A Dynkin game on assets with incomplete information on the return
Tiziano De Angelis, Fabien Gensbittel, St\'ephane Villeneuve

TL;DR
This paper analyzes a zero-sum Dynkin game for option pricing with incomplete return information, using filtering to reduce the problem and characterizing Nash equilibria with moving boundary stopping strategies.
Contribution
It introduces a novel approach to model and solve a Dynkin game under incomplete information by reducing it to a bi-dimensional diffusion and characterizing pure strategy equilibria.
Findings
Existence of Nash equilibrium in pure strategies with hitting times
Characterization of stopping sets with moving boundaries
Global $C^1$ regularity of the value function
Abstract
This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion . Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of to a set with moving boundary. A detailed description of the stopping sets for the two players is provided along with global regularity of the value function.
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A Dynkin game on assets with incomplete information on the return
Tiziano De Angelis, Fabien Gensbittel, Stéphane Villeneuve
T. De Angelis: School of Mathematics, University of Leeds, Woodhouse Lane, LS2 9JT Leeds, UK.
F. Gensbittel and S. Villeneuve: Toulouse School of Economics (TSE-R, Université Toulouse 1 Capitole), 21 allée de Brienne, 31000 Toulouse, France.
Abstract.
This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion . Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of to a set with moving boundary. A detailed description of the stopping sets for the two players is provided along with global regularity of the value function.
Key words and phrases:
Zero-sum games; Nash equilibrium; incomplete information; free boundaries;
Acknowledgments: T. De Angelis was partially supported by the EPSRC grant EP/R021201/1.
We thank an anonymous referee whose insightful comments contributed to the discussion in Section 8.
1. Introduction
Zero-sum optimal stopping games (Dynkin games) have received a lot of attention since the seminal paper by Dynkin [13], see also the classical references [2] and [27]. In particular, these games have found applications in mathematical finance where the arbitrage-free pricing of American options with early cancellation (game options) relies on the computation of the value of a zero-sum game of optimal stopping between the buyer and the seller (see [24],[26]). A common assumption in the financial application of Dynkin games is that the players have complete information about the parameters of the underlying stochastic process. In practice, however, there are many situations in which parameters are difficult to estimate and in particular this is true for the drift of the process.
Our work is inspired by the real option literature, where the value of an investment (like the beginning of the extraction of a natural resource or the investment in a R&D programme) is a contingent asset, depending on the price of some underlying asset, and it is computed by using arbitrage arguments (see [12]). It is known that the problem itself boils down to an optimal timing decision, hence optimal stopping is the key mathematical tool. Following [12], we assume that the price process evolves according to a geometric Brownian motion
[TABLE]
where is the log-return on the so-called risk-adjusted asset price.
The capital asset pricing model allows us to determine the risk-adjusted discount rate which is used to discount future cashflows (notice that this is in general larger than the risk-free rate, see, e.g., [12, p. 178]). In line with [12] we assume and denote the difference by . The condition avoids that the value of an investment project whose payoff is linear in becomes unbounded (which would lead the investor to delay the investment forever). It is known that estimating the return of the risk-adjusted price of an asset is a challenging task and we embed this feature in our model by considering an asset with a partially unobservable drift .
A typical problem that we have in mind is the one of a firm holding a concession to drill oil wells. Being aware of the social costs and benefits of the oil field development, a public authority would like to sign a contract where the concession rights can be cancelled at any time, pending the payment of a contractual penalty. From the investor’s point of view (and simplifying the model for the benefit of tractability) the decision to invest would be profitable only if the value of the underlying commodity can compensate for the fixed cost of investment . In this sense one can interpret the option to invest as a Call option on the price of the commodity, with strike equal to . A cancellation of the agreement would require a payment equal to the Call payoff plus a penalty (i.e., to compensate for the lost investment opportunity).
Motivated by the above considerations, in this paper we study zero-sum optimal stopping games with incomplete information about the return of the underlying asset. We are interested in the existence of the value as well the existence and characterization of Nash equilibria for the game. To enable a detailed theoretical analysis, we shall keep the real option model simple while, at the same time, drawing from the vast literature on Israeli options (initiated by [24]).
We assume that the buyer (player 1) and the seller (player 2) of a Call option on an asset agree on a constant risk-adjusted discount rate , which is used to discount future payoffs in the game (i.e., we assume that players have the same belief on the future of the economy). Moreover we model the uncertainty on the the asset return by assuming that the adjusted log-return is random and only partially observable. To avoid confusion with the previously introduced notation we denote it by (as opposed to in the previous page). In particular we assume , where is a constant and is random and unobservable to the players.
Our choice for ties up nicely with the usual concept of net return on a stock paying dividends at a rate . Although other choices for are clearly possible, we shall see below that this basic model already poses significant mathematical challenges. To the best of our knowledge this is the first paper addressing a zero-sum game with partial information via a probabilistic analysis of the related free boundary problem, hence we leave other parameter choices for future work.
The asset in our model evolves, on a probability space , according to
[TABLE]
where is a Brownian motion and is the volatility. The random variable takes the values [math] or with and it is assumed to be independent of . We denote by the filtration generated by the observed process and by its augmentation with -null sets (see further details in Section 2). Then we define by the set of -stopping times.
In our game we fix and and let
[TABLE]
be the payoff for player 1 (the option holder) and the cost of cancellation for player 2 (the seller), respectively. Then the formulation of our game is the following: the expected discounted payoff of the game is
[TABLE]
where . In particular the option holder picks , in order to exercise the option, and the seller picks , in order to cancel it. The holder aims at maximising her revenue while the seller wants to minimise costs. By convention, we set
[TABLE]
The notation accounts for the dependence of the stopping functional on the initial asset value and on the a-priori probability of the event . This notation will be fully justified and explained in Section 2 below.
As usual we define the upper value and the lower value of the stopping game, respectively by
[TABLE]
When , the game has a value . Moreover, if there exist two stopping times such that
[TABLE]
for all stopping times and , the pair is a saddle point or a Nash equilibrium for the optimal stopping game and in that case the game has a value with .
In the context of Israeli options one has or (the non-dividend case). Explicit computations have been established by [17] and [35] in the perpetual case. Both papers show that the dividend parameter plays an important role for the existence of an equilibrium in the game and this will be the case also in the present work.
We recall now some results from the existing literature so that we can later discuss the mathematical novelty of our work. The existence of the value for optimal stopping games with multi-dimensional Markov processes was proved in [16] using martingale methods and by Bensoussan and Friedman [2] via variational inequalities. These methods require suitable integrability of the payoff processes, i.e., in our notation, the processes , must be uniformly integrable. When such condition is not fulfilled, the existence of the value was proven in [17] but only for one-dimensional diffusions. Results in [17] rely upon a generalized type of concavity introduced in [14] and brought up to date in [7].
On the other hand sufficient conditions for the existence of Nash equilibria in Markovian setting have been studied in [16] and [17]. For a rather general class of Markov processes these conditions include the above mentioned uniform integrability of the payoff processes. In the special case of one-dimensional diffusions weaker integrability may instead be sufficient (see [17], Proposition 4.3).
In our setting we are faced with two main technical difficulties in establishing existence of the value and of a Nash equilibrium: (i) the process is not Markovian and (ii) it fails to fulfil the condition of uniform integrability (see Remark 2.1), in particular for any initial condition we have
[TABLE]
To overcome the first difficulty we rely upon filtering theory and increase the dimension of our state space. Informally we could say that we take into account the progressive update of the players’ estimate on , based on the observation of . This approach leads us to study a two dimensional Markovian system which we denote by , where (at least formally) and . On the other hand, to tackle the lack of uniform integrability and prove the existence of the value of the game, we adapt methods developed by Lepeltier-Maingueneau [27] and Ekstrom-Peskir [16].
After we prove existence of the value, we are then in the position to carry out a detailed analysis of the structure of the stopping sets for the two players, i.e. the sub-sets of the state space in which , . Denoting , we study properties of the boundaries of and which we subsequently use to state conditions for the existence of a saddle point (Nash equilibrium). The latter is provided in terms of hitting times to and .
In our analysis we use two equivalent representations of the two-dimensional dynamics. These are linked to one another by a deterministic transformation – the so-called reduction of second order PDEs to normal form (observe that a similar transformation was already used by several papers like [11],[15] and [20] among others). Indeed we first observe that the process is driven by only one Brownian motion and it is therefore degenerate; then we perform a change of coordinates to obtain a new process . Here is deterministic and either increasing or decreasing, depending on the choice of parameters in the problem. Effectively the process plays the role of a ‘time’ process.
We would like to emphasize that the probabilistic study of free boundary problems related to zero-sum Dynkin games on two dimensional diffusions has not received much attention so far. Works in this direction but in a parabolic setting are [9] and [35]. Our analysis here goes beyond results in those papers by showing for example that the value of the game is a globally function of the state variables . This type of regularity is much stronger than the well-known smooth-fit, which gives continuity of one directional derivative with respect to one state variable. Related work on regularity is contained in [10], which however does not cover our game setting.
The outline of the paper is as follows. In Section 2, we specify the model and provide a Markovian formulation of the zero-sum game. Existence and continuity of the value for the game (1.3) is obtained in Section 3. The geometry of the stopping sets is obtained in Section 4, in the -plane, and in Section 5, in the -plane (parabolic formulation). Hitting times to those sets are used in Section 6 to prove higher regularity of the value, e.g. its global regularity, and in Section 7 we obtain sufficient conditions for the existence of a saddle point. Finally, in Section 8 we collect some concluding remarks and discuss possible directions for further research. A few technical results are given in Appendix.
2. Dynamics of the underlying asset
We begin by considering the probability space endowed with a sigma algebra and the product probability where is the standard Wiener measure and . Let us denote a canonical element of and let , and be fixed. Then the asset’s value (with uncertain return rate) which is described by (1.1) has an explicit expression in terms of the couple , i.e.
[TABLE]
The process is a geometric Brownian motion whose drift parameter depends on the unobservable random variable . We recall that the latter is independent of the Brownian motion . As discussed in the introduction a technical difficulty arising in our model is the lack of uniform integrability of the process .
Remark 2.1**.**
If , the process is not uniformly integrable because
[TABLE]
whereas
[TABLE]
Hence, by linearity of the payoffs , in (1.2) we obtain (1.5).
We aim at giving a rigorous formulation for the game call option (1.3). One way to do it is to replace and in (1.3) by and defined above. Let be the filtration generated by and be its augmentation with -null sets. Then we denote the set of -stopping times and the optimisation is taken over stopping times . The disadvantage of this formulation is that the dynamics of is not Markovian and therefore for the solution of the problem we cannot rely upon free boundary methods. To overcome this difficulty we want to reduce our problem to a Markovian framework by using filtering techniques.
According to [1, Thm. 2.35, p. 40] the filtration is right continuous and therefore satisfies the usual assumptions. Thus, we define the process as an -càdlàg version of the martingale . Notice that is a bounded martingale that converges almost surely to . The latter is -measurable because
[TABLE]
According to Chapter 9 in Liptser-Shiryaev [28] (see also Chapter 4.2 in Shiryaev [34]), the process is the unique strong solution to the following SDE,
[TABLE]
where
[TABLE]
is an -adapted Brownian motion under . The couple is therefore adapted to the augmentation of the filtration generated by , which we denote by . This implies in particular and because is -adapted. Notice also that the process is adapted to the filtration by construction, so that it is no surprise that the new Brownian motion is also adapted to .
Above we have obtained on the space which depends on the probability distribution of the random variable . We prefer to get rid of such dependence and consider another process , having the same law than , but defined below on a new probability space.
Take a probability space , denote by a Brownian motion on this space and by the augmentation of the filtration that it generates. For , let be the unique strong solution of the bi-dimensional SDE
[TABLE]
To keep track of the initial point we use the notation and notice that by standard theory is indeed continuous -almost surely. Notice also that the second equation is independent of the first one and therefore its solution, , is independent of .
Since the processes and have the same law then the game option is more conveniently formulated using the former since it is Markovian and the probability measure is independent of . This will be done in the next section.
Often in what follows we use the notation and drop the apex in the couple . Before closing the section we notice that for all
[TABLE]
Moreover we recall that since is a continuous super-martingale, with last element , the optional sampling theorem guarantees (see [22, Thm.1.3.22])
[TABLE]
for all stopping times .
3. The game and its value
The payoffs , in (1.2) are non-decreasing and -Lipschitz continuous on with . It is also clear that
[TABLE]
for any , due to the first formula in Remark 2.1. We now recall the formulation of the game expected payoff (1.3) given in the Introduction and notice that, thanks to the equivalence explained in the previous section, we can rewrite it as
[TABLE]
The stopping times are drawn from the set of -stopping times and the dependence of on is clearly expressed. Thanks to (3.1) on the event we simply get a zero payoff for both players.
We recall here that player 1 (the buyer) picks in order to maximise (3.2), whereas player 2 (the seller) chooses in order to minimise (3.2). The upper value and the lower value of the game are expressed as in (1.4). We spend the rest of this section proving that these functions indeed coincide so that the game has a value .
We start by proving some regularity result of and .
Lemma 3.1**.**
The functions and are:
- (i)
non-decreasing with respect to (w.r.t.) and non-increasing w.r.t.
- (ii)
-Lipschitz w.r.t. , uniformly w.r.t.
- (iii)
locally Lipschitz w. r. t. , i.e. for or and a given constant we have
[TABLE]
Proof of Lemma 3.1.
Without loss of generality, we only provide full details for .
[Proof of (i)] Let us first prove monotonicity with respect to . Fix and , then for any , there exist a couple such that
[TABLE]
Therefore we also have
[TABLE]
where the last inequality follows by observing that , -a.s. for thanks to (2.8). Since was arbitrary we have non-decreasing.
To prove monotonicity with respect to we argue in a similar way. We fix and , and for any we can find a couple such that
[TABLE]
For the last inequality this time we have used the comparison principle for SDEs, which guarantees , -a.s. for , and (2.8), which gives , -a.s. for . By arbitrariness of we obtain the claim.
[Proof of (ii)] As above we fix and so that . For any we can find a couple such that
[TABLE]
where the second inequality uses the Lipschitz property of the call payoff. From (2.8) we have
[TABLE]
Since , is a positive supermartingale, we deduce that
[TABLE]
and Lipschitz continuity in follows from (3) since is arbitrary.
[Proof of (iii)] Now we use the equivalence between the couple on the space and the couple on the space (see explanation in Sec. 2 and (1.3) and (3.2)) to write
[TABLE]
for any couple . Set
[TABLE]
and notice that conditionally on , the law of is independent of , so denoting the expectation under the Wiener measure we get
[TABLE]
Now we use the above representation of the game payoff as follows. Fix and , then for any we find such that
[TABLE]
For any stopping time and for we have as in (2.9). Moreover have linear growth so that the Lipschitz property of follows. ∎
Now we can prove the existence of the value for the game. As explained in the introduction, the main difficulty comes from the fact that we are working with a bi-dimensional stopping game with a lack of uniform integrability on the stopping payoff.
Theorem 3.2**.**
The game with payoff (3.2) has a value for all . Moreover player 2, i.e. the minimiser (seller), has an optimal strategy
[TABLE]
with the convention and the process
[TABLE]
is a closed supermartingale.
Finally, if we define
[TABLE]
with the convention and the process
[TABLE]
is a (not necessarily closed) submartingale.
Proof: The proof of Theorem 3.2 is postponed to the Appendix.
Remarks 3.3**.**
According to Lemma 3.1, the value function is non-increasing with respect to . Therefore for each , we have
[TABLE]
where is the game value when . According to [35], Theorem 2.1, the value function is strictly positive therefore is also strictly positive.
4. Properties of the stopping regions
Having established that the game has a value we can introduce the so-called continuation region
[TABLE]
and the stopping regions for the two players, i.e.
[TABLE]
for player 1, and
[TABLE]
for player 2. It is clear that is open and , are closed, because is jointly continuous (see Lemma 3.1), and obviously .
These sets are important because, according to theory on zero-sum Dynkin games, the only candidate to be a Nash equilibrium is the pair given by (3.6) and (3.7) (see [32]). Under complete information the perpetual game call option has been studied in [17] for and in [35] for . Those papers analyse the geometry of the continuation and stopping regions and for completeness we account for a summary of their results in appendix. For future reference here we only note that [17, Sec. 5.1] obtain
[TABLE]
In the rest of this section we study the shape of the stopping regions. For that we need to introduce the infinitesimal generator of the two-dimensional diffusion , i.e. for any
[TABLE]
Let us also introduce the sets
[TABLE]
and notice that indeed and . We denote the complements of these sets by , and define .
Proposition 4.1**.**
We have,
[TABLE]
Proof.
It is sufficient to prove the first inclusion (i.e. for ) because arguments for the second one (i.e. for ) are analogous.
Because is strictly positive (Remark 3.3), it is clear that . Fix , then it is possible to find an open neighbourhood of such that , i.e. on . Let be the exit time of from and let , then Theorem 3.2 guarantees that
[TABLE]
Using this property and Itô’s formula we obtain
[TABLE]
which implies . ∎
Our next lemma shows that the stopping region is up and right-connected while the region is down and left-connected on .
Lemma 4.2**.**
The following properties hold
- (i)
* for .*
- (ii)
* for .*
- (iii)
* for .*
- (iv)
* for .*
Proof.
The two first properties follow directly from the fact that is non-increasing. To prove (iii) let us fix (notice that in particular ). Since is -Lipschitz w.r.t. and non-decreasing (see (i)-(ii) in Lemma 3.1) then for all , we have
[TABLE]
where we have used that for , and that by assumption. Clearly (4.8) implies as claimed. Similar arguments give (iv). ∎
Lemma 4.3**.**
For , . Hence .
Proof.
Notice that for and therefore on . Let be an open set and fix . Denote and let be defined by (3.7). Notice also that , -a.s. because player 1 does not stop in .
Then using Theorem 3.2 and Itô formula we obtain
[TABLE]
∎
The next Lemma shows that if the penalty for cancellation does not exceed the strike price, i.e. , then the stopping region is non-empty and unbounded.
Lemma 4.4**.**
If then the set is non-empty for all .
Proof.
We argue by contradiction and assume that is empty for some . Fix and denote
[TABLE]
then clearly almost surely. Theorem 3.2 therefore implies that
[TABLE]
For any stopping time we have
[TABLE]
Using Lipschitz continuity (Lemma 3.1) and (4.4), we also have
[TABLE]
Plugging the latter into (4.9) to estimate , recalling and using that is a positive, bounded, supermartingale we obtain
[TABLE]
Since was arbitrary we then have where
[TABLE]
The same arguments as in the proof of Lemma 3.1 allow us to prove that
[TABLE]
We can now use the above to obtain
[TABLE]
Next we want to find a lower bound for . Notice that for
[TABLE]
For , setting
[TABLE]
we can rely on standard formulae for the Laplace transform of and to obtain
[TABLE]
Letting it is easy to check that
[TABLE]
where the final inequality uses . The latter and (4.10) imply that for sufficiently small, and thus a contradiction. ∎
Thanks to above lemmas we can define boundaries of the stopping regions as follows
[TABLE]
with the usual convention that and . Notice that if and it is empty otherwise. From Lemma 4.2 and because the sets are closed, we deduce the next corollary
Corollary 4.5**.**
The functions and are non-increasing on their respective domains and determine the stopping sets as follows:
[TABLE]
Moreover is lower semi-continuous (hence right-continuous) whereas is upper-semi-continuous (hence left-continuous). Finally, thanks to Proposition 4.1 and the definition of and we have on .
Next we show is a well-defined function on .
Lemma 4.6**.**
For all , .
Proof.
Arguing by contradiction let us assume that there exists such that . Then by monotonicity of ((i) and (iii) of Lemma 4.2) and lower semi continuity, it holds on .
Denote . We thus have , -a.s. for any starting point with . From now on fix . Theorem 3.2 guarantees that
[TABLE]
Therefore, using also that , for any we have
[TABLE]
with
[TABLE]
According to the last two expressions above, for fixed , we get
[TABLE]
which contradicts . ∎
Lemma 4.7**.**
If then
[TABLE]
Proof.
From Corollary 4.5 we have for all . Since Lemma 4.4 holds, then it must be . The latter also gives .
∎
From now on, whenever we refer to properties of and its boundary, we tacitly assume that . We recall that indeed this is always true for , thanks to Lemma 4.4. In this context we also denote
[TABLE]
and notice that Lemma 4.4 (with ) implies that the set is non-empty.
5. A parabolic formulation of the problem
In order to study existence of Nash equilibria and regularity of the value function of the game (beyond continuity) it is useful to introduce a deterministic transformation of the process . Such transformation also unveils a parabolic nature of the problem.
Given , let us define and the process such that and:
[TABLE]
Then setting
[TABLE]
it is not hard to check, by using Itô’s formula, that evolves according to
[TABLE]
From (5.1) we observe that -almost surely
[TABLE]
with defined by
[TABLE]
Notice that is on . The process is indeed deterministic and of bounded variation, hence it plays the role of a “time” process. Whether is increasing or decreasing depends on the sign of . In the rest of the paper we study the case which is truly two-dimensional. We leave aside the case that reduces to a one-dimensional problem parametrised in the variable .
Remark 5.1**.**
In the new coordinates it becomes clear that the law of is supported on the curve , which is a set of null Lebesgue measure in .
We can now look at our game in the new coordinates and consider the functions given by
[TABLE]
By construction, we have and is equal to the value of the stopping game
[TABLE]
For this new parametrization of the game we naturally introduce the continuation and stopping regions
[TABLE]
Using Lemma 3.1 it is immediate to verify that is locally Lipschitz continuous in so that is open and , are closed. Moreover, it is clear that and as in (3.6)-(3.7) are the entry times of into and , respectively.
The infinitesimal generator associated with is defined by
[TABLE]
for . One advantage of this formulation is that is a parabolic operator and it is non-degenerate on , so that the associated Cauchy-Dirichlet problems admit classical solutions under standard assumptions on the boundary conditions.
Since is continuous then is a continuous martingale for (the latter follows from Theorem 3.2 and the fact that is linked to by a deterministic map). We can use results of interior regularity for solutions to parabolic PDEs (see, e.g., [25, Corollary 2.4.3]) and Itô’s formula to deduce that any solution to with on is and coincides with . Therefore, and thus it satisfies
[TABLE]
As a consequence, as well.
We denote by the closure in of the set in which , i.e.
[TABLE]
where . According to Proposition 4.1 and Lemma 4.3, the stopping regions and lie in . Notice that since is increasing, if then any pair belongs to for . Somewhat in analogy with (4.13) we also define
[TABLE]
In the new coordinates the sets and are connected with respect to the variable, as illustrated in the next lemma.
Lemma 5.2**.**
Let .
- (i)
* for all , such that ,*
- (ii)
* for all .*
Proof.
Using that is increasing for each it is not difficult to show (by direct comparison) that is also non-decreasing. To prove take , then of Lemma 3.1 implies
[TABLE]
If then yielding . With an analogous argument we can prove . ∎
The stopping sets are not necessarily connected with respect to the variable and indeed we only have connected sets for some values of the rate and volatility of . In particular in the rest of the paper we make the following standing assumption (unless otherwise specified).
Assumption 5.3**.**
We assume .
For the sets , enjoy the next desired property.
Lemma 5.4**.**
Let , then
- (i)
* for all such that ,*
- (ii)
* for all .*
Moreover it also holds
- (iii)
* for , .*
Proof.
Take , fix and let . Let be optimal for player 2 in the game started at and an -optimal stopping time for player 1 in the game started at . Recall also that on both players have zero payoff due to (3.1). Then using that for all and we obtain
[TABLE]
Now we notice that, since , -a.s. for all and is decreasing, then
[TABLE]
and the right-hand side of the inequality is positive. Therefore we can use Fatou’s lemma and (5.15) to obtain
[TABLE]
Setting , for any and , Itô formula gives
[TABLE]
Hence substituting the above into (5) and noticing that , we can use the optional sampling theorem to obtain
[TABLE]
Using now that, for , the map is non-increasing with
[TABLE]
and recalling once again that , we see that (5) implies
[TABLE]
Since is arbitrary then is non-decreasing and therefore and easily follow.
The proof of follows from the fact that is decreasing for . ∎
The next corollary is a simple consequence of Lemma 5.2 and 5.4. We recall that as no player stops for (see Remark 3.3 and Lemma 4.3).
Corollary 5.5**.**
There exists non-decreasing functions , , with on and on , such that
[TABLE]
Next we provide continuity of the boundaries and , .
Proposition 5.6**.**
The stopping boundaries and are continuous.
Proof.
Step 1. First we prove the claim for . Since the proofs are similar for the two boundaries, we only provide details for . According to Corollary 4.5, the boundary is left-continuous. To show the right-continuity, we will argue by contradiction.
Assume that there exists such that and fix . Next define by and notice that since , then and therefore . We take a decreasing sequence with as so that Lemma 5.4 implies that for all . Equivalently so that taking limits and using that is continuous, we obtain . The latter is a contradiction.
Step 2. Now we show continuity of . Let us start from and fix . Take a sequence as so that , where and the limit exists by monotonicity. Since is closed we have and therefore , hence implying left-continuity.
To prove that is also right-continuous we use Theorem 3.3 in [8]. Since the latter theorem is not given in our game context we repeat here some arguments for completeness. Let us assume and denote , for simplicity. Fix such that the open rectangle with vertices , , and is contained in . Let and , then results of interior regularity for solutions to PDEs (see e.g. [25, Corollary 2.4.3]) imply that and, by deriving (5.8) with respect to , it turns out that
[TABLE]
where and
[TABLE]
Let be positive and such that . Multiply (5.22) by and integrate by parts over to obtain
[TABLE]
where is the adjoint operator of . Now, taking limits as , we can use dominated convergence in the right hand side of the above equation and the fact that on , to find
[TABLE]
In the final inequality we set and notice that due to (see (5.18)).
By its definition and (5.24) implies that its right limit at exists and it is strictly negative. Then for some , using integration by parts and Fubini’s theorem, we have
[TABLE]
where the last inequality follows because is non-decreasing as shown in the proof of Lemma 5.4. Therefore we reach a contradiction and must be continuous at . By arbitrariness of we conclude that is continuous.
To prove continuity of we simply refer to [8, Thm. 3.1]. The latter is not obtained in a game context but arguments as above allow a straightforward extension to it. We also notice that in applying that theorem we use that is locally Lipschitz on . ∎
6. Regularity across the boundaries
In this section we show that the value function is indeed in . The key to this result is the so-called regularity of the optimal boundaries. Roughly speaking this means that the process immediately enters the interior of the sets and upon hitting their boundaries and . Analogous considerations apply to the process and the sets , .
We recall that we work under Assumption 5.3. Let us introduce the hitting times
[TABLE]
The next lemma provides a clear statement of the regularity of the optimal boundaries for the diffusions and . Its proof is postponed to the end of the section so that we can move quickly towards the main result, i.e. Proposition 6.4.
Lemma 6.1**.**
If (resp. ) then
[TABLE]
Similarly, if (resp. ) then
[TABLE]
Notice that if (6.4) holds with (resp. ).
Adopting the convention that for and for , we can use Corollary 5.5 and write -a.s.
[TABLE]
To avoid further technicalities we assume that
[TABLE]
however all the results of this section can be easily adapted to the case in which for some (i.e. for some ).
We consider hitting times to the interior of the stopping sets, i.e. we define -a.s.
[TABLE]
Notice that for each line, the second equality follows from the continuity of the optimal boundaries. Precisely, for all , we have the equivalences
[TABLE]
We remark that if on an interval then should account also for the first crossing time of .
An argument used in [5], Corollary 8 (see eq. (2.39) therein) allows us to obtain the next useful lemma. The proof, originally developed in [5] is given in Appendix B for the reader’s convenience.
Lemma 6.2**.**
For any we have
[TABLE]
Equivalently for any we have
[TABLE]
The above lemma says that the process (or equivalently ), upon hitting the optimal boundaries, will immediately enter the interior of the stopping set. This has the following important consequence
Proposition 6.3**.**
Let be a sequence in and let and denote the corresponding hitting times for the process . It follows that
- (i)
If as , then , -a.s.
- (ii)
If as , then , -a.s.
Notice that if the above holds with .
Proof.
Let us consider (ii) and with no loss of generality let (arguments as below apply also to ). Denote . Since by Lemma 6.2, it is sufficient to prove that . In particular , -a.s. by Lemma 6.2 and Lemma 6.1. Hence there exists a set of null measure such that and is continuous, for all . Fix and an arbitrary . We can find such that . It follows that for all sufficiently large because and is continuous. Therefore . Since is arbitrary and the argument holds for a.e. we obtain (ii).
The proof of (i) follows from an analogous argument. ∎
Now we can use the result above to obtain continuous differentiability of the value function. In preparation for that we need to recall some results concerning differentiability of the stochastic flow. In particular by [33], Theorem 39, Chapter V.7 we can define the process
[TABLE]
which is continuous in both and and solves the SDE
[TABLE]
Notice that the couple forms a Markov process and that is an exponential local martingale. Moreover, since the process is bounded, it is not difficult to see that Novikov condition holds and is indeed an exponential martingale. Finally we also remark here that is a strong solution of a SDE and notice that, using the explicit representation (2.8), we also have
[TABLE]
For all we set
[TABLE]
and define the process as
[TABLE]
Then from the semi-harmonic characterisation of the value function provided in Theorem 3.2, we obtain for any
[TABLE]
For future reference we also introduce
[TABLE]
and denote by the closure of .
Proposition 6.4**.**
The value function is in (possibly with the exception of the point if ). Moreover (see (5.6)) is continuous on (possibly on if ).
Proof.
The value function is inside the continuation set by simply recalling that in (see the free boundary problem (5.8)–(5.10)). Therefore we only need to prove the property across the optimal boundaries. We provide full details for the continuity of as the continuity of follows analogous arguments up to trivial modifications.
Let us start by looking at points of , i.e. the boundary of the stopping region for the buyer. Let us fix and let us pick inside the continuation set . Later we will take limits and use Proposition 6.3.
Denote by the first entry time of into and by the first entry time of into for some . From of Lemma 3.1 and (6.13) we know that since is non-increasing in . In order to find a lower bound for we want to use the semi-harmonic property of . For that we introduce the stopping time where is fixed and . Notice that since (see (2.8)) then . Now, using (6.17) and (6.18) we obtain
[TABLE]
Notice that on and
[TABLE]
so that
[TABLE]
Using this fact in (6.20) we get
[TABLE]
Now we use that (see (2.8)) and that is non-increasing (as shown in the proof of and of Lemma 4.2). Therefore from the right-hand side of the above inequality we easily get
[TABLE]
Lower bounds can be provided for both terms on the right-hand side of the above expression. For the first term we recall of Lemma 3.1 and get
[TABLE]
where , and the final inequality follows by observing that and that the quantity under expectation is positive. For the integral term in (6.22) we argue in a similar way and obtain
[TABLE]
Collecting (6.22), (6) and (6) we find
[TABLE]
and we now aim at taking limits as . In order to apply dominated convergence, it is sufficient to prove that the family of random variables is uniformly bounded in when ranges through all valued stopping times and . Indeed, by Cauchy-Schwarz inequality, this will imply that is bounded in uniformly with respect to and .
The bound for follows directly from the explicit expression (2.8). Note then that is an exponential martingale. Indeed, denoting , we have
[TABLE]
It follows that
[TABLE]
Since is uniformly bounded by , the second term in the above expression is a martingale, and we deduce that for any stopping time taking values in
[TABLE]
Using that almost surely for all , as , we conclude
[TABLE]
In the above estimate we have used that
[TABLE]
which follows from the continuity of and the fact that (see Proposition 4.1).
Notice that the above estimates also imply that is bounded in and is bounded in , uniformly with respect to stopping times and .
It remains to take limits as with . By continuity of the sample paths for . We use of Proposition 6.3, dominated convergence and (6.25) (along with the fact that ) to obtain
[TABLE]
The latter implies continuity of at .
To prove that is also continuous across we need to argue in a slightly different way. Fix with and pick . With no loss of generality we consider as the proof requires minor changes for . We set the first entry time of into and denote by the first entry time of into for some . Then we define for some . Again we recall that .
We know that from of Lemma 3.1 and (6.13). In order to find a lower bound we use (6.17) and (6.18) and get
[TABLE]
From this point onwards we can repeat the arguments used above up to trivial modifications. These allow us to conclude that is continuous across with the possible exception of , because Proposition 6.3 does not hold at that point if .
As already mentioned, analogous arguments allow to prove that is also continuous everywhere with the possible exception of . It follows that on and on (see (5.6)). The latter and (5.8) imply that is continuous on as claimed. ∎
It remains to prove Lemma 6.1 and for that it is convenient to change variables to the coordinate system . We set
[TABLE]
with the notation , and . In these variables from (6.19) reads
[TABLE]
Notice that for the boundary is non-decreasing and the stopping set lies below it. Hence (6.3) is a consequence of standard arguments involving the law of iterated logarithm. Showing (6.4) for is instead more difficult because is also non-decreasing but lies above the boundary. A symmetric situation occurs for .
In what follows we first show that the classical smooth-fit condition holds and then prove that under our assumptions this implies Lemma 6.1. In the next lemma we only consider smooth-fit in those cases when the monotonicity of the boundary does not allow a direct proof of (6.3) or (6.4) based on the law of iterated logarithm.
Lemma 6.5**.**
If and , then . Analogously if , with and , and then . Finally, if , with and then .
Proof.
We carry out the proof under the assumption of (see (5.2)). This induces no loss in generality as symmetric arguments hold for .
Let with . Notice that for we have . Also we know from the proof of in Lemma 5.4 that locally at . We argue by contradiction and assume . The latter limit exists because is locally bounded (see (5.13)) and in due to (5.8), for a suitable .
Fix , consider the open rectangle and let . With no loss of generality we assume and from (6.17) we obtain
[TABLE]
Since is non-increasing (see (5.13)) and is bounded we can find a constant depending on and such that
[TABLE]
Recalling that is bounded on , we can apply Itô-Tanaka formula to get
[TABLE]
Boundedness of and the assumption give
[TABLE]
for some positive . For , Burkholder-Davis-Gundy inequality and some algebra give
[TABLE]
with depending on and . Plugging the latter inside (6.32) and letting we reach a contradiction. Therefore it must be .
The proof is entirely analogous for and . It is also worth noticing that for with , the smooth-fit condition amounts to because the stopping payoff is . Using in Lemma 5.4 and arguments similar to those above we can prove that holds. ∎
Proof of Lemma 6.1.
Here we only consider the case but the same results hold for and these can be proven by symmetric arguments. It is convenient to recall the function from (6.28).
The proof of (6.3), which we omit for brevity, is a straightforward consequence of the fact that is non-decreasing and is non-degenerate away from [math] and , so that the law of iterated logarithm can be applied. The same rationale allows to prove that (6.4) holds for for .
To prove (6.4) with and let us argue by contradiction and assume that is not regular or equivalently is not regular (with ), i.e. (6.4) does not hold. Pick and such that . Denote , , , . Notice that , then from (6.17) and (6.18) and setting we obtain
[TABLE]
where in the last inequality we have used that is non-decreasing as shown in the proof of Lemma 5.4. Recall that is strictly negative (see (5.18)) so that almost surely and for all we have
[TABLE]
As in the proof of Proposition 6.4 we have as . Moreover increases111Notice that, due to the geometry of , can only enter by hitting . as , hence , -a.s. To prove the reverse inequality we fix and pick such that . Then in particular we have
[TABLE]
for some . Recall that is continuous, hence bounded on by a constant . Using that we find
[TABLE]
from (6.35). This implies that for all sufficiently small . Since was arbitrary we conclude . The argument holds for a.e. hence we obtain
[TABLE]
Convergence of and imply
[TABLE]
Dividing (6) by and taking limits as , we may use Fatou’s theorem and the expression (5.18) for to obtain
[TABLE]
Now we let and use that -a.s. the following limits hold
[TABLE]
In particular we notice that for the convergence of we can use the same arguments as those used above for the convergence of . Clearly
[TABLE]
and by assumption, . Using again Fatou’s lemma, taking limits in (6.36) the stopping time converges to a stopping time , -a.s. Hence , which contradicts the smooth-fit principle proven in Lemma 6.5. In conclusion must be regular for , i.e. (6.4) holds. ∎
7. Existence of a Nash equilibrium
Building on the results of the previous sections, we can prove the existence of a Nash equilibrium for our game with incomplete information. We recall here that the two main difficulties for such existence arise from the lack of uniform integrability of the stopping payoffs and the fact that the problem is bi-dimensional. In the rest of this section we make the next standing assumption.
Assumption 7.1**.**
We assume .
The next result will allow us to circumvent the lack of uniform integrability and it shows that the boundary of is always strictly positive.
Lemma 7.2**.**
For every we have .
Proof.
Arguing by contradiction we assume that there exists such that . Hence
[TABLE]
Since is increasing, properties of studied in Section 4 imply that for fixed we may define a strip
[TABLE]
and .
In particular if we pick and then, assuming without loss of generality that (see (5.2)), we have , -a.s. The latter follows by the fact that for all the couple lies in because its joint distribution is supported along a curve (see Remark 5.1). Notice that for and with , monotonicity of and (7.1) imply -a.s.).
Theorem 3.2 gives
[TABLE]
where . We aim at showing that for sufficiently close to zero we get
[TABLE]
or equivalently
[TABLE]
The latter and (7) lead to , hence a contradiction.
Defining the probability measure by
[TABLE]
by Girsanov’s theorem we have that , is a Brownian motion under . Moreover under the new measure evolves according to
[TABLE]
From the above dynamics it follows immediately that for all and
[TABLE]
Using the inequality valid for , we have
[TABLE]
where for the last inequality we used (7.4) and Cauchy-Schwarz inequality. We aim at showing that
[TABLE]
To see this, we observe that is a supermartingale under the probability measure . Indeed, applying Itô’s formula we get
[TABLE]
and the drift part of the SDE is non-positive because . Thus (7.5) holds as claimed.
Finally we obtain
[TABLE]
Recalling that , for sufficiently small we have . Moreover when it is immediate to check that as (see (5.5)). In conclusion the right-hand side in (7.6) diverges, yielding the desired contradiction. ∎
We can now prove existence of a saddle point for our game.
Proposition 7.3**.**
If the pair defined in Theorem 3.2 is a saddle point.
Proof.
Since Theorem 3.2 guarantees the optimality of , i.e.
[TABLE]
it remains to prove the optimality of , that is
[TABLE]
Let be fixed and set . Invoking Theorem 3.2 and observing that for any fixed and ,
[TABLE]
we obtain
[TABLE]
for any stopping time .
We now prove that the last term of the expression above converges to zero as . Notice first that is non-decreasing (see Corollary 5.5) and therefore is non-increasing due to Corollary 4.5. For we have , which implies that after the change of variables. Then and we have the uniform bound for . Notice that thanks to Lemma 7.2, so that we also have .
Using such bound we get
[TABLE]
Next, the monotone convergence theorem yields
[TABLE]
that is, is optimal for the buyer. ∎
Let us now analyze the case , for which we prove existence of a Nash equilibrium under stronger assumptions on the parameters. We start with an auxiliary lemma, which will require the following assumption (recall also that by Assumption 7.1).
Assumption 7.4**.**
We take such that
[TABLE]
Notice that (7.7) indeed implies .
Lemma 7.5**.**
Under Assumption 7.4 it holds that:
[TABLE]
Proof.
First note that
[TABLE]
Then recall that (see (5.11)) and since then as . It is therefore sufficient to prove that as
[TABLE]
for some constants and .
Define . Let , and . Note that since is non-decreasing and , we have -almost surely. Therefore, for all Theorem 3.2 gives
[TABLE]
On the event we have and , -a.s., so that , -a.s. (as in the proof of Proposition 7.3). The latter implies
[TABLE]
and hence
[TABLE]
Taking limits in (7.9) as and using monotone convergence, we deduce that
[TABLE]
In order to compute the Laplace trasform of we need to recall the fundamental solutions of , where denotes the infinitesimal generator of the diffusion . Letting be the unique positive, increasing solution and the unique positive, decreasing one, we have
[TABLE]
where is the largest solution of
[TABLE]
In terms of the Laplace transform of reads (recall that )
[TABLE]
In conclusion, for any and , taking we have (recall (5.6))
[TABLE]
Now we fix and pick such that . Since for all we can use the latter and (7.10), replacing therein by \big{(}z+kt,a\,c_{1}(z+kt)\big{)}, to estimate
[TABLE]
Simple algebra gives
[TABLE]
for some constant depending on , and , and with . Now Assumption 7.4 implies that as required in (7.8). ∎
Proposition 7.6**.**
Under Assumption 7.4 the pair is a saddle point.
Proof.
As in Proposition 7.3, we only have to prove the optimality of and we argue in a similar way. Let be fixed and set , then as in the proof of Proposition 7.3 we find
[TABLE]
for any stopping time and any . Under we have and for we have , which implies . The latter gives
[TABLE]
which goes to zero according to Lemma 7.5. Then taking limits as in (7) and using also monotone convergence we conclude the proof. ∎
8. Concluding remarks
Our approach relies on the ability to obtain a two-dimensional Markovian dynamic for the stock price and the expected value of the dividend rate (given the observations of the stock). This stems from fact that the dividend rate has a two-point distribution. Similarly, if has a discrete distribution taking values, we can use filtering methods to reduce the problem to a stopping game on a -dimensional degenerate diffusion. However, it should be clear at this point that a free boundary analysis of such problem is likely to be extremely convoluted. An even more complex situation arises when the dividend rate is allowed to take infinitely many values. In that case the dynamics obtained via our filtering approach may easily lead to a formulation of the game which is intractable with free boundary methods.
An alternative approach relies on the use of a Girsanov transformation. While it falls outside the scope of the present paper to perform a fully rigorous analysis of this method, we believe it may be useful for future research to outline the main ideas of this approach and point out some questions that arise naturally.
Letting a process be defined by
[TABLE]
we have that the stock price in (2.1) reads
[TABLE]
Moreover, the process is an -Brownian motion for , under the measure defined by
[TABLE]
for all , where is the (augmented) filtration generated by and .
Although it is not possible, in general, to perform the change of measure on (see, e.g., [22, pp. 192-193]), let us set this problem aside for now and assume that the distribution of is sufficiently ‘nice’ to allow the use of (8.2) in order to rewrite (1.3) as
[TABLE]
Now we define a process with . Thanks to (8.2), using the fact that and are independent under , and expressing in terms of (see (8.1)), we have for some function , depending on the specific distribution of . Then, the game’s payoff reads
[TABLE]
where we note that can be computed explicitly in some cases222For example, in the simple case of we have
with and ..
The construction above holds for any law of that allows to justify the change of measure on (a seemingly non-trivial task). However, under the expectation, the resulting game’s payoff depends explicitly on the initial value of the stock price . One way to circumvent this issue would be to consider as a ‘parameter’ in the game formulation (8.3), and treat it independently of the initial value of the process . That is, we would fix an arbitrary and study the game with payoff
[TABLE]
where the process starts from an arbitrary point , possibly different from . Now, for each one must solve the Dynkin game with payoff as in (8.4), which remains a challenging task due to the (generally) convoluted expression of . Moreover, the shapes of the continuation and stopping region need to be studied not only as functions of time but also as functions of the parameter .
It is interesting to notice that the approach outlined above corresponds to the study of pre-commitment strategies in the closely related literature on time-inconsistent control/stopping problems. (The interested reader may consult, e.g., [6] and references therein, for a recent detailed study on a class of time-inconsistent stopping problems where time inconsistency stems from a ‘parametric’ dependence of the gain function on the starting point of the process, i.e., the analogue of our ). To the best of our knowledge, time-inconsistent Dynkin games have never been addressed in the literature. Moreover, there seems to be no clear consensus, as to whether the pre-commitment strategy is conceptually the best way forward in time-inconsistent stochastic optimisation problems. This interesting question is left for future research.
Appendix A Proof of Theorem 3.2
The main idea of the proof is to approximate our game by a sequence of games with bounded stopping payoffs indexed by . For each approximating problem we can apply the results of [16] regarding existence of the value and of a saddle point. Eventually we pass to the limit as to obtain the existence of the value for the game with unbounded payoffs.
For let us define the functions , . Next for let us introduce the the associated payoff
[TABLE]
According to Theorem 2.1. in [16], the game with payoff (A.1) has a value, i.e.
[TABLE]
Moreover, the stopping times
[TABLE]
[TABLE]
form a Nash equilibrium. Since
[TABLE]
and for , then for . The latter implies that
[TABLE]
and therefore
[TABLE]
Concerning the value of the approximating game, it is easy to check that Lemma 3.1 holds for with the same proof. Moreover the sequence is non-decreasing in and it is bounded from above by . Hence the sequence is non-decreasing in with
[TABLE]
for all . In particular implies
[TABLE]
Since is non-decreasing in then is non-increasing and we set .
Now we aim at showing that so that (A.2) implies and therefore the value exists and it coincides with . For all , we have
[TABLE]
Observe that
[TABLE]
and recall that by (2.9). Using dominated convergence in (A.3) we obtain
[TABLE]
On the other hand, Fatou’s Lemma implies
[TABLE]
Collecting the above limits we deduce that
[TABLE]
Now, for , let be such that
[TABLE]
Using optimality of in the approximating problem, and (A.4) we obtain
[TABLE]
Finally, letting and recalling (A.2), we obtain
[TABLE]
and hence the existence of the value . As a byproduct we also obtain that is optimal for player 2, that is
[TABLE]
Next we want to prove optimality of and super/sub-martingale properties of . For all and any we have (see [16, Thm. 2.1.])
[TABLE]
where in the second inequality we used that . Now we take limits as . Recalling that , that and that is integrable, the second term in the last expression above converges to zero by dominated convergence. Moreover, Fatou’s Lemma yields,
[TABLE]
Since was arbitrary the process , is a super-martingale. Noticing that and choosing in (A.5) for some , we see that also the process , is a super-martingale as claimed. As it is a non-negative super-martingale, Fatou’s lemma gives
[TABLE]
hence the super-martingale is closed.
Finally, we prove that is optimal for the seller, i.e. player 2. We have
[TABLE]
Taking the supremum over gives the optimality of strategy for player 2.
It remains to prove the sub-martingale property. Let us denote
[TABLE]
the stopping region of player . Notice that an analogous set can be defined relatively to and (see (4.2)). In Section 4 properties of are proven in Lemma 4.2 by using continuity and monotonicity of . The same methodology can be applied to to prove analogous properties for . To be precise it is worth noticing that (4.8) holds for provided that therein. The rest of (iii) in Lemma 4.2 follows by recalling that . The analogy holds with Corollary 4.5 as well. In particular there exists a non-increasing lower-semi-continuous map such that for it holds .
Observe that if is such that , we have
[TABLE]
which implies . Together with the fact that , this implies that . By the same arguments, we prove that . We deduce that the sequence is non-decreasing and that is a non-decreasing sequence of stopping times such that . Moreover, if is such that , then and, for sufficiently large , we have , implying that . We deduce that converges to on pointwise.
Now, we prove that . Since , it is sufficient to show that the equality holds -almost surely on . For the claim is trivial. Fix and . Since the sequence is non-decreasing then for fixed and any we have . The latter implies
[TABLE]
by using that as well. Taking the supremum over in the right-hand side of the above expression and recalling that pointwise we conclude
[TABLE]
Since , -a.s. for all , using continuity of paths and (A.7), we also find
[TABLE]
which implies , -a.s. as requested.
Finally we notice that the process is a sub-martingale for all . Since for all , we deduce that
[TABLE]
Letting , monotone convergence implies that
[TABLE]
for all . Taking and recalling that , bounded convergence implies
[TABLE]
The above result and the Markov property imply that is a sub-martingale as claimed.
Appendix B Proof of Lemma 6.2
The proof is more easily carried out considering the boundaries and rather than and . However we incur no loss of generality thanks to the equivalence of the problem formulation with respect to the coordinates and . We provide a full argument for but a completely symmetric proof holds for .
Since is non-decreasing and is non-degenerate at all points of , the law of iterated logarithm implies that , -a.s. Similarly if hits the line from above then it will immediately cross it downwards.
For the same result relative to the boundary we repeat the steps in [5, Cor. 8]. In particular let us introduce some notation
[TABLE]
so that and . We have and
[TABLE]
Assume that for any we have
[TABLE]
so that , -a.s. Then
[TABLE]
where the last limit is easily verified by definition of and we could swap the limits because is non-decreasing in both and .
Now it remains to verify (B.3). We start by noticing that any interval of the form may be decomposed into the union of countably many intervals over which is either strictly increasing or flat. Consider the latter, i.e. let be an interval such that for and a fixed . Fix also , then it is immediate to check that on the event one has , -a.s., because immediately crosses after reaching it. This in particular implies that
[TABLE]
Next we fix so that for we have , -a.s., because and are non-decreasing. Moreover the inequality is strict whenever is strictly increasing. Hence, the latter consideration and (B.4) imply
[TABLE]
where in the last expression we have expressed explicitly so that it can be treated effectively as a ‘time’ variable.
We now denote by and the probability transition density and the speed measure of , respectively. Then by using the Markov property of we obtain
[TABLE]
Scheffé’s theorem (see page 224 in [4]) guarantees that
[TABLE]
thus implying that taking limits as we obtain
[TABLE]
Letting now we find (B.3) as claimed, because it is easy to verify that .
Appendix C Game with complete information: summary of results
In this appendix we provide a short summary of existing results concerning the stopping regions in the game call option problem with perfect information, i.e. when is either [math] or . The material below is based on results contained in [35], for , and [17], for .
We recall as in (1.3) and emphasise that here . Denote by the value of the optimal stopping problem for the buyer when there is no possible seller’s cancellation (i.e. when ):
[TABLE]
and by the value of the problem when , i.e. the hitting time of by :
[TABLE]
We also define the critical dividend levels by
[TABLE]
For the process we recall that the fundamental solutions of
[TABLE]
are and , with increasing (notice that ) and decreasing, and where solve
[TABLE]
The next Proposition summarises results of [35] and [17, Sec. 5.1].
Proposition C.1**.**
The following four cases hold
- •
Case 1:* If we have*
- –
,
- –
* and .*
- •
Case 2:* If and we have*
- –
* and .*
- –
* and .*
- •
Case 3:* If and we have*
- –
* and .*
- –
* and , where is the unique solution of*
[TABLE]
- •
Case 4:* If and we have*
- –
* and *
- –
* and where is the unique solution of the system of equations*
[TABLE]
For all the above cases, the pair defined in Theorem 3.2 is not a saddle point if , and is a saddle point if .
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- 7[7] Dayanik, S., and Karatzas, I. On the optimal stopping problem for one-dimensional diffusions. Stochastic Process. Appl., 107(2), 173-212, 2003.
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