A Dynkin Game Under Knightian Uncertainty ∗

We study a zero-sum Dynkin game under Knghtian uncertainty. The associated Hamiton-Jacobi-Bellman-Isaacs equation takes the form of a semi-linear backward stochastic partial differential variational inequality (SBSPDVI). We establish existence and uniqueness of a strong solution by using the Banach fixed point theorem and a comparison theorem. A solution to the SBSPDVI is used to construct a saddle point of the Dynkin game. In order to establish this verification we use the generalized Itó-Kunita-Wentzell formula developed by Yang and Tang (2011).

In the classical Dynkin game the objective functional is an expectation taken with respect to an objective probability measure. In the real world, however, people usually face situations where it is impossible to assess unique probability measures for the future events. For instance, we do not know the exact probability of a stock index to take a value in a specific interval. Knight [21] called such a situation where one cannot assess a unique probability measure for future events uncertainty in contrast to risk where the future outcome cannot be predicted but one can assess a unique probability for each possible event. Ellsberg [13] provided an experimental evidence that people typically exhibit aversion toward Knightian uncertainty. Gilboa and Schmeidler [15] gave an axiomatic model of aversion to Knightian uncertainty (or ambiguity aversion) where a set of probability measures, instead of a unique probability measure, exists and utility is expressed as a minimum of expectations taken with respect to the probability measures, i.e., where Φ is the set of probability measures and v(·) is the cardinal utility function. The model is called a multiple priors model. Chen and Epstein [3] have extended the model to continuous time and shown that utility can be expressed as a solution to a backward stochastic differential equation (BSDE). Maccheroni, Marinacci, and Rustichini [22] have extended the multiple priors model to include the robust control framework proposed by Hansen and Sargent [17]. Their preference, which is called a variational preference, admits the following representation: where c is a function defined on the set Φ of probability measures which they call the index of ambiguity aversion. In Chen and Epstein [3] the set Φ of probability measures is usually defined as that of probability measures Q whose densities with respect to P are given by dQ dP := exp − 1 2 T 0 θ 2 s ds + T 0 θ s dW s with an adapted process θ being square-integrable on [0, T ] almost surely and taking values in a given set K. If, as assumed in Cheng and Riedel [5], the index c takes the following form c(Q) := T 0 g(s, θ s ) ds for some random function g, then a dynamic variational preference is, in general, given by a solution to the following BSDE (as in Tang and Koo [25]): where η(s, z) := inf θ∈K [g(s, θ) − z, θ ].
In this paper we study a Dynkin game with the functional being specified by a general nonlinear BSDE, which incorporates Knightian uncertainty and the dynamic variational preference (1.2). Such a kind of Dynkin game has been addressed by Cvitanic and Karatzas [7], but their saddle point was of an open form. In the following, we are interested in constructing the saddle point of a closed form, in spirits of Yang and Tang [27].
Our approach appears to be straightforward and intuitive. Time consistency of the assumed preference enables us to apply the dynamic programming principle. By the principle we derive its associated Hamiton-Jacobi-Bellman-Isaacs equation, which takes the form of a semi-linear backward stochastic partial differential variational inequality (SBSPDVI). We show existence and uniqueness of a strong solution to the SBSPDVI. The non-linearity of the problem stems from two aspects: the non-linear differential operator and the variational inequality, which cause difficulties. We use the Banach fixed point theorem to establish the existence of a strong solution. But we deal with the space of weak solutions rather than that of strong solutions, because it is highly difficult to deduce the estimates in the space of strong solutions due to the non-linearity. Via proving a comparison theorem for SBSPDVI, we show existence of a weak solution to the SBSPDVI by iteration. Then we show that the weak solution is in fact regular enough to be a strong one by means of the relationship between SBSPDVI and a backward stochastic partial differential equation(BSPDE). Uniqueness is a consequence of the comparison theorem for SBSPDVI.
The verification theorem is important to connect Dynkin games and the SBSPDVI. We prove that the strong solution to the SBSPDVI is the value of the Dynkin game and give a pair of optimal stopping times. There are at least two difficulties because the problem is investigated with a strong solution in a Non-Markov and Knightian uncertain framework. One difficulty is about how to use the Itô formula. As we all know, the value function, V , is required to be deterministic in order to apply the Itô formula to it. The requirement is loosened to allow a stochastic function in the Itó-Kunita-Wentzell formula, but the function needs be C 2 with respect to state variable, x. In our problem, the strong solution V is random, and only has weak second-order derivatives. We overcome the difficulty by applying the generalized Itó-Kunita-Wentzell formula developed by [27]. The other difficulty arises because the payoff can't be described as a linear conditional expectation due to Knightian uncertain, so that the problem has a strong non-linear property. Fortunately, the payoff can be represented as a solution to the BSDE in (1.2). So, we can use the properties of BSDEs to overcome the difficulty.
Moreover, we show explicitly that the optimal stopping problem by one agent is an extreme case of a Dynkin game. Optimal stopping problems emerge naturally in many fields of economics, finance and statistics. Individual career changes and voluntary retirements, firms' capital investments, firms' entry and exist decisions, exercise of American options or conversion of bonds into shares, sample selections involve optimal stopping [6,9,14,24]. An optimal stopping problem can be considered a Dynkin game where a fictitious second player exists who proposes a value which is beyond the reach of the agent. So, our result generalizes those of previous studies on optimal stopping with ambiguity aversion ( [5,24]).
The paper proceeds as follows. In section 2 we explain our assumptions and notation. In section 3 we provide the verification theorem. In section 4 we prove existence and uniqueness of a strong solution to the SBSPDVI. In section 5 we show that an optimal stopping problem is an extreme case of a Dynkin game. In section 6 we provide two examples, one concerning a convertible bond and the other concerning optimal retirement.
2 Problem, notation and some assumptions.
In our problem, we suppose that the real probability property of the market can be described as (Ω, F , {F t } t≥0 , P), on which two independent standard Brownian motions: are defined, which drive the risk factor.
Denote by F W {F W t } t≥0 and F B {F B t } t≥0 the natural filtrations generated by W and B, respectively. Assume that they contain all P-null sets in F . Define F F W ∨ F B . Denote by P and P B the σ-algebras of predictable sets in Ω × [ 0, T ] associated with F and F B , respectively. Denote by B(D) the Borel σ-algebra of the domain D in R n .
Suppose that the state process (for example, the price of the risk asset) X = (X 1 , · · ·, X n ) is governed by the following stochastic differential equation (SDE): where i = 1, · · ·, n and x = (x 1 , · · ·, x n ) ∈ R n . Note that here and in the following we use repeated indices for summation. For example, the repeated subscript l implies summation over l = 1, · · ·, d 1 and the repeated subscript k implies summation over k = 1, · · ·, d 2 . And the coefficients β, γ, and θ in SDE (2.3) satisfy Assumptions (D1) and (D2) (see below in this section). Introducing the ambiguity averse (or averse to Knightian uncertainty) utility representation by the BSDE in (1.2) into the objective functional (1.1), we get the following BSDE for the payoff: T is the class of all F-stopping times which take values in [ t, T ] and (τ 1 , τ 2 ) ∈ U t,T × U t,T , which consists of the strategies of both agents. The above BSDE has a nonlinear generator G with the terminal value being the sum of the last three terms on the right-hand side of the equation. The solution (R, Y 1 , Y 2 ) obviously depends on the pair of stopping times (τ 1 , τ 2 ), and its first component R t,x t will be denoted by R t (x; τ 1 , τ 2 ). Consider the following Dynkin game under Knightian uncertainty: each player tries either to minimize or to maximize R t (x; τ 1 , τ 2 ).
Throughout this paper, we assume that the running payoff G is a P B ×B(R n+1+(d 1 +d 2 ) )measurable random field taking values in R, and satisfies Assumption (D3) λ (see below in this section). The terminal payoffs V and V are P B × B(R n )-measurable random fields taking values in R, and ϕ is an F B T × B(R n )-measurable random field taking values in R. They satisfy some proper assumptions.
In this paper, we consider the following Nash equilibrium of the non-Markovian zerosum Dynkin game (denoted by D tx hereafter): Find a pair (τ * 1 , τ * 2 ) ∈ U t,T × U t,T such that the following inequalities hold Such a pair (τ * 1 , τ * 2 ), if it exists, is called a Nash equilibrium point or saddle point of Problem D tx , and the random variable V t (x) R t (x; τ * 1 , τ * 2 ) is called the value of Problem D tx . In this case, we have The value V t (x) is unique if it exists. In general, a Nash equilibrium point may not be unique, and here it always means the smallest one.
In order to facilitate the discussion, we introduce the following notation: Denote by N and N + the set of all nonnegative and positive integers, respectively. Denote by E a Euclidean space like R or R n , R n×d 1 , R n×d 2 , and R n×n . Moreover, for any x ∈ R n , γ ∈ R n×d 1 , θ ∈ R n×d 2 and a ∈ R n×n , define Denote by Dη the gradient of the function η : E → R.
For an integer k ∈ N, p ∈ [1, +∞), q ∈ [1, +∞), λ ∈ [ 0, +∞), and a positive number T , we introduce the following spaces: • C k , the set of all functions η : R n → E such that η and D α η are continuous for all 1 ≤ |α| ≤ k; • C k 0 , the set of all functions in C k with compact support in R n ; • H k, p λ , the completion of C k under the norm • L k,p λ , the set of all H k, p λ -valued and F B T -measurable random variables such that E (| ϕ| p k,p; λ ) < ∞; • L p , the set of all P-predictable stochastic processes taking values in E with the norm • S p , the set of all path continuous processes in L p with the norm • H k, p λ , the set of all P B -predictable stochastic processes with values in H k, p λ with the norm • S k, p λ , the set of all path continuous stochastic processes in H k, p λ equipped with the norm The space notation H k, p 0 , L k,p 0 , H k, p 0 , S k, p 0 , |η| k, p; 0 , V k, p; 0 , and | V | k, p; 0 will be abbreviated as H k, p , L k,p , H k, p , S k, p , |η| k, p , V k, p and | V | k, p if there is no confusion.
The assumption notation (D3) 0 will be abbreviated as (D3) if there is no confusion.

Verification theorem.
In this section, we prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of SBSPDVI: where the repeated superscript k is summed from 1 to d 2 , and Remark 3.1. In fact, the free term f can be included in F . But we keep it because it is convenient to give some estimates for the SBSPDVI.
In this paper, we maybe consider the following assumptions: Assumption (V1) (Boundedness). Functions a, b, c, σ, µ are P B × B(R n )-measurable, and are bounded by a constant K, taking values in the set of real symmetric n×n matrices, in the spaces R n , R, R n×d 2 , R d 2 , respectively. Da and Dσ exist almost everywhere and are bounded by K.
Assumption (V2) (Superparabolicity). There exist two positive constants κ and K such that: Assumption (V3) λ (Regularity). The non-linear function F is uniformly Lipschitz in (v, y, z) ∈ R 1+n+d 2 , i.e., there exists a positive constant K such that And V and V are continuous semimartigales of the following form hold in the sense of distribution. That is, for any nonnegative function η ∈ C 2 0 ( R n ), we have λ , and V and V have the following representation: The strong solution of SBSPDVI (3.1) is defined as follows: with some nonnegative number λ, and satisfies: a.e. x ∈ R n for all t ∈ [0, T ] and a.s. in Ω; (3.5) Recalling the generalized Itô formula in [27], we have is a continuous semimartigale of form: for every t ∈ [ 0, T ] and almost surely ω ∈ Ω, such that V ∈ H 2, 2 , Z ∈ H 1, 2 , and U ∈ H 0, 2 . Let X be a continuous semi-martingale X of form (2.3), and Assumptions (D1) and (D2) be satisfied. Then we have where the repeated superscript l is summed from 1 to d 1 and the repeated superscript k is summed from 1 to d 2 , and for any l = 1, 2, · · ·, d 2 , k = 1, 2, · · ·, d 1 , and the repeated superscript i, j is summed from 1 to n.
We have the following verification theorem for Problem D tx .
and a ij 1 2 for any i, j = 1, · · · , n and k = 1, · · · , d 2 . Then V (t, x) is the value of Problem D tx . Define Proof of Theorem 3.2. It is sufficient to prove the following for any τ 1 , τ 2 ∈ U t,T , with the equality holding true in the first inequality if τ 2 = τ * 2 and in the second inequality if In what follows, we only prove the second inequality and the first one can be proved in a symmetric way.
According to [27,Theorem 3.1], we deduce that V (X) − V (X) and V (X) − V (X) are stochastic processes with continuous paths. So, we have that almost surely in Ω. Moreover, according to the method in [27], we can deduce the following equalities from the equalities on the sixth line in Definition 3.1, in Ω × (0, T ), and for any τ 1 , τ 2 ∈ U t, T satisfying τ 1 ≤ τ * 1 , τ 2 ≤ τ * 2 , the following hold where we have used (3.7), (3.8) and (3.9), and Recalling k + ≥ 0, we have with the equality holding true if τ 1 = τ * 1 , which follows from τ * 2 ≤ τ * 1 and (3.9). On the event {τ ∈ U t, T : with the equality holding true if τ 1 = τ * 1 . So, we obtain that almost surely with the equality holding true if τ 1 = τ * 1 . This means that almost surely with the equality holding true if τ 1 = τ * 1 . From the comparison principle for BSDE, we deduce that with the equality holding true if τ 1 = τ * 1 . The proof is then complete. In the section, we use the Banach fixed point theory and the comparison theory for the SBSPDVI to established the existence and uniqueness of the strong solution of SB-SPDVI (3.1). First, we review an existing estimate for the following semi-linear backward stochastic partial differential equation (SBSPDE, for short) [10, Theorem 5.3]: (4.1) Lemma 4.1. Let the differential operators L and M satisfy Assumptions (V1) and (V2), the nonlinear term F satisfy Assumption (V3), the free term f ∈ H 0, 2 , and the terminal value ϕ ∈ L 1, 2 . Then SBSPDE (4.1) has a unique strong solution (V, Z). Moreover, it satisfies the following estimate: where C(κ, K, T ) is a constant depending on κ and K, T .
Then we recall an estimate for linear backward stochastic partial differential equations (BSPDE, for short) [27, Lemma 5.1]: Lemma 4.2. Let the differential operators L and M satisfy Assumptions (V1) and (V2), and F ≡ 0, f ∈ H 0, 2 , ϕ ∈ L 1, 2 . Then the strong solution (V, Z) of BSPDE (4.1) satisfies the following estimate: where C(κ, K, T ) is a constant increasing with respect to K, T , and decreasing with respect to κ.
We then have the following estimate for SBSPDVI (3.1): Then we have that where C(κ, K, T ) is a constant increasing with respect to K, T , and decreasing with respect to κ.
Then ∆V satisfies the following BSPDE: In view of Lemma 4.2 and k ± 1 , k ± 2 ≥ 0, we have that Next, we estimate the last three terms on the right-hand side of the last inequality.
On the other hand, in the domain {V 1 < V 2 }, we have Hence, we deduce that in Ω×Q, then the last two equalities in (3.5) imply that From (4.5)-(4.8), we conclude that Consider the above estimates only on the time interval [ t, T ], then we have Applying the Gronwall inequality, we have (4.2). Repeating the similar argument, we can derive that So, (4.3) is obvious. And (4.4) follows from the Hölder inequality. 2 From Lemma 4.3, we can deduce the following comparison theorem for SBSPDVI: In order to apply the Banach fixed point theorem, we need define the weak solution of SBSPDVI (3.1) as follows: in Ω × R n . The difference between strong solutions and weak solutions of SBSPDVI is only in the first equation, that is just the difference between strong solutions and weak solutions of BSPDEs. So, according to [10,Proposition 2.2] or the proof of [27,Theorem 5.3], we have the following proposition: Proposition 4.6. Let Assumptions (V1)-(V3) be satisfied. If (V, Z, k + , k − ) ∈ H 2, 2 × H 1, 2 ×H 0, 2 ×H 0, 2 is the strong solution of SBSPDVI (3.1), then it is also a weak solution. Moreover, if the weak solution of SBSPDVI (3.1) satisfies V ∈ H 2, 2 and Z ∈ H 1, 2 , then it is also a strong solution.
Moreover, we have the following equivalence between the strong solution and the weak solution under Assumptions (V1)-(V5). Proof. According to Proposition 4.6, it is sufficient to prove that V ∈ H 2, 2 and Z ∈ H 1, 2 if (V, Z, k + , k − ) is the weak solution of SBSPDVI (3.1).
In view of the first equation in (4.10), we know that (V, Z) is a weak solution of the following BSPDE: According to [10,Corollary 3.4] or [27, Lemma 2.2], we deduce that BSPDE (4.11) has a unique strong solution such that (V * , Z * ) ∈ H 2, 2 × H 1, 2 , which is also a weak solution of BSPDE (4.11).
On the other hand, [23,Corollary 3.4] implies that the weak solution of BSPDE (4.11) is unique. So, (V, Z) = (V * , Z * ) ∈ H 2, 2 × H 1, 2 . And the result in this lemma is obvious.2 Thank to the above preparations, we can establish the existence and uniqueness of the solution of SBSPDVI (3.1).
Theorem 4.8. Let Assumptions (V1)-(V5) be satisfied. Then SBSPDVI (3.1) has a unique strong solution (V, Z, k + , k − ) such that Proof. We will apply the Banach fixed point theorem to establish the existence of the strong solution of BSPDVI (3.1). The proof is divided into five steps: Step 1. Construct the proper space and mapping for applying the Banach fixed point theorem. Let D H 1, 2 × H 0, 2 with the norm Let the mapping A be defined as follows. Given a function (v, z) ∈ D, then it is clear that |F (·, v, Dv, z)| ≤ |F (·, 0, 0, 0)| + K |v| + |Dv| + |z| ∈ H 0, 2 .
Hence, the following linear BSPDVI has a unique strong solution (V, Z, k + , k − ) and ). According to Lemma 4.7, (V, Z, K + , k − ) is also the unique weak solution of BSPDVI (4.13). Define A by Step 2. Testify that the mapping A is a strict contraction provided T is small enough. That means there exists a ς ∈ (0, 1) such that In fact, from (4.3), we deduce that Hence, if T is small enough such that CT < 1/2, then we have (∆V, ∆Z) D ≤ 1 2 (∆v, ∆z) D .
Step 3. We prove that SBSPDVI (3.1) has a unique strong solution provided CT < 1/2. Fix any point (V 0 , Z 0 ) ∈ D and thereafter iteratively define (V m+1 , Z m+1 ) = A(V m , Z m ) for m = 0, 1, ···. Applying the Banach fixed point theorem, we deduce that {(V m+1 , Z m+1 )} is a cauchy sequence in the space D. Then there exists a (V, Z) ∈ D such that Denoted by (V m , Z m , k + m , k − m ) the strong solution of BSPDVI (4.13) with (v, z) (V m−1 , Z m−1 ). Then Lemma 4.5 implies that So, there exist k + , k − and a subsequence of {(V m , Z m , k + m , k − m )}, still denoted by itself, such that as m → ∞, Moreover, we computer Step 4. Finally, we prove that SBSPDVI (3.1) has a unique strong solution for any given T > 0. We select T 1 > 0 so small such that CT 1 ≤ 1/2. We can find a strong solution existing on time interval [ T − T 1 , T ].
Since V t ∈ L 1, 2 a.e. in [ 0, T ], upon redefining T 1 if necessary, we may assume V T −T 1 ∈ L 1, 2 . We can then repeat the argument above to extend our solution to the time interval [ T − 2T 1 , T − T 1 ]. Continuing, after finitely many steps we can prove the strong solution existing on the full interval [ 0, T ].
The uniqueness of the solution for SBSPDVI (3.1) comes from Theorem 4.4.
Step 5. Prove the estimate (4.12). Assume (V 1 , Z 1 , k + 1 , k − 1 ) is the strong solution of SBSPDVI (3.1) with F ≡ 0. Then we can rewrite the first equation in the definition 3.1 as the following . According to the estimate (4.4) in Lemma 4.3 and Lemma 4.5, we have V 1, 2 + Z 0, 2 ≤ ∆V 1 1, 2 + ∆Z 1 0, 2 + V 1 1, 2 + Z 1 0, 2 ≤ C(κ, K, T ) F (·, V 1 , DV 1 , Z 1 ) 0, 2 + V 1 1, 2 + Z 1 0, 2 ≤ C(κ, K, T ) F (·, 0, 0, 0) 0, 2 + V 1 1, 2 + Z 1 0, 2 ≤ C(κ, K, T ) L In view of Lemma 4.5, we have So, we have proved the estimate (4.12). 2 In fact, the assumptions imposed on the payoff function in Theorem 4.8 is too strong, and many financial models don't satisfy them. For example, if let X t be the logarithmic function of the stock's price, then the terminal payoff of the American call option is (e x − K) + , where K is the strike price. In this case, (e x − K) + / ∈ H 1, 2 . But based on Theorem 4.8, we can loosen the assumptions, which are proper for most of the financial models.  Proof. In the following, we apply a proper transformation to SBSPDVI (3.1) so that the new problem satisfies the assumptions in Theorem 4.8. Denote It is not difficult to check that ψ ∈ C 2 and where It is not difficult to check that SBSPDVI (4.15) satisfies all assumptions in Theorem 4.8. So, SBSPDVI (4.15) has a unique strong solution (v, z, k + , k − ) satisfying the corresponding estimate. Hence, SBSPDVI (3.1) has a unique strong solution (V, Z, k + , k − ) satisfying (4.14). Repeating an argument similar to the above and recalling Theorem 3.2, we can deduce the remain results in this theorem.
where P t (x; τ ) is just P t,x t , the diffusion term Y = (Y 1 , Y 2 ). They are similar to those in Dynkin game problem D tx . The coefficients satisfy Assumption (D1) and (D2). The running payoff G satisfies Assumption (D3), the terminal payoffs V and ϕ are similar to those in Dynkin game problem D tx , satisfying some proper assumptions.
The optimal stopping problem O tx , associated to the initial data (t, x), is to find a stopping time τ * ∈ U t,T such that The HJB equation for Problem O tx is the following BSPDVI with one obstacle: where the operators L and M are defined by (3.2).
is called a strong solution of BSPDVI (5.2) if it satisfies the following: Next, we will consider the following assumptions: with some nonnegative number λ. The lower obstacle V is a continuous semimartigales of the following form The assumption notation (O1) 0 will be abbreviated as (O1) if there is no confusion. Identical to the proof of Lemma 4.3 and Theorem 4.9, we have the following comparison theorem: Theorem 5.1. Let Assumptions (V1), (V2) and (V3) λ and Assumptions (O1) λ and (O2) be satisfied. Let (V i , Z i , k + i ) be the strong solution of BSPDVI (5.1) associated with (f i , ϕ i , V i ) for i = 1, 2. If f 1 ≥ f 2 , ϕ 1 ≥ ϕ 2 , and V 1 ≥ V 2 , then V 1 ≥ V 2 a.e. in Ω × Q.
The following lemma gives the relationship between Problems D tx and O tx , and between BSPDVIs (3.1) and (5.1).
Let ( V , Z) be the strong solution of the following BSPDE: where L and M are defined by (3.6) and According to Lemma 4.1, BSPDE (5.4) has a strong solution ( V , Z) ∈ H 2, 2 × H 1, 2 . Moreover, the comparison theorem for BSPDEs in [10] implies that V ≥ 0. Since Then the comparison theorem for BSPDEs in [10] implies that V ≥ V ≥ V .
In the following we prove that Problems O tx and D tx are equivalent to each other. We first claim R t (x; τ 1 , τ 2 ) ≥ P t (x; τ 1 ) for any τ 1 , τ 2 ∈ U t, T . (5.5) Applying Theorem 3.1 and repeating the method in the proof of Theorem 3.2, we deduce that According to the comparison principle for BSDE, we can deduce that Hence, we derive that P t,x τ 1 ∧τ 2 ≤ R t, x . Again applying the comparison principle for BSDE, we can achieve (5.5).
If Problem D tx has a saddle point (τ * 1 , τ * 2 ), then we have where τ 1 is an arbitrary stopping time in U t, T and we have used (5.5) in the last inequality. Hence, Problem O tx has an optimal stopping time τ * 1 ∈ U t, T . Suppose that Problem O tx has an optimal stopping time τ * 1 ∈ U t, T . Then we choose τ * 2 = T . We see that R t (x; τ 1 , τ * 2 ) = P t (x; τ 1 ) for any τ 1 ∈ U t, T . Hence, we have On the other hand, according to (5.5), we have Hence, (τ * 1 , τ * 2 ) is a saddle point of Problem D tx . Until now we have proved that Problems O tx and D tx are equivalent.
Suppose now that Assumptions (V1)-(V3), (O1) and (O2) are satisfied. Denote by ( V , Z) the solution of the following BSPDE: where L and M are defined in (3.2) and f is defined as Moreover, we define Repeating the above argument, we derive that BSPDE (5.6) has a strong solution ( V , Z) ∈ H 2, 2 × H 1, 2 . Moreover, we have V ≥ V + , V > V , and V , V and ϕ satisfy Assumptions (V4) and (V5). The estimate (5.3) follows from Lemma 4.1.
On the other hand, since V ≤ V < V and ( V , Z ) is a strong solution of BSPDE (5.6), then ( V , Z, 0, 0) is a strong solution of the following BSPDVI: In view of Theorem 4.4, V ≥ V and V > V . So, we deduce that k − = 0 a.e. in Ω × Q and (V, Z, k + ) is a strong solution of BSPDVI (5.2).
On the other hand, in view of Theorem 5.1, the strong solution of BSPDVI (5.2) is unique. So, the unique strong solutions of BSPDVI (5.2) and (3.1) coincide.
Moreover, Let X be a solution of SDE (2.3), and V (X), V (X) ∈ S 2 and ϕ(X T ) ∈ L 2 . Assume that Assumptions (D1)-(D2), (D3) λ are satisfied, and equations (3.7) and (3.8) hold. Then V (t, x) is the value of Problem O, and its optimal stopping time τ * can be described as 6 Two examples.
In this section, we give two financial examples.
Example 6.1. The convertible bond pricing model Suppose that the underlying asset of the convertible bond is the price S of the issuer's stock, which is governed by the following SDE: where W and B are one-dimensional independent standard Brownian motions. γ is a deterministic function of (u, S), and represents the volatility of the return on the stock. r u is the appreciation rate of the stock, which is a Jacobi stochastic process (a meanreverting diffusion) with upper and low bounds for any fixed S, and governed by the second SDE (for example, [8,20]). Then under proper assumptions, the stochastic process X = ln S can be described as (2.3) with n = k = l = 1, and the parameters in (2.3) are as follows: Moreover, if we imposed some proper assumptions on the coefficient γ, α, γ, then the coefficient functions β, γ satisfy Assumptions (D1) and (D2). The issuer of the convertible bond issues the bond to raise their capital, and the bondholder of the bond buys the bond to gain the return.
The bondholder buys a share of convertible bond from the issuer, then he continuously receives coupon paid from the issuer at a constant rate ̺ before maturity T and converting the bond. Prior to maturity, the bondholder has the right to convert the bond into the firm's stock with the constant conversion factor ρ ∈ (0, 1), then gets ρ S stock gain from the firm after converting. Moreover, the firm has the right to call the bond and force the bondholder to either surrender it to the firm for a previously agreed price K or convert it into the stock with the conversion factor ρ. If neither the bondholder nor the firm exercises their right before maturity, the bondholder must sell the bond to the firm at a preset value L ≤ K or convert it into the firm's stock at expiry date T . So, the bondholder receives max{L, ρ S} from the firm at maturity.
So, in the life of the convertible bond, the discounted value of the overall payoff of the bondholder can be described as the following: where D t s is the discounted process, and c u is the discounted rate process, bounded and P B -measurable. And (t, x) means that logarithmic value of the stock's price is equal to x at time t, and τ 1 , τ 2 ∈ U t, T represents the bondholder's conversion strategy and the issuer's calling strategy, respectively.
Since the bondholder's gain comes from the issuer. Then the discounted value of the issuer's overall cost is R(x, t; τ 1 , τ 2 ), too. Now, we consider a model of ambiguity aversion. As we all know, ambiguity (or Knightian uncertainty) can be represented as a g-evaluation in a general context (see [3]).
Fix a positive parameter κ and take a set, Θ, of probability measures such that (see [4,24]) where ζ = (ζ 1 , ζ 2 ). Set Θ represents the κ−ignorance measure (see [4]). The worse-case expectation min Q∈Θ E[ξ|F t ] is the same as the g-evaluation E g [ξ|F t ], which is a solution of the following BSDE where ξ is a random variable such that ξ ∈ F T , E P [|ξ| 2 ] < +∞. It is reasonable that the bondholder chooses his convertible strategy to maximize his payoff, whereas, the issuer chooses his calling strategy to minimize his cost. So, the convertible bond pricing model can be formulated as Problem D tx , and the price of the convertible bond is equal to the value function V t (x). Where G(t, x, r, y) = ̺ + κ|y|, V t (x) = ρ e x , V t (x) = max{K, ρ e x }, ϕ(x) = max{L, ρ e x }.
Moreover, there exists a discounted rate process c u in this problem, which is equal to zero in D tx .
It is not difficult to check the problem satisfies Assumptions (D1), (D2) and (D3) 1 . Moreover, V = V in the domain {x ≥ ln K − ln ρ}, and Assumptions (V4) 2 and (V5) 2 hold true in the domain D {x < ln K − ln ρ}. If we make a slight modification in the previous proof, then we deduce from Theorem 4.9 that in the domain Ω × [ 0, T ) × D, the price V t satisfies the following BSPDVI: Moreover, the above BSPDVI has a strong solution. The optimal conversion strategy τ * 1 and the optimal calling strategy τ * 2 are given by τ * 1 inf s ∈ [ t, T ] : V s (X s ) = ρ e Xs ∧ inf {s ∈ [ t, T ] : X s = ln K − ln ρ} ∧ T, Remark 6.1. It is not difficult to show the discounted rate process, c t , coincides with the coefficient, c t , in the differential operator L if we make a slight modification in the proof of Theorem 3.2.
Example 6.2. The optimal retirement and portfolio selection model in a finite horizon Suppose there are a riskless asset and d 1 risky assets in the market, where the risk-free rate is a positive constant r.
The price P 0 of the riskless asset and the price P i of the i−th risky asset are governed by the following SDE where α = (α 1 , · · ·, α d 1 ) * , Σ = ( σ ij ) d 1 ×d 1 represents appreciation rate and volatility, respectively. They are deterministic continuous functions of time u with values in R d 1 or R d 1 ×d 1 . Moreover, Σ is non-degenerate, i.e., there exists a positive constant κ such that ξ * Σ(u)ξ ≥ κ|ξ| 2 for any ξ ∈ R d 1 , u ∈ [ 0, T ]. And W is a d 1 -dimensional standard Brownian motion. In the following, we keep the previous notations. Denoted by π = (π 1 , · · ·, π d 1 ) * and c, the amount of money invested in the risky assets and the consumption rate, respectively, which are F W t −progressively measurable stochastic processes, and c is nonnegative. Let τ be the time of retirement from labor, which belongs to U t, T .
Suppose that the agent can gain labor income at a nonnegative constant rate ̺ until retirement. So, the agent's wealth process Y is governed by where 1 d 1 is the d 1 −dimensional column vector of 1's. We only consider the triple of control (τ, c, π) such that T t c s + π s 2 ds < ∞ subject to Y (t,y) which is called admissible policy. Denote by A(t, y) the set of all admissible policies. We assume the utility function U(c) be strictly concave and satisfies For example, U is the constant relative risk aversion (CRRA), i.e., Our aim is to maximize the following utility where l is a constant disutility (or a utility loss) due to labor, and (t, y) ∈ [ 0, T ]×(0, +∞). Concretely speaking, we find an optimal strategy (τ * , c * , π * ) ∈ A 1 (t, y) such that, J(t, y; τ * , c * , π * ) = V (t, y) sup J(t, y; τ, c, π) : (τ, c, π) ∈ A 1 (t, y) , where We have used the strong Markovian property of Y in the last second equality. On the other hand, for any n ∈ N + , there exist strategies (τ n , c n 1 , π n 1 ) ∈ A 1 (t, y) and (τ n , c n (τ n , Y t, y τ n ), π n (τ n , Y t, y τ n )) ∈ A 1 τ n (τ n , Y t, y τ n ) such that Define τ n = τ n , c n = c n 1 I {t≤τ n } + c n (τ n , Y t, y τ n )I {t>τ n } , π n = π n 1 I {t≤τ n } + π n (τ n , Y t, y τ n )I {t>τ n } . We can check that ( τ n , c n , π n ) ∈ A 1 (t, y) and J(t, y; τ n , c n , π n ) ≥ V * (t, y) + 2 n .
Hence, we have subject to (6.1), where V is the value function of the following stochastic control problem subject to (6.1).
Following the idea in [18,19], we use the martingale and dual method to transform the original problem into a standard stopping time problem.
Define the discount process D(s), the relative risk R(s), the exponential martingale M s and the state-price-density process S s as D(s) = e −r(s−t) , and We have used the fact S t = 1 in the first equality. Since the left-hand sides of the equalities are nonnegative and the right-hand sides are local martingales. Applying the optimal sampling theorem, we have Remark 6.2. According to [18,19], the equalities hold in (6.4) and (6.5) in some cases, and they are equivalent to (6.1) in some senses.
First, we analyze the property of V . For a Lagrange multiplier z ≥ 0, we transform the stochastic control problem (6.3) subject to (6.5) into sup (τ,c,π)∈A 1 t (t,y) E T t e −β(u−t) U(c u ) − z S u c u du + (0 − zS T Y T ) .
According to [18,Theorem 6.11], the above problem can be described as with K u = z e β(u−t) S u , where U 1 is the convex dual functions of the concave function U, i.e., U 1 (z) = sup y≥0 U(y) − zy , ∀ z ≥ 0.
Moreover, we have used the fact that the convex dual function of 0 is 0, and V is concave and nondecreasing. As in Example 6.1, we apply the logarithmic transformation to V and K u , and define X u = ln K u , x = ln z, G 1 (x) = U 1 (e x ), V 1 (t, x) = V 1 (t, e x ).
It is not difficult to check that X s is governed by and V 1 (t, x) = E T t e −β(u−t) G 1 (X u )du .
In view of Feynman-Kac formula, V 1 satisfies the following PDE: From the regularity of PDE and the comparison principle, it is not difficult to deduce that V 1 ∈ H 2, 2 2 , ∂ t V 1 ∈ H 0, 2 2 . Moreover, we can prove that V 1 ∈ W 2, 1 p, loc for any fixed p > 1, which means that V 1 , ∂ t V 1 , ∂ x V 1 , ∂ xx V 1 ∈ L p ([ 0, T ] × [ −m, m ]) for any m ∈ N + . Next, we use the similar method in [19] to transform problem (6.2) subject to (6.4) into a standard stopping time problem.
According to (6.2), for any z ≥ 0, we have where K u is same as the above. Similarly as in [19], we have with the equality holding if V (t, z) is differentiable with respect to z. So, it is sufficient to consider the optimal problem of V (t, z). As the above, we apply the following transformation and let X u = ln K u , x = ln z, G(x) = U (e x ), V (t, x) = V (t, e x ), V (t, x) = V (t, e x ), then V (t, x) is the value function of the following standard optimal stopping time problem, where the state equation is (6.6), and the payoff function is P (t, x; τ ) E τ t e −β(u−t) G(X t,x u )du + e −β(τ −t) V (τ, X t,x τ ) .
In view of Theorem 5.3, V satisfies the following PDVI: where L is defined in (6.7). Moreover, the above BSPDVI has a strong solution V ∈ H 2, 2 2 . Since, it is a deterministic PDVI, then V ∈ W 2, 1 2, loc . So, the above PDVI is written into the following: a.e. in [ 0, T ] × R.
According to the regularity theorem of PDE, the solution V ∈ W 2, 1 p, loc for any p > 1. Then the Sobolev embedding theorem implies that ∂ x V is continuous, and V (t, z) is differentiable in z. Hence, the value function V of the original problem is given by V (t, y) = inf z≥0 V (t, z) + z y − ̺ r e rt−rT − 1 .