Nash Equilibrium and the Legendre Transform in Optimal Stopping Games with One Dimensional Diffusions

Transcription

1 Nash Equilibrium and the Legendre Transform in Optimal Stopping Games with One Dimensional Diffusions J. L. Seton This version: 9 Januar 2014 First version: 2 December 2011 Research Report No. 9, 2011, Probabilit and Statistics Group School of Mathematics, The Universit of Manchester

2 Nash equilibirum and the Legendre transform in optimal stopping games with one dimensional diffusions Jenn Seton Januar 9, 2014 Abstract We show that the value function of an optimal stopping game driven b a one-dimensional diffusion can be characterised using the etension of the Legendre transform introduced in 19]. It is shown that under certain integrabilit conditions, a Nash equilibrium of the optimal stopping game can be derived from this etension of the Legendre transform. This result is an analtical complement to the results in 19] where the dualit between a concave biconjugate which is modified to remain below an upper barrier and a conve biconjugate which is modified to remain above a lower barrier is proven b appealing to the probabilistic result in 18]. The main contribution of this paper is to show that, for optimal stopping games driven b a one-dimensional diffusion, the semiharmonic characterisation of the value function ma be proven using onl results from conve analsis. Kewords: optimal stopping, conve analsis. Mathematics Subject Classification 2000): 26A51, 60G40, 91A15, 91G80 1 Introduction This paper eamines the connection between conve analsis and optimal stopping of one dimensional diffusions. The connection between the stud of superharmonic functions and optimal stopping problems dates back to Dnkin 7] where the solution to an optimal stopping problem of the tpe V ) = sup E G X τ )] τ where X is a Markov process) was first characterised as the smallest superharmonic majorant of the gains function G. When X is a one-dimensional diffusion, the corresponding superharmonic functions can been characterised in terms of a generalised tpe of concavit see 10] pp. 115). This so called superharmonic characterisation of the value function is discussed in further detail in 20] Chapter IV Section 9, to illustrate the properties of the solution to certain freeboundar problems associated with optimal stopping problems. More recentl, the change of time and scale technique underpinning the superharmonic characterisation was re-introduced School of Mathematics, Universit of Manchester. Oford Road, Manchester, M13 9PL, United Kingdom. jennifer.seton@postgrad.manchester.ac.uk 1

3 in 6]. Furthermore, the connection between the superharmonic characterisation and conve analsis has been highlighted in 19]. The minima version of this problem is referred to as an optimal stopping or Dnkin) game. The sup-plaer selects a stopping time τ with the aim of maimising the functional R τ, σ) = E G Xτ ) I τ σ] + X σ ) I τ>σ] ] while the inf-plaer selects a stopping time σ with the aim of minimising the same functional. Optimal stopping games are eplained in more detail in Section 2 below. A variant of this optimal stopping game was first studied b Dnkin 9] using martingale based methods. These problems have also been approached via variational inequalities in 2] and 3]. General conditions which ensure that such a saddle point eists have been studied in 11] and 12]. Optimal stopping games have been applied to solve problems in finance, see for eample, 1], 13], 14], 15], 16] or 17]. In 18], the superharmonic characterisation of the solution to optimal stopping problems has been etended to optimal stopping games where it is shown that the value function is a semiharmonic function. The value function is shown to be the smallest function which is superharmonic and dominates G on the set {V < } as well as the largest function which is subharmonic and minorises on the set {V > G}. It is shown in 18] that a single function with both these properties eists if and onl if the optimal stopping game ehibits a saddle point. When the underling Markov process is a one dimensional diffusion, these dual problems have been formulated in terms of an etension of the Legendre transform in 19]. The purpose of this paper is to establish the semiharmonic characterisation of the value function of optimal stopping games driven b one dimensional diffusions using purel analtical techniques. This requires us to modif some of the basic objects of conve analsis to account for the additional constraint on the domain of the smallest superharmonic majorant resp. largest subharmonic minorant). The net section formall introduces optimal stopping games and signposts the contents of the rest of this paper. 2 Semiharmonic characterisation Consider a stochastic basis Ω, F, F, {P } I ) supporting a one dimensional diffusion X = X t ) t 0 with X 0 = I R under P. The level passage times of X will be denoted b T = inf {t 0 X t = }. The state space I of X is an open interval I = a, b) and the boundaries of I are natural, i.e. P T a < ) = P T b < ) = 0 for all I. The diffusion X is also assumed to be regular in the sense that for all I, P T < ) > 0 for some, b). For a given discount rate r > 0, a Borel measurable function U : I R is r-superharmonic with respect to X if U ) E e rτ U X τ ) I τ< ] ] for all I and all stopping times τ. The function U is r-subharmonic with respect to X if U ) E e rτ U X τ ) I τ< ] ] for all I and all stopping times τ. Moreover, U is referred to as r-harmonic, if it is both r-superharmonic and r-subharmonic. The generator of X is denoted L X and under some additional regularit conditions see 4] Section 4.6) can be epressed as L X f ) = µ ) d d2 f ) + D ) d d 2 f ) 2

4 for I where µ is the drift and D > 0 is the diffusion coefficient of X. Take a constant r > 0 and consider the ODE L X f ) = rf ) 2.1) for I. This ODE has two linearl independent positive solutions, denoted ϕ and ψ. As the boundaries of I are natural, ϕ and ψ ma be taken such that ϕ is increasing and ψ is decreasing with ϕ a+) = ψ b ) = 0, ϕ b ) = ψ a+) = +. The functions ϕ and ψ are shown in 23] V.50 to be continuous, strictl monotone and strictl conve. Furthermore, for I, the process e rt T) ϕx t T )) t 0 is a P, F t T )-martingale for all > while e rt T) ψx t T )) t 0 is a P, F t T )-martingale for all. For, I the Laplace transforms of the first passage times ma be epressed as E e rt ] I T< ] = ψ) ψ) for >, ϕ) ϕ) for. Definition 1. Let J : I R be a monotone function. A Borel measurable function f : I R is J-concave if J z) J ) J ) J ) f ) f ) + f z) J z) J ) J z) J ). for, z] I. The J-derivative of f is denoted d f ) f ) f ) := lim dj J ) J ). Define a pair of strictl increasing functions, F : I 0, + ) and F : I, 0) using F ) = ϕ ) ψ ), F ) = ψ ) ϕ ) = 1 F ). 2.2) It is well-known see 8] Theorem 16.4) that a Borel measurable function U is r-superharmonic if and onl if U/ψ is F -concave, or equivalentl, U/ϕ is F -concave. For a < z < b the Laplace transforms of the eit times from the open set, z) I are E e rt ] ψ ) I T<Tz] = ψ ) E e rt z ] ψ ) I T>Tz] = ψ z) F z) F ) F z) F ) = ϕ ) ϕ ) F ) F ) F z) F ) = ϕ ) ϕ z) F z) F ) F z) F ), F ) F ) F z) F ). 2.3) Take a gains function G : I R which is upper semi-continuous and such that for all I, ) P lim t e rt G X t ) = c = 1, E e rt G X t ) ] < + 2.4) for some c R. Section 3 eamines the solution to the discounted optimal stopping problem sup t 0 V ) = sup E e rτ G X τ ) ] 2.5) τ 3

5 for I. Theorem 3.2 in 19] shows that when the value function V defined in 2.5) coincides with the solution to the dual problem ˆV = inf U 2.6) U SupG] where Sup G] := {U : I G, + ) U is continuous and r-superharmonic} the function V can be epressed in terms of the concave biconjugate of W := G/ψ) F 1. The observation that when a finite optimal stopping time in 2.5) eists, the solutions to 2.5) and 2.6) coincide is often referred to as the superharmonic characterisation of the value function see Theorem 2.4 in 20] for a probabilistic proof). Theorem 3 is the converse of Theorem 3.2 in 19] in the sense that function V is shown to be related to the concave biconjugate of W without assuming that V solves 2.6). Furthermore, Theorem 3 provides a new proof that 2.5) and 2.6) coincide. The remainder of this section introduces optimal stopping games and the semiharmonic characterisation of the solution to such games as well as outlining the contents of the rest of this paper. Take a pair of gains functions G, that are continuous and for all I satisf: and G ) ), E sup e rt G X t ) t 0 ] < +, E sup t 0 e rt X t ) ] < + 2.7) ) P lim t e rt G X t ) = lim e rt X t ) = ) t Section 5 studies the solution to the infinite time horizon optimal stopping game with lower value V defined as V ) = sup inf E e rτ σ) ) ] G X τ ) I τ σ] + X σ ) I τ>σ], 2.9) τ σ and upper value V defined as V ) = inf σ sup E e rτ σ) ) ] G X τ ) I τ σ] + X σ ) I τ>σ]. 2.10) τ where G and satisf the assumptions outlined above. For ease of notation, the objective function is denoted R τ, σ) := E e rτ σ) G X τ ) I τ σ] + X σ ) I τ>σ] ) ]. 2.11) The optimal stopping game has a Stackelberg equilibrium if the upper and lower values coincide, i.e. V ) = V ) =: V ) for all I. In 11] and 12] it is shown via probabilistic means that this game ehibits a Stackelberg equilibrium when both 2.7) and 2.8) hold. The assumptions in 12] are slightl more general, in which case the Stackelberg equilibrium is determined b how the objective function is specified at the natural boundaries. A saddle point is a pair of stopping times τ, σ ) such that for an other stopping times τ, σ R τ, σ ) R τ, σ ) R τ, σ) I. 4

6 The optimal stopping game ehibits a Nash equilibrium if the game has saddle point. In particular, eistence of a Nash equilibrium implies a Stakelberg equilibrium eists but the converse is not necessaril true. The result in 11] which applies to more general processes) shows that, under the assumptions 2.7) and 2.8), the optimal stopping game described above has a Nash equilibrium. Introduce a pair of dual problems ˆV := where the admissible sets of functions are inf U, ˇV := sup U 2.12) U SupG,) U SubG,] Sup G, ) = {U : I G, ] U is continuous and r-superharmonic on {U < }}, Sub G, ] = {U : I G, ] U is continuous and r-subharmonic on {U > G}}. An function U Sup G, ) SubG, ] is referred to as a r-semiharmonic. It has been shown in 18] Theorem 2.1 that when 2.7) and 2.8) hold ˆV = ˇV and that the value of the dual problems coincides with the value of the optimal stopping game with upper and lower values 2.9)-2.10) if and onl if the optimal stopping game has a Nash equilibrium. The joint solution to the dual problems 2.12) is referred to in 18] as the semiharmonic characterisation of the value function. When the driving process X is a one dimensional diffusion absorbed upon eit from a compact set, it has been shown in 19] that the solutions to the dual problems 2.12) can be epressed in terms of an etension of the Legendre transform. It then follows from Theorem 2.1 in 18] that the Nash and Stackelberg equilibrium of the optimal stopping game 2.9)-2.10) can be epressed in terms of this etension of the Legendre transform. The main contribution of Section 4 is to establish equalit between the two etension of the Legendre transform introduced in 19] without reference to the probabilistic results in 18] and 11]. In particular, we construct an etension of the concave biconjugate of W G := G/ψ) F 1 of the form W G ) ) = inf sup z ) + W G )) z R A G z) for > 0 where the set mapping z A G z) is defined in such a wa as to ensure that W G ) ) /ψ) F 1 ) =: W ) for all > 0. Similarl, we construct an etension of the conve biconjugate of W of the form W ) G ) = sup inf z R A z) z ) + W )) for > 0 where the set mapping z A z) is defined in such a wa as to ensure that W ) G ) W G ) for all > 0. In particular, Theorem 14 is a converse to 19] Theorem 4.1 as it shows that W G ) ) = W ) G ) for all > 0 using a purel analtical approach. Section 5 establishes that under the assumptions 2.7) and 2.8) the infinite time horizon optimal stopping game 2.9)-2.10) has both a Stackelberg and Nash equilibrium. In Theorem 18 the optimal stopping game 2.9)-2.10) is shown to have a Stackelberg equilibrium which can be epressed in terms of the etensions of the Legendre transform introduced in Section 4. It is then shown that the functions G), ), : I G, ] defined as G) ) = W G ) F ))ψ), ) G ) = W ) G F ))ψ) 5

7 are r-semiharmonic and solve the dual problems 2.12). Consequentl, it follows from the results in Section 4 that the solutions to the dual problems 2.12) coincide. Finall in Theorem 22 it is shown that the F -concave/f -conve structure of W G ) can be used to characterise a Nash equilibrium of the optimal stopping game 2.9)-2.10). 3 Optimal stopping using the Legendre transformation Before proceeding to solve the optimal stopping problem 2.5), we shall first recall the definition and some properties of the concave biconjugate. Let f : dom f) R be a finite, measurable function on the domain dom f), + ]. The concave conjugate of f, denoted f, is defined for c R as f c) = inf c f )). domf) The concave biconjugate of f is defined as f ) = inf f domf )) = ) inf sup domf ) c domf) c) + f c)). 3.13) The epigraph of a function f is the set of all points above the graph of f, that is epi f) := {, µ) R 2 µ f ) }. The conve hull of the set epi f) is the intersection of all conve sets containing epi f) and is denoted conv f). With a slight abuse of notation, let conv f) ) := inf { R, ) conv f)}, then f is the upper semi-continuous modification of conv f) ), see 22] Theorem 12.2 and Corollar A constant c R is a subgradient of the function f at R if The set of all such subgradients, denoted f z) f ) + c z ) z R. f ) := {c R f ) f ) c ) R} 3.14) is referred to as the subdifferential. The function f is referred to as concave at R when f ). When f is concave and differentiable at then f ) = {f )}. The net lemma characterises the set upon which f coincides with f. The proof illustrates how f ) can be constructed using a spike variation. Lemma 2. An function f : domf) R coincides with its concave biconjugate on the set { R f ) = f )} = { R f ) }. 3.15) Proof. Suppose that f ) and take c f ) then it follows from the definition 3.14) that f c) := inf c f )) = c f ). R 6

8 Consequentl, f can be written as f ) = inf c ) + f )) = f ). c R so { R f ) } { R f ) = f )}. To show this inclusion holds with equalit assume that f ) = and for a fied c R let It follows that and f can be written as Let ε ; c) := inf {ε 0 f ) + c ) f ) + ε R}. f c) = sup f ) c) = f ) + ε ; c) c, R f ) = inf c f c)) = inf f ) + ε ; c)). 3.16) c R c R ε ) := inf {ε 0 c R s.t. f ) + c ) f ) + ε R} 3.17) so that { R ε ) > 0} = { R f ) = }. ence we ma conclude from 3.16) that when f ) =, f ) = f ) + ε ) > f ). Let W : 0, ) R be defined as W ) := ) G F ) 1 ) 3.18) ψ where F was defined in 2.2). Similarl, let W :, 0) R be defined via ) G W ) := ϕ F ) 1 ) 3.19) The net result is the main result in this section and it shows that the value function V defined in 2.5) is such that V/ψ coincides with W F where W is the concave biconjugate of the function W. The value function V is also such that V/ϕ coincides with W F where W is the concave biconjugate of the function W. The case that both boundaries are absorbing has been handled b showing that W F )) ψ ) solves 2.6) in 19] Theorem 3.2. Theorem 3. Assume that G : I R is an upper-semicontinuous function satisfing the assumption 2.4) and such that W 0+) = W 0 ) = 0. Consider the stopping problem 2.5), then V ) = W F )) ψ ) = W F ))ϕ ) for all I. The stopping time which attains the supremum in 2.5) is τ = T a T b where a = sup {c a, ] W F c)) = W F c))}, 3.20) b = inf {c, b) W F c)) = W F c))}, and we use the convention that sup = a and inf = b. Furthermore V solves 2.6). 7

9 Proof. It is assumed that W 0+) = W 0 ) = 0 without loss of generalit as when this is not true it can be achieved b subtracting the constant c introduced in 2.4) from the value function V. Let b n ) n 1 be a sequence of real numbers such a, b n ] a, b), b n b n+1 for all n 1 and a, b n ] a, b) as n +. Fi a, b) then for some sufficientl large N, a, b n ] for all n N. Let X n t := X t Tbn and consider the following famil of optimal stopping problems where G satisfies 2.4). Consider the sets of functions V n ) = sup E e rτ G Xτ n ) ] 3.21) τ Sup n G] = {U : a, b n ] G, + ) U is continuous and r superharmonic} for each n 1. Take an U 1, U 2 Sup n G], then U 1 /ψ and U 2 /ψ are F -concave on a, b n ] see 8] Theorem 16.4). Since U 1 /ψ and U 2 /ψ are F -concave, the functions W 1 = U 1 /ψ) F 1 and W 2 = U 2 /ψ) F 1 are concave. Consequentl, W 3 = W 1 W 2 is a concave function and reversing this argument implies U 3 = U 1 U 2 Sup n G]. ence the set Sup n G] is downwards directed so Ṽ n := inf U Sup n G] U eists. In fact this infimum is attained since, when X is a one dimensional diffusion, all r-superharmonic functions are continuous see 20] Sections and replacing S with F ). This shows that the conditions of 20] Theorem 2.17 hold so if an optimal stopping time eists it is of the form τn = inf{t 0 X t D n } where D n := { I G) = Ṽ n )}. The set D is a closed set and C n := { I G) < Ṽ n )} is open since G is upper-semicontinuous and Ṽ n is continuous on a, b n ]. ence, for each n 1 there is a collection of disjoint open intervals i, i )) i 1 such that C n = i 1 i, i ). Suppose that X 0 i, i ) for some i 1, as the paths t X t are continuous, T i T i T j T j for all j i. Consequentl we can restrict attention to candidate stopping times which are eit times from open intervals. If an optimal stopping time for 3.21) eists we ma write the optimal stopping time for 3.21) as τn = T a n T b n for some a a n b n b n. Furthermore, each τn T bn, τn τn+1 for all n 1 and lim n + τn = τ P -a.s. for all I. For all n 1, Xτ n n = X τn and the process X is left-continuous over stopping times so lim n + Xτ n n = X τ P -a.s for all I. The assumption 2.4) implies that e rt G X t ) ) is uniforml integrable so it follows that t 0 lim V n ) = lim E e rτ ngx n n n τn )] = V ). The process e rt T bn ) ϕx n t )) t 0 is a P, F t Tbn )-martingale for each b n > so appling the optional sampling theorem ields E e rτ ϕ X n τ )] = ϕ ) for all a, b n ] and all τ T bn. ence for an arbitrar c R V n ) = sup τ E e rτ G Xτ n ) ] = cϕ ) + sup E e rτ G Xτ n ) cϕ Xτ n )) ]. τ Thus, V n ) = inf cϕ ) inf E e rτ cϕ X n c R τ τ ) G Xτ n )) ]). 3.22) The net step is to epand the right hand side of this epression in such a wa that it converges to the concave biconjugate of W as n +. Using 2.3), the inner infimum in 3.22) can be written as ] inf E e rt Tz) cϕxt n a< z b T z ) GXT n T z )) n ) cϕ ) G ) F z) F ) cϕ z) G z) F ) F ) = inf + ψ ). a< z b n ψ ) F z) F ) ψ z) F z) F ) 8

10 We claim that the right hand side of this epression converges to cϕ ) G ) inf a,b) ψ ) as n. To this end, take ) ψ ) = inf >0 c W ) )ψ ) =: W c) ψ ) z + = sup {z I W c) ψ z) = cϕ z) G z)} = inf { I W c) ψ ) = cϕ ) G )} with the convention that sup = a and inf = b. In general for all a z b n, W c) λc W )) + 1 λ)cz W z)) for all λ 0, 1]. In particular, taking λ = F z) F ))/F z) F )) we obtain W c) cϕ ) G ) F z) F ) ψ ) F z) F ) cϕ z) G z) F ) F ) + ψ z) F z) F ). 3.23) When z ) holds with equalit for = and z = z +. When, let m ) m 1 be such that m and lim m m = a. For = m and z = the inequalit in 3.23) is strict, however, since we assumed W 0+) = 0 it follows from the definition of F that the as m the right hand side of 3.23) converges to W c). When z + the inequalit is strict as cϕ ) G ) F z + ) F ) ψ ) F z + ) F ) + cϕ z+ ) G z + ) ψ z + ) F ) F ) F z + ) F ) > cϕ z+ ) G z + ) ψ z + ) 3.24) for all, b n ]. We ma switch to the other ratio of the fundamental solutions using 2.3) on the left hand side of 3.24) and use that ϕ ) + as b to conclude lim inf cϕ ) G ) F z + ) F ) n,b n] ϕ ) F z + ) F ) + cϕ z+ ) G z + ) F ) F ) ) ϕ z + ) F z + ) F = W c). ) ence letting n on both sides of 3.22) ields V ) = inf c ϕ ) ) c R ψ ) + W c) ψ ). The argument using the other ratio of fundamental solutions follows analogousl. The statement about the optimal stopping time follows from the form of the value function provided as the stopping region D and continuation region C for 2.5) are C = { I V ) > G )} = { I W F )) ψ ) > G )}, D = { I V ) = G )} = { I W F )) ψ ) = G )}. The stopping time τ := {t 0 X t D} = T a T b attains the supremum in 2.5) although it ma occur that P τ < + ) < 1. Finall, W = cl conv W )) is the smallest concave function dominating W, so V ) = W F )) ψ ) is the smallest r-superharmonic function dominating the gains function G ) and hence V ) solves 2.6). 9

11 In Theorem 3 it has been shown that when W 0+) = W 0 ) = 0 the value function of 2.5) coincides with the solution to 2.6). Whether the optimal stopping time is obtained in finite time depends both on the function G and the behaviour of the diffusion X t as t. In Theorem 3 we can rela the assumptions 2.4) and instead impose that W 0+) < and W 0 ) < which is the assumption used in 6]. The case that W 0+) = + is degenerate as V ) lim c a E GX Tc )e rtc I Tc< ] = lim c 0 W c)ψ) = W 0+)ψ) =. Remark 4. The assumption in Theorem 3 that W 0+) = W 0 ) = 0 ensures that V a+) = V b ) = 0. In 6] and 12] this condition is imposed b etending all Borel measurable functions F on I onto a, b] b setting F X τ ω)) = 0 on {τ = + }. If we use this convention and the weaker assumption that W 0+) < and W 0 ) < rather than 2.4) the optimal stopping time ma not eist. To see this, suppose that a sequence a n ) n 1 such that a n a as n is optimising in the sense that E GX Tan )e rtan ] E GX Tan+1 )e rta n+1 ] and V ) = lim n E GX Tan )e rtan ] = Ga+)ψ)/ψa+) = W 0+). When W 0+) > 0, this limit is not obtained as τ = lim n T an = P -a.s. for all I so E GX τ )e rτ ] = 0. For fied I let { ) l c, p) = sup G ) ψ { ) r c, p) = inf G ) ψ p ψ ) } = c F ) F )) p = c F ) F )) ψ ) with the usual convention that sup = and inf = +. For a given I consider the sets A 1 ) = {p G ), + ) c R s.t. l c, p), r c, p)) I }, 3.25), }, A 2 ) = {p G ), + ) c R s.t. r c, p) = l c, p) = + }. In Lemma 2 it was shown that W ) can be constructed b minimising over the F - tangents of the form p/ψ ) + c F ) F )) which strictl dominate W on dom W ), i.e. W ) = inf A 2 ). The net corollar provides a short proof of the dual interpretation provided in 19] which claims that W ) can be also constructed b maimising over the F -tangents which intercept W on both sides of the point in dom W ), i.e. W ) = sup A 1 ). The latter formulation facilitates the construction of a maimising sequence of stopping times. Corollar 5. Let ε ; c) := sup {ε 0 a l c, G ) + ε) r c, G ) + ε) b} and set A c ) := F l c, G ) + ε ; c))), F r c, G ) + ε ; c)))], then the value function of the optimal stopping problem 2.5) can be represented as V ) = sup sup c R A c) c F ) ) + W )) ψ ). Moreover, V ) = sup A 1 ) for each I where A 1 ) is the set defined in 3.25). 10

12 ε,c) ε,c) G) Figure 1: This diagram illustrates ε ; c) and ε ; c) for a fied c in the case that r = 0 and X is a standard Brownian motion i.e. ψ 1 and ϕ) = hence F ) = ) Proof. For the first statement, note that for each c R, b definition G ) + ε ; c) ψ ) + c F )) W ) for A c ) and equalit holds for = F l c, G ) + ε ; c))). ence, and it follows that Let sup ε ; c) sup W ) c) = W F )) + cf ) A c) ψ ) sup c R A c) ε ; c) := inf c F ) ) + W )) ψ ) = sup G ) + ε ; c)) c R { ε 0 ) G ) + c F ) ) G) + ε ψ ψ) } R. For each c R, ε ; c) ε ; c) as illustrated in Figure 1 but to avoid contradicting their definitions inf c R ε ; c) = sup c R ε ; c) and hence the result follows from Lemma 2. Furthermore, it follows from the definition of the set A 1 ) that sup A 1 ) = G ) + sup c R ε ; c) = W F ))ψ) = V ). The net corollar shows that Theorem 3 is capable of handling the case when one or more) of the boundaries is absorbing. Corollar 6. Take α, β) a, b) and consider the stopped diffusion X α,β t := X t Tα,β where T α,β = T α T β and the corresponding optimal stopping problem ] V α,β ) = sup E e rτ GXτ α,β ) 3.26) τ 11

13 for G : a, b) R satisfing the assumptions in Theorem 3. Then V α,β ) = W α,β] ) F )) ψ ) = W α,β] ) F ))ϕ) where W α,β] ) := W ) δ F 1 ) I \ α, β]), W α,β] ) := W ) δ F 1 ) I \ α, β]). Moreover, the stopping time which attains the supremum in 2.5) is τ = T a T b where a = sup{ α, ] W α,β] F )) = W α,β] ) F ))}, b = inf{, β] W α,β] F )) = W α,β] ) F ))}. Proof. An optimal stopping time for a problem of 3.26) eists see 20] Chapter 1 Theorem 2.7) and is the eit time from a open set containing so we ma write τ = T a T b for some α a b β. Suppose that β < b then as shown in the proof of Theorem 3, the value function of the optimal stopping problem 3.26) can be written as where R c ) := inf a z b V α,β ) = inf c R cϕ ) R c )). 3.27) cϕ ) G ) F z) F ) ψ ) F z) F ) + cϕ z) G z) ψ z) ) F ) F ) ψ) F z) F ) and G ) := G α) I a,α] + G ) I α,b) + G β) I β,b). As ϕ is r-harmonic on, b) for all > a, it follows that for an I cϕ ) G ) ψ ) = E e rt cϕ ) ) X T G XT )] ψ ) for all a, α). Likewise for β, b), cϕ ) G ) ψ ) = cϕ ) E e rt G X T ) ] > cϕ ) G α) E e rt α ] cϕ α) G α) = ψ ) ψ α) ] ψ ) > cϕ ) G β) E e rt β cϕ α) G β) = ψ ). ψ β) ence, an argument similar to that presented in Theorem 3 can be used to show that cϕ ) R c ) = inf G ) ) ) cϕ ) G ) ψ ) = inf ψ ) =: W c). a,b) ψ ) α,β] ψ ) Thus 3.27) reads V α,β ) = inf cϕ ) c R inf α,β] cϕ ) G ) ψ ) )) ψ ) = W α,β] F )) ψ ). ) The case β = b can be handled using an approimating sequence of domains as in Theorem 3. The argument using the other ratio of fundamental solutions follows analogousl. 12

14 This section concludes with a basic eample, the perpetual American put option. Eample 7. Take σ > 0, r > 0 and define X to be the solution to dx t = rx t dt + σx t dw t with X 0 = > 0 where W is a standard Brownian motion. Consider the perpetual American put option, the gains function of which is G) = K ) + for a strike price K > 0. The risk neutral value of this option is V ) := sup E e rτ K X τ ) +]. 3.28) τ The infinitesimal generator associated with the geometric Brownian motion X is L X u := r du d σ2 2 d2 u d 2 and the ODE L X u = ru has two fundamental solutions ϕ ) = and ψ ) = 2r/σ2. Let F ) = ψ/ϕ) ) = 1/α where α := 1/ 1 + 2r/σ 2) 0, 1]. The rescaled gains function associated with the problem 3.28) is G ) := K α 1) +. The function G ) has G ) 0 for all R + and hence to find G ) we need onl to find 0 such that G ) G ) = ) The unique solution to 3.29) is = 1/ K 1 α)) 1/α and { G G ) for 0, ], ) = G ) for >. Appling Theorem 3, V ) = G F ))ϕ ) = σ 2 2r K 1+σ 2 /2r) ) 1+2r/σ 2 2r/σ2 for K for 0, ] 3.30) where = ) α = K/ 1 + σ 2 /2r ). Figure 2 illustrates the original and transformed paoff functions, the concave biconjugate of the transformed paoff and the corresponding value function. Moreover, the continuation and stopping region for the problem 3.28) are D := { R + V ) = G )} = 0, ], C := { R + V ) > G )} =, + ). which coincides with the established solution see for eample 20] Section 25.1). 4 An etension of the Legendre transform The purpose of this section is to etend the Legendre transform so that the observations about optimal stopping in the previous section can be applied to optimal stopping games. The function G represents the paoff of the maimising agent while the function represents the paoff to 13

15 G) G ) K1 α)) 1/α K ) + V) K 1 + σ2 ) 2r K Figure 2: In the left figure, the transformed paoff function G is drawn along with its concave biconjugate. The original paoff G) = K ) + and the corresponding value function V is in the figure on the right hand side. the minimising agent. The two continuous gains functions G, are such that G) ) for all I and satisf the assumptions 2.7) and 2.8) unless otherwise stated. Inspired b the transformation used in Theorem 3 introduce a pair of rescaled gains functions W G : 0, ) R and W : 0, ) R defined via ) G W G ) := F 1 ) 4.31) ψ and W ) := Under the assumptions 2.7) and 2.8) we have ) F 1 ). 4.32) ψ limw ) W G )) = limw ) W G )) = ) 0 This assumption can be relaed a little as discussed in Remark 10 below. These definitions could equall well be formulated with respect to the other ratio of the fundamental solutions but for clarit we shall focus onl on these two epressions. The aim of this section is to define and describe a version of the conve biconjugate of the function W which is modified to ensure that it remains inside epiw G ). At the same time, we define a version of the concave biconjugate of the function W G which is modified to ensure that it remains inside clr 2 \ epiw )). This is achieved b defining an etension of the ε-sub/superdifferential tpicall used in conve analsis. The main result in this section is Theorem 14 which is the purel analtical version of 19] Theorem 4.1. Theorem 14 shows that the conve biconjugate respecting the lower barrier W and the concave biconjugate respecting the upper barrier W G coincide. This dualit result naturall follows from the spike-variations we use to construct these etensions of the Legendre transform. Moreover, the dualit between these two etensions of the Legendre transform are used in Section 5 to provide a new purel analtical) proof that the optimal stopping game 2.9)-2.10) ehibits both a Stackelberg and a Nash equilibrium. 14

16 For a given function f : R + R, let lf c, p) = sup {z fz) p = cz )}, rf c, p) = inf { f) p = c )}, with the convention that sup = 0 and inf = +. For ease of notation, let l G c, p) := l W G c, p) and l c, p) := l W c, p). The point l G c, W )) resp. r G c, W ))) is the last resp. first) time the line passing through, W )) with slope c intercepts W G before resp. after). Define the sperbdifferential of W in the presence of the lower boundar W G as G ) := {c R W ) W ) c ) l Gc, W )), r Gc, W )))}. 4.34) If the tangent W ) + c ) minorises W ) prior to intercepting the lower boundar W G ), then we refer to c as a supergradient of W in the presence of W G at, i.e. c G ), as shown in Figure 3. Similarl, the subdifferential of W G in the presence of the upper boundar W is G) := { c R W G ) W G ) c ) l c, W G )), r c, W G ))) }. 4.35) If the tangent W G ) + c ) majorises W G ) prior to intercepting the upper boundar W ), then we refer to c as a subgradient of G in the presence of at, i.e. c G), as illustrated in Figure 3. W ) W G ) c G 1 ) c G 2 ) Figure 3: In this figure W G ) = W ) for all 3. The slope of the dashed black line, c, is a supergradient of W G in the presence of W at 1 and a subgradient of W in the presence of W G at 2. The δ-superdifferential of W in the presence of the lower boundar W G is defined as G δ ) := {c R W ) W ) + δ c ) l Gc, W ) δ), r Gc, W ) δ))}. 4.36) When c is a δ-supergradient of W in the presence of the lower boundar W G, i.e. c G δ ), it is possible to draw a line with slope c through the point, W ) δ) which minorises 15

17 W ) prior to intercepting the lower boundar W G ). For eample 0 G δ G 2) in Figure 4. Similarl, the ε-subdifferential of W G in the presence of the upper boundar W is defined as ε G) := { c R W G ) W G ) ε c ) l c, W G ) + ε), r c, W G ) + ε)) }. 4.37) When c is an ε-subgradient of W G in the presence of the upper boundar W, i.e. c ε G), it is possible to draw a line with slope c through the point, W G ) + ε) which dominates W G ) prior to intercepting the upper boundar W ). For eample 0 ε G 1 ) in Figure 4. W ) ε 0 ε G 1 ) W G ) δ 0 G δ 2 ) Figure 4: c = 0 is a ε-subgradient of W G in the presence of W at 1 and a δ-supergradient of W in the presence of W G at 2. and Let ε c) := inf { ε 0, W ) W G )] c ε G) } 4.38) δ c ) := inf { δ 0, W ) W G )] c G δ )} 4.39) which are the smallest spike variations that can be made in W G, resp. W at such that c ε G), resp. c δ G ). The quantities 4.38) and 4.39) are illustrated in Figure 5. Remark 8. It follows from these definitions that for all δ > δc ) W ) > W ) δ + c ) l Gc, W ) δ), r Gc, W ) δ)]. Whereas, for δ < δ c ) there eists l G c, W ) δ), r G c, W ) δ)] such that W ) < W ) δ + c ). These two statements impl that either: a) There eists z l G c, W ) δ c )), r G c, W ) δ c ))] such that c G z) and/or b) δ c z) = W z) W G z) for z = l G c, W ) δ c )) and/or z = r G c, W ) δ c )). 16

18 The top left panel of Figure 5 shows a point where onl condition a) holds, whereas the bottom left panel illustrates a situation where onl condition b) holds. Similarl, for all ε > ε c) W G ) > W G ) + ε + c ) l c, W G ) + ε), r c, W G ) + ε)]. Whereas, for ε < ε c) there eists l c, W G ) ε), r c, W G ) ε)] such that W G ) < W G ) + ε + c ). These two statements impl that either: a) There eists z l c, W G )+ε c)), r c, W G )+ε c))] such that c Gz) and/or b) δ c z) = W z) W G z) for z = l G c, W ) δ c )) and/or z = r G c, W ) δ c )). The top right panel of Figure 5 shows a point where onl condition a) holds, whereas the bottom right panel illustrates a situation where onl condition b) holds. W ) W ) W G ) W G ) δ c ) ε c ) z z W ) δ c ) W ) W G ) ε c ) W G ) l G r G l r Figure 5: The two figures on the left illustrate δ c ) and the two figures on the right illustrate ε c). For ease of notation, in the bottom left figure l G := l G c, W ) δ c )) and r G := r G c, W ) δ c )) and in the bottom right figure l := l c, W G ) + ε c)) and r := r c, W G ) + ε c)). Furthermore, let ε ) := inf c R ε c), δ ) := inf c R δ c ) 4.40) which are the smallest spike variation that can be made in W G resp. W ) at such that the set ε G) resp. δ G )) is non-empt. The first result in this section shows that there is no 17

19 gap between the minimum spike variation in W downwards admitting a δ-supergradient and the minimum spike variation in W G upwards admitting a ε-subgradient. Lemma 9. Suppose that W G : 0, ) R and W : 0, ) R are as defined in 4.31) and 4.32) and that 4.33) holds, then: for all > 0 and c R W G ) + ε c) W ) δ c ) 4.41) and W G ) + ε ) = W ) δ ). 4.42) Proof. Fi > 0, c R and take δ > δ c ). It follows from the definition of δ c ) that W ) > W ) δ + c ) l Gc, W ) δ), r Gc, W ) δ)] and l Gc, W ) δ), r Gc, W ) δ)] l c, W ) δ), r c, W ) δ)] 4.43) where the inequalit and inclusion are strict to avoid contradicting that δ > δ c ). Due to the properties of the line W ) δ c ) + c ) described in Remark 8, it follows from the inclusion 4.43) and the continuit of G that l c, W ) δ), r c, W ) δ)] s.t. W ) δ + c ) < W G ) 4.44) Let ε := W ) W G ) δ so that 4.44) is equivalent to l c, W ) δ), r c, W ) δ)] s.t. W G ) ε + c ) < W G ). It follows from the definition of ε c) that ε ) < ε c) or equivalentl W ) δ < W G ) + ε c) δ > δ c ). 4.45) Take an sequence of real numbers δ n ) n 1 such that δ n δc ), W ) W G )) for all n 1 and lim n δ n = δc ) then 4.45) shows that W ) δ n < W G ) + ε c) for all n 1. Taking the limit as n of both sides of this inequalit we obtain 4.41). Suppose that for some c R that W G ) + ε c) = W ) δc ). When the assumption 4.33) holds the line g) := W G ) + ε c) + c ) has both sets of properties described in Remark 8 so is the onl line passing through, p) for some p W G ), W )) which both dominates W G ) on l c, p), r c, p)] and minorises W ) on lg c, p), r G c, p)]. When 4.33) fails, g) ma not be the onl line with these properties and 4.42) ma not hold, this case is eamined further in Remark 10 below. Consequentl, it follows from the continuit of G and and the assumption 4.33) that for an c c the line h) = W G ) + ε c) + c ) must either satisf both l Gc, W ) δ c )) < l c, W ) δ c )), r Gc, W ) δ c )) < r c, W ) δ c )), 4.46) 18

20 c) W ) W ) b) W G ) W G ) a) a) c) b) Figure 6: These figures illustrate the properties used in the proof of Lemma 9. or l Gc, W ) δ c )) > l c, W ) δ c )), r Gc, W ) δ c )) > r c, W ) δ c )). 4.47) In the right panel of Figure 6 the line marked b) has the properties 4.46) whereas in the left panel the line marked b) has the properties 4.47). When 4.33) fails one or more) of the inequalities in both of these statements can hold with equalit which invalidates the following argument. Net the line h) is shifted upwards so that it passes through, W ) δ) for some δ 0, δ c )]. As is illustrated b the lines marked c) in Figure 6 the properties 4.47) impl that l c, ) δ) l c, ) δ c )), ] and hence l c, ) δ) > l G c, ) δ). Whereas if 4.46) holds then r c, ) δ), r c, ) δ c ))]. which implies that r c, ) δ) < rg c, ) δ). In both cases c / δ G ) for arbitrar δ < δc c). Moreover, as c was arbitrar we have shown that δ G) = for all δ < δ c c). We ma conclude that when for some c R equalit holds in 4.41) that δc ) = δ ). A smmetric argument can be used to show that when equalit holds for some c R equalit holds in 4.41) that ε c) = ε ) from which we deduce 4.42) holds. Remark 10. In the previous lemma it is essential to assume that 4.33) holds. When this is not the case it ma be the case that W G ) + ε ) < W ) + δ ) as is illustrated in Figure 7. Suppose that as in the illustration W G 0+) < W 0+) and there is a > 0 such that for some c R, c G), lg c, W G )) = l c, W G )) = 0 and W G 0+) < W G ) c < W 0+). Take a W G ) c, W 0+)) such that b < c with the properties: i) a+b W G ), W )), ii) rg b, a+b) < r b, a+b) and iii) b GrG b, a+b)). We have chosen a, b and such that δb ) = W ) a + b) hence δ ) W ) a + b) < W ) W G ). owever, b construction G) so ε ) = 0 which implies W G ) + ε ) < W ) δ ), 19

21 W ) δ b ) f) =a +b r G b,f)) W G ) Figure 7: When W 0+) > W G 0+) it ma be the case that condition 4.42) in Lemma 9 fails. so in this case 4.42) fails. Remark 11. The assumption that 4.33) holds in Lemma 9 can be relaed in the following wa. When W G 0+) < W 0+), take an w 0 W G 0+), W 0+)] and etend W G respectivel W ) onto 0, ) b allowing W G to be multivalued at zero taking all values W G 0+), w 0 ] respectivel w 0, W 0+)]). If we use this etension of W G and W and the intervals in the definitions of ε G ) and δ G) are take to be closed as apposed to open) then 4.42) in Lemma 9 holds without the need to assume 4.33) as the problem discussed in Remark 10 can not occur. owever, the choice of w 0 not onl completel determines δ 0+) and ε 0+) but the values of the functions δ ) and ε ) on 0, z) for some z 0. W )  G c) W G )  c) Figure 8: The sets  G c) and  c) are defined in 4.48) and 4.49) are marked when c is the slope of the pair of parallel lines. For a given > 0, define two sets of admissible neighborhoods using  Gc) := { > 0 lc, p) lgc, p) rgc, p) rc, p)}, 4.48)  c) := p W G ),W )] p W G ),W )] { > 0 l Gc, p) l c, p) r c, p) r Gc, p)}. 4.49) 20

22 These admissible neighborhoods are illustrated in Figure 8. The net result describes a pair of functions which coincide with W G ) + ε ) and W ) δ ). Proposition 12. For > 0, define a pair of functions W G ) ) := sup sup z ) + W G ) ), 4.50) z R  G z) W ) G ) := inf inf z R  z) then W G ) ) = W ) δ ) and W ) G ) = W G ) + ε ). z ) + W )), 4.51) Proof. Fi a > 0 and c R, it follows from the definition of δ c ) and the continuit of G that the set 4.48) satisfies  Gc) = l Gc, W ) δ c )), r Gc, W ) δ c ))]. B definition c G δ ) for all δ > δ c ), which is equivalent to W ) δ c ) + c ) W G )  Gc) 4.52) and equalit holds in 4.52) for = l G c, W ) δ c )) and = r G c, W ) δ c )). ence, sup W G ) c) = W G rgc, W ) δc ))) crgc, W ) δc ))  G c) and W G ) ) := sup sup c ) + W G ) ) c R  G c) = sup c r G c, W ) δc ))) + W G rgc, W ) δc ))) ) c R = sup W ) δc )) = W ) δ ). c R Thus we have shown that W G ) ) = W ) δ ) for all > 0. A smmetric argument can be used to show that W ) G ) = W G ) + ε ). The previous result holds without the need to assume that 4.33) holds. At first glance, the notation used in the previous result ma appear to be the wrong wa around but when 4.33) holds Lemma 9 implies that W G ) ) = W G ) + ε ) and W ) G ) = W ) δ ). Remark 13. For a given > 0 consider the sets { } A 1 ) := p W G ), W )] c R s.t. lgc, p), rgc, p)]  Gc), { } A 2 ) := p W G ), W )] c R s.t. lc, p), rc, p)]  c). The dual interpretation provided in 19] illustrates that W G ) ) can be constructed b maimising over functions of the form p + c ) which hit W G before W, i.e. W G ) ) = sup A 1 ) as shown in Proposition 12 or b minimising over the functions of the form p + c ) which hit W before W G i.e. W G ) ) = inf A 2). 21

23 W )  c) W G ) A G c) W ) A c)  G c) W G ) Figure 9: The left figure shows a situation when  G c) A G c) whereas the figure on the right shows a situation when  c) A c). In view of the previous remark, Proposition 12 can be viewed as a preliminar version of the net result which shows that the functions 4.50) and 4.51) can be viewed as an etension of the concave biconjugate of W G ) and the conve biconjugate of W ). To this end, let L c, p) = sup{z p + cz ) / clepiw G ) \ epiw ))}, R c, p) = inf{ p + c ) / clepiw G ) \ epiw ))}. For a given > 0, let p) = W ) W G ) and define two sets of admissible neighborhoods which are larger than 4.48) and 4.49) using A Gc) := { 0, ) L c, W ) δ) R c, W ) δ)}, 4.53) δ 0,p)] c δ G) A c) := { 0, ) L c, W G ) + ε) R c, W G ) + ε)}. 4.54) ε 0,p)] c ε G) These admissible neighborhoods are illustrated in Figure 9. These neighborhoods coincide with those used in 19] and are used in the net result which is an analtical complement to 19] Theorem 4.1. Theorem 14. Suppose that 4.33) holds, then the functions W G ) ) and W ) G ) defined in 4.50) and 4.51) satisf W G ) ) = inf z R W ) G ) = sup sup z ) + W G ) ), 4.55) A G z) inf z R A z) z ) + W )). 4.56) for all > 0. Moreover, W G ) ) = W ) G ) for all > 0 and { 0, ) W G ) ) = W G ) } = { 0, ) G) }, { 0, ) W G ) ) = W ) } = { 0, ) G ) }. 22

24 Proof. Fi > 0 and c R and define W G ) c) := inf A c) c W G )). B definition, the line f) := W G ) + ε c) + c ) dominates W G ) on  c) = l c, W G )), r c, W G ))] A c). If there eists z A c) \  c) such that fz) = W G z) then either c Gz) or z is on the boundar of A c), i.e. z = L c, W G )) or z = R c, W G )). Thus we ma conclude that W G ) c) = c W G ) for some such that ence c ) + W G ) = W G ) + ε c). inf sup z ) + W G ) ) = inf W G ) ε z)) z R A G z) z R so it follows from the definition of ε ) and Proposition 12 that inf sup z ) + W G ) ) = W G ) + ε ) = W G ) z R A G z) ). A smmetric argument can be used to show that inf sup z ) + W G ) ) = W ) δ ) = W ) G ). z R A G z) It follows from Lemma 9 that W G ) ) = W ) G ) and the final statement follows from the definition of ε ) and δ ). Theorem 14 can be reformulated in terms of the F -concavit resp. F -conveit) of the functions G and determining W G and W. To see this observe that it follows from the definitions of W G ) and W ) in 4.31) and 4.32) that G G) ) := W G ) F ))ψ) = inf ) G ) := W ) G F ))ψ) = sup sup z R F ) A G z) inf z R F ) A z) zf ) F )) + zf ) F )) + ψ ψ ) ) ) ψ), 4.57) ) ) ) ψ), 4.58) which are the modified F -concave/f -conve biconjugates used in 19]. It is shown in the net section that 4.57)-4.58) are r-semiharmonic functions which solve the dual problems 2.12). We began constructing an etension of the conve/concave biconjugates of W G and W due to the fact that G is r-superharmonic with respect to the diffusion X if and onl if G/ψ is F -concave or equivalentl W G is concave. For studing optimal stopping games, we shall also use that G is r-superharmonic with respect to the diffusion X if and onl if G/ϕ is F -concave or equivalentl G/ϕ) F ) 1 is concave on 0, ). With this in mind, Theorem 14) can be formulated with respect to the other ratio of the fundamental solutions. Corollar 15. Let W G := G/ϕ) F ) 1 and W := /ϕ) F ) 1. Assuming that 2.7)- 2.8) hold. Then for all > 0 W G ) ) := inf sup z ) + W ) G ) = sup inf z ) + W ) ) =: W ) G ). z R à G z) 23 z R à z)

25 where the sets à G z), à z) are defined as in 4.53) and 4.54) replacing W G, W with W G, W G. Proof. In this section we have not used an properties of the diffusion X, nor the specific form of the functions W G, W. Consequentl, the previous Theorem holds for an two continuous functions g, h : 0, ) R such that g h and lim 0 g) h) = lim g) h) = 0. In, particular, replacing W G, W with W G, W G does not alter the result. It was observed in Remark 11 that the assumption 2.8) ma be relaed b etending the functions W G, W onto R + in a manner which ensures that W G ) + ε ) = W ) δ ) for all > 0. This approach can be used to rela the assumptions of Theorem 14. Corollar 16. Take w 0 W G 0+), W 0+)] and etend the functions W G, W onto R + b setting W G 0) = W G 0+), w 0 ] and W 0) = w 0, W 0+)]. Suppose that the intervals in the definitions 4.36)-4.37) are take to be closed as apposed to open) then W G ) ) = inf for all 0. z R sup z ) + W G ) ) = sup inf z ) + W )) = W ) G ) A G z) z R A z) Proof. Under these assumptions, Remark 11 shows that Lemma 9 holds. Furthermore, under these assumptions the functions W G ) and W ) G defined in 4.51)-4.50) are defined on R + and W G ) 0) = W ) G 0) = w 0. Thus with this etended definition, the corollar follows from Theorem 14. Remark 17. The function W G ) ) can be viewed as the concave biconjugate of W G in the presence of W ) or constrained to remain within clr 2 \epiw ))) for the following three reasons: i) Theorem 14 illustrates that the function W G ) ) can be epressed as W G ) ) = inf z inf z W G ) + δ R + \ A Gz)) )), z R R where δ R + \ A G z)) is the characteristic function of the set R + \ A G z), i.e. δ R + \ A G z)) = 0 if A G z) and δ R + \ A G z)) = otherwise. The difference with the standard concave biconjugate is that this characteristic function depends on the choice of z and. owever, this dependence seems essential to ensure that epiw ) epiw G ) ). ii) For an measurable function f : 0, ) R, the superdifferential of f is as defined in 3.14) and the ε-superdifferential is defined as ε f) = {c R f) f) ε c ) 0, )}. ence f ) f) = inf{ε 0 ε f) }. This definition of a ε-superdifferential coincides with the definition 4.37) when W is taken to be the multivalued function W G 0), + ] = 0, W ) = 0, + ), W G ), + ] = +. 24

26 Moreover, when 4.33) holds, it follows from Theorem 14 that W G ) ) W G ) = ε ) where ε ) is as defined in 4.40)) so W G ) ) and f ) can both be characterised in terms of a smallest spike variation. iii) In Corollar 5 it was shown that the concave biconjugate of G/ψ) F 1 ) is such that G/ψ) F 1 ) ) = sup A 1 ) where the set A 1 ) is defined in 3.25). This characterisation of the concave biconjugate uses the same method of construction as is used in Proposition 12 to define W G ) ). The function W ) G ) will be referred to as the conve biconjugate of W G in the presence of W ) or constrained to remain within epiw G )) for directl similar reasons. 5 Nash equilibrium in optimal stopping games In this section Theorem 14 is applied to show that the infinite time horizon optimal stopping game 2.9)-2.10) ehibits both a Stackelberg and a Nash equilibrium. The first Theorem of this Section shows that under the assumptions 2.7) and 2.8) upper and lower values of the optimal stopping game 2.9) and 2.10) satisf V G) and V )G respectivel. Consequentl, it hence the game has a Stackelberg equilibrium. In fact the game has a value without the need to assume the boundar condition 2.8) holds, but we need to account for the fact that when both minimising and maimising agent select the same stopping time R τ, τ) = E e rτ GX τ )]. follows from Theorem 14 that the game has a value, namel V = V = G) Theorem 18. Suppose that 2.7) holds and consider the optimal stopping game 2.9)-2.10). The optimal stopping game has a Stackelberg equilibrium, i.e. the games has a value V which can be represented as V ) = G) ) = W G ) F ))ψ) for all I. Moreover, when 2.8) hold, it follows that V = G) = )G without the need to etend W G and W onto R +. Proof. To accomodate the asmmetr in the objective function discussed above, etend the functions W G, W onto R + b setting W G 0) := W G 0+) and W 0) = W G 0+), W 0+)]. As noted in Corollar 16, these etended definitions of W G, W ensure that W ) G 0) = W G ) 0) = W G 0+) = Ga+)/ϕa+). Let F a) := F a+) = 0 and ϕa) := ϕa+) = + then ) G ) := W ) G F ))ϕ), G) ) := W G ) F ))ϕ), for all a, b) and ) G a) = G) a) = Ga+). It follows from Corollar 16 or Theorem 14 if 2.8) holds) that it is sufficient to show that ) G V V G) 5.59) for all I where V and V were defined in 2.9) and 2.10) respectivel. The second inequalit follows from the definitions of V and V. For the first inequalit in 5.59) let  = z Z  G z) and τ = inf{t 0 X t / Â}. Let a = inf  and b = sup  and suppose that b < b. It follows that V ) inf σ τ E e rσ Ĥ X σ ) 25 ] =: V ).

27 where Ĥ ) := ) I a,b ) + G ) I =a ] =b ]. The stopping time which attains this infimum is of the form σ = T a T b for some a a b b. As we have assumed that sup  < b the process e rt σ) ϕ X t σ ) ) t 0 is a P, F t σ )-martingale. The optional sampling theorem implies that E e rσ ϕ X σ )] = ϕ ) for all  and all σ τ. ence for arbitrar c R, ] ] V ) = inf E e rσ Ĥ X σ ) = cϕ ) + inf E e rσ Ĥ X σ) cϕ X σ )). 5.60) σ τ σ τ Since 5.60) holds for all c R, it follows that V ) = sup c R cϕ ) + inf E e rσ Ĥ X σ) cϕ X σ ))] ). 5.61) σ τ The optimal stopping problem on the right hand side of 5.61) can be epressed as ] R c ) := inf E e rσ Ĥ X σ) cϕ X σ )) σ τ ) Ĥ ) cϕ ) F z) F ) Ĥ z) cϕ z) F ) F ) = inf + ψ ) a z b ψ ) F z) F ) ψ z) F z) F ) ) Ĥ ) cϕ ) inf  ψ ) ence, from 5.61) we obtain Let V ) sup c R ψ ). c ϕ ) ψ) + inf  )) Ĥ ) cϕ ) ψ). ψ ) c) = inf{c R inf  = inf  Gc)}, c) = sup{c R sup  = sup  Gc)}. Recall that δc ) and δ ) were defined in 4.39) and 4.40). Due to the definition of  it follows that for all c R ) Ĥ) cϕ) cf ) + inf ) ψ) ψ) δ c F )).  and equalit holds for all c c), c)]. In particular, it follows from the definition of  that δ c F )) = δ F )) for some c c), c)] so we ma conclude that V ) ) δ F ))ψ) = W ) G F ))ψ) = ) G ) which is the first inequalit in 5.59). The case that b = b can be handled b using an approimating sequence of domains as in Theorem 3. For the final inequalit in 5.59), let A = z Z  z) and let σ = inf {t 0 X t / A }. Let a = inf A and b = sup A and suppose that b < b. There is a natural asmmetr in the value function defined in 2.11) as R τ, τ) = E e rτ GX τ )]. Thus V ) = inf σ sup τ σ sup E e rτ σ) ) ] G X τ ) I τ σ] + X σ ) I τ>σ] τ E e rτ G X τ ) ] =: V + ). 26