BSDE Approach for Dynkin Game and American Game Option

Abstract

Consider a Dynkin game with payoff

$$ J(\lambda , {\upsigma }) = F\bigg [U_{\lambda }1_{\{\lambda< {\upsigma }\}} + L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + \xi 1_{\{ {\upsigma }= \lambda = T \}}\bigg ], $$

where $F : \mathbb {R}\longrightarrow \mathbb {R}$ is a continuous nondecreasing function and $\lambda , {\upsigma }$ are stopping times valued in [0, T]. We show the existence of a value as well as a saddle-point for this game using the theory of BSDE with double reflecting barriers. An American game option pricing problem is also discussed.

1 Introduction

Stochastic game was first introduced by Dynkin and Yushkevich [6] and later studied, in different contexts, by several authors, including Neveu [18], Bensoussan and Friedman [2], Bismut [3], Morimoto [17], Alario-Nazaret, Lepeltier and Marchal [1], Lepeltier and Maingueneau [16], Cvitanic and Karatzas [4], Touzi and Vieille [19] and others, such stochastic games are known as Dynkin games.

Considerable attention has been devoted to studying the association between backward stochastic differential equations (BSDEs for short) and stochastic differential games. Among others, Cvitanic and Karatzas showed in [4] existence and uniqueness of the solution to the BSDE with double reflecting barriers, and associated their equation to a stochastic games. Hamadène [9] and Hamadène and Hassani [11] studied the mixed zero-sum stochastic differential game problem using the notion of a local solution of BSDEs with double reflecting barriers. Hamadène and Lepeltier [10] added controls to the Dynkin game studied by Cvitanic and Karatzas in [4]. Karatzas and Li [14] studied a non-zero-sum game with features of both stochastic control and optimal stopping, for a process of diffusion type via the BSDE approach. Dumitrescu et al. [5] introduced a generalized Dynkin game problem associated with a BSDE with jumps.

Consider the Dynkin game, associated with processes L, U, $\xi$ and Q, with payoff:

$$ J(\lambda , {\upsigma }) = F\bigg [U_{\lambda }1_{\{\lambda< {\upsigma }\}} + L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + \xi 1_{\{ {\upsigma }= \lambda = T \}}\bigg ], $$

where $F : \mathbb {R}\longrightarrow \mathbb {R}$ is a continuous nondecreasing function and $\lambda , {\upsigma }$ are stopping times valued in [0, T]. In the direction of connection between BSDE with two reflecting barriers and Dynkin games, in order to prove that this game has a saddle point, which is a pair of stopping $(\lambda ^*, {\upsigma }^*)$ such that for any stopping times $\lambda$ and ${\upsigma }$ one has

$$ \mathbb {E}\bigg (J(\lambda ^*, {\upsigma })\bigg )\le \mathbb {E}\bigg (J(\lambda ^*, {\upsigma }^*)\bigg ) \le \mathbb {E}\bigg (J(\lambda , {\upsigma }^*)\bigg ), $$

all the works [4, 9–11, 14] have considered the case of bounded or square integrable processes $F(\xi )$, F(Q), F(L) and F(U). Moreover, they have assumed that the barriers F(L) and F(U) have to satisfy one of the conditions:

• 1. The so-called Mokobodski condition which turns out into the existence of a difference of nonnegative supermartingales between F(L) and F(U).

• 2. The complete separation i.e. $F(L)<F(U)$.

One of the main objective of this work is to weaken the assumptions assumed on the data $F(\xi )$, F(Q), F(L) and F(U) in the case of association between BSDE with two reflecting barriers and Dynkin games. Yet, checking Mokobodski’s condition appears as a difficult question. So, instead of assuming the Mokobodski’s condition on the barriers F(L) and F(U), we suppose only that there exists a semimartingale between them. It should be also noted here that if the barriers are completely separated this implies that there exists a semimartingale between them (see [8]). Actually, if we assume the following conditions:

• 1. There exists a semimartingale between L and U and for every semimartingale S such that $L\le S\le U$, F(S) is a also a semimartingale.

• 2. $\mathbb {E} [F(L_{{\upsigma }})^-] < +\infty ,$ for all stopping time $0\le {\upsigma }\le T$, where $F(L)^-= \sup (-F(L), 0)$.

• 3. $\displaystyle \liminf _{r\rightarrow +\infty }r\,P\bigg [\displaystyle \sup _{s\le T} F(U_s)^+ >r\bigg ] =0$, where $F(U)^+ = \sup (F(U), 0)$.

• 4. $\displaystyle \liminf _{r\rightarrow +\infty }r\,P\bigg [\displaystyle \sup _{s\le T} F(L_s)^- >r\bigg ] =0$,

then the pair of stopping times $(\lambda ^*, {\upsigma }^*)$ defined by

$$\lambda ^* = \inf \{s\ge 0\,\, :\,\, Y_s = F(U_s)\}\wedge T \quad \text{ and }\quad {\upsigma }^* = \inf \{s\ge 0\,\, :\,\, Y_s = F(L_s)\}\wedge T,$$

is a saddle-point for the game, where Y is the solution of the following BSDE with double reflecting barriers F(L) and F(U) (see Definition 2):

$$ \left\{ \begin{array}{ll} (i) &{} Y_{t}=F(\xi ) +\int _{t}^{T}dK_{s}^+ -\int _{t}^{T}dK_{s}^- -\int _{t}^{T}Z_{s}dB_{s}\,, t\le T, \\ ({ {ii}})&{} Y \text{ between } F(L) \text{ and } F(U),\, \, i.e.\,\, \forall t\le T,\,\, F(L_t) \le Y_{t}\le F(U_{t}),\\ ({ {iii}}) &{} \text{ the } \text{ Skorohod } \text{ conditions } \text{ hold: }\\ &{} \int _{0}^{T}( Y_{t}-F(L_{t})) dK_{t}^+= \int _{0}^{T}(F(U_{t})-Y_{t}) dK_{t}^-=0,\,\, \text{ a.s. }. \\ \end{array} \right. $$

We should mention here that if F(L) and F(U) are $L^1$—integrable, i.e. $\mathbb {E}\displaystyle \sup _{t\le T}(|F(U_t)|+|F(L_t)|) <+\infty$, then the above assumptions 2–4 are satisfied and then the Dynkin game has a saddle point. This corresponds to the main assumption assumed in the general context of Dynkin games.

An American option is a contract which enables its buyer (holder) to exercise it at any time up to the maturity. An American game option gives additionally the right to the option seller (writer, issuer) to cancel it early paying for this a prescribed penalty. American game option was first introduced by Kifer [15] and studied later by several authors, see for example Hamadène [9], Hamadène and Zhang [13] and the references therein. The second aim of this work is to prove, under weaker conditions than the square integrability assumed on the data in [9], that the value of the option at any time $t\in [0,T]$ is given by $e^{rt}Y_t$, where Y is the solution of some BSDE with two reflecting barriers. Moreover, we also show that a hedge after t, against the game option, exists.

2 Preliminaries

2.1 Notations and Assumptions

Let $(\varOmega , {\mathscr {F}}, ({\mathscr {F}}_t)_{t\le T}, P)$ be a stochastic basis on which is defined a Brownian motion $(B_t)_{t\le T}$ such that $({\mathscr {F}}_t)_{t\le T}$ is the natural filtration of $(B_t)_{t\le T}$ and ${\mathscr {F}}_0$ contains all P-null sets of $\mathscr {F}$. Note that $({\mathscr {F}}_t)_{t\le T}$ satisfies the usual conditions, i.e. it is right continuous and complete.

Let us now introduce the following notations:

• $\mathscr {P}$ the sigma algebra of ${\mathscr {F}}_t$-progressively measurable sets on $\varOmega \times [0,T].$

• ${\mathscr {C}}$ the set of $\mathbb {R}$-valued $\mathscr {P}$-measurable continuous processes $(Y_t)_{t\le T}$.

• ${\mathscr {L}}^{2,d}$ the set of $\mathbb {R}^d$-valued and $\mathscr {P}$-measurable processes $(Z_t)_{t\le T}$ such that

  $$\int _{0}^{T}|Z_s|^2ds<\infty , P- a.s.$$

• ${\mathscr {K}}$ the set of ${\mathscr {P}}$-measurable continuous nondecreasing processes $(K_t)_{t\le T}$ such that $K_0 = 0$ and $K_T <+\infty ,$ P– a.s.

Throughout the paper, we introduce the following data:

• $L:=\left\{ L_{t},\,0\le t\le T\right\}$ and $U:=\left\{ U_{t},\,0\le t\le T\right\}$ are two real valued barriers which are $\mathscr {P}$-measurable and continuous processes such that $L_t \le U_t,\,\,\forall t\in [0,T]$.

• Q be a process such that, $\forall t\in [0,T]$ $L_t\le Q_t\le U_t,\,P-a.s.$

• $\xi$ is an ${\mathscr {F}}_T$-measurable one dimensional random variable such that

  $$L_T\le \xi \le U_T.$$

• $F : \mathbb {R}\longrightarrow \mathbb {R}$ is a continuous nondecreasing function.

We assume the following assumptions:

$(\mathbf {A.1})$ There exists a continuous semimartingale $S_. = S_0 + V_.^+ -V_.^- +\int _0^. \alpha _s dB_s$, with $S_0\in \mathbb {R}, V^+, V^-\in \mathscr {K}$ and $\alpha \in {\mathscr {L}}^{2,d}$, such that

$$L_t \le S_t\le U_t, \,\, \forall t\in [0,T].$$

$(\mathbf {A.2})$ For every semimartingale S such that $L\le S\le U$, F(S) is a also a semimartingale.

2.2 Existence of Solution for BSDE with Double Reflecting Barriers

In view of clarifying this issue, we recall some results concerning BSDEs with double reflecting barriers with two continuous barriers (see Essaky and Hassani [8] for more details). Let us recall first the following definition of two singular measures.

Definition 1

Let $K^1$ and $K^2$ be two processes in ${\mathscr {K}}$. We say that:

$K^1$ and $K^2$ are singular if and only if there exists a set $D\in {\mathscr {P}}$ such that

$$ \mathbb {E}\int _0^T 1_D(s,\omega ) dK^1_s(\omega ) = \mathbb {E}\int _0^T 1_{D^c}(s,\omega ) dK^2_s(\omega ) =0. $$

This is denoted by $dK^1 \perp dK^2$.

Let us now introduce the definition of a BSDE with double reflecting obstacles L and U.

Definition 2

• 1. We call $(Y,Z,K^+,K^-):=( Y_{t},Z_{t},K_{t}^+,K_{t}^-)_{t\le T}$ a solution of the GBSDE with two reflecting barriers L and U associated with a terminal value $\xi$ if the following hold:

  $$\begin{aligned} \left\{ \begin{array}{ll} (i) &{} Y_{t}=\xi +\int _{t}^{T}dK_{s}^+ -\int _{t}^{T}dK_{s}^- -\int _{t}^{T}Z_{s}dB_{s}\,, t\le T, \\ ({ {ii}})&{} Y \text{ between } L \text{ and } U,\, \, i.e.\,\, \forall t\le T,\,\, L_t \le Y_{t}\le U_{t},\\ ({ {iii}}) &{} \text{ the } \text{ Skorohod } \text{ conditions } \text{ hold: } \\ &{} \int _{0}^{T}( Y_{t}-L_{t}) dK_{t}^+= \int _{0}^{T}( U_{t}-Y_{t}) dK_{t}^-=0,\,\, \text{ a.s. }, \\ ({ {iv}})&{} Y\in {\mathscr {C}} \quad K^+, K^-\in {\mathscr {K}} \quad Z\in {\mathscr {L}}^{2,d}, \\ ({ v})&{} dK^+\perp dK^-. \end{array} \right. \end{aligned}$$

  (1)

• 2. We say that the BSDE (1) has a maximal (resp. minimal) solution $(Y ,Z, K^+ , K^- )$ if for any other solution $(Y^{'} ,Z^{'} ,K'^{+} , K'^{-} )$ of (1) we have for all $t \le T$, $Y_t^{'}\le Y_t$, $P-{\text {a.s.}}$ (resp. $Y_t^{'}\ge Y_t$, $P-{\text {a.s.}}$).

The following theorem has already been proved in [8].

Theorem 1

Let assumption $(\mathbf {A.1})$ holds true. Then there exists a maximal (resp. minimal) solution for BSDE with double reflecting barriers (1).

3 Dynkin Game

Our purpose in this section is to show that the existence of a solution $(Y, Z, K^+, K^-)$ to the BSDE (1) implies that Y is the value of a certain stochastic game of stopping.

Consider the payoff

$$ J(\lambda , {\upsigma }) = F\bigg (U_{\lambda }1_{\{\lambda< {\upsigma }\}} + L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + \xi 1_{\{ {\upsigma }= \lambda = T \}}\bigg ). $$

The setting of our problem of Dynkin game is the following. There are two players labeled player 1 and player 2. Player 1 chooses the stopping time $\lambda$, player 2 chooses the stopping time ${\upsigma }$, and $J(\lambda , {\upsigma })$ represents the amount paid by player 1 to player 2. It is the conditional expectation $\mathbb {E}\bigg (J(\lambda , {\upsigma })\bigg )$ of this random payoff that player 1 tries to minimize and player 2 tries to maximize. The game stops when one player decides to stop, that is, at the stopping time $\lambda \wedge {\upsigma }$ before time T, the payoff is then equals

$$ J(\lambda , {\upsigma }) =\left\{ \begin{array}{ll} &{} F(U_{\lambda }) \,\,\,\hbox {if player 1 stops the game first}\\ &{} F(L_{{\upsigma }}) \,\,\,\hbox {if player 2 stops the game first} \\ &{} F(Q_{{\upsigma }}) \,\,\,\hbox {if players stop the game simultaneously before time T} \\ &{} F(\xi ) \,\,\,\hbox {if neither have exercised until the expiry time}\,\,\,{ T}. \end{array} \right. $$

It is then natural to define the lower and upper values of the game:

$$ \underline{V} := \displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda , {\upsigma })\bigg ]\le \overline{V} := \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda , {\upsigma })\bigg ], $$

where ${\mathscr {M}}_{t,T}$ is the set of stopping times valued between t and T. If it happens that $\underline{V} = \overline{V}$, then the above Dynkin game is said to have a value. A pair $(\lambda ^*_0, {\upsigma }^*_0 )$ is called a saddle point if

$$ \mathbb {E}\bigg (J(\lambda ^*_0, {\upsigma })\bigg )\le \mathbb {E}\bigg (J(\lambda ^*_0, {\upsigma }_0^*)\bigg ) \le \mathbb {E}\bigg (J(\lambda , {\upsigma }^*_0)\bigg ). $$

Our objective is to show the existence of a saddle-point for the game and to characterize it. This implies that this game has a value.

Let assumptions $(\mathbf {A.1})$ and $(\mathbf {A.2})$ hold true. Let $(Y, Z, K^+, K^-)$ be the solution, which is exists according to Theorem 1, of the following BSDE with double reflecting barriers:

$$\begin{aligned} \left\{ \begin{array}{ll} (i) &{} Y_{t}=F(\xi ) +\int _{t}^{T}dK_{s}^+ -\int _{t}^{T}dK_{s}^- -\int _{t}^{T}Z_{s}dB_{s}, t\le T, \\ ({ {ii}})&{} \forall t\le T,\,\, F(L_t) \le Y_{t}\le F(U_{t}),\\ &{} \int _{0}^{T}( Y_{t}-F(L_{t})) dK_{t}^+= \int _{0}^{T}( F(U_{t})-Y_{t}) dK_{t}^-=0,\,\, \text{ a.s. }, \\ ({ {iv}})&{} Y\in {\mathscr {C}} \quad K^+, K^-\in {\mathscr {K}} \quad Z\in {\mathscr {L}}^{2,d}, \\ (v)&{} dK^+\perp dK^-. \end{array} \right. \end{aligned}$$

(2)

Let $\lambda ^*_t$ and ${\upsigma }^*_t$ be the stopping times defined as follows:

$$\lambda ^*_t = \inf \{s\ge t\,\, :\,\, Y_s = F(U_s)\}\wedge T \quad \text{ and }\quad {\upsigma }^*_t = \inf \{s\ge t\,\, :\,\, Y_s = F(L_s)\}\wedge T.$$

The main result of this section is the following.

Theorem 2

Assume the following assumptions:

• 1. $\mathbb {E} F(L_{{\upsigma }})^- < +\infty ,$ for all stopping time $0\le {\upsigma }\le T$, where $F(L)^- = \sup (-F(L), 0)$.

• 2. $\displaystyle \liminf _{r\rightarrow +\infty }r\,P\bigg [\displaystyle \sup _{s\le T} F(U_s)^+ >r\bigg ] =0$, where $F(U)^+ = \sup (F(U), 0)$.

• 3. $\displaystyle \liminf _{r\rightarrow +\infty }r\,P\bigg [\displaystyle \sup _{s\le T} F(L_s)^- >r\bigg ] =0$.

Then

$$\begin{aligned} \begin{array}{ll} Y_t &{}= \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ]\\ &{} =\displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma })\mid {\mathscr {F}}_t\bigg ] = \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\mathbb {E}\bigg [J(\lambda , {\upsigma }^*_t)\mid {\mathscr {F}}_t\bigg ] \\ &{} = \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ] = \displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {T}}_t}\mathbb {E}\bigg [J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ], \end{array} \end{aligned}$$

(3)

where ${\mathscr {M}}_{t,T}$ is the set of stopping times valued between t and T. $Y_0$ can be interpreted as the value of the game and $(\lambda ^*_0, {\upsigma }^*_0 )$ as the fair strategy for the two players (or a saddle point for the game).

Proof

Let $(a_n^{+})_n$ and $(a_n^{-})_n$ be two nondecreasing sequences such that

$$\begin{aligned} \displaystyle \liminf _{n\rightarrow +\infty }a_n^+ P\bigg [\displaystyle \sup _{s\le T} F(U_s)^+>a_n^+\bigg ] =0, \,\,\, \displaystyle \liminf _{n\rightarrow +\infty }a_n^- P\bigg [\displaystyle \sup _{s\le T} F(L_s)^- >a_n^-\bigg ] =0. \end{aligned}$$

(4)

Let also $(\alpha _i)_{i\ge 0}$ and $(\upsilon _i^{\pm })_{i\ge 0}$ be families of stopping times defined by

$$\alpha _i =\inf \{s\ge t : \int _t^s \mid Z_r\mid ^2dr \ge i\}\wedge T,\,\,\,\, \upsilon _i^{\pm } =\inf \{s\ge t : Y_s^{\pm } > a_i^{\pm }\}\wedge T. $$

It follows from Eq. (2) that for every stopping time ${\upsigma }\in {\mathscr {M}}_{t,T}$

Then for every stopping time ${\upsigma }\in {\mathscr {M}}_{t,T}$

$$ \begin{array}{ll} Y_t &{}\ge \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \alpha _i \wedge \upsilon _n^+\wedge \upsilon _m^-}\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \alpha _i \wedge \upsilon _n^+\wedge \upsilon _m^-}^+\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \alpha _i \wedge \upsilon _n^+\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg ). \end{array} $$

In view of passing to the limit on i and n respectively and using Fatou’s lemma for $Y^+$ and dominated convergence theorem for $Y^-$ since it is bounded, we have

$$ \begin{array}{ll} Y_t&\ge \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^+\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg ). \end{array} $$

Now taking the upper limit on m we get

$$ \begin{array}{ll} Y_t &{}\ge \displaystyle \limsup _{m}\bigg [\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^+\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg )\bigg ] \\ &{} = \displaystyle \limsup _{m}\bigg [\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^+1_{\lambda _t^*\wedge {\upsigma }\le \upsilon _m^-}\mid {\mathscr {F}}_t\bigg )+\mathbb {E} \bigg (Y_{\upsilon _m^-}^+ 1_{\lambda _t^*\wedge {\upsigma }> \upsilon _m^-}\mid {\mathscr {F}}_t\bigg ) \\ &{} \quad -\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg )\bigg ] \\ &{} \ge \displaystyle \limsup _{m}\bigg [\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^+1_{\lambda _t^*\wedge {\upsigma }\le \upsilon _m^-}\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg )\bigg ] \\ &{} = \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^+\mid {\mathscr {F}}_t\bigg )-\displaystyle \liminf _{m}\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg ). \end{array} $$

In view of using the limit appearing in (4), we obtain

$$ \begin{array}{ll} &{}\displaystyle \liminf _{m}\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }\wedge \upsilon _m^-}^-\mid {\mathscr {F}}_t\bigg )\\ &{} \le \displaystyle \liminf _{m}\bigg [\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^- 1_{\lambda _t^*\wedge {\upsigma }\le \upsilon _m^-}\mid {\mathscr {F}}_t\bigg )+ a_m^-\mathbb {E}\bigg (1_{\lambda _t^*\wedge {\upsigma }>\upsilon _m^-}\mid {\mathscr {F}}_t\bigg )\bigg ] \\ &{} =\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^-\mid {\mathscr {F}}_t\bigg ) + \displaystyle \liminf _{m\rightarrow +\infty }a_m^-\mathbb {E}\bigg (1_{\lambda _t^*\wedge {\upsigma }>\upsilon _m^-}\mid {\mathscr {F}}_t\bigg ) \\ &{} \le \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^-\mid {\mathscr {F}}_t\bigg ) + \displaystyle \liminf _{m\rightarrow +\infty }a_m^-\mathbb {E}\bigg (1_{\{\displaystyle {\sup _{s\le T}} F(L_s)^- >a_m^-\}}\mid {\mathscr {F}}_t\bigg ) \\ &{}= \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^-\mid {\mathscr {F}}_t\bigg ), \end{array} $$

it follows then that for all stopping time ${\upsigma }\in {\mathscr {M}}_{t,T}$,

$$\begin{aligned} \begin{array}{ll} Y_t &{}\ge \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^+\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}^-\mid {\mathscr {F}}_t\bigg ) = \mathbb {E} \bigg (Y_{\lambda ^*_t\wedge {\upsigma }}\mid {\mathscr {F}}_t\bigg ) \\ &{} \ge \mathbb {E}\bigg (F(U_{\lambda ^*_t})1_{\{\lambda ^*_t< {\upsigma }\}} + F(L_{{\upsigma }})1_{\{\lambda ^*_t > {\upsigma }\}}+ F(Q_{{\upsigma }})1_{\{ {\upsigma }=\lambda ^*_t < T \}} + F(\xi ) 1_{\{ {\upsigma }= \lambda ^*_t = T \}}\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E}\bigg (J(\lambda ^*_t, {\upsigma })\mid {\mathscr {F}}_t\bigg ). \end{array} \end{aligned}$$

(5)

Now it follows from Eq. (2) that for every stopping time $\lambda \in {\mathscr {M}}_{t,T}$

$$ \begin{array}{ll} Y_t &{}\le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \alpha _i \wedge \upsilon _m^-\wedge \upsilon _n^+}\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \alpha _i \wedge \upsilon _m^-\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) - \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \alpha _i \wedge \upsilon _m^-\wedge \upsilon _n^+}^-\mid {\mathscr {F}}_t\bigg ). \end{array} $$

In view of passing to the limit on i and m respectively and using dominated convergence theorem for $Y^+$ since it is bounded, we have

$$ \begin{array}{ll} &{} Y_t \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) - \displaystyle {\limsup _{m}}\,\,\mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _m^-\wedge \upsilon _n^+}^-\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) - \displaystyle {\limsup _{m}}\bigg [\mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^- 1_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+\le \upsilon _m^-}\mid {\mathscr {F}}_t\bigg ) \\ &{} \quad + \mathbb {E} \bigg (Y_{ \upsilon _m^-}^- 1_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+> \upsilon _m^-}\mid {\mathscr {F}}_t\bigg )\bigg ] \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) - \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^-\mid {\mathscr {F}}_t\bigg )-\displaystyle {\limsup _{m}}\,\,\mathbb {E} \bigg (Y_{ \upsilon _m^-}^- 1_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+ > \upsilon _m^-}\mid {\mathscr {F}}_t\bigg ) \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) - \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^-\mid {\mathscr {F}}_t\bigg ). \end{array} $$

By using Fatou’s lemma and assumption 1. Of Theorem 2 we get

$$ \begin{array}{ll} Y_t + \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^-\mid {\mathscr {F}}_t\bigg ) &{} \le Y_t +\displaystyle \liminf _{n} \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^-\mid {\mathscr {F}}_t\bigg )\\ &{}\le \displaystyle \liminf _{n}\mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t\wedge \upsilon _n^+}^+\mid {\mathscr {F}}_t\bigg ) \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^+\mid {\mathscr {F}}_t) +\displaystyle \liminf _{n\rightarrow +\infty }a_n^+\mathbb {E}\bigg (1_{\lambda \wedge {\upsigma }_t^*>\upsilon _n^+}\mid {\mathscr {F}}_t\bigg ) \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^+\mid {\mathscr {F}}_t)+\displaystyle \liminf _{n\rightarrow +\infty }a_n^+ \mathbb {E}\bigg (1_{\{\sup _{s\le T} F(U_s)^+ >a_n^+\}}\mid {\mathscr {F}}_t\bigg ) \\ &{} \le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^+\mid {\mathscr {F}}_t\bigg ), \end{array} $$

where we have used the limit appeared in (4).

It follows that for every stopping time $\lambda \in {\mathscr {M}}_{t,T}$

$$\begin{aligned} \begin{array}{ll} Y_t &{}\le \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^+\mid {\mathscr {F}}_t\bigg )-\mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}^-\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E} \bigg (Y_{\lambda \wedge {\upsigma }^*_t}\mid {\mathscr {F}}_t\bigg ). \\ &{}\le \mathbb {E}\bigg (F(U_{\lambda })1_{\{\lambda< {\upsigma }^*_t\}} + F(L_{{\upsigma }^*})1_{\{\lambda > {\upsigma }^*_t\}}+ F(Q_{{\upsigma }^*})1_{\{ {\upsigma }^*_t=\lambda < T \}} + F(\xi ) 1_{\{ {\upsigma }^*_t = \lambda = T \}}\mid {\mathscr {F}}_t\bigg ) \\ &{} = \mathbb {E}\bigg (J(\lambda , {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ). \end{array} \end{aligned}$$

(6)

In force of inequalities (5) and (6) we obtain that for all ${\upsigma }, \lambda \in {\mathscr {M}}_{t,T}$

$$ \mathbb {E}\bigg (J(\lambda ^*_t, {\upsigma })\mid {\mathscr {F}}_t\bigg )\le Y_t = \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ]\le \mathbb {E}\bigg (J(\lambda , {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ). $$

Then it is immediately checked that

$$ \begin{array}{ll} \displaystyle {\inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {T}}_t}}\mathbb {E}\bigg (J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ) &{}\le \displaystyle {\sup _{{\upsigma }\in {\mathscr {T}}_t}}\mathbb {E}\bigg (J(\lambda ^*_t, {\upsigma })\mid {\mathscr {F}}_t\bigg ) \\ &{}\le Y_t = \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ]\\ &{}\le \displaystyle {\inf _{\lambda \in {\mathscr {M}}_{t,T}}}\mathbb {E}\bigg (J(\lambda , {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg )\\ &{}\le \displaystyle {\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {M}}_{t,T}}}\mathbb {E}\bigg (J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ). \end{array} $$

Since $\displaystyle {\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {M}}_{t,T}}}\mathbb {E}\bigg (J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg )\le \displaystyle {\inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}}\mathbb {E}\bigg (J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg )$, we have

$$ \begin{array}{ll} Y_t &{}= \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma }_t^*)\mid {\mathscr {F}}_t\bigg ]\\ &{} =\displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda ^*_t, {\upsigma })\mid {\mathscr {F}}_t\bigg ] = \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\mathbb {E}\bigg [J(\lambda , {\upsigma }^*_t)\mid {\mathscr {F}}_t\bigg ] \\ &{} = \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}\bigg [J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ] = \displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {T}}_t}\mathbb {E}\bigg [J(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ], \end{array} $$

Theorem 2 is then proved. $\square$

Remark 1

We should remark here that:

• 1. If F(L) and F(U) are $L^1-$integrable, i.e. $\mathbb {E}\displaystyle \sup _{t\le T}(|F(U_t)|+|F(L_t)|) <+\infty$, then the assumption of Theorem 2 are satisfied.

• 2. If we suppose that $F(x) = e^{\theta x}$ (or $F(x) = -e^{-\theta x}$), $\theta >0$, we have an utility function which is of exponential type and then our result can give, in particular, a solution to the existence a saddle point for the risk-sensitive problem (see [7] for more details).

4 American Game Option

4.1 Problem Formulation

We deal with American game option or a game contingent claim which is a contract between a seller A and a buyer B at time $t = 0$ such that both have the right to exercise at any stopping time before the maturity time T . If the buyer exercises at time t he receives the amount $L_t \ge 0$ from the seller and if the seller exercises at time t before the buyer he must pay to the buyer the amount $U_t \ge L_t$ so that $U_t - L_t$ is viewed as a penalty imposed on the seller for cancellation of the contract. If both exercise at the same time t before the maturity time T then the buyer may claim $Q_t$ and if neither have exercised until the expiry time T then the buyer may claim the amount $\xi$. In short, if the the seller decides to exercise at a stopping time $\lambda \le T$ and the buyer exercises at a stopping time ${\upsigma }\le T$ then the former pays to the latter the amount:

$$ J^1(\lambda , {\upsigma }) = U_{\lambda }1_{\{\lambda< {\upsigma }\}} + L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + \xi 1_{\{ {\upsigma }= \lambda = T \}}. $$

Such game option is considered in a standard securities market consisting of a non-random component $S^0_t$ representing the value of a savings account at time t with an interest rate r and of a random component $S_t$ representing the stock price at time t. More precisely and following the same idea as in Hamadène [9], we consider a security market $\mathscr {M}$ that contains, say, one bond and one stock and we suppose that their prices are subject to the following system of stochastic differential equations:

Let X be an ${\mathscr {F}}_t$-measurable random variable such that $X\ge 0$. The classical approach suggests that valuation of options should be based on the notions of a self-financing portfolio and on hedging. For this reason, we give the following definitions.

Definition 3

A self-financing portfolio after t with endowment at time t is X, is a $\mathscr {P}$-measurable process $\pi =(\beta _s, \gamma _s)_{t\le s\le T}$ with values in $\mathbb {R}^2$ such that:

(i) $\int _t^T (\mid \beta _s\mid + (\gamma _sS_s)^2)ds <\infty$.

(ii) If $\varDelta _s^{\pi , X} = \beta _sS_s^0 +\gamma _sS_s, \quad s\le T$, then $\varDelta _s^{\pi , X} =X+ \int _t^s\beta _u dS_u^0 +\int _t^s\gamma _u dS_u, \forall s\le T$.

Definition 4

A hedge against the game with payoff

$$ J^1(s, \lambda ) := U_{\lambda }1_{\{\lambda< s\}} + L_{s}1_{\{s<\lambda \}}+ Q_{s}1_{\{ s=\lambda < T \}} + \xi 1_{\{s= \lambda =T \}}, $$

after t whose endowment at t is X is a pair $(\pi , \lambda )$, where $\pi$ is self-financing portfolio after t whose endowment at t is X and a stopping time $\lambda \in {\mathscr {M}}_{t,T}$, satisfying: P-a.s. $\forall s\in [t, T]$,

$$ \varDelta _{s\wedge \lambda }^{\pi , X}\ge J^1(s, \lambda ). $$

Definition 5

The fair price of a contingent claim game is the infimum of capitals X for which the hedging strategy exists. It is defined by

$$ V_t:= \inf \{X\ge 0, \,\,\exists (\pi , \lambda ) \,\,\text{ such } \text{ that }\,\, \varDelta _{s\wedge \lambda }^{\pi , X}\ge J^1(s, \lambda ),\,\,\forall t\le s\le T, \,\,P-a.s.\}. $$

4.2 Fair Price of the Game as a Solution of BSDE with Two Reflecting Barriers

Now, let $P^*$ be the probability on $(\varOmega , \mathscr {F})$ under which the actualized price of the asset is a martingale, i.e.

$$ \frac{dP^*}{dP}:= exp\bigg (-\delta ^{-1}(b-r)B_t -\frac{1}{2}(\delta ^{-1}(b-r))^2t\bigg ),\quad t\le T. $$

Hence the process $W_t = B_t +\delta ^{-1}(b-r)t$ is an $({\mathscr {F}}_t, P^*)$-Brownian motion.

Let $\xi , L, U$ and Q be as in the beginning such that: $0\le L\le U.$ Assume moreover that assumption $(\mathbf {A.1})$ holds true and consider, on the probability space $(\varOmega , {\mathscr {F}}, P^*)$, the following BSDE with two reflecting barriers whose solution exists according to Theorem 1

$$\begin{aligned} \left\{ \begin{array}{ll} (i) &{} Y_{t}=e^{-rT}\xi +\int _{t}^{T}dK_{s}^+ -\int _{t}^{T}dK_{s}^- -\int _{t}^{T}Z_{s}dW_{s}, t\le T, \\ ({ {ii}})&{} \forall t\le T,\,\, e^{-rt}L_t \le Y_{t}\le e^{-rt} U_{t},\\ &{} \int _{0}^{T}( Y_{t}-e^{-rt}L_{t}) dK_{t}^+= \int _{0}^{T}( e^{-rt}U_{t}-Y_{t}) dK_{t}^-=0,\,\, \text{ a.s. }, \\ ({ {iv}})&{} Y\in {\mathscr {C}} \quad K^+, K^-\in {\mathscr {K}} \quad Z\in {\mathscr {L}}^{2,d}, \\ ({ v})&{} dK^+\perp dK^-. \end{array} \right. \end{aligned}$$

(7)

Let $\varrho ^*_t$ and $\vartheta ^*_t$ be the stopping times defined as follows:

$$\varrho ^*_t = \inf \{s\ge t\,\, :\, Y_s = e^{-rt}U_s\}\wedge T \quad \text{ and }\quad \vartheta ^*_t = \inf \{s\ge t\,\, :\, Y_s = e^{-rt}L_s\}\wedge T.$$

If we suppose that $\displaystyle {\liminf _{r\rightarrow +\infty }rP^*(\sup _{s\le T} U_s >r) =0}$, it follows then from Theorem 2, since $L\ge 0$, that for all ${\upsigma }, \lambda \in {\mathscr {M}}_{t,T}$, $Y_t$ solution of BSDE (7) is given by

$$\begin{aligned} \begin{array}{ll} Y_t &{}= \mathbb {E}^*\bigg [\overline{J}(\varrho ^*_t, \vartheta ^*_t)\mid {\mathscr {F}}_t\bigg ] \\ &{} = \displaystyle \inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}} \mathbb {E}^*\bigg [\overline{J}(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ] = \displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\inf _{\lambda \in {\mathscr {T}}_t}\mathbb {E}^*\bigg [\overline{J}(\lambda , {\upsigma })\mid {\mathscr {F}}_t\bigg ], \end{array} \end{aligned}$$

(8)

where

$$ \overline{J}(\lambda , {\upsigma }) = e^{-r\lambda }U_{\lambda }1_{\{\lambda< {\upsigma }\}} + e^{-r{\upsigma }}L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ e^{-r\lambda }Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + e^{-rT}\xi 1_{\{ {\upsigma }= \lambda = T \}}. $$

The main result of this section is the following.

Theorem 3

Assume that $\displaystyle {\liminf _{r\rightarrow +\infty }rP^*(\sup _{s\le T} U_s >r) =0}$. Then, the fair price of our game is given by $V_t = e^{rt}Y_t$, for any $t\le T$. Moreover, a hedge after t against the option exists and it is given by:

$$\gamma _s = \frac{e^{rs}Z_s}{\delta S_s}1_{\{ s\le \vartheta ^*_t\}}\,\,\text{ and }\,\, \beta _s = \bigg ( e^{rs}(Y_t +\int _t^{s}Z_u dW_u ) -\gamma _sS_s\bigg )(S_s^{0})^{-1},\,\,\forall s\in [t, T]. $$

Proof

Let $(\pi , \lambda )$ a hedge after t against the option. Therefore $\lambda \in {\mathscr {M}}_{t,T}$ and $\pi =(\beta _s, \gamma _s)_{t\le t\le T}$ is a self-financing portfolio whose value at t is X satisfying $\varDelta _{s\wedge \lambda }^{\pi , X}\ge J^1(s, \lambda ),\,\,\forall t\le s\le T$. But

$$ e^{-r(s\wedge \lambda )}\varDelta _{s\wedge \lambda }^{\pi , X} = e^{-rt}X +\delta \int _t^{s\wedge \lambda }\gamma _u S_ue^{-ru}dW_u\ge e^{-r(s\wedge \lambda )} J^1(s, \lambda ), \,\,\forall t\le s\le T. $$

Let ${\upsigma }\ge t$ be a stopping time. Putting $s={\upsigma }$ and taking the conditional expectation we obtain

$$ e^{-rt}X \ge \mathbb {E}^*\bigg (e^{-r({\upsigma }\wedge \lambda )}J^1({\upsigma }, \lambda )\mid {\mathscr {F}}_t\bigg ). $$

In view of relation (8) we have

$$ \begin{array}{ll} e^{-rt}X &{}\ge {\displaystyle \sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\mathbb {E}^*\bigg (e^{-r({\upsigma }\wedge \lambda )}J^1({\upsigma }, \lambda )\mid {\mathscr {F}}_t\bigg )}\\ &{} \ge \displaystyle {\inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\mathbb {E}^*\bigg (e^{-r({\upsigma }\wedge \lambda )} J^1({\upsigma }, \lambda ) \mid {\mathscr {F}}_t\bigg ) } \\ &{} = \displaystyle {\inf _{\lambda \in {\mathscr {M}}_{t,T}}\sup _{{\upsigma }\in {\mathscr {M}}_{t,T}}\mathbb {E}^*\bigg (\overline{J}({\upsigma }, \lambda ) \mid {\mathscr {F}}_t\bigg )} \\ &{} =Y_t. \end{array} $$

Henceforth $V_t \ge e^{rt} Y_t$. Let us now prove the converse inequality. It follows that for every $t\le s\le T$,

$$ \begin{array}{ll} &{} Y_t +\int _t^{s\wedge \vartheta ^*_t}Z_u dW_u \\ &{} \le Y_{s\wedge \vartheta ^*_t}- \int _t^{s\wedge \vartheta ^*_t}dK^-_u \\ &{} \le Y_{s\wedge \vartheta ^*_t} \\ &{} \le e^{-rs}U_{s}1_{\{s< \vartheta ^*_t\}} + e^{-r\vartheta ^*_t}L_{\vartheta ^*_t}1_{\{s > \vartheta ^*_t\}}+ e^{-rs}Q_{\vartheta ^*_t}1_{\{ \vartheta ^*_t=s < T \}} + e^{-rT}\xi 1_{\{ \vartheta ^*_t = s= T \}} \\ &{}= e^{-r(s\wedge \vartheta ^*_t)} J^1(s, \vartheta ^*_t). \end{array} $$

Hence for every $t\le s\le T$,

$$ J^1(s, \vartheta ^*_t) \ge e^{r(s\wedge \vartheta ^*_t)} (Y_t +\int _t^{s\wedge \vartheta ^*_t}Z_u dW_u). $$

Now if we put for all $s\in [t, T],$ $\gamma _s = \frac{e^{rs}Z_s}{\delta S_s}1_{\{ s\le \vartheta ^*_t\}}$ and $\beta _s = \bigg ( e^{rs}(Y_t +\int _t^{s}Z_u dW_u ) -\gamma _sS_s\bigg )(S_s^{0})^{-1}$.

Hence $(\beta _s, \gamma _s)_{t\le s\le T}$ is a self-financing portfolio whose value at t is $e^{rt} Y_t$. On other hand we have

$$e^{r(s\wedge \varrho ^*_t)}(Y_t +\int _t^{s\wedge \varrho ^*_t}Z_u dW_u)\ge J^1(s, \varrho ^*_t), \,\, \forall s\in [t,T].$$

Hence $((\beta _s, \gamma _s)_{t\le t\le T}, \varrho ^*_t)$ is a hedge against the game option. Then $e^{rt}Y_t\ge V_t$. Henceforth $e^{rt}Y_t = V_t$. The proof of Theorem 3 is then achieved. $\square$