Regularity of the Optimal Stopping Problem for Jump Diffusions
aa r X i v : . [ m a t h . O C ] M a r REGULARITY OF THE OPTIMAL STOPPING PROBLEM FOR JUMP DIFFUSIONS
ERHAN BAYRAKTAR AND HAO XING
Abstract.
The value function of an optimal stopping problem for jump diffusions is known to be a generalizedsolution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and amild assumption on the singularity of the L´evy measure, this paper shows that the value function of this optimalstopping problem on an unbounded domain with finite/infinite variation jumps is in W , p,loc with p ∈ (1 , ∞ ). As aconsequence, the smooth-fit property holds. Introduction
On a probability space (Ω , ( F t ) t ∈ R + , P ), consider a one-dimensional jump diffusion process X = { X t ; t ≥ } whose dynamics is governed by the following stochastic differential equation:(1.1) dX t = b ( X t , t ) dt + σ ( X t , t ) dW t + Z R h ( X t − , y, t ) (cid:0) N ( dt, dy ) − {| y |≤ } dt ν ( dy ) (cid:1) , in which W = { W t ; t ≥ } is a 1-dimensional Wiener process, N , independent of the Wiener process, is a Poissonrandom measure on R + × R with its mean measure dt × ν ( dy ), and ν is a L´evy measure on R . The coexistence ofdiffusions and infinite activity jumps is motivated by recent studies of A¨ıt-Sahalia and Jacod in [1] and [2].This paper studies the problem of maximizing the discounted terminal reward g by optimally stopping theprocess X before a fixed time horizon T . The value function of this problem is defined as(OS) u ( x, t ) = sup τ ∈T t,T E t,x h e − ( τ − t ) r ( X t ) g ( X τ ) i , in which T t,T is the set of all stopping times valued between t and T . A specific example of such an optimal stoppingproblem is the American option pricing problem, where X models the logarithm of the stock price process and g represents the pay-off function.The value function u is expected to satisfy a variational inequality with a nonlocal integral term (see e.g. Chapter3 of [7]). Different concepts of solutions were employed to characterize the value function: Pham used the notionof viscosity solution in [23]. Also see [3], [4] for more recent results in this direction. Lamberton and Mikou workedwith L´evy processes and showed in [19] that the value function can be understood in the distributional sense.When the diffusion component in X is nondegenerate, the value function is expected to have higher degreeof regularity. Sections 1-3 in Chapter 3 of [7] and [15] analyzed the Cauchy problems for second order partialintegro-differential equations and showed the existence and uniqueness of solutions in both Sobolev and H¨olderspaces. Also see [20]. The intuition is that the diffusions component dominates the contribution from jumps indetermining the regularity of solutions, no matter whether jumps have finite variation or not. However this intuitionis only a folklore theorem for obstacle problems. There are some limited results available whose assumptions on Key words and phrases.
Optimal stopping, variational inequality, L´evy processes, regularity of the value function, smooth fitprinciple, Sobolev spaces.The first author is supported in part by the National Science Foundation under an applied mathematics research grant and a Careergrant, DMS-0906257 and DMS-0955463, respectively, and in part by the Susan M. Smith Professorship. The second author is supportedin part by STICERD at London School of Economics. obstacles, domains, and the structure of the jumps may not be appropriate for financial applications. For example,Bensoussan and Lions analyzed an obstacle problem for jump diffusions where jumps may have finite/infiniteactivity with finite/infinite variation; see in Theorem 3.2 in [7] on pp. 234. However, their assumption on theobstacle may not be satisfied by option payoffs. In the mathematical finance literature, when irregular obstaclesare considered, the jumps are usually restricted to finite activity or infinite activity with finite variation cases.Zhang studied in [27] an obstacle problem for a jump diffusion with finite active jumps. Also see [22], [26], [5], and[6] for further developments. More recently, Davis et al. in [11], generalizing the results in [17] for the diffusioncase, analyzed an impulse control problems for jump diffusions with infinite activity but finite variation jumps.A regularity result which treats obstacle problems with irregular obstacles and infinite variation jumps has beenmissing in the literature.In this paper, we allow for infinite activity, infinite variation jumps. We show in Theorem 2.5 that the valuefunction of an obstacle problem solves a variational inequality for almost all points in the domain, and that it isan element in W , p,loc with p ∈ (1 , ∞ ) (see later this section for the definition of this Sobolev space). This regularityresult directly implies that the smooth fit property holds and the value function is C , inside the continuationregion. These results confirm the intuition that the nondegenerate diffusions components dominate any type ofL´evy jumps in determining the regularity of the value function for obstacle problems. We also develop a non-localversion of the interior Schauder estimate in Proposition 3.5, which could be useful to study other integro-differentialequations with irregular initial conditions.The remainder of the paper is organized as follows. After introducing notation at the end of this section, mainresults are presented in Section 2. Regularity properties of the infinitesimal generator of X are analyzed in Section 3.Then main results are proved in Section 4.1.1. Notation.
For a given open interval D = ( ℓ, r ) with −∞ ≤ ℓ < r ≤ ∞ , let us define the δ -neighborhoodof D as D δ := ( ℓ − δ, r + δ ) for δ >
0. We will also denote D s := D × (0 , s ), D δs := D δ × (0 , s ) for any s > E s := R × [0 , s ], and by A the closure of the indicated set A . Let us recall definitions of Sobolev spaces and H¨olderspaces in what follows; see [18] pp. 5-7 for further details. Definition 1.1. C , ( D s ) denotes the class of continuous functions on D s with continuous classical time andspatial derivatives up to the first and second order respectively.For any positive integer p ≥ W , p ( D s ) is the space of functions v ∈ L p ( D s ) with generalized derivatives ∂ t v , ∂ x v , ∂ xx v , and a finite norm k v k W , p ( D s ) := k ∂ t v k L p ( D s ) + k ∂ x v k L p ( D s ) + k ∂ xx v k L p ( D s ) . The space W , p, loc ( D s )consists of functions whose W , p -norm is finite on any compact subsets of D s .For any positive nonintegral real number α , H α,α/ (cid:0) D s (cid:1) is the space of functions v that are continuous in D s with continuous classical derivatives ∂ rt ∂ sx v for 2 r + s < α , and have finite norm k v k ( α ) D s := | v | ( α ) x + | v | ( α/ t + P r + s ≤ [ α ] k ∂ rt ∂ sx v k (0) , in which k v k (0) = max D s | v | , | v | ( α ) x = P r + s =[ α ] sup | x − x ′ |≤ ρ | ∂ rt ∂ sx v ( x,t ) − ∂ rt ∂ sx v ( x ′ ,t ) || x − x ′ | α − [ α ] , and | v | ( α/ t = P α − < r + s<α sup | t − t ′ |≤ ρ | ∂ rt ∂ sx v ( x,t ) − ∂ rt ∂ sx v ( x,t ′ ) || t − t ′ | ( α − r − s ) / , for a constant ρ . The space H α (cid:0) Ω (cid:1) is the H¨olderspace when only the spatial variable is considered.2. Main results
Model.
Let us first specify the jump diffusion X in (1.1). We assume that the drift and the volatility of X ,the discounting factor r , and the jump size h satisfy the following set of assumptions: Assumption 2.1.
Let a := σ . Coefficients a, b, r ∈ H ℓ, ℓ ( E T ) for some ℓ > r ( x, t ) ≥
0. Moreover, thereexist a strictly positive constant λ such that a ( x, t ) ≥ λ for all ( x, t ) ∈ E T . The jump size h ( x, y, t ) is continuously EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 3 differentiable in x and ∂ x h ( x, y, t ) is H¨older continuous in ( x, t ), moreover there exists a constant C such that(2.1) | h ( x, y, t ) | ≤ C | y | and | h ( x , y, t ) − h ( x , y, t ) | ≤ C | y | ( | x − x | + | t − t | ) , for ( x, t ) , ( x i , t i ) ∈ R × [0 , T ], i = 1 or 2, and y ∈ R .Without loss of generality, we will take C in (2.1) to be equal to one, otherwise we would rescale the process X .For the pure jump component in (1.1), we assume that ν is a L´evy measure on R . See [24] for this terminology.In particular, we require that R R ( y ∧ ν ( dy ) < ∞ . When h ≡ y , the jump component of (1.1) is a L´evy process.The aforementioned assumptions on coefficients and the jump component ensure that (1.1) admits a unique strongsolution (see [16]), which we denote by X . This jump diffusion process X is said to have finite activity , if ν is afinite measure on R , otherwise it is said to have infinite activity . We say that the jumps of X have finite variation ,if R R | y | ν ( dy ) < ∞ , otherwise we say that they have infinite variation .Among all possible L´evy measures, we consider the following large subclass in this paper: Assumption 2.2.
The L´evy measure satisfies R | y | > | y | ν ( dy ) < ∞ . Moreover it has a density, which we denoteby ρ , and this density satisfies ρ ( y ) ≤ M | y | α on | y | ≤
1, for some constants
M > α ∈ [0 , | y | ≤ Remark . Virtually all L´evy processes used in the financial modeling satisfy above assumption. For jumpdiffusions models, ν is a finite measure as in Merton’s and Kou’s model. For normal tempered stable processes, ρ has a power singularity 1 / | y | β at y = 0, with 0 ≤ β <
1; see (4.25) in [10]. In particular, this classcontains Variance Gamma and Normal Inverse Gaussian where β = 0 or 1 / ρ ( y ) = C − | y | α − e − λ − | y | { y< } + C + | y | α + e − λ + y { y> } , with α − , α + < λ − , λ + >
0. In particular, CGMY processes in [9] and regular L´evy processes of exponential type (RLPE) in [8]are special examples of this class.Having introduced the jump diffusion process X , let us discuss the problem (OS). We assume that the payofffunction g satisfies the following set of assumptions: Assumption 2.4.
The payoff function g is positive, bounded and Lipschitz continuous on R . That is, thereexists positive constants K and L such that 0 ≤ g ( x ) ≤ K for any x ∈ R and | g ( x ) − g ( y ) | ≤ L | x − y | forany x, y ∈ R . Moreover g satisfies ∂ xx g ≥ − J for some positive constant J in the distributional sense, i.e., R R g ( x ) ∂ xx φ ( x ) dx ≥ − J R R φ ( x ) dx for any compactly supported smooth function φ on R .A typical example, where these assumptions holds, is the American put option payoff g ( x ) = ( K − e x ) + for some K ∈ R + .For the problem (OS), we define its continuation region C and stopping region D as usual: C := { ( x, t ) ∈ R n × [0 , T ) : u ( x, t ) > g ( x ) } and D := { ( x, t ) ∈ R n × [0 , T ) : u ( x, t ) = g ( x ) } . Main regularity results.
Intuitively, one can expect from Itˆo’s formula that the value function u satisfiesthe following variational inequality :min { ( − ∂ t − L + r ) u, u − g } = 0 , ( x, t ) ∈ R × [0 , T ) ,u ( x, T ) = g ( x ) , x ∈ R . (2.2) ERHAN BAYRAKTAR AND HAO XING
Here, the integro-differential operator L is the infinitesimal generator of X . Its application on a smooth testfunction φ is(2.3) L φ := L D φ + Iφ, where L D φ ( x, t ) := a ( x, t ) ∂ xx + b ( x, t ) ∂ x and the integral term(2.4) Iφ ( x, t ) := Z R (cid:2) φ ( x + h ( x, y, t ) , t ) − φ ( x, t ) − h ( x, y, t ) ∂ x φ ( x, t ) 1 {| y |≤ } (cid:3) ν ( dy ) . In what follows we will not write down the arguments of h explicitly or only indicate the argument that we arefocusing in order to keep the notation simple.In general, one does not know a priori whether u is sufficiently smooth so that it solves (2.2) in the classical sense.Moreover, it is not even clear whether Iu is well defined in the classical sense. When φ ( · , t ) is Lipschitz continuouson R with a Lipschitz continuous derivative ∂ x φ ( · , t ) in a neighborhood of x , it can be shown that Iφ ( x, t ) is welldefined in the classical sense. Indeed, Iφ ( x, t ) = I ǫ φ ( x, t ) + I ǫ φ ( x, t ) < ∞ , where I ǫ φ ( x, t ) := Z | y | >ǫ [ φ ( x + h, t ) − φ ( x, t )] ν ( dy ) − ∂ x φ ( x, t ) Z ǫ< | y |≤ h ν ( dy ) ≤ C Z | y | >ǫ | y | ν ( dy ) + | ∂ x φ ( x, t ) | Z ǫ< | y |≤ | y | ν ( dy ) ,I ǫ φ ( x, t ) := Z | y |≤ ǫ [ φ ( x + h, t ) − φ ( x, t ) − h ∂ x φ ( x, t )] ν ( dy )(2.5) = Z | y |≤ ǫ h ( ∂ x φ ( z, t ) − ∂ x φ ( x, t )) ν ( dy ) ≤ C Z | y |≤ ǫ y ν ( dy ) . Here, the first inequality follows from the Lipschitz continuity of φ ( · , t ) and the assumption that | h | ≤ | y | ; the meanvalue theorem implies the second equality in (2.5) where z satisfies | z − x | < | h | ; the last inequality holds due tothe Lipschitz continuity of ∂ x φ ( · , t ) in an ǫ -neighborhood of x . However, the value function u , in general, doesnot have these regularity properties mentioned above. We only know from Lemma 3.1 in [23] that u is Lipschitzcontinuous in x and 1 / − H¨older continuous in t . Nevertheless, we will see that the integral term Iu is well definedin the classical sense in Lemma 3.2 below. In fact, more is true as we show in the next theorem, which is the mainresult of the paper. Theorem 2.5.
Let Assumptions 2.1, 2.2, and 2.4 hold. Then u ∈ W , p,loc ( R × (0 , T )) for any integer p ∈ (1 , ∞ ) .Moreover, u solves (2.2) for almost every point in E T . The following corollary is of special interest for the American option problem.
Corollary 2.6.
Under the assumptions of Theorem 2.5, (i) ∂ x u ∈ C ( R × [0 , T )) , i.e., the smooth-fit holds; (ii) u ∈ C , in the region where u > g .Remark . When jumps of X have finite variation, i.e., R R | y | ∧ ν ( dy ) < ∞ , the proof of the main result is muchsimpler. This is because, when jumps of X have finite variation, the infinitesimal generator L can be rewritten sothat its integral component has a reduced form. For any test function φ that is Lipschitz continuous in its firstvariable, L φ can be decomposed as L φ = L fD φ + I f φ , in which L fD φ = a ∂ xx φ + [ b − R | y |≤ h ν ( dy )] ∂ x φ and(2.6) I f φ ( x, t ) := Z R [ φ ( x + h, t ) − φ ( x, t )] ν ( dy ) . EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 5
The previous integral is clearly well defined. Indeed | I f φ ( x, t ) | ≤ C R R | y | ν ( dy ) < + ∞ follows from the Lipschitzcontinuity of φ ( · , t ) and | h | ≤ | y | . Moreover, I f φ is also H¨older continuous in its both variables; see Lemma 3.1below. Since the value function u is known to be Lipschitz continuous in its first variable (see Lemma 3.1 in [23]), I f u is already well defined and H¨older continuous. Therefore, in order to study the regularity of u , I f u can betreated as a driving term in (2.2). However, this simplification cannot be applied when jumps of X have infinitevariation, i.e., R R ( | y | ∧ ν ( dy ) = ∞ .3. Regularity properties of the integro-differential operator
The integral operator.
The integral operator I has two basic features. First, ν has a singularity at y = 0.As a result, I maps functions with certain degree of regularity to functions with less regularity. This is contrastto the case in which ν is a finite measure. In that case R R φ ( x + h, t ) ν ( dy ) is already well defined, for any φ withat most linear growth, and this integral has the same regularity as φ ; see [26]. Second, I is a nonlocal operator.Therefore, regularity of Iφ on a given interval D depends on φ outside D . In this subsection, we shall study thesetwo features in detail and analyze the regularity of Iφ when φ is either a function in certain H¨older or Sobolevspaces.Consider I as an operator between H¨older spaces. When jumps of X have finite variation, we can work with thereduced integral operator I f in (2.6). It has the following regularity property. Lemma 3.1.
Let Assumption 2.2 hold with ≤ α < and s > . For any φ which is Lipschitz continuous in itsfirst variable and / -H¨older continuous in its second variable, I f φ ∈ H − γ, − γ ( D s ) ∀ γ ∈ (0 , , when α = 0; I f φ ∈ H − α, − α ( D s ) , when < α < . However when jumps of X have infinite variation, the integral term I f φ is no longer well defined for Lipschitzcontinuous functions. Hence we work with L and its integral part I in the forms of (2.3) and (2.4). We will seethat if we choose an appropriate test function φ , Iφ is still well defined and H¨older continuous in both its variables.Regularity estimates of the following type have been obtained in [25] and [21]. Lemma 3.2.
Let Assumption 2.2 hold with α ∈ [1 , and s > . (i) Suppose that φ satisfies | φ ( x , t ) − φ ( x , t ) | ≤ L ( | x − x | + | t − t | ) for some L > and any ( x , t ) , ( x , t ) ∈ E s . If, moreover, φ ∈ H β, β ( D s ) for some β ∈ ( α, , then Iφ ∈ H β − α − γ, β − α − γ (cid:0) D s (cid:1) and (3.1) k Iφ k ( β − α − γ ) D s ≤ C (cid:16) L + k φ k ( β ) D s (cid:17) , for a positive constant C depending on D , α , and β . (ii) If φ ∈ H β, β ( E s ) for some β ∈ ( α, , then Iφ ∈ H β − α − γ, β − α − γ ( E s ) and (3.2) k Iφ k ( β − α − γ ) E s ≤ C k φ k ( β ) E s , for a positive constant C depending on α and β .Here γ = 0 when α ∈ (1 , ; γ is an arbitrary number in (0 , β − α ) when α = 1 . Since the proofs of Lemmas 3.1 and 3.2 are similar, we only present the proof of Lemma 3.2.
Proof of Lemma 3.2.
Statement (ii) is a special case of Statement (i) when the domain is taken to be R , insteadof D . In particular, k · k ( β ) E s ≥ L ; see Definition 1.1. It then suffices to prove statement (i). For notational simplicity, C represents a generic constant throughout the rest of proof. ERHAN BAYRAKTAR AND HAO XING
Step 1: Estimate max D s | Iφ | . For any ( x, t ) ∈ D s , | Iφ ( x, t ) | ≤ Z | y |≤ | φ ( x + h, t ) − φ ( x, t ) − h ∂ x φ ( x, t ) | ν ( dy ) + Z | y | > | φ ( x + h, t ) − φ ( x, t ) | ν ( dy ) ≤ Z | y |≤ | h | | ∂ x φ ( z, t ) − ∂ x φ ( x, t ) | ν ( dy ) + L Z | y | > | h | ν ( dy ) ≤ k φ k ( β ) D s Z | y |≤ | y | β ν ( dy ) + L Z | y | > | y | ν ( dy ) ≤ C (cid:16) L + k φ k ( β ) D s (cid:17) , where the second inequality follows from the mean value theorem with | z − x | ≤ | h | ≤ | y | ≤
1; the third inequalityis the result of the ( β − ∂ x φ on D s and | h | ≤ | y | ; the fourth inequality holds thanks toAssumption 2.2. Step 2: Show that Iφ is H¨older continuous in x . For x , x ∈ D and t ∈ [0 , s ], we break up | Iφ ( x , t ) − Iφ ( x , t ) | into three parts: | Iφ ( x , t ) − Iφ ( x , t ) | ≤ I + I + I , in which I ( x, t ) := Z | y |≤ ǫ [ | φ ( x + h ( x ) , t ) − φ ( x , t ) − h ( x ) ∂ x φ ( x , t ) | + | φ ( x + h ( x ) , t ) − φ ( x , t ) − h ( x ) ∂ x φ ( x , t ) | ] ν ( dy ) ,I ( x, t ) := Z ǫ< | y |≤ [ | φ ( x + h ( x ) , t ) − φ ( x , t ) − φ ( x + h ( x ) , t ) + φ ( x , t ) | + | h ( x ) ∂ x φ ( x , t ) − h ( x ) ∂ x φ ( x , t ) | ] ν ( dy ) ,I ( x, t ) := Z | y | > [ | φ ( x + h ( x ) , t ) − φ ( x + h ( x ) , t ) | + | φ ( x , t ) − φ ( x , t ) | ] ν ( dy ) , where variables y and t are ignored in h and the constant ǫ ≤ I ≤ k φ k ( β ) D s R | y |≤ ǫ | y | β ν ( dy ) = C k φ k ( β ) D s ǫ β − α . Second, it follows from the Lipschitz continuity of φ ( · , t ), ( β − ∂ x φ , and | h | ≤ | y | that | φ ( x + h ( x ) , t ) − φ ( x , t ) − φ ( x + h ( x ) , t ) + φ ( x , t ) |≤ | φ ( x + h ( x ) , t ) − φ ( x , t ) − φ ( x + h ( x ) , t ) + φ ( x , t ) | + | φ ( x + h ( x ) , t ) − φ ( x + h ( x ) , t ) |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z h ( x )0 | ∂ x φ ( x + z, t ) − ∂ x φ ( x + z, t ) | dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + C | x − x | | y |≤ k φ k ( β ) D s | x − x | β − | y | + C | x − x | | y | . Similarly, | h ( x ) ∂ x φ ( x , t ) − h ( x ) ∂ x φ ( x , t ) | ≤ k φ k ( β ) D s | y | ( | x − x | + | x − x | β − ). Therefore, I ≤ Z ǫ< | y |≤ C (cid:16) k φ k ( β ) D s (cid:17) | x − x | β − | y | ν ( dy ) ≤ C (cid:16) k φ k ( β ) D s (cid:17) | x − x | β − · ( ǫ − α − < α < − log ǫ when α = 1 , where the second inequality follows from Assumption 2.2. Third, it is clear from the Lipschitz continuity of φ and(2.1) that I ≤ C | x − x | R | y | > (1 + | y | ) ν ( dy ). EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 7
Now pick ǫ = | x − x | ∧
1. Since 1 ≤ α < β > α , we have ǫ β − α ≤ | x − x | β − α , ǫ − α − ≤ | x − x | − α , ǫ − α − ≤ | x − x | − α and − log ǫ ≤ α + γ − | x − x | − α − γ . All above estimates combined imply that | Iφ ( x , t ) − Iφ ( x , t ) | ≤ C (cid:16) k φ k ( β ) D s (cid:17) | x − x | β − α − γ , for a constant C independent of x , x , and t . Step 3: Show that Iφ is H¨older continuous in t . The proof is similar to that in Step 2. First we separate | Iφ ( x, t ) − Iφ ( x, t ) | into three parts as above. Then using | ∂ x φ ( x, t ) − ∂ x φ ( x, t ) | ≤ k φ k ( β ) D s | t − t | β − (seeDefinition 1.1), together with semi-H¨older continuity of h ( x, · ) in (2.1), and choosing ǫ = | t − t | ∧
1, we canobtain | Iφ ( x, t ) − Iφ ( x, t ) | ≤ C (cid:16) k φ k ( β ) D s (cid:17) | t − t | β − α − γ , for a constant C independent of x , t , and t . (cid:3) When I is considered as an operator between Sobolev spaces, it maps W , p − functions to L p − functions on asmaller domain. Lemma 3.3.
Let Assumption 2.2 hold. Consider a function φ ∈ W , p ( D × ( t , t )) such that φ is bounded and ∂ x φ is locally bounded on R × [ t , t ] . Then for any η > and α ∈ [0 , , (3.3) k Iφ k L p ( D × ( t ,t )) ≤ Cη − α k φ k W , p ( D η × ( t ,t )) + C (cid:18) max R × [ t ,t ] | φ | + max D × [ t ,t ] | ∂ x φ | (cid:19) · ( (1 + η − α ) , α = 1(1 − log η ) , α = 1 , for some constant C depending on D , t , and t .Remark . When X has finite variation jumps, i.e., 0 ≤ α < η in (3.3) can be chosen as zero. Hence L p − normof Iφ only depends on max R × [ t ,t ] | φ | and max D × [ t ,t ] | ∂ x φ | . Proof.
Utilizing truncation and smooth mollification, one can construct a sequence of smooth function ( φ ǫ ) ǫ> suchthat φ ǫ converges to φ in W , p and Iφ ǫ converges to Iφ in L p as ǫ → φ .To this end, observing that φ ( x + h, t ) − φ ( x, t ) − h∂ x φ ( x, t ) = h R (1 − z ) ∂ xx φ ( x + zh, t ) dz , the integral Iφ can be bounded above by three integral terms: | Iφ ( x, t ) | ≤ Z | y |≤ η h ν ( dy ) Z dz (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) + Z η< | y |≤ ν ( dy ) | φ ( x + h, t ) − φ ( x, t ) − h∂ x φ ( x, t ) | + Z | y | > ν ( dy ) | φ ( x + h, t ) − φ ( x, t ) | =: I + I + I . ERHAN BAYRAKTAR AND HAO XING
In the rest of proof, the L p -norm of each above term is estimated respectively. First, k I ( · , t ) k pL p ( D ) = Z D dx "Z | y |≤ η h ν ( dy ) Z dz (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) p ≤ Z D dx Z dz "Z | y |≤ η ν ( dy ) h (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) p ≤ C Z D dx Z dz "Z | y |≤ η dy | y | − α (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) p ≤ C Z D dx Z dz Z | y |≤ η dy | y | − α ! pq · Z | y |≤ η dy | y | − α (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) p ≤ C Z | y |≤ η dy | y | − α ! pq · Z | y |≤ η dy | y | − α Z dz Z D dx (cid:12)(cid:12) ∂ xx φ ( x + zh, t ) (cid:12)(cid:12) p ≤ C η (2 − α ) p (cid:13)(cid:13) ∂ xx φ ( · , t ) (cid:13)(cid:13) pL p ( D η ) , where the first inequality follows from Fubini’s theorem and Jensen’s inequality since p >
1; the second inequalityis a result of the assumption that | h | ≤ | y | and Assumption 2.2; the third inequality follows from H¨older inequalitywith 1 /p + 1 /q = 1; the fourth inequality utilizes Fubini’s theorem; and the fifth inequality holds since x + zh ∈ D η for any | h | ≤ | y | ≤ η and z ∈ [0 , x + h ∈ D for x ∈ D , and | h | ≤ | y | ≤
1, it follows that k I ( · , t ) k L p ( D ) ≤ C max D × [ t ,t ] | ∂ x φ | · Z η ≤| y |≤ | y | ν ( dy ) ≤ C max D × [ t ,t ] | ∂ x φ | · ( (1 + η − α ) , α = 1(1 − log η ) , α = 1Third, it is clear that k I φ ( · , t ) k L p ( D ) ≤ C · max R × [ t ,t ] | φ | , since φ is bounded.Now, recall k Iφ k L p ( D × ( t ,t )) := hR t t k Iφ ( · , t ) k L p ( D ) dt i p . The statement then follows from above L p -normestimates on I k , k = 1 , , (cid:3) An interior estimate.
The L p − norm estimate of the integral term in Lemma 3.3 helps to derive the following W , p − norm estimate for solutions of the Cauchy problem below. This estimate is a nonlocal version of the parabolicCalderon-Zygmund estimate (c.f. Theorem 9.1 in [18] pp.341). Proposition 3.5.
Suppose that Assumptions 2.1 and 2.2 are satisfied. Let v be a W , p,loc − solution of the followingCauchy problem: ( ∂ t − L D − I + r ) v = f ( x, t ) , ( x, t ) ∈ R × (0 , T ] ,v ( x,
0) = g ( x ) , x ∈ R , where f ∈ L p,loc ( E T ) . If v is bounded and ∂ x v is locally bounded on E T , then for any s ∈ (0 , T ) , there exist δ ∈ (0 , s ) and C δ , depending on δ , such that (3.4) k v k W , p ( D × ( s,T )) ≤ C δ (cid:20) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | + k f k L p ( D δ/ × ( δ ,T )) (cid:21) . Remark . The main idea of the following proof is to treat Iv as a driving term and utilize the classical Calderon-Zygmund estimate for local PDEs. However, as we have seen in Lemma 3.3, W , p − norm of v controls L p − norm of Iv , which in turn bounds the W , p − norm of v via the Calderon-Zygmund estimate. Therefore, a careful balancebetween extending domains and controlling W , p − norm of v needs to be maintained in the following proof. Thisis contrast to the case where only finite variation jumps are considered. As we have seen in Remark 3.4, max | ∂ x v | EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 9 and max | v | control the L p − norm of Iv which bounds the W , p -norm of v . Hence, in the case of finite variationjumps, (3.4) can be obtained directly from the classical Calderon-Zygmund estimate for local PDEs. Proof.
The constant C denotes a generic constant throughout this proof. Domains used in this proof are displayedin Figure 1.For a constant δ ∈ (0 , s ) which will be determined later, let us choose a cut-off function ζ δ such that 0 ≤ ζ δ ≤ ζ δ = 1 inside D × ( δ, T ) and ζ δ = 0 outside D δ/ × ( δ/ , T ). Moreover ζ δ can be chosen to satisfy(3.5) (cid:12)(cid:12) ∂ x ζ δ (cid:12)(cid:12) ≤ Cδ , (cid:12)(cid:12) ∂ xx ζ δ (cid:12)(cid:12) ≤ Cδ , and (cid:12)(cid:12) ∂ t ζ δ (cid:12)(cid:12) ≤ Cδ .
The function w := ζ δ v satisfies( ∂ t − L D + r ) w = ζ δ Iv ( x, t ) + ζ δ f ( x, t ) + h ( x, t ) , ( x, t ) ∈ D δ/ × (0 , T ) ,w ( x, t ) = 0 , ( x, t ) ∈ ∂D δ/ × [0 , T ) ,w ( x,
0) = 0 , x ∈ D δ/ , in which h := ∂ t ζ δ v − a (cid:0) ∂ xx ζ δ v + 2 ∂ x ζ δ ∂ x v (cid:1) − b ∂ x ζ δ v . Appealing to Theorem 9.1 in [18] pp.341, we can find aconstant C such that k w k W , p ( D δ/ × (0 ,T )) ≤ C h(cid:13)(cid:13) ζ δ Iv (cid:13)(cid:13) L p + (cid:13)(cid:13) ζ δ f (cid:13)(cid:13) L p + k h k L p i , (3.6)where all L p -norms on the right-hand-side are taken on D δ/ × (0 , T ).In what follows, we will estimate the terms on the right-hand-side of (3.6) respectively. First, when α = 1, (cid:13)(cid:13) ζ δ Iv (cid:13)(cid:13) L p ( D δ/ × (0 ,T )) ≤ k Iv k L p ( D δ/ × ( δ ,T )) ≤ C (cid:18) δ (cid:19) − α k v k W , p ( D δ/ × ( δ ,T )) + C (cid:18) δ (cid:19) − α ! (cid:20) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | (cid:21) , where the first inequality follows from the choice of ζ δ ; the second inequality follows from Lemma 3.3 with η = δ/ t = δ/
2, and t = T . When α = 1, a similar estimate can be obtained. In that case, the rest of proof issimilar to that for α = 1 case, hence we only present the proof for α = 1 henceforth. Second, it is clear that Figure 1.
Domains used in this proof δ δ T B ρ B ρ + δ B ρ + δ (cid:13)(cid:13) ζ δ f (cid:13)(cid:13) L p ( D δ/ × (0 ,T )) ≤ k f k L p ( D δ/ × ( δ ,T )) . Third, we will estimate the L p − norm of h . To this end, let us derive abound for k ∂ t ζ δ v k L p ( D δ × (0 ,T )) in what follows. It follows from (3.5) that (cid:13)(cid:13) ∂ t ζ δ v (cid:13)(cid:13) L p ( D δ × (0 ,T )) ≤ C max E T | v | δ − Area (cid:16) D δ/ × ( δ/ , T ) \ D × ( δ, T ) (cid:17) p ≤ C max E T | v | δ − pp , where Area ( · ) is the Lebesgue measure. Estimates on other terms of h can be performed similarly to obtain k h k L p ( D δ/ × (0 ,T )) ≤ C (cid:16) δ − pp + δ − pp (cid:17) (cid:18) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | (cid:19) . Utilizing above estimates on the right-hand-side of (3.6), we obtain k v k W , p ( D × ( δ,T )) ≤ k w k W , p ( D δ/ × (0 ,T )) ≤ C (cid:18) δ (cid:19) − α k v k W , p ( D δ/ × ( δ ,T )) + C (cid:16) δ − α + δ − pp + δ − pp (cid:17) (cid:18) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | (cid:19) + k f k L p ( D δ/ × ( δ ,T )) . Multiplying δ on both hand sides of the previous inequality, δ k v k W , p ( D × ( δ,T )) ≤ C (cid:18) δ (cid:19) − α (cid:18) δ (cid:19) k v k W , p ( D δ/ × ( δ ,T )) + K ( δ ) , where K ( δ ) = C (cid:16) δ + δ − α + δ pp + δ p (cid:17) (cid:0) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | (cid:1) + δ k f k L p ( D δ/ × ( δ ,T )) . Denote F ( τ ) := τ k v k W , p ( D δ − τ × ( τ,T )) . The previous inequality gives the following recursive inequality F ( δ ) ≤ C (cid:18) δ (cid:19) − α F (cid:18) δ (cid:19) + K ( δ ) . Now choosing a sufficiently small δ ∈ (0 , s ) such that 4 C ( δ/ − α ≤ , we obtain from the above inequality that F ( δ ) ≤ F (cid:18) δ (cid:19) + K ( δ ) . Note that F ( δ ) is finite for any δ , since the W , p − norm of v is finite in any compact domain of R × (0 , T ), and K ( δ ) is increasing in δ . We then obtain from iterating the previous inequality that F ( δ ) ≤ ∞ X i =0 i K (cid:18) δ i (cid:19) ≤ ∞ X i =0 i K ( δ ) = 2 K ( δ ) . In terms of W , p,loc − norms, the previous inequality reads k v k W , p ( D × ( s,T )) ≤ C h δ − α + δ − pp + δ − pp i (cid:20) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | (cid:21) + 2 k f k L p ( D δ/ × ( δ ,T )) ≤ C δ (cid:20) max E T | v | + max D δ/ × [0 ,T ] | ∂ x v | + k f k L p ( D δ/ × ( δ ,T )) (cid:21) . (cid:3) EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 11 Proof of main results
The penalty method.
We use the penalty method (see e.g. [14] and [26]) to analyze the following variationalinequality: min { ( ∂ t − L D − I + r ) v, v − g } = 0 , ( x, t ) ∈ R × (0 , T ] ,v ( x,
0) = g ( x ) , x ∈ R . (4.1)The nonlocal integral term introduces several technical difficulties in applying the penalty method. In this section,we will focus on the case where X has infinite variation jumps, i.e., Assumption 2.2 holds with 1 ≤ α <
2. When X has finite variation jumps, i.e., 0 ≤ α <
1, the integral operator has the reduced form I f in (2.6), see Remark2.7. Then all proofs are similar but easier than those in the infinite variation case.For each ǫ ∈ (0 , ∂ t − L D − I + r ) v ǫ + p ǫ ( v ǫ − g ǫ ) = 0 , ( x, t ) ∈ R × (0 , T ] ,v ǫ ( x,
0) = g ǫ ( x ) , x ∈ R , (4.2)Here { g ǫ } ǫ ∈ (0 , is a mollified sequence of g such that ∂ xx g ǫ ( x ) ≥ − J , 0 ≤ g ≤ K , and | ( g ǫ ) ′ ( x ) | ≤ L for any x ∈ R ;see [14] pp.27 for its construction. The mollified sequence can be chosen such that constants J, K , and L , appearingin Assumption 2.4, are independent of ǫ . The penalty term p ǫ ( y ) ∈ C ∞ ( R ) is chosen to satisfy following properties:( i ) p ǫ ( y ) ≤ , ( ii ) p ǫ ( y ) = 0 for y ≥ ǫ, ( iii ) p ǫ (0) = − a (0) J − | b | (0) L − r (0) K − J Z | y |≤ | y | ν ( dy ) − K Z | y | > ν ( dy ) , ( iv ) p ′ ǫ ( y ) ≥ , ( v ) p ′′ ǫ ( y ) ≤ , and ( vi ) lim ǫ ↓ p ǫ ( y ) = ( , y > −∞ , y < , (4.3)where a (0) = max E T a , | b | (0) = max E T | b | , and r (0) = max E T r are finite thanks to Assumption 2.1. Indeed, p ǫ canbe chosen as a smooth mollification of the function min {− p ǫ (0) x/ǫ + p ǫ (0) , } .Now we show that each penalty problem (4.2) has a classical solution. To this end, let us first recall the Schauderfixed point theorem (see Theorem 2 in [13] pp. 189). Lemma 4.1.
Let Θ be a closed convex subset of a Banach space and let T be a continuous operator on Θ suchthat T Θ is contained in Θ and T Θ is precompact. Then T has a fixed point in Θ . Lemma 4.2.
Let Assumptions 2.1, 2.2 with ≤ α < , and 2.4 hold. Then for any ǫ ∈ (0 , and β ∈ ( α, , (4.2) has a solution v ǫ ∈ H β − α − γ, β − α − γ ( E T ) . Here γ = 0 when < α < ; γ is an arbitrary number in (0 , β − α ) when α = 1 .Proof. We will first prove the existence on a sufficiently small time interval [0 , s ] via the Schauder fixed pointtheorem, then extend this solution to the interval [0 , T ].Let us consider the set Θ := n v ∈ H β, β ( E s ) with its H¨older norm k v k ( β ) E s ≤ U o , where s and U will be deter-mined later. It is clear that Θ is a bounded, closed and convex set in the Banach space H β, β ( E s ). For any v ∈ Θ,consider the following Cauchy problem for u − g ǫ :( ∂ t − L D + r ) ( u − g ǫ ) = Iv − p ǫ ( v − g ǫ ) + ( L D − r ) g ǫ , ( x, t ) ∈ R × (0 , s ] ,u ( x, − g ǫ ( x ) = 0 , x ∈ R . (4.4)We define an operator T via u = T v using the solution u of (4.4). Let us check the conditions for the Schauderfixed point theorem are satisfied in the following four steps: Step 1:
T v is well defined.
Since v ∈ H β, β ( E s ) with β ∈ ( α, Iv ∈ H β − α − γ, β − α − γ ( E s ) with k Iv k ( β − α − γ ) E s ≤ C k v k ( β ) E s . On the other hand, using properties of v , g ǫ and p ǫ , one cancheck that − p ǫ ( v − g ǫ ) + ( L D − r ) g ǫ ∈ H β − α − γ, β − α − γ ( E s ). Therefore, Theorem 5.1 in [18] pp. 320 implies that(4.4) has a unique solution u − g ǫ ∈ H β − α − γ, β − α − γ ( E s ). Hence u = T v ∈ H β − α − γ, β − α − γ ( E s ), since g ǫ issmooth. Step 2. T Θ ⊂ Θ . It follows from Lemma 2 in [13] pp. 193 that there exists a positive constant A β , dependingon β , such that k u − g ǫ k ( β ) E s ≤ A β s ξ h k Iv k (0) E s + k p ǫ ( v − g ǫ ) k (0) E s + k ( L D − r ) g ǫ k (0) E s i ≤ A β Cs ξ k v k ( β ) E s + e A, (4.5)where ξ = − β , C is the constant in Step 1, and e A is a sufficiently large constant. Let s be such that τ := A β Cs ξ < / U := max { e A − τ , k g ǫ k ( β ) E s } . Since k v k ( β ) E s ≤ U , it then follows from (4.5) that(4.6) k u k ( β ) E s ≤ k u − g ǫ k ( β ) E s + k g ǫ k ( β ) E s ≤ τ U + e A + U ≤ τ U + 1 − τ U + U U . This confirms that u = T v ∈ Θ. Step 3. T Θ is a precompact subset of H β, β ( E s ) . For any η ∈ ( β, v ∈ Θ, k T v k ( η ) E s ≤ U for some constant U depending on U and s . Since bounded subsets of H η, η ( E s ) areprecompact subsets of H β, β ( E s ) (see Theorem 1 in [13] pp.188), then T Θ is a precompact subset in H β, β ( E s ). Step 4. T is a continuous operator. Let v n be a sequence in Θ such that lim n →∞ k v n − v k ( β ) E s = 0, we will showlim n →∞ k T v n − T v k ( β ) E s = 0. From (4.4), w , T v n − T v satisfies the Cauchy problem( ∂ t − L D + r ) w = I ( v n − v ) − [ p ǫ ( v n − g ǫ ) − p ǫ ( v − g ǫ )] , ( x, t ) ∈ R × (0 , s ] ,w ( x,
0) = 0 , x ∈ R . It follows again from Lemma 2 in [13] pp. 193 that kT v n − T v k ( β ) E s = k w k ( β ) E s ≤ A β s γ h k I ( v n − v ) k (0) E s + k p ǫ ( v n − g ǫ ) − p ǫ ( v − g ǫ ) k (0) E s i ≤ A β s γ (cid:20) C k v n − v k ( β ) E s + max E s ,n (cid:12)(cid:12)(cid:12) p ′ ǫ ( v n − g ǫ ) (cid:12)(cid:12)(cid:12) k v n − v k (0) E s (cid:21) → n → ∞ . Now all conditions of the Schauder fixed point theorem are checked, hence T has a fixed point in H β, β ( E s ),which is denoted by v ǫ . Moreover, it follows from results in Step 1 that v ǫ = T v ǫ ∈ H β − α − γ, β − α − γ ( E s ).Finally, let us extend v ǫ to the interval [0 , T ]. We can replace g ǫ ( · ) by v ǫ ( · , s ) in (4.4), since k v ǫ ( · , s ) k (2+ β − α − γ ) R isfinite thanks to the result after Step 4 and because the choice of s in Step 2 only depends on β and C . If we choosea sufficiently large U , depending on k v ǫ ( · , s ) k (2+ β − α − γ ) R , such that (4.6) holds on [ s, s ], then k v ǫ ( · , s ) k (2+ β − α − γ ) R is finite thanks to the argument after Step 4. Now one can repeat this procedure to extend the time interval by s each time, until it contains [0 , T ]. (cid:3) After the existence of classical solutions for (4.2) is established, we will study properties of the sequence ( v ǫ ) ǫ ∈ (0 , in the rest of this subsection. The following maximum principle is a handy tool in our analysis. Lemma 4.3.
Suppose that a > , a and b are bounded and the Levy measure ν satisfies R | y | > | y | ν ( dy ) < ∞ .Assume also that we are given a function c bounded from below on E T . If v ∈ C ( E T ) ∩ C , ( E T ) satisfies ( ∂ t − L D − I + c ) v ( x, t ) ≥ and v is bounded from below on E T , then v ( x, ≥ for x ∈ R implies that v ≥ on E T . EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 13
Proof.
Let v ≥ − m and c ≥ − C on E T for some positive constants m and C . For any positive R , consider thefollowing function: w ( x, t ) := mf ( R ) ( f ( | x | ) + C t ) e C t , ( x, t ) ∈ E T , where C will be determined later and f : R + → R + is an increasing C function such that f = 0 in a neighborhoodof 0 and f ( R ) = R R for sufficiently large R . It is clear that lim R → + ∞ f ( R ) = ∞ and derivatives f ′ and f ′′ arebounded. Then If ( | x | ) is bounded on R . Indeed, there exists a constant C such that (cid:12)(cid:12) If ( | x | ) (cid:12)(cid:12) ≤ Z | y |≤ ν ( dy ) Z dz (1 − z ) h (cid:12)(cid:12) ∂ xx f ( | x + zh | ) (cid:12)(cid:12) + Z | y | > ν ( dy ) | f ( | x + h | ) − f ( | x | ) |≤ C Z | y |≤ y ν ( dy ) + Z | y | > | y | ν ( dy ) ! < + ∞ . Combining above estimate with c + C ≥
0, one can find a sufficient large constant C such that( ∂ t −L D − I + c ) w = e C t mf ( R ) (cid:2) C + ( c + C )( f ( | x | ) + C t ) − a ∂ xx f ( | x | ) − b ∂ x f ( | x | ) − If ( | x | ) (cid:3) > C m, on E T . Now define ˜ v := v + w . The previous estimate gives(4.7) ( ∂ t − L D − I + c + C )˜ v > C v + C m ≥ , for any ( x, t ) ∈ E T . On the other hand, ˜ v ( x,
0) = mf ( R ) f ( | x | ) + v ( x, ≥ v ( x, ≥
0, moreover ˜ v ( x, t ) ≥ m + v ( x, t ) ≥ | x | ≥ R because f is increasing and v ≥ − m . Therefore, we claim that ˜ v ≥ x, t ) ∈ [ − R , R ] × [0 , T ]. Indeed,if there exists ( x, t ) ∈ [ − R , R ] × (0 , T ] such that ˜ v ( x, t ) <
0, ˜ v must take its negative minimum at some point( x , t ) ∈ [ − R , R ] × (0 , T ]. Note that this is also a global minimum for ˜ v on E T , hence I ˜ v ( x , t ) ≥ ∂ t ˜ v ( x , t ) ≤ ∂ x ˜ v ( x , t ) = 0, ∂ xx ˜ v ( x , t ) ≥
0, and ( c + C )˜ v ( x , t ) ≤
0. As a result, ( ∂ t − L D − I + c + C )˜ v ( x , t ) ≤ , which contradicts with (4.7). Now for fixed point ( x, t ), the statement follows from sending the constant R in ˜ v to ∞ . (cid:3) This maximum principle implies the uniqueness of classical solutions for the penalty problem (4.2).
Corollary 4.4.
Under assumptions of Lemma 4.2, v ǫ is the unique bounded classical solution of (4.2) .Proof. Lemma 4.2 and the definition of H¨older spaces combined imply that v := v ǫ is a bounded classical solution.Now suppose there exists another solution v , then v − v satisfies( ∂ t − L D − I + r ) ( v − v ) + p ǫ ( v − g ǫ ) − p ǫ ( v − g ǫ ) = 0 , ( x, t ) ∈ R × (0 , T ] , ( v − v )( x,
0) = 0 , x ∈ R . It follows from the mean value theorem that p ǫ ( v − g ǫ ) − p ǫ ( v − g ǫ ) = p ′ ǫ ( y )( v − v ) for some y ∈ R , where p ′ ǫ ( y ) ≥ c = r + p ′ ǫ ( y ) that v ≥ v on R × (0 , T ]. Thesame argument applied to v − v gives the reverse inequality. (cid:3) Utilizing the maximum principle, we will analyze properties of the sequence ( v ǫ ) ǫ ∈ (0 , in the following results. Lemma 4.5.
Let Assumptions 2.1, 2.2 with ≤ α < , and 2.4 hold. Then for any ǫ ∈ (0 , , ≤ v ǫ ≤ K + 1 on E T . Proof.
It follows from Lemma 4.2 that v ǫ is bounded on E T for each ǫ ∈ (0 , ǫ . First, it follows from (4.3) part (i) that ( ∂ t − L D − I + r ) v ǫ = − p ǫ ( v ǫ − g ǫ ) ≥ v ǫ ( x,
0) = g ǫ ( x ) ≥ x ∈ R . Then first inequality in the statement follows from Lemma 4.3 directly.Second, consider w = K + 1 − v ǫ , it satisfies( ∂ t − L D − I + r ) w = r ( K + 1) + p ǫ ( v ǫ − g ǫ ) , ( x, t ) ∈ R × (0 , T ] . Combining (4.3) part (ii) and g ǫ ≤ K , we have p ǫ ( K + 1 − g ǫ ) = 0. Hence,(4.8) ( ∂ t − L D − I + r ) w + p ǫ ( K + 1 − g ǫ ) − p ǫ ( v ǫ − g ǫ ) = h ∂ t − L D − I + r + p ′ ǫ ( y ) i w = r ( K + 1) ≥ , where the first equality follows from the mean value theorem. Now applying Lemma 4.3 to above equation with c = r + p ′ ( y ) ≥ w ( x, t ) = K + 1 − v ǫ ( x, t ) ≥ E T for any ǫ ∈ (0 , (cid:3) Lemma 4.6.
Let Assumptions 2.1, 2.2 with ≤ α < , and 2.4 hold. Then for any ǫ ∈ (0 , , | ∂ x v ǫ | ≤ C on E T , in which C depends on T and L .Proof. Formally differentiating (4.2) with respect to x gives the following equation: " ∂ t − a∂ xx − ( b + ∂ x a ) ∂ x − ˆ I + r − ∂ x b − Z | y | > ∂ x h ν ( dy ) ! w + v ǫ ∂ x r + p ′ ǫ ( v ǫ − g ǫ ) (cid:16) w − ( g ǫ ) ′ (cid:17) = 0 , ( x, t ) ∈ R × (0 , T ] ,w ( x,
0) = ( g ǫ ) ′ ( x ) , x ∈ R . (4.9)Here ˆ Iφ := Iφ + R R [ φ ( x + h, t ) − φ ( x, t )] ∂ x h ν ( dy ), where the second integral is well defined for Lipschitz boundedfunction φ because | ∂ x h | ≤ | y | from (2.1) and R | y | > | y | ν ( dy ) < ∞ from Assumption 2.2. We will show that ∂ x v ǫ isindeed a classical solution of (4.9). To this end, let us consider the equation " ∂ t − a∂ xx − ( b + ∂ x a ) ∂ x − ˆ I + r − ∂ x b − Z | y | > ∂ x h ν ( dy ) ! w = − ∂ x r v ǫ − p ′ ǫ ( v ǫ − g ǫ ) (cid:16) ∂ x v ǫ − ( g ǫ ) ′ (cid:17) , ( x, t ) ∈ R × (0 , T ] ,w ( x,
0) = ( g ǫ ) ′ ( x ) , x ∈ R . Using Assumption 2.1 and Lemma 4.2, one can check that the driving term − ∂ x r v ǫ − p ′ ǫ ( v ǫ − g ǫ ) (cid:16) ∂ x v ǫ − ( g ǫ ) ′ (cid:17) and all coefficients of the previous equation are H¨older continuous. It then follows from Theorem 3.1 in [15] onpp. 89 that the last equation has a classical solution, say w . Define v ( x, t ) := R x w ( z, t ) dz + v ǫ (0 , t ). It is straightforward to check that v is a classical solution of the following equation( ∂ t − L D − I + r ) v = − p ǫ ( v ǫ − g ǫ ) , ( x, t ) ∈ R × (0 , T ] ,v ( x,
0) = g ǫ ( x ) , x ∈ R . Since g ǫ and v ǫ are both bounded, then − p ǫ ( v ǫ − g ǫ ) is also bounded. As a result, estimate (3.6) in Theorem 3.1of [15] on pp. 89 implies that v is bounded solution of the last equation. However, Corollary 4.4 already showsthat v ǫ is the unique bounded solution of the last solution, therefore v = v ǫ , hence ∂ x v ǫ = w on E T and ∂ x v ǫ is aclassical solution of (4.9). EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 15
Now we shall show ∂ x v ǫ is bounded uniformly in ǫ . Consider ˜ v = e γt L + ∂ x v ǫ , where L is given by Assumption 2.4and γ > v satisfies the following equation " ∂ t − a∂ xx − ( b + ∂ x a ) ∂ x − ˆ I + r − ∂ x b − Z | y | > ∂ x h ν ( dy ) + p ′ ǫ ( v ǫ − g ǫ ) ˜ v = γ + r − ∂ x b − Z | y | > ∂ x h ν ( dy ) ! e γt L − ∂ x r v ǫ + p ′ ǫ ( v ǫ − g ǫ ) (cid:16) e γt L + ( g ǫ ) ′ (cid:17) , ( x, t ) ∈ R × (0 , T ] , ˜ v ( x,
0) = e γt L + ( g ǫ ) ′ ( x ) , x ∈ R . (4.10)Recall that ∂ x b and ∂ x r are bounded from Assumption 2.1. Observe that R | y | > ∂ x h ν ( dy ) < ∞ because | ∂ x h | ≤ | y | and R | y | > | y | ν ( dy ) < ∞ . Moreover, v ǫ is bounded uniformly in ǫ thanks to Lemma 4.5. Therefore, one can find asufficiently large γ , independent of ǫ , such that (cid:16) γ + r − ∂ x b − R | y | > ∂ x h ν ( dy ) (cid:17) e γt L − ∂ x r v ǫ >
0. On the otherhand, p ′ ǫ ( v ǫ − g ǫ ) (cid:16) e γt L + ( g ǫ ) ′ (cid:17) is also positive due to (4.3)-(iv) and | ( g ǫ ) ′ | ≤ L . As a result, the right-hand-sideof (4.10) is positive. Now since r − ∂ x b − R | y | > ∂ x h ν ( dy ) + p ′ ǫ ( v ǫ − g ǫ ) is bounded from below, moreover the L´evymeasure (1 + ∂ x h ) ν ( dy ) associated to ˆ I satisfies R | y | > | y | (1 + | ∂ x h | ) ν ( dy ) ≤ R | y | > ( | y | + | y | ) ν ( dy ) < ∞ (see (2.1)and Assumption 2.2), we then have from Lemma 4.3 with I = ˆ I that ˜ v ≥ E T . Hence ∂ x v ǫ ≥ − e γT L on E T , forsome positive γ independent of ǫ . The upper bound can be shown similarly by working with ˜ v = e γt L − ∂ x v ǫ . (cid:3) Lemma 4.7.
Let Assumptions 2.1, 2.2 with ≤ α < , and 2.4 hold. Then for any ǫ ∈ (0 , , v ǫ ≥ g ǫ on E T . Proof.
Let us first show that Ig ǫ is uniformly bounded from below. Indeed, Ig ǫ ( x ) = Z | y |≤ ν ( dy ) Z dz (1 − z ) h ∂ xx g ǫ ( x + zh ) + Z | y | > [ g ǫ ( x + h ) − g ǫ ( x )] ν ( dy ) ≥ − J Z | y |≤ | y | ν ( dy ) − K Z | y | > ν ( dy ) , where the inequality follows from ∂ xx g ǫ ≥ − J and 0 ≤ g ǫ ≤ K . As a result, ( ∂ t − L D − I + r ) g ǫ is bounded fromabove. This is because( ∂ t − L D − I + r ) g ǫ ( x ) = − a ( x, t ) ∂ xx g ǫ ( x ) − b ( x, t ) ∂ x g ǫ ( x ) + r ( x, t ) g ǫ ( x ) − Ig ǫ ( x ) ≤ a (0) J + | b | (0) L + r (0) K + J Z | y |≤ | y | ν ( dy ) + K Z | y | > ν ( dy )= − p ǫ (0) , where the second equality follows from (4.3) part (iii). Therefore,( ∂ t − L D − I + r ) ( v ǫ − g ǫ ) = − p ǫ ( v ǫ − g ǫ ) − ( ∂ t − L D − I + r ) g ǫ ≥ − p ǫ ( v ǫ − g ǫ ) + p ǫ (0) . The previous inequality and the mean value theorem combined imply that (cid:16) ∂ t − L D − I + r + p ′ ǫ ( y ) (cid:17) ( v ǫ − g ǫ ) ≥ , for some y ∈ R . Hence the statement of the lemma follows applying Lemma 4.3 to the previous inequality andchoosing c = r + p ′ ǫ ( y ) ≥ (cid:3) Corollary 4.8.
Let assumptions of Lemma 4.7 hold. Then p ǫ ( v ǫ − g ǫ ) is bounded uniformly in ǫ ∈ (0 , . Proof.
Lemma 4.7 and (4.3)-(i)&(iv) together imply that p ǫ (0) ≤ p ǫ ( v ǫ − g ǫ ) ≤ . Then the statement follows since p ǫ (0) is independent of ǫ ; see (4.3) part (iii). (cid:3) Proof of Theorem 2.5 and Corollary 2.6.
Proof of Theorem 2.5.
The proof consists of two steps. First, we show that there exists a function v ∗ which solves(4.1) and v ∗ ∈ W , p ( B × ( s, T )) for any integer p ∈ (1 , ∞ ), compact domain B ⊂ R , and s ∈ [0 , T ). Second, weconfirm that u ∗ ( x, t ) := v ∗ ( x, T − t ) is the value function for the problem (OS). Step 1:
First, it follows from Lemma 4.2 that ∂ t v ǫ , ∂ x v ǫ , and ∂ xx v ǫ are continuous, hence locally bounded on R × (0 , T ). Therefore v ǫ ∈ W , p,loc ( R × (0 , T )) for each ǫ ∈ (0 , v ǫ and ∂ x v ǫ are bounded on E T , uniformly in ǫ . Moreover, the penalty term p ǫ ( v ǫ − g ǫ ) is also bounded uniformly in ǫ due to Corollary 4.8. Therefore these boundedness properties and Proposition 3.5 with f = − p ǫ ( v ǫ − g ǫ ) togetherimply that(4.11) k v ǫ k W , p ( B × ( s,T )) ≤ C, for some constant C independent of ǫ. Thanks to the weak compactness of the Sobolev space W , p , 1 < p < ∞ , we can then find a subsequence ( ǫ k ) k ≥ converging to zero and a function v ∗ , such that v ǫ k ⇀ v ∗ ∈ W , p ( B × ( s, T )). Here “ ⇀ ” represents the weakconvergence; c.f. Appendix D.4. in [12] pp. 639. In fact this convergence can be shown to be pointwise and uniformin the index. Indeed, (4.11) and the Sobolev embedding theorem (c.f. Lemma 3.3 in [18] pp. 80) combined implythat k v ǫ k ( β ) B × [ s,T ] ≤ C, where β = 2 − p and C is some constant independent of ǫ. Choosing p > β > ǫ k ) k ≥ , which is still denoted by ( ǫ k ) k ≥ , such that ( v ǫ k ) k ≥ converge to v ∗ uniformly on B × [ s, T ]. Since each v ǫ k is continuous, v ∗ is also continuous on B × [ s, T ].Let us show that v ∗ solves (4.1). On the one hand, since p ǫ ( v ǫ k − g ǫ k ) ≤
0, we have ( ∂ t − L D − I + r ) v ǫ k ≥ ǫ k . Hence Z ( ∂ t − L D − I + r ) v ∗ φ dxdt = lim ǫ k → Z ( ∂ t − L D − I + r ) v ǫ k φ dxdt ≥ , for any compactly supported smooth function φ . Here the identity above follows from applying the dual operatorof ∂ t − L D − I + r to φ and utilizing the dominated convergence theorem. The previous inequality then yields( ∂ t − L D − I + r ) v ∗ ≥ B × [ s, T ] in the distributional sense, which implies the same inequality on R × (0 , T ]in the distributional sense, since the choices of B and s are arbitrary. On the other hand, Lemma 4.7 shows that v ǫ k ≥ g ǫ k . Then v ∗ ≥ g after sending ǫ k →
0. Therefore, we obtain min { ( ∂ t − L D − I + r ) v ∗ , v ∗ − g } ≥ R × (0 , T ] in the distributional sense. It then remains to show ( ∂ t − L D − I + r ) v ∗ = 0 when v ∗ > g .To this end, take any ( x, t ) such that v ∗ ( x, t ) > g ( x ). Since both v ∗ and g are continuous, one can find asufficiently small δ > x, t ), such that v ∗ (˜ x, ˜ t ) ≥ g (˜ x ) + 2 δ for any (˜ x, ˜ t ) inside thisneighborhood. Utilizing the uniform convergence of ( v ǫ k ) k ≥ and ( g ǫ k ) k ≥ , we can then find sufficiently small ǫ k such that v ǫ k (˜ x, ˜ t ) ≥ g ǫ k (˜ x )+ δ in the aforementioned neighborhood. Hence p ǫ k ( v ǫ k − g ǫ k )( x, t ) = 0, due to (4.3)-(ii),which induces ( ∂ t − L D − I + r ) v ǫ k ( x, t ) = 0. After sending ǫ k →
0, we conclude that ( ∂ t − L D − I + r ) v ∗ = 0 inthe distributional sense when v ∗ > g . Finally, since v ∗ ∈ W , p,loc , v ∗ also solves (4.1) at almost every point in E T . Step 2:
Let us first show that v ∗ is a viscosity solution of (4.1). We will use the definition of viscosity solutionsin [23]. Denote by C ( E T ) the class of functions which have at most linear growth, i.e., | φ ( x, t ) | ≤ C (1 + | x | ) forsome C and any ( x, t ) ∈ E T . Then viscosity solutions of (4.1) are defined as follows: Any v ∈ C ( E T ) is a viscosity EGULARITY OF THE OPTIMAL STOPPING PROBLEMS 17 supersolution (subsolution) of (4.1) ifmin { ∂ t φ − L D φ − Iφ + rv, v − g } ≥ ≤ , ( x, t ) ∈ R × (0 , T ] ,v ( x, ≥ g ( x ) ( ≤ g ( x )) , x ∈ R , for any function φ ∈ C , ( R × (0 , T )) ∩ C ( E T ) such that v ( x, t ) = φ ( x, t ) and v (˜ x, ˜ t ) ≥ φ (˜ x, ˜ t ) ( v (˜ x, ˜ t ) ≤ φ (˜ x, ˜ t )) forany other point (˜ x, ˜ t ) ∈ R × (0 .T ). The function v is said to be a viscosity solution of (4.1) if it is both supersolutionand subsolution.Let us show that v ∗ is a viscosity subsolution of (4.1). Fix ( x, t ) ∈ R × (0 , T ], consider v ∗ ( x, t ) > g ( x ), otherwisemin { ∂ t φ − L D φ − Iφ + rv ∗ , v ∗ ( x, t ) − g ( x ) } ≤ x, t ) is the strict maximum of v ∗ − φ in a neighborhood B ( x, t ; δ ), otherwise the test function can bemodified appropriately. On the other hand, since ( v ǫ k ) k ≥ converges to v ∗ uniformly in compact domains, we canfind sufficiently small ǫ k such that v ǫ k − φ attains its maximum over B ( x, t ; δ ) at ( x k , t k ) ∈ B ( x, t ; δ ). Moreover,( x k , t k ) → ( x, t ) as ǫ k →
0. Since v ǫ k is a classical solution of (4.2) (see Lemma 4.2), it is also a viscosity solution.Hence ( ∂ t − L D − I + r ) φ ( x k , t k ) + p ǫ k ( v ǫ k ( x k , t k ) − g ǫ ( x k )) ≤
0. Now, since v ∗ ( x, t ) > g ( x ) and v ǫ k ( x k , t k ) − g ( x k )converges to v ∗ ( x, t ) − g ( x ), we obtain lim ǫ k → p ǫ k ( v ǫ k ( x k , t k ) − g ǫ ( x k )) = 0. As a result, ( ∂ t − L D − I + r ) φ ( x, t ) ≤ ǫ k →
0. This confirms that v ∗ is a viscosity subsolution of (4.1).For the supersolution property, since v ∗ ≥ g , it suffices to show that ( ∂ t − L D − I + r ) φ ( x, t ) ≥ φ . This follows from the similar arguments which we used for the subsolution property in the previousparagraph.Define u ∗ ( x, t ) = v ∗ ( x, T − t ). It is clear that u ∗ is a viscosity solution of (2.2). Then the statement follows fromTheorem 4.1 in [23], which states that the value function u is the unique viscosity solution of (2.2) when the L´evymeasure satisfies R | y | > | y | ν ( dy ) < ∞ . (cid:3) Proof of Corollary 2.6. (i) Combining Theorem 2.5 and the Sobolev embedding theorem (c.f. Lemma 3.3 in [18]pp. 80), we have u ∈ H β, β ( D × [0 , T − s ]), where β = 2 − p and s < T . Choosing p > β >
1, thecontinuity of ∂ x u follows from Definition 1.1.(ii) Let us first show that Iu is well defined and H¨older continuous. Since u ∈ H β, β ( D × [0 , T − s ]) (whichfollows due to (i)), choosing sufficiently large p so that β > α , Iu ∈ H β − α − γ, β − α − γ ( D T − s ) by Lemma 3.2 part (i).Now, for B ⊂ R and t , t ∈ [0 , T ) such that B × ( t , t ) ⊂ C , consider the following boundary value problem:( − ∂ t − L D + r ) v = Iu, ( x, t ) ∈ B × [ t , t ) ,v ( x, t ) = u ( x, t ) , ( x, t ) ∈ ∂B × [ t , t ) ∪ B × t . (4.12)It is straightforward to show that u is the unique viscosity solution for the previous problem using the fact that u is the unique viscosity solution for (2.2). On the other hand, since the boundary and terminal values of (4.12)are continuous and the driving term Iu is H¨older continuous, it follows from Theorem 9 in [13] pp. 69 that (4.12)has a classical solution u ∗ ∈ C , ( B × ( t , t )). Hence u = u ∗ on B × ( t , t ), since u ∗ is also a viscosity solution.Therefore, u ∈ C , ( B × ( t , t )). The statement now follows, since B × ( t , t ) is an arbitrary subset of C . (cid:3) References [1]
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