aa r X i v : . [ m a t h . A P ] D ec THE CLASSICAL STOCHASTIC IMPULSE CONTROL PROBLEM
ROHIT JAIN
Abstract.
In this paper we study regularity estimates for the solution to anobstacle problem arising in stochastic impulse control theory. We prove usingelementary methods the known sharp C , loc estimate for the solution. The newproof is also easily generalizable to stochastic impulse control problems withfully noninear operators. Moreover we obtain new regularity estimates for thefree boundary in the classical case. Introduction
We consider an implicit constraint obstacle problem arising in impulse controltheory. Stochastic impulse control problems ([2], [15], [16], [9]) are control problemsthat fall between classical diffusion control and optimal stopping problems. In theseproblems the controller is allowed to instantaneously move the state process by acertain amount every time the state exits the non-intervention region. This allowsfor the controlled process to have sample paths with jumps. There is an enormousliterature studying stochastic impulse control models and many of these models havefound a wide range of applications in electrical engineering, mechanical engineering,quantum engineering, robotics, image processing, and mathematical finance. Someclassical references are [15], [9], [2]. A key operator in stochastic impulse controlproblems is the intervention operator,(1) M u ( x ) = inf ξ ≥ ( u ( x + ξ ) + 1) . The operator represents the value of the control policy that consists of taking thebest immediate action in state x and behaving optimally afterwards. Since it is notalways the case that the optimal control requires intervention at t = 0, this leadsto the quasi-variational inequality,(2) u ( x ) ≤ M u ( x ) ∀ x ∈ R n . From the analytic perspective one obtains an obstacle problem where the obstacledepends implicitly and nonlocally on the solution. More precisely we can considerthe classical stochastic impulse control problem as a boundary value problem,(3) Lu ≤ f ( x ) ∀ x ∈ Ω .u ( x ) ≤ M u ( x ) ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . Here we let, Lu ≡ − P ni,j =1 a ij ( x ) ∂ u∂x i ∂x j + P ni =1 b i ( x ) ∂u∂x i + c ( x ) u with suitableregularity assumptions on the data and,(4) M u ( x ) = 1 + inf ξ ≥ x + ξ ∈ ¯Ω ( u ( x + ξ )) . In this paper, we present a new proof for the sharp C , loc (Ω) estimate for thesolution to (1.3). We point out that the sharp C , loc estimate has been previouslyobtained (see [5], [6]). As a corollary of our proof we also obtain a direct proof ofthe fact that the nonlocal obstacle, M u ( x ) is C , loc on the contact set { u = M u } .Since the obstacle depends on the solution, the strategy is to improve the regulari-ity of the solution and use it to improve the regularity of the obstacle. We start byfirst proving continuity of the solution and then proceeding to prove a semiconcav-ity estimate for the obstacle. In the following section we use the semiconcavity ofthe obstacle and the superhamonicity of the solution to produce the C , estimate.The new idea to prove the C , estimate in the classical case has been subsequentlygeneralized to the fully nonlinear case with more general semiconcave obstacles [14].In the last section we study regularity estimates for the free boundary ∂ { u < M u } . We first observe that the set of free boundary points can be structured according towhere inf u ( x + ξ ) is realized. If the infimum is realized in the interor of the positivecone then we conclude that the obstacle is locally constant in a neighborhood ofa free boundary point. This gives us regularity estimates of the free boundary atregular points and singular points as defined in the classical obstacle problem. Ifthe infimum is realized on the boundary of the cone then under the assumption that f is analytic we conclude that the free boundary is contained in a finite collectionof C ∞ submanifolds. In particular we prove the following theorem, Theorem 1.
Consider the classical stochastic impulse control problem (5) Lu ≤ f ( x ) ∀ x ∈ Ω .u ( x ) ≤ M u ( x ) ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . Assume that all coefficients in L are analytic, f is analytic and f ( x ) ≤ f ( x + ξ ) ∀ ξ ≥ . Then it follows that, ∂ { u < M u } = Γ r ( u ) ∪ Γ s ( u ) ∪ Γ d ( u ) where,1. ∀ x ∈ Γ r ( u ) there exists some appropriate system of coordinates in which thecoincidence set { u = M u } is a subgraph { x n ≤ g ( x , . . . , x n − ) } in a neighborhoodof x and the function g is analytic.2. ∀ x ∈ Γ s ( u ) , x is either isolated or locally contained in a C submanifold.3. Γ d ( u ) ⊂ Σ( u ) where Σ( u ) is a finite collection of C ∞ submanifolds.Acknowledgements I would like to express my sincerest gratitude and deepestapprecation to my thesis advisors Professor Luis A. Caffarelli and Professor AlessioFigalli. It has been a truly rewarding experience learning from them and havingtheir guidance.
HE CLASSICAL STOCHASTIC IMPULSE CONTROL PROBLEM 3 Basic Definitions and Assumptions
Let Ω ⊆ R n be a bounded domain with C ,α boundary ∂ Ω. Assume c ( x ) ≥ c > a ij , b i , c , ∈ C α ( ¯Ω) for 0 < α <
1, and the matrix ( a ij ) is positive definite forall x ∈ ¯Ω. Furthermore let f ∈ C α ( ¯Ω). For any ξ = ( ξ , . . . , ξ n ) we let ξ ≥ ξ i ≥ ∀ i . Consider,(6) Lu ≡ − n X i,j =1 a ij ( x ) ∂ u∂x i ∂x j + n X i =1 b i ( x ) ∂u∂x i + c ( x ) u. Define the operator:(7) M u ( x ) = 1 + inf ξ ≥ x + ξ ∈ Ω u ( x + ξ ) . We introduce the bilinear form a ( u, v ) associated to our operator L,(8) a ( u, v ) = ( Lu, v ) ∀ u, v ∈ C ∞ (Ω) . Furthermore assume that our bilinear form is coercive,(9) a ( u, u ) ≥ γ ( k u k W , (Ω) ) ∀ u ∈ W , (Ω) , γ > . We consider the quasi-variational inequality: u ∈ W , (Ω) u ≤ M u, (10) a ( u, v − u ) ≥ ( f, v − u ) ∀ v ∈ W , (Ω) v ≤ M u. We list a few properties of our operator M u that will be useful for the remainingparts of this chapter, u ( x ) ≤ u a.e. ⇒ M u ( x ) ≤ M u ( x ) a.e. M : L ∞ → L ∞ . M : C ( ¯Ω) → C ( ¯Ω) . Furthermore we assume that f ≥ − c . This implies that the solution ¯ u to thevariational equation L ¯ u = f in Ω, ¯ u ∈ H (Ω) satisfies the property ¯ u ≥ −
1. Thisin particular implies that the set of solutions to v ∈ H (Ω) v ≤ M¯ u is nonempty.Without loss of generality we assume that ¯ u < Existence and Uniqueness Theory
We now proceed to prove the existence of a unique continuous solution to (3.5).We follow closely the proof in [13].
Lemma 1.
There exists a unique solution u ∈ C (Ω) of (3.5).Proof. From standard elliptic theory we know that there exists a unique solution u ∈ C (Ω) of(11) ( a ( u, v ) = ( f, u − v ) ∀ x ∈ Ω ,u = 0 ∀ x ∈ ∂ Ω . ROHIT JAIN
Since M u is continuous we know from the theory of variational inequalities thatthere exists a unique solution u ∈ C (Ω) of(12) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω ,u ≤ M u ∀ x ∈ Ω ,u = 0 ∀ x ∈ ∂ Ω . Moreover for n = 2 , , . . . we obtain u n ∈ C (Ω) satisfying,(13) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω ,u ≤ M u n − ∀ x ∈ Ω ,u = 0 ∀ x ∈ ∂ Ω . Since u is a subsolution of (11), by the comparison principle, we know that u ≤ u . We also know that 0 is a subsolution of (12), hence the comparison impliesthat 0 ≤ u . Moreover it follows from the properties of M u that 0 ≤ M u ≤ M u .This implies in particular that u is an admissable subsolution to (12). Arguingas before we see that 0 ≤ u ≤ u . We can continue this process and obtain asequence of functions(14) 0 ≤ . . . ≤ u n ≤ . . . ≤ u ≤ u . Now we look to prove an upper bound on the sequence. Consider µ ∈ (0 ,
1) suchthat µ k u k C (Ω) ≤
1. Assume there exists θ n ∈ (0 ,
1] such that ∀ n ∈ N ,(15) u n − u n +1 ≤ θ n u n . We claim that this implies(16) u n +1 − u n +2 ≤ θ n (1 − µ ) u n +1 . With this claim we are able to almost conclude the proof of the theorem. Inparticular the positivity of u n implies that u − u ≤ u . We can set θ = 1.Moreover from (16) it follows that u − u ≤ (1 − µ ) u . Hence θ = (1 − µ ).Therefore setting θ n = (1 − µ ) n − we find(17) u n +1 − u n +2 ≤ (1 − µ ) n u n +1 ≤ (1 − µ ) n k u k C (Ω) . Combining (17) with (14) we see that there exists a function u ∈ C (Ω) suchthat k u n − u k C (Ω) → n → ∞ . Moreover from the estimate k M u − M v k C (Ω) ≤k u − v k C (Ω) it follows that u is a solution to the classical stochastic impulse controlproblem. Hence we are reduced to proving (16) and establishing uniqueness of thesolution. By the concavity of M u and (15) it follows,(*) ψ = (1 − θ n )M u n + θ n ≤ (1 − θ n )M u n + θ n M0 ≤ M(1 − θ n u n ) ≤ M u n +1 . We consider the continuous solutions to the following obstacle problems. Let w ∈ C (Ω) solve,(18) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω .u ≤ ψ ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . HE CLASSICAL STOCHASTIC IMPULSE CONTROL PROBLEM 5
Let z ∈ C (Ω) solve,(19) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω .u ≤ ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . From (*) and the comparision theorem for variational inequalities it follows that w ≤ u n +2 . Moreover it follows that θ n z solves,(20) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω .u ≤ θ n ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . Observing that ψ ≥ θ n , it follows from comparision that θ n w ≥ θ n z . Next weobserve that (1 − θ n ) u n +1 is a subsolution and (1 − θ n ) w is a solution of the followingobstacle problem,(21) a ( u, v ) ≥ ( f, u − v ) ∀ x ∈ Ω .u ≤ (1 − θ n ) ψ ∀ x ∈ Ω .u = 0 ∀ x ∈ ∂ Ω . Hence we find, (1 − θ n ) u n +1 ≤ (1 − θ n ) w . Putting this together we obtain,(**) (1 − θ n ) u n +1 + θ n z ≤ (1 − θ n ) w + θ n w = w ≤ u n +2 . Recall that ∀ n , µu n +1 ≤
1. This implies that µu n +1 is a subsolution of (19). So inparticular, µu n +1 ≤ z . Putting this into (**) we obtain our desired estimate (16), u n +1 − u n +2 ≤ θ n (1 − µ ) u n +1 . Finally to prove uniqueness, suppose u and ¯ u are distinct solutions. The positivityof the solution implies u − ¯ u ≤ u . Hence arguing as above we find u − ¯ u ≤ (1 − µ ) n u ,for all n ≥
0. Letting n → ∞ we find that u − ¯ u ≤
0. Interchanging u and ¯ u weconclude u = ¯ u . (cid:3) Using the improved regularity on the solution u , we now proceed to prove thatthe obstacle M u ( x ) is semi-concave with semi-concavity modulus ω ( r ) = Cr . Westate the following theorem which is proven in more generality in [5], [14]. Theorem 2.
Let ϕ ( x ) ∈ C , (Ω) , strictly positive, bounded, and decreasing in thepositive cone ξ ≥ . Then the obstacle M u ( x ) = ϕ ( x ) + inf ξ ≥ x + ξ ∈ Ω u ( x + ξ ) is locally semi-concave with a semi-concavity modulus ω ( r ) = Cr . Optimal Regularity for the Stochastic Impulse Control Problem
In the previous section we proved that the unique bounded solution to the classi-cal stochastic impulse control problem is continuous and that our implicit constraintobstacle is locally semi-concave. We now consider the sharp C , estimate for thesolution. Theorem 3.
Let u be the unique continuous solution of the classical stochasticimpulse control problem. Then u ∈ C , loc (Ω) . ROHIT JAIN
We set
M u ( x ) = ϕ u ( x ). Recall that a function v is semi-concave with semi-concavity modulus ω ( r ) if a vector p ∈ R n belongs to D + v ( x ) if and only if v ( y ) − v ( x ) − h p, y − x i ≤ ω ( | x − y | ). Fix x ∈ { u = ϕ u } . Define the linearpart of the obstacle, L x ( x ) = ϕ u ( x ) + h p, x − x i . We consider w ( x ) = u ( x ) − L x ( x ) . We observe that in B r ( x ), w ( x ) has a modulus of semi-concavity ω ( r ) = Cr , i.e. w ( x ) ≤ Cr . We now state our main lemma. Lemma 2.
There exists universal constants
K, C > , such that ∀ x ∈ B r/ ( x ) , (22) − K ≤ ∆ w ≤ C. Before proving this lemma we make a few observations. Fix Φ ∈ C ∞ ( B r ( x )).We recall the following fact from the theory of distributions: If u is a negativedistribution in X with u ( Φ ) ≤ Φ ∈ C ∞ ( X ), then u is anegative measure. In particular we have,(23) 0 ≥ Z B r Φ dµ = Z B r ∆ u Φ. We consider ∀ ρ < r ,(24) µ ( B ρ ( x )) | B ρ ( x ) | = 1 α ( n ) ρ n Z B ρ dµ = 1 α ( n ) ρ n Z B ρ ∆ u. A straightforward application of the Gauss-Green Formula gives to us the followingidentity,(25) 1 α ( n ) ρ n Z B ρ ∆ w = nρ ddρ Ψ ( ρ ) . Where Ψ ( ρ ) = nα ( n ) ρ n − R ∂B ρ w . Before proving the main lemma we will first provethe following claim. Claim 1.
Let w = u − L x be defined as before. Then for some universal constant K ( n ) > , nρ ddρ Ψ ( ρ ) ≥ − K. HE CLASSICAL STOCHASTIC IMPULSE CONTROL PROBLEM 7
Proof.
We expand the derivative and compute. nρ ddρ Ψ ( ρ ) = nρ − nnα ( n ) ρ n Z ∂B ρ ( x ) w ( y ) dS ( y ) + nnα ( n ) ρ n ddρ Z ∂B ρ ( x ) w ( y ) dS ( y )= nρ n − nα ( n ) ρ n Z ∂B ρ ( x ) − w ( y ) dS ( y ) + 1 α ( n ) ρ n ddρ ρ n − Z ∂B (0) w ( x + ρz ) dS ( z )= nρ n − nα ( n ) ρ n Z ∂B ρ ( x ) − w ( y ) dS ( y ) + ρ n − ( n − α ( n ) ρ n ρ n − ρ n − Z ∂B (0) w ( x + ρz ) dS ( z )+ ρ n − α ( n ) ρ n ddρ Z ∂B (0) w ( x + ρz ) dS ( z )= nρ n − nα ( n ) ρ n Z ∂B ρ ( x ) − w ( y ) dS ( y ) + ( n − α ( n ) ρ n +1 Z ∂B ρ ( x ) w ( y ) dS ( y )+ ρ n − α ( n ) ρ n ddρ Z ∂B (0) w ( x + ρz ) dS ( z )Now we proceed to estimate each integral. By the modulus of semi-concavity onthe ball we have, nρ n − nα ( n ) ρ n Z ∂B ρ ( x ) − w ( y ) dS ( y ) ≥ n ( n − α ( n ) nρ n +1 | ∂B ρ ( x ) | ( − Cρ ) = − C ( n − n ) . By the mean value theorem for subharmonic functions we have,( n − α ( n ) ρ n +1 Z ∂B ρ ( x ) w ( y ) dS ( y ) ≥ ( n − α ( n ) ρ n +1 w ( x ) = 0 . By the nondecreasing property for the average integral we have: ρ n − α ( n ) ρ n ddρ Z ∂B (0) w ( x + ρz ) dS ( z ) ≥ . Hence for K = C ( n − n ) we obtain the desired estimate. (cid:3) Proof. (Lemma 3) From the claim we obtain the estimate,1 α ( n ) ρ n Z B ρ ∆ w ≥ − K. Moreover from (23) and the semi-concavity estimate from above we know, C ≥ µ ( B ρ ( x )) | B ρ ( x ) | ≥ − K. Letting ρ → ∀ x ∈ B r ( x ), C ≥ ∆ u ( x ) ≥ − K. (cid:3) We now state and prove the sharp estimate for the solution.
Theorem 4.
Let u be a solution to the classical stochastic impulse control problem.Then, (26) k u k C , ( B r/ ) ≤ C ROHIT JAIN
Proof.
We recall some basic notions and definitions for convenience. For furtherdetails refer to ([4]). We say that P is a parabaloid of opening M whenever, P ( x ) = l + l ( x ) ± M | x | . We define, Θ( u, A )( x ) , to be the infimum of all positive constants M for which there is a conex parabaloidof opening M that touches u from above at x in A . Similarly one can definethe infimum of all positive constants M for which there is a convex parabaloid ofopening − M that touches u from below at x in A ,Θ( u, A )( x ) . We further define,Θ( u, A )( x ) = sup { Θ( u, A )( x ) , Θ( u, A )( x ) } ≤ ∞ . As before we fix x ∈ { u = M u } . We consider the second incremental quotients of u and M u , ∆ h u ( x ) = u ( x + h ) + u ( x − h ) − u ( x ) | h | . ∆ h M u ( x ) = M u ( x + h ) + M u ( x − h ) − M u ( x ) | h | . We make the following observations,1. ∆ h u ( x ) ≤ ∆ h M u ( x ) .
2. 0 ≤ Θ( u, B ρ )( x ) = Θ( M u, B ρ )( x ) ≤ C .3. 0 ≤ Θ( u, B ρ )( x ) = Θ( M u, B ρ )( x ) ≤ K .Putting the estimates together we obtain, − K ≤ − Θ( u, B ρ )( x ) ≤ ∆ h u ( x ) ≤ ∆ h M u ( x ) ≤ Θ( M u, B ρ )( x ) ≤ C. In particular ∀ x ∈ B ρ , − K ≤ − Θ( u, B ρ )( x ) ≤ ∆ h u ( x ) ≤ Θ( u, B ρ )( x ) ≤ C. This follows from choosing ∀ x ∈ B ρ , the lower parabaloid and upper parabaloid tobe respectively, P ( y ) = u ( x ) + h p , y − x i − K | y | .P ( y ) = u ( x ) + h p , y − x i + C | y | . Hence we obtain, Θ( u, ǫ ) = Θ( u, B ρ ∩ B ǫ ( x ))( x ) ∈ L ∞ ( B ρ ) . This implies, k D u k L ∞ ( B ρ ) ≤ C. In particular we obtain our desired estimate, k u k C , ( B ρ ) ≤ C. HE CLASSICAL STOCHASTIC IMPULSE CONTROL PROBLEM 9 (cid:3) Regularity Estimates for the Free Boundary
In this section we prove a structural theorem for the free boundary Γ = ∂ { u < M u } . Theorem 5.
Consider the classical stochastic impulse control problem (27) ∆ u ( x ) ≥ f ( x ) ∀ x ∈ Ω ,u ( x ) ≤ M u ( x ) = 1 + inf ξ ≥ x + ξ ∈ Ω u ( x + ξ ) ∀ x ∈ Ω ,u = 0 ∀ x ∈ ∂ Ω . Moreover assume that f is analytic and f ( x ) ≤ f ( x + ξ ) ∀ ξ ≥ . Then it followsthat, ∂ { u < M u } = Γ r ( u ) ∪ Γ s ( u ) ∪ Γ d ( u ) where,1. ∀ x ∈ Γ r ( u ) there exists some appropriate system of coordinates in which thecoincidence set { u = M u } is a subgraph { x n ≤ g ( x , . . . , x n − ) } in a neighborhoodof x and the function g is analytic.2. ∀ x ∈ Γ s ( u ) , x is either isolated or locally contained in a C submanifold.3. Γ d ( u ) ⊂ Σ( u ) where Σ( u ) is a finite collection of C ∞ submanifolds.Proof. Recall Σ x = { u ( x + ξ ) = M u ( x ) } and Σ ≥ x = { x + ξ : ξ ≥ } . We definethe following sets1. Σ ≥ x = { ξ ∈ Σ ≥ x | ξ i > ∀ i = 1 , . . . , n } . ∂ i Σ ≥ x = { ξ ∈ Σ ≥ x | ξ i > ξ k = 0 ∀ k = 1 , . . . , i − , i + 1 , . . . , n } .
3. Σ x = { ξ ∈ Σ x | ξ ∈ Σ ≥ x } . ∂ i Σ x = { ξ ∈ Σ x | ξ ∈ ∂ i Σ ≥ x } . We note that Σ ≥ x = Σ ≥ x ∪ ( n [ i ∂ i Σ ≥ x ) , Σ x = Σ x ∪ ( n [ i ∂ i Σ x ) . Fix x ∈ ∂ { u < M u } and let ξ be the positive vector such that,inf ξ ≥ x + ξ ∈ Ω u ( x + ξ ) = 1 + u ( x + ξ ) . Case 1 : ξ ∈ Σ x . Then it follows from Claim 4 in [14], that ∀ x ∈ B δ ( x ) , ξ ∈ Σ x . In particular for a fixed constant C , M u = C in B δ ( x ) . Without loss of generalitywe take C = 0 . Furthermore it follows that at a contact point x we have thefollowing chain of inequalities, f ( x ) ≤ ∆ u ( x ) ≤ ∆M u ( x ) ≤ f ( x + ξ ) . In particular, f ( x ) ≤ ∆ u ( x ) ≤ . We make the following claim,
Claim 2. f ( x ) < . Proof.
Suppose by contradiction that f ( x ) = 0 . By analyticity of f , it follows thatΩ = { f > } satisfies an interior sphere condition. Hence, ∀ z ∈ ∂ Ω there exists, y ∈ Ω and open ball B r ( y ) such that B r ( y ) ∩ Ω = { z } . In particular consider z = x and y = y . Observe that ∀ x ∈ B r ( y ) \ { x } , it follows that w = u − M u < w = ∆ u = f >
0. Hence by the Hopf Boundary point lemma, ∂w∂ν ( x ) > . But w ∈ C , ( x ) . A contradiction. (cid:3)
From the claim it follows that in a small neighborhood B η ( x ), we can study thefollowing problem,(28) ∆ w ( x ) = f ( x ) < ∀ x ∈ { w < } ∩ B η ( x ) ,w ( x ) ≤ ∀ x ∈ B η ( x ) ,w ∈ C , ∀ x ∈ B η ( x )Hence w is a normalized solution and the conclusion follows for,Finally to conclude we define,Γ r ( u ) = { x ∈ Γ | Σ x = Σ x and x is a Regular Point } . Γ s ( u ) = { x ∈ Γ | Σ x = Σ x and x is a Singular Point } . Case 2 : ξ ∈ ∂ i Σ x . We consider the setΣ( u ) = n [ i { u x i = 0 } × R n − . By analyticity of f it follows that { u x i = 0 } is a finite set ∀ i = 1 , . . . , n. HenceΣ( u ) is a finite collection of hyperplanes { l j } kj =1 ⊂ R n − . We defineΓ d ( u ) = { x ∈ Γ( u ) | ∃ ¯ ξ ∈ ∂ i Σ x } . Finally to conclude we observe, Γ d ( u ) ⊂ Σ( u ) . (cid:3) References [1] Bensoussan, A.,
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