cid:2) ¯ s λ , ∞ (cid:1) , we always have L λ ( s ) ≥ U λ so v λ ( s ) ≤ L λ ( s ) ≤ v λ ( s ) . Thus, the value of the convertible bond exists, and v λ ( s ) = ¯ v λ ( s ) = v λ ( s ) = L λ ( s ) , with the optimal stopping strategy τ ∗ ,λ = σ ∗ ,λ = T .In the domain s ∈ (cid:0) , ¯ s λ (cid:1) , we have L λ ( s ) < U λ . Introduce an F -stopping time θ λ := inf { u ≥ S su ≥ ¯ s λ } . Then, it follows from the dynamic programming principle that v λ ( s ) = inf σ ∈ ˜ R T ( λ ) sup τ ∈ ˜ R T ( λ ) E "Z σ ∧ τ ∧ θ λ e − ru c du + e − rθ λ v λ ( S sθ λ ) { σ ∧ τ ≥ θ λ } + (cid:0) e − rτ γS sτ { τ ≤ σ } + e − rσ K { σ<τ } (cid:1) { σ ∧ τ<θ λ } ] . Gechun Liang and Haodong Sun By the definition of the stopping time θ λ , v λ (cid:0) S sθ λ (cid:1) = v λ (cid:0) ¯ s λ (cid:1) = L λ (¯ s λ ) = U λ . Thus,in the domain s ∈ (0 , ¯ s λ ), (6.3)-(6.4) are equivalent to(6.5) v λ ( s ) = inf σ ∈ ˜ R T ( λ ) sup τ ∈ ˜ R T ( λ ) E h ˜ P ( s ; σ, τ ) i , (6.6) v λ ( s ) = sup τ ∈ ˜ R T ( λ ) inf σ ∈ ˜ R T ( λ ) E h ˜ P ( s ; σ, τ ) i , where the payoff ˜ P ( s ; σ, τ ) is Z σ ∧ τ ∧ θ λ e − ru c du + e − rθ λ U λ { σ ∧ τ ≥ θ λ } + e − rτ γS sτ { τ<θ λ ,τ ≤ σ } + e − rσ K { σ<θ λ ,σ<τ } . Note that if we introduce the G -stopping time(6.7) T M := inf { T N ≥ θ λ : N ≥ } , since the payoff function ˜ P ( s ; σ, τ ) does not change after T M , we may replace thecontrol set ˜ R T ( λ ) in (6.5)-(6.6) with R T ( λ ), the latter of which consists of G -stoppingtimes T , T , · · · , T M .Now, we apply Theorem 2.3 with T = θ λ , L t = γS st , U t = K , f t = c and ξ = U λ to (6.5)-(6.6), and obtain the existence of the value of the convertible bond in thedomain s ∈ (0 , ¯ s λ ).Thanks to the above proposition, we focus our analysis to the domain s ∈ (cid:0) , ¯ s λ (cid:1) in the rest of this section . We characterize the value of the convertible bond and thecorresponding optimal stopping strategy via the solution of ODEs and the associatedfree boundaries, respectively. Proposition 6.3. Suppose that Assumption 6.1 holds. Define the infinitesimalgenerator L = σ s ∂ ss + ( r − q ) s∂ s − r. For s ∈ (0 , ¯ s λ ) , the value of the convertiblebond v λ ( s ) is the unique solution to the following ODEs:(i) If c > qK , then v λ ( s ) > γs , and (6.8) − L v λ = c − λ ( v λ − K ) + with the boundary condition v λ (¯ s λ ) = U λ ;(ii) If c < rK , then v λ ( s ) < K , and (6.9) − L v λ = c + λ ( γs − v λ ) + with the boundary condition v λ (¯ s λ ) = U λ .Proof . It is immediate from Theorem 2.3 and (6.5)-(6.6) that the convertible bondvalue is v λ ( s ) = V λ,s , for s ∈ (0 , ¯ s λ ), where V λ,s is the first component of the solutionto the penalized BSDE(6.10) V λ,st ∧ θ λ = U λ + Z θ λ t ∧ θ λ h c + λ (cid:0) γS su − V λ,su (cid:1) + − λ (cid:0) V λ,su − K (cid:1) + − rV λ,su i du − Z θ λ t ∧ θ λ Z λ,su dW u . Moreover, the optimal stopping strategy is(6.11) ( σ ∗ ,λ = inf { T N ≥ T : V λ,sT N ≥ K } ∧ T M ; τ ∗ ,λ = inf { T N ≥ T : V λ,sT N ≤ γS sT N } ∧ T M , ynkin games with Poisson random intervention times T M given in (6.7).On the other hand, by the Markov property of the stock price S , V λ,st = v λ ( S st ).In turn, Itˆo’s formula further implies that(6.12) v λ ( S sθ λ ) − v λ ( S st ∧ θ λ ) = Z θ λ t ∧ θ λ (cid:2) L v λ ( S su ) + rv λ ( S su ) (cid:3) du + Z θ λ t ∧ θ λ σs∂ s v λ ( S su ) dW u . It then follows from (6.10) and (6.12) that v λ ( s ), for s ∈ (0 , ¯ s λ ), solves the ODE(6.13) − L v λ = c + λ ( γs − v λ ) + − λ ( v λ − K ) + , with the boundary condition v λ (¯ s λ ) = U λ . Note that if c < rK , Proposition 6.2 yields v λ ( s ) ≤ U λ = c + λKr + λ < rK + λKr + λ = K, and if c > qK , it follows that v λ ( s ) ≥ L λ ( s ) = cr + λ + λq + λ γs > qKr + λ + λq + λ γs > γs. The ODEs (6.8)-(6.9) then follow immediately.The rest of this section is devoted to the characterization of the optimal stoppingstrategy of the constrained convertible bond via its associated free boundaries. qK < c < rK . From Proposition 6.3, when qK < c < rK ,we always have γs < v λ ( s ) < K. Thus, following from (6.11), the optimal stoppingstrategy is τ ∗ ,λ = σ ∗ ,λ = T M . Intuitively, when the coupon rate c satisfies c < rK , i.e. cr < K , the firm shallnever spend K to call the bond back, since it only needs to pay the coupon rate c asa perpetual bond, whose value is cr . Thus, the firm shall never call until T M .When the coupon rate c satisfies c > qK , i.e. c > qK > q r + λq + λ γs > qγs , theinvestor shall never convert her bond into stocks, since the stock dividends she willreceive by holding γ shares of the stock are no more than what she would otherwisereceive from the bond coupons. Thus, the investor shall never convert until T M .In Figure 7.1, the bold horizontal line ¯ s λ represents the conversion and callingboundary. We simulate three Poisson times T = 0 . T = 0 . T = 0 . 8, and twostock price paths. The investor (and the firm) will convert (and call) the bond at T for the stock path 1. They will continue at T and T , and terminate the contract at T for the stock path 2.We further calculate the convertible bond value by solving the corresponding ODEexplicitly. Note that in such a situation, v λ = v ,λ solves(6.14) −L v ,λ − c = 0 , for 0 < s < ¯ s λ ; v ,λ (0+) = cr ; v ,λ (¯ s λ ) = U λ . We put the perpetual bond value cr at the boundary v ,λ (0+) := lim s ↓ v ,λ ( s ), be-cause in such a situation, there is no motivation for the firm to call or for the investorto convert the bond.2 Gechun Liang and Haodong Sun The general solution of (6.14) has the form v ,λ ( s ) = A + s α + + A − s α − + cr , for0 < s < ¯ s λ , where(6.15) α ± = − ( r − q − σ ) ± q ( r − q − σ ) + 2 rσ σ . Since α − < 0, we obtain A − = 0 by the boundary condition at v ,λ (0+). Using theother boundary condition, we further obtain(6.16) v ,λ ( s ) = A ,λ s α + cr , where α = α + and A ,λ = λr + λ rK − cr (cid:0) ¯ s λ (cid:1) − α . In Figure 2, we further plot the value function v ,λ ( s ), which always stays between[ L λ ( s ) , U λ ] for s ∈ (0 , ¯ s λ ). Since L λ > γs and U λ < K , the value function also staysbetween ( γs, K ), which means it is never optimal for the firm or the investor to stopin the region s ∈ (0 , ¯ s λ ). c ≥ rK . It is obvious that c > qK if c ≥ rK . Thus, fromProposition 6.3, we always have v λ ( s ) > γs, and following from (6.11), the optimalconversion strategy for the investor is τ ∗ ,λ = T M , i.e. it is never optimal for the investor to convert until T M . Instead, the investor’soptimal strategy is to keep the convertible bond to receive its coupons (up to T M ).On the other hand, following from (6.8), v λ = v ,λ solves(6.17) −L v ,λ − c + λ ( v ,λ − K ) + = 0 , for 0 < s < ¯ s λ ; v ,λ (0+) = U λ ; v ,λ (¯ s λ ) = U λ . We put U λ at the boundary v ,λ (0+) := lim s ↓ v ,λ ( s ). In this situation, since thecoupon rate c is too large, the firm would prefer to convert as soon as possible to stoppaying the bond coupons. It is clear that v ,λ ( s ) = U λ ≥ K. In turn, by (6.11), it isoptimal for the firm to call as soon as possible, i.e. at the first Poisson arrival time σ ∗ ,λ = T . In Figure 3, the bold horizontal line ¯ s λ represents the conversion boundary for theinvestor. Once again, we simulate three Poisson times T = 0 . T = 0 . T = 0 . T firstly, and for the stock price path 2, both the firm and the investor will terminatethe contract at T .Figure 4 further plots the value function v ,λ , which is a constant U λ for s ∈ (0 , ¯ s λ ). Since the value function always stays above K , and therefore also above γs ,it is never optimal for the investor to convert in the region (0 , ¯ s λ ). c ≤ qK . It is obvious that c < rK if c ≤ qK . Thus, fromProposition 6.3, we always have v λ ( s ) < K, and following from (6.11), the optimalcalling time for the firm is σ ∗ ,λ = T M , ynkin games with Poisson random intervention times T M . Furthermore, following from(6.9), v λ = v ,λ solves(6.18) −L v ,λ − c − λ ( γs − v ,λ ) + = 0 , for 0 < s < ¯ s λ ; v ,λ (0+) = cr ; v ,λ (¯ s λ ) = U λ . Next, we solve (6.18) explicitly. Since c ≤ qK , the intersection point of the lowerbound L λ ( s ) of the convertible bond value and the investor’s payoff function γs is nogreater than ¯ s λ (so γs is no less than L λ ( s ) between this intersection point and ¯ s λ ).Thus, it may happen that, in the region s ∈ (0 , ¯ s λ ), the investor converts the bondearlier than T M . Since v ,λ ( s ) > γs when s ↓ 0, and v ,λ ( s ) ≤ γs for s = ¯ s λ , we define(6.19) x ∗ ,λ = inf (cid:8) s ∈ (0 , ¯ s λ ] : v ,λ ( s ) ≤ γs (cid:9) . By definition it is obvious v ,λ > γs for s ∈ (0 , x ∗ ,λ ), and by the continuity of v ,λ ( · ), v ,λ ( x ∗ ,λ ) = γx ∗ ,λ . Let us at the moment assume that v ,λ ≤ γs for s ∈ ( x ∗ ,λ , ¯ s λ ].Later, we will verify this condition. If this condition holds, (6.18) is equivalent to thefollowing free boundary problem −L v ,λ − c = 0 , for 0 < s < x ∗ ,λ ;(6.20) −L v ,λ − c + λ ( v ,λ − γs ) = 0 , for x ∗ ,λ < s < ¯ s λ ;(6.21) v ,λ (0+) = cr ;(6.22) v ,λ (¯ s λ ) = U λ ;(6.23) v ,λ ( x ∗ ,λ − ) = γx ∗ ,λ ;(6.24) v ,λ ( x ∗ ,λ +) = γx ∗ ,λ ;(6.25) (cid:0) v ,λ (cid:1) ′ ( x ∗ ,λ − ) = (cid:0) v ,λ (cid:1) ′ ( x ∗ ,λ +) . (6.26)We first observe that, with the boundary condition (6.22), ODEs (6.20)-(6.21) imply(6.27) v ,λ ( s ) = ( A ,λ s α + cr , if s ∈ (0 , x ∗ ,λ ); B + s β + + B − s β − + cr + λ + λq + λ γs, if s ∈ ( x ∗ ,λ , ¯ s λ ) , where α = α + is given in (6.15),(6.28) β ± = − ( r − q − σ ) ± q ( r − q − σ ) + 2( r + λ ) σ σ , and four unknowns ( A ,λ , B + , B − , x ∗ ,λ ) are to be determined. Using the continuityacross x ∗ ,λ , i.e. (6.24)-(6.25), the smooth pasting across x ∗ ,λ , i.e. (6.26), and theboundary condition at s = ¯ s λ , i.e. (6.23), we obtain that x ∗ ,λ ∈ (cid:0) , ¯ s λ (cid:3) is the (unique)solution to the following algebraic equation(6.29) C x β + − β − +1 + C x β + − β − + C x + C = 0 , with(6.30) C = (cid:16) α − λq + λ − qq + λ β + (cid:17) γ ; C = − (cid:16) α cr − cr + λ β + (cid:17) ; C = − (cid:16) α − λq + λ − qq + λ β − (cid:17) (¯ s λ ) β + − β − γ ; C = (cid:16) α cr − cr + λ β − (cid:17) (¯ s λ ) β + − β − , Gechun Liang and Haodong Sun and the coefficients are determined by(6.31) A ,λ = (cid:0) x ∗ ,λ (cid:1) − α (cid:0) γx ∗ ,λ − cr (cid:1) ; B + = qq + λ γx ∗ ,λ − cr + λ ( x ∗ ,λ ) β + − (¯ s λ ) β + − β − ( x ∗ ,λ ) β − ; B − = qq + λ γx ∗ ,λ − cr + λ ( x ∗ ,λ ) β − − (¯ s λ ) β −− β + ( x ∗ ,λ ) β + . It remains to verify the condition v ,λ ≤ γs for s ∈ ( x ∗ ,λ , ¯ s λ ]. Indeed, since A ,λ > α > B + < β + > B − > β − < 0, it is clear that v ,λ is convex in the interval (0 , x ∗ ,λ ) and concave in the interval ( x ∗ ,λ , ¯ s λ ]. Moreover, (cid:0) v ,λ (cid:1) ′ ( x ∗ ,λ ) < γ . This verifies the condition.The optimal conversion time for the investor is therefore given as τ ∗ ,λ = inf { T N : S sT N ≥ x ∗ ,λ } ∧ T M . In Figure 5, the top bold horizontal line ¯ s λ represents the calling boundary forthe firm, and the bottom bold horizontal line x ∗ ,λ represents the conversion boundaryfor the investor. Once again, we simulate three Poisson times T = 0 . T = 0 . T = 0 . 8, and two stock price paths. For the stock price path 1, both the investorand the firm will terminate the contract at T ; and for the stock path 2, the investorwill continue at T and convert at T , while the firm will not call the bond back atneither T nor T .In Figure 6, we further plot the value function v ,λ , which crosses the payofffunction γs in the region (0 , ¯ s λ ], so the crossing point x ∗ ,λ is the optimal conversionboundary for the investor. Furthermore, the value function v ,λ is strictly dominatedby K for s ∈ (0 , ¯ s λ ), so the firm never calls the bond back in this region. 7. Asymptotics as λ → ∞ . We study the asymptotic behavior of the convert-ible bond price and its associated free boundaries when the Poisson intensity λ → ∞ .Intuitively, they will converge to their continuous time counterparts. We prove thisintuition in this section. The setting is the same as insection 6 except that both the investor and the firm choose their respective optimalstopping strategies as F -stopping times taking values in [0 , ∞ ]. Then, the upper andlower values of the standard convertible bond are given by(7.1) v = inf σ ∈ ˜ R sup τ ∈ ˜ R E [ P ( s ; σ, τ )] , (7.2) v = sup τ ∈ ˜ R inf σ ∈ ˜ R E [ P ( s ; σ, τ )] , and the control set ˜ R is defined as˜ R = { F -stopping time τ for τ ≥ } . We say this game has value v if v = v = v , and has a saddle point ( σ ∗ , τ ∗ ) ∈ ˜ R × ˜ R if E [ P ( s ; σ ∗ , τ )] ≤ E [ P ( s ; σ ∗ , τ ∗ )] ≤ E [ P ( s ; σ, τ ∗ )] for every ( σ, τ ) ∈ ˜ R × ˜ R .The proof of the following result follows along the similar arguments in [35] andis thus omitted. We refer to [35] for its further details. Proposition 7.1. Suppose that Assumption 6.1 holds. Let ¯ s := Kγ , and define an F -stopping time θ = inf { u ≥ S su ≥ ¯ s } . Then, the value of the standard convertiblebond v ( s ) is given as follows: ynkin games with Poisson random intervention times (i) The Case I: qK < c < rK , (7.3) v ( s ) = (cid:26) A s α + cr , if s ∈ (0 , ¯ s ); γs, if s ∈ [¯ s, ∞ ) , with α = α + as in (6.15) and A = rK − cr (¯ s ) − α . The optimal stopping strategy is givenby (7.4) σ ∗ = τ ∗ = θ. (ii) The Case II: c ≥ rK , (7.5) v ( s ) = (cid:26) K, if s ∈ (0 , ¯ s ); γs, if s ∈ [¯ s, ∞ ) . The optimal stopping strategy is given by (7.6) σ ∗ = 0; τ ∗ = θ. (iii) The Case III: c ≤ qK , (7.7) v ( s ) = (cid:26) A s α + cr , if s ∈ (0 , x ); γs, if s ∈ [ x , ∞ ) , with α = α + and A = (cid:0) γx − cr (cid:1) ( x ) − α . The optimal stopping strategy is given by (7.8) σ ∗ = θ, τ ∗ = inf { t ≥ S st ≥ x } , where x = (cid:26) x ∗ := αα − cγr , if c ≤ α − α rK ;¯ s, if c > α − α rK. We conclude the paper by studying, when λ → ∞ , (i) theconvergence of the constrained convertible bond price v λ to its continuous-time coun-terpart v ; (ii) the convergence of the optimal conversion/calling boundaries for theconstrained convertible bond to its continuous-time counterparts.It is easy to check that ¯ s λ → ¯ s , A ,λ → A , L λ ( s ) → γs and U λ → K with theconvergence rate 1 /λ by using their explicit forms. As a consequence, we have v ,λ ( s ) → v ( s ); v ,λ ( s ) → v ( s ) , with the convergence rate 1 /λ . Hence, we only need to establish the convergenceresults for Case III when c ≤ qK . To this end, we first establish the monotonicproperty of x ∗ ,λ , as defined in (6.19), with respect to λ in the following lemma. Proposition 7.2. Suppose that Assumption 6.1 holds and that c ≤ qK . Then, x ∗ ,λ is non-decreasing with respect to λ .Proof . By the definition of x ∗ ,λ in (6.19), it is sufficient to prove v ,λ is non-decreasing in λ . Recall that v ,λ is the solution to the ODE (6.18) in the domain s ∈ (0 , ¯ s λ ), and v ,λ = L λ in the domain s ∈ [¯ s λ , ∞ ).6 Gechun Liang and Haodong Sun Let us suppose λ < λ and it is easy to check that ¯ s λ < ¯ s λ . For s ≥ ¯ s λ , wehave v ,λ = L λ . Then, v ,λ ( s ) − v ,λ ( s ) ≤ L λ ( s ) − L λ ( s )= c ( λ − λ )( r + λ )( r + λ ) − q ( λ − λ )( q + λ )( q + λ ) γs ≤ ( q − r ) qK ( λ − λ )( r + λ )( q + λ )( r + λ ) < . On the other hand, for s < ¯ s λ , note that v ,λ (0+) = v ,λ (0+) = cr and v ,λ (¯ s λ ) < v ,λ (¯ s λ ). Define the set N = (cid:8) s ∈ (cid:0) , ¯ s λ (cid:1) : v ,λ ( s ) > v ,λ ( s ) (cid:9) , andsuppose that N 6 = ∅ . Then on N , we have (cid:26) −L v ,λ = c + λ ( γs − v ,λ ) + ; −L v ,λ = c + λ ( γs − v ,λ ) + , which implies −L ( v ,λ − v ,λ ) = λ ( γs − v ,λ ) + − λ ( γs − v ,λ ) + ≤ λ (cid:2) ( γs − v ,λ ) + − ( γs − v ,λ ) + (cid:3) ≤ . Hence, we have v ,λ ≤ v ,λ on N , which is in contradiction with the definition of N . Since x ∗ ,λ is bounded by ¯ s λ ( ≤ ¯ s ), Proposition 7.2 then implies that lim λ →∞ x ∗ ,λ exists, denoted by x ∞ . Moreover, by Proposition 7.1, we have x ∞ ≤ x ∗ if c ≤ α − α rK ,and x ∞ ≤ ¯ s if c > α − α rK .On the other hand, by (6.29), x ∗ ,λ is the solution to the following allergic equation (cid:20)(cid:16) x ¯ s λ (cid:17) β + − β − − (cid:21) (cid:20)(cid:18) α − λq + λ (cid:19) γx − α cr − β + (cid:18) qq + λ γx − cr + λ (cid:19)(cid:21) (7.9) = ( β + − β − ) (cid:18) qq + λ γx − cr + λ (cid:19) . Sending λ → ∞ in (7.9), since the right hand side of (7.9) has the limit 0, we obtainlim λ →∞ "(cid:18) x ∗ ,λ ¯ s λ (cid:19) β + − β − − I λ (cid:20)(cid:18) α − λq + λ (cid:19) γx ∗ ,λ − α cr − β + (cid:18) qq + λ γx ∗ ,λ − cr + λ (cid:19)(cid:21)| {z } II λ = 0 . This implies at least one of I λ and II λ has the limit 0.If c < α − α rK , we have lim λ →∞ I λ = − 1, sincelim λ →∞ x ∗ ,λ ¯ s λ = x ∞ ¯ s ≤ x ∗ ¯ s = αα − crK < . This implies lim λ →∞ II λ = 0, i.e. x ∞ = x ∗ .If c > α − α rK , we havelim λ →∞ II λ = ( α − γx ∞ − α cr < ( α − γx ∞ − K ) ≤ , ynkin games with Poisson random intervention times λ →∞ I λ = 0, i.e. x ∞ = ¯ s .If c = α − α rK , it is easy to check that x ∞ = x ∗ = ¯ s .Hence, we have established the convergence of x ∗ ,λ → x as λ → ∞ . As a con-sequence, it also follows that v ,λ ( s ) → v ( s ). However, due to the lack of explicitsolutions for Case III, it is unclear what is the corresponding convergence rate. Acknowledgments . The author would like to thank the editor, associate editor,and two referees for their valuable comments and suggestions on the manuscript. REFERENCES[1] M. Alario-Nazaret, J. P. Lepeltier, and B. Marchal. Dynkin games, stochastic differentialsystems. in Proceedings of the 2nd Bad Honnef Workshop on Stochastic Processes, LectureNotes in Control and Information Sciences , pp. 23–32, Springer, 1982.[2] E. J. Baurdoux and A. E. Kyprianou. 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Initial Stock Price s B ond P r i c e ¯ s λ ¯ s KL λ (s) γ Sv λ U λ Fig. 7.2 . The value function v ,λ for Case I. Gechun Liang and Haodong Sun Time S t o ck P r i c e ¯ s λ Stock Path 1Stock Path 2 Optimal Calling TimeOptimal Conversion and Calling Time Fig. 7.3 . Scenario Simulation for Case II. The figure shows two simulated stock price paths inthe case of c ≥ rK . The parameters are the same as those in Figure 7.1.The bold horizontal linedescribes the conversion boundary ¯ s λ . Given the Poisson times T =0.25, T =0.5 and T =0.8, thefirm will call the bond at T (marked square) for the stock price path 1; and both the firm and theinvestor will terminate the contract at T (marked square) for the stock price path 2. Initial Stock Price s B ond P r i c e ¯ s λ ¯ s KL λ (s) γ S v λ U λ Fig. 7.4 . The value function v ,λ for Case II. ynkin games with Poisson random intervention times Time S t o ck P r i c e ¯ s λ x λ Stock Path 1Stock Path 2 Optimal Conversion TimeOptimal Conversion and Calling Time Fig. 7.5 . Scenario Simulation for Case III. The figure shows two simulated stock price pathsin the case of c ≤ qK . The parameters are the same as those in Figure 7.1. The top bold horizontalline is the calling boundary ¯ s λ , and the bottom bold horizontal line is the conversion boundary x ∗ ,λ .Given the Poisson times T =0.3, T =0.5 and T =0.8, both the investor and the firm will terminatethe contract at T (marked square) for the stock price path 1; and the invertor will convert her bond T (marked square) for the stock price path 2. Initial Stock Price s B ond P r i c e ¯ s λ ¯ sx ∗ , λ KL λ (s) γ Sv λ U λ Fig. 7.6 . The value function v ,λ,λT } (cid:12)(cid:12)(cid:12) G T n − i = E h e − λ ( T − T n − ) ˜ ξ (cid:12)(cid:12)(cid:12) G T n − i , and E h (cid:16) { Q λTn ≥ ˜ U Tn } ˜ U T n + { Q λTn ≤ ˜ L Tn } ˜ L T n + { ˜ L Tn
T } + ˆ Q λT n { T n ≤ T } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G T n − , ˜ L T n − )) , which also admits a unique solution since we can calculate its solution backwards in arecursive way. We show that ˆ Q λT n − is the value of another constrained Dynkin game.Introduce the upper and lower values of an auxiliary constrained Dynkin game as(3.9) ˆ q λT n − = ess inf N σ ∈N n − ( λ ) ess sup N τ ∈N n − ( λ ) E h ˜ R n − ( T N σ , T N τ ) |G T n − i , (3.10) ˆ q λT n − = ess sup N τ ∈N n − ( λ ) ess inf N σ ∈N n − ( λ ) E h ˜ R n − ( T N σ , T N τ ) |G T n − i , where˜ R n − ( σ, τ ) = Z σ ∧ τ ∧ TT n − ∧ T ˜ f s ds + ˜ ξ { σ ∧ τ ≥ T } + ˜ L τ { τ