A bank salvage model by impulse stochastic controls
aa r X i v : . [ q -f i n . M F ] O c t A bank salvage model by impulse stochastic controls
Francesco Cordoni a Luca Di Persio a Yilun Jiang b Abstract
The present paper is devoted to the study of a bank salvage model with finite timehorizon and subjected to stochastic impulse controls. In our model, the bank’s defaulttime is a completely inaccessible random quantity generating its own filtration, thenreflecting the unpredictability of the event itself. In this framework the main goal isto minimize the total cost of the central controller who can inject capitals to save thebank from default. We address the latter task showing that the corresponding quasi–variational inequality (QVI) admits a unique viscosity solution, Lipschitz continuousin space and H¨older continuous in time. Furthermore, under mild assumptions on thedynamics the smooth-fit W (1 , ,ploc property is achieved for any 1 < p < + ∞ . AMS Classification subjects:
Keywords or phrases:
Bank salvage model, stochastic impulse control, viscositysolution, inaccessible bankruptcy time, smooth-fit property.
Mainly motivated by the recent financial credit crisis, starting from 2008-2009 creditcrunch, the financial and mathematical community started investigating and generalize ex-isting models, since previous events have shown that financial models used prior to the crisiswhere inadequate to describe and capture main features of financial markets. Thereforethe mathematical and financial communities have focus on developing general and robustmodels that are able to properly describe financial markets and their main peculiarities.From a purely mathematical perspective the above mentioned attention led, among manyother research topics, for instance into the study of general stochastic optimal control prob-lems, where instead of classical type of controls, some more realistic controls have beenconsidered. Among the most studied type, impulse type controls have to be mentioned,and regained attention in last decades also due to the many application in finance and eco-nomics. In this setting, the controller can intervene on the system at some random timewith a discrete type control, where in this case the control solution is represented by thecouple u = ( τ n , K n ) n , where τ n represents the decision time at which the ocntroller inter-vene and K n instead denotes the action taken by the controller. Above type of controlimplies that at the intervention time τ n the system jumps from the state X ( τ − n ) to the newstate X ( τ n ) = Γ( X ( τ n ) , K n ), for a suitable function Γ. Therefore, as standard in optimalcontrol theory, using the dynamic programming principle, it can be shown that stochasticimpulse control problems can be associated to a quasi–variational Hamilton–Jacobi–Bellmanequation (HJB) of the form min (cid:20) − ∂∂t V − L V − f, V − H V (cid:21) , (1)where above f is the running cost, L is the infinitesimal generator for the process X and V is the value function solution to the above HJB equation. Further, H is the nonlocal a Department of Computer Science, University of Verona, Strada le Grazie, 15, Verona, 37134, ItalyE-mail addresses: [email protected] (Francesco Cordoni), [email protected] (Luca DiPersio) b Department of Mathematics, Penn State University, University Park, PA 16802, USAE-mail addresses: [email protected] (Yilun Jiang) impulse operator that characterize HJB equation for impulse type of control. The particularform for the HJB implies that two regions can be retrieved, the continuation region where V > H V and therefore no impulse control is used, and the impulse region where on thecontrary V = H V and the controller intervenes. Solution to equation (1) can be formallydefined so that the value function is in fact a viscosity solution, in a sense to be properlydefined later on, to equation (1). It is clear that, following the above characterization ofthe domains for the HJB equation, particular attention must be given to the interventionboundary . In fact, particular attention is usually given in this field to proving that theboundary is regular enough; this regularity is referred to in literature as smooth–fit principle .Several results exist in the smooth–fit principle where the terminal horizon for the controlproblem, whereas instead finite horizon problem, and in particular the terminal conditionof the problem, makes less straightforward the derivation of the smooth–fit principle.At last, we stress that impulse type stochastic control is strictly connected to optimalstopping problems and optimal switching. The literature on the topic is wide, we refer theinterested reader to [26, 45], or also to [4, 5, 12, 13, 21, 25, 38, 42, 46] for other relatedresults.A second crucial financial aspect that emerged to be fundamental in a general financialformulation after recent crisis there is possible failures of financial entities. In fact, oneof the major lack of classical financial models is that no risk of failure is considered intothe general setting. Recent financial event has shown that no financial operator can beconsidered immune from bankruptcy. Therefore it has emerged in last decade an extensiveliterature that focus on credit risk modeling, assessing as main object the risk that financialentities has to face borrowing or lending money to other players that might fail, see, e.g.[10, 17, 18].Along aforementioned lines, two main approaches have been developed in literature: structural approach and intensity–based approach , see, e.g. [7]. Mathematically speaking,the first scenario consits in considering some default event that can be triggered by theunderlyng process. Typical example are default triggered by some stopping time definedas a hitting time. Such an approach has been for instance considered in [14, 15, 35]. Thelatter instead considers a default event which is completely inaccessible for the probabilisticreference filtration, so that in order to solve the problem the typical approach is to rely on filtration enlargment techniques, see, e.g., [6, 40].The present paper is devoted to study a stochastic optimal control problem of impulsetype, where a financial supervisor controls a system, such as financial operators or alsosome banks. The final goal of the controller is to prevent failures, injecting capital introthe system according to a given criterion to be maximized. The controller has no perfectinformation regarding the failure of the bank, so that mathematically speaking the failurecannot be foreseen by the controller. The supervisor, which can be though for instance asa central bank, can intervene with some impulse type controls over a finite horizon, so thatthe optimal solution is represented by both am intervention time and the quantity to injectinto the system.Our approach will be based on a intensity–based approach , so that we will assume thedefault event to be totally inaccessible from the reference filtration, assuming only a typicaldensity assumption. This assumptions will allows us to rewrite the system as determinis-tic finite horizon impulse problem, using the density distribution of the default event, via enlargement of filtrations techniques. We stress that, due to the terminal condition to be im-posed, typically a finite horizon stochastic impulse control problem is more difficult to solvethan infinite horizon impulse control problems. In fact, it exists an exhaustive literature onstochastic impulse control on infinite time horizon, see, e.g. [4, 5, 12, 21, 25, 38, 42, 46],whereas very few results exist for the finite dimensional case, see, e.g. [13, 26, 45].A more financially oriented motivation of the control problem considered in the presentwork, has often arise in the last decade, mostly as a consequence of the 2007–2008 creditcrunch. This has been for instance the case of Lehman Brothers failure, which has shownthe cascade effect triggered by the default of a sufficiently large and interconnected financialinstitution, see, e.g., [29, 30] and references therein. We stress that, particular attentionhas to be given not only towards the magnitude of the stressed bank’s financial assets, butalso to its interconnection grade. Indeed, while the exposure with few financial institutions,he general setting provided its magnitude is reasonable, can be managed by ad hoc politics established on a one to one relationship basis, the situation could be simply ungovernable in case of a highnumber of connections, hidden links and over-structured contracts.Since above mentioned financial crisis, it has became typical, within the financial orientedstochastic optimal control theory, to model a given problem up to a random terminal timeinstead of considering a fixed, even infinite, horizon. From a modelling point of view, theaforementioned scenario has lead to consider the stochastic optimal control approach tomodel such situations by considering random terminal times, instead of considering a fixed,or infinite, horizon. Analogously, data analysts as well as mathematicians, have startedto consider problems of bank bailouts, where bank’s default and the consequent contagionspreading inside the network, may induce serious consequences for decades, see, e.g., [23].From a government perspective, such type of likely high financial fall out, have pushedseveral central banks to establish specific economic actions to help those sectors of thebanking sector of (at least) national interest, under concrete failure risks. As an example, thelatter has been the case of the pro bail-in procedures followed in agreement with the Directive2014/59/UE (approved last 1 st of January, 2016 by the European Union Parliament), andthen applied, e.g., in Italy, Ukraine, etc., see, e.g. [33, 44]. It is relevant to underline thatsuch actions rely also on the following grades of freedom: the possibility, as an alternativeto internal rescue, to relocate goods as well as legal links to a third party, often calledbridge-bank, or to a bad bank which will collect only a part of assets aiming at maximizingits long-term value; the hierarchical order of those who are called to bear the bail-in, whichmeans that the government can decide to put small creditors on the safe side; and theprinciple that no shareholder, or creditor, has to bear greater losses than would be expectedif there was an administrative liquidation, namely the no worse off creditor idea.Similar situations have been recently taken into consideration by a series the CentralEuropean Bank procedures, with particular reference to the well known quantitative easing ,as well as in agreement to the creation of injected currency , see, e.g., [2, 3, 8, 19]. We wouldlike to underline that quantitative easing type procedures have been experienced also outsidethe European Union, as in the case of the actions undertaken by the Japanese Central Bank,whose intervention has lasted over years, see, e.g. [9, 47, 37], or how has been done by theUS Federal Reserve not only starting from 2008, but also during the Great Depression ofthe 1930s, see, e.g., [48, 49, 50, 51, 52].The main contribution of the present paper is to develop a concrete financial setting thatmodels the evolution of a financial entity, controlled by an external supervisor who is willingto lend money in order to maximize a given utility function; see also [11, 15, 20, 35, 43]for setting in which a financial supervisor aims at controlling a system of banks of generalfinancial entities. In compete generality, we will assume that the financial entity may failat some random time that is inaccessible to the reference filtration, which represents thecontroller knowledge. Also, we consider a controller that can act on a system with animpulse–type control, so that the optimal solution consists in both a random time at whichinjecting money into the system and the precise amount of money to inject. We characterizethe value function of the above problem, showing that it must solve in a given viscositysense a certain quasi–variational inequality (QVI). At last we will prove that above QVIadmits a unique solution in a viscosity sense and also we provide a regularity results for theintervention boundary, known in literature as smooth fit principle .The paper is so organized, Section 2 introduces the general financial and mathematicalsetting; then Section 3 prove some regularity results for the value function and Section 4address the problem of existence and uniqueness of a solution. At last Section 5 is devotedto the smooth fit principle. We will in what follows consider a complete filtered probability space (cid:16) Ω , F , ( F t ) t ∈ [0 ,T ] , P (cid:17) ,( F t ) t ∈ [0 ,T ] being a filtration satisfying the usual assumptions, namely right–continuity andsaturation by P –null sets. Let T < ∞ be a fixed terminal time, and let x , resp. y , denotesthe total value of the investments of a given bank, resp. the total amount of deposit of thehe general setting same bank. We assume that x and y evolve according to the following system of SDEs ( dx ( t ) = c ˙ y ( t ) dt + ˜ µ ( t ) x ( t ) dt + ˜ σ ( t ) x ( t ) dW ( t )˙ y ( t ) = λ (cid:16) x ( t ) y ( t ) (cid:17) y ( t ) , (2)where W ( t ) is assumed to be a standard Brownian motion adapted to the aforementionedfiltration. In particular, the first term in equation (2) accounts for the increase in X dueto the fact that new deposits are made, where c ∈ [0 ,
1] denotes the fractions of depositswhich are actually invested in more or less risky financial operations. We stress that by arescaling argument, with no less of generality c = 1 it can be assumed. Moreover we definethe value over liability ratio X ( t ) := x ( t ) y ( t ) . Then, according to eq. (2) and exploiting theItˆo-D¨oblin formula, we have ( dX ( t ) = (( c − X ( t )) λ ( X ( t )) + ˜ µ ( t ) X ( t )) dt + ˜ σ ( t ) X ( t ) dW ( t ) x (0) = x . (3)We assume the process X to be stopped at completely inaccessible random time τ , notadapted to the reference filtration ( F t ) t ∈ [0 ,T ] . From a financial point of view, assumingthat X represents the financial value of an agent, above assumption reflects the fact that abank’s failure cannot be predicted. In particular, let us introduce the filtration ( H t ) t ∈ [0 ,T ] generated by the stopping time τ , namely H t := { τ ≤ t } . Then we define the augmentedfiltration ( G t ) t ∈ [0 ,T ] , where G t := F t ∨ H t .Within this setting it is interesting to consider an external controller , e.g., a central bank ,or an equivalent financial agent acting as a governance institution, with suitable surveillancerights. Such controller can inject capital in the bank, at random times τ n . Then, at thattime τ n , the state process X ( t ) jumps , in particular we have X ( τ − n ) = X ( τ n ) = X ( τ − n ) + K n , therefore X ( t ) evolves according to ( dX ( t ) = (( c − X ( t )) λ ( X ( t )) + ˜ µ ( t ) X ( t )) dt + ˜ σ ( t ) X ( t ) dW ( t ) + P n : τ n ≤ t K n X (0) = x . (4)The solution to the aforementioned system is represented by a couple u = ( τ n , K n ) n ≥ ,where ( τ n ) n ≥ is a non–decreasing sequence of stopping times representing the interventiontimes , while ( K n ) n ≥ is a sequence of ( G t )–adapted random variables taking values in A ⊂ [0 , ∞ ). In particular the sequence ( K n ) n ≥ indicates the financial actions taken at time τ n .The following is the definition of admissible impulse strategy u . Definition 2.0.1 (Admissible impulse strategy) . The admissible control set U consists ofall the impulse controls u = ( τ n , K n ) n ≥ such that { τ i } i ≥ are G t adapted stopping times and increasing, i.e τ < τ < · · · < τ i < · · · ,K i ∈ A and K i ∈ G τ i , ∀ i ≥ . (5) Remark . Equivalently, we will use a different notation, ξ t ( · ) to express the same space,i.e. for all 0 ≤ t ≤ s ≤ T , ξ t ( s ) = X t ≤ τ i
0, is a suitable constant defining the cost required by the capital injection. Above, wehave denoted by τ the bank default time , with respect to the process X ( t ). We assume, asspecified above, that τ is a completely inaccessible random time , and it is not adapted tothe reference filtration ( F t ) t ∈ [0 ,T ] . Also, recall that ( H t ) t ∈ [0 ,T ] is the filtration generatedby the stopping time τ , namely H t := { τ ≤ t } , whilst ( G t ) t ∈ [0 ,T ] is the filtration, namely G t := F t ∨ H t .Following the standard literature, see, e.g., [32] both the running and terminal costs areusually given in terms of suitable utility functions representing the utility gains from thebank’s value. A typical example is f ( x ) = x p p , p ∈ (0 , K + κ , itreflects the fact that injecting an amount K of capital to increase the bank’s liquidity level,implies a non negligible cost, otherwise such a financial help would be always profitable.Throughout the work we will make the following assumptions: Hypothesis . (i) the function λ : R → R is Lipschitz continuous, namely there exists aconstant L λ > | λ ( x ) − λ ( y ) | ≤ L λ | x − y | , ∀ x, y ∈ R ; (ii) the functions f , g , g are Lipschitz continuous, namely there exist constants L f , l g and L g > | f ( x ) − f ( y ) | ≤ L f | x − y | , ∀ x, y ∈ R ; | g ( x ) − g ( y ) | ≤ L g | x − y | , ∀ x, y ∈ R ; | g ( x ) − g ( y ) | ≤ L g | x − y | , ∀ x, y ∈ R ; (9)We also assume that there exist constants C f , C g and C g > f ( x ) < C f , g ( x ) < C g , g ( x ) < C g ; (iii) the functions µ ( t ) , σ ( t ) ∈ C ([0 , T ]). (iv) No terminal impulse, i.e. g ( x ) ≥ sup K> g ( x + K ) − K − κ. he general setting The boundedness properties for the running and terminal cost can be interpreted in thefollowing sense: since we are seeking the optimal capital injection strategy for the governmentover a finite time horizon, we may think that there is a healthy level
U > U ( x ) = U − e − x . Remark . A further generalization of the above optimal control problem, consists inconsidering a controller having two different ways to influence the evolution of the stateprocess x , namely(1) an impulse type control ( τ n , K n ) n , hence as in equation (4) by injecting capital atrandom times τ n ;(2) a continuous type control α ( t ), by choosing at any time t the rate at which x is growing.In particular an action of type 2 implies that eq. (4) can be reformulated as follows ( dX ( t ) = (( c − X ( t )) λ ( X ( t )) + ( µ ( t ) − α ( t )) X ( t )) dt + σ ( t ) X ( t ) dW ( t ) + P n : τ n ≤ t K n X (0) = x , where α represents the continuous control variable α ( t ) ∈ [0 , ¯ r ], for a suitable constant ¯ r ,where α = 0 stands for higher returns and α = ¯ r denotes lower returns. This reflectsthe financial assumption that the controller, e.g. a central bank , can change the interestrate according to macroeconomic variables, as the country inflation level, the forecast ofsupranational interest rates, the the markets’ belief about the health of the financial sectorunder the central bank control, etc. In fact, choosing α = 0 the bank value grows at rate µ ( t ), which is strictly greater than µ ( t ) − α ( t ) for a given control 0 < α ( t ) ≤ ¯ r . We referto the above discussion, see also, e.g., [2, 3, 8, 9, 47, 37, 19], for more financially orientedideas supporting the latter setting. Accordingly, we can assume that the controller aims atmaximizing a functional of the following type J u,a ( t, x ) = E t (cid:20) Z τ ∧ Tt f ( X u,at,x ( s ) , α ( s )) ds + g ( X u,at,x ( T )) { τ ≥ T } + − g ( X u,at,x ( τ )) { τ For any F T -measurable random variable X it holds E [ X T ≤ τ | G t ] = { τ>t } E [ X τ>T | F t ] E [ τ>t | F t ] = { τ>t } e Γ t E (cid:2) Xe − Γ T (cid:12)(cid:12) F t (cid:3) , (12) with Γ t := − ln (1 − P ( τ ≤ t | F t )) . he general setting A typical example, which will be used in what follows, consists in considering a Coxprocess, hence taking ρ to be an exponential function of the form ρ t ( s ) := e − R st β ( r ) dr , for a suitable function β . In this particular case we have thatΓ s := − ln (cid:16) e − R st β ( r ) dr (cid:17) = Z st β ( r ) dr , so that the equation (12) reads E [ X T ≤ τ | G s ] = { τ>s } e R st β ( r ) dr E h Xe − R Tt β ( r ) dr (cid:12)(cid:12)(cid:12) F s i . We can thus prove the following result. Hypothesis . Let us assume that τ is a Cox process, namely it is of the form ρ t ( s ) := e − R st β ( r ) dr , (13)with intensity given by β . Remark . Notice that we could have assumed a more general assumption, often denotedin literature as density hypothesis , requiring that there exists a process β such that P ( τ ∈ ds | F t ) = β ( s ) , see, e.g. [7]. Theorem 2.8. Let F be a G -adapted process and let us assume τ to be a Cox process definedas in equation (13) , then it holds E " Z τ ∧ Tt F r dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t = { τ>t } Z Tt E h e − R rt β ( s ) ds F r (cid:12)(cid:12)(cid:12) F t i dr . Proof. Exploiting (2.5) together with (13) we have that E " Z τ ∧ Tt F r dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t = Z Tt E (cid:2) { τ>r } { τ>t } F r (cid:12)(cid:12) G t (cid:3) dr == { τ>t } Z Tt E h { τ>r } F r e R t β ( s ) ds (cid:12)(cid:12)(cid:12) F t i dr == { τ>t } Z Tt e R t β ( s ) ds E (cid:2) E (cid:2) { τ>r } (cid:12)(cid:12) F r (cid:3) F r (cid:12)(cid:12) F t (cid:3) dr == { τ>t } Z Tt e R t β ( s ) ds E h e − R r β ( s ) ds F r (cid:12)(cid:12)(cid:12) F t i dr == { τ>t } Z Tt E h e − R rt β ( s ) ds F r (cid:12)(cid:12)(cid:12) F t i dr , and this completes the proof.Let us then denote the impulse control for this system by u = ( τ , τ , ..., τ j , ... ; K , K , ..., K j , ... ) ∈ U , where 0 ≤ τ ≤ τ ≤ ... are G t stopping times and K j ∈ A is G τ j - measurable for all j, forany u ∈ U , then, using (2.6) together with (2.8), the corresponding functional in equation(7) can be rewritten as J u ( t, x ) = E t (cid:20) Z Tt ρ t ( s ) (cid:16) f ( X ( s )) − β ( s ) g ( X ( s )) (cid:17) ds + ρ t ( T ) g ( X ( T ))+ (14) − X t ≤ τ n ≤ T ρ t ( τ n ) ( K n + κ ) (cid:21) , so that the original stochastic control problem, with random terminal time, turns out to bea stochastic control problem with deterministic terminal time.n the regularity of the value function Remark . A different approach would be to consider τ to be F t –adapted, for instance ofthe form τ = inf { t : x ( t ) ≤ } , which implies that the hypothesis (11) is no longer satisfied and, consequently, the abovementioned techniques cannot be exploited any longer. Nevertheless, under this setting it ispossible to recover a HJB equation endowed with suitable boundary conditions. We refer to[24, 39], for a mathematical treatment of this type of stochastic control problems, while in[35, 36] one can find applications to the mathematical finance scenario. Theorem 2.10. (Dynamic programming principle) Let ( t, x ) ∈ [0 , T ] × R , then it holds V ( t, x ) = sup u ∈ U [ t,T ] E (cid:20) Z θt ρ t ( s ) (cid:16) f ( X ( s )) − β ( s ) g ( X ( s )) (cid:17) ds + − X t ≤ τ n ≤ θ ρ t ( τ n ) ( K n + κ ) + ρ t ( θ ) V ( θ, X ut,x ( θ )) (cid:21) , for any stopping time θ valued in [ t, T ] .Proof. See, e.g., [39, 40].For simplicity, we define the following functions c ( t, s, x ) = ρ t ( s )( f ( x ) − β ( s ) g ( x )) with s ≥ t,g ( t, x ) = ρ t ( T ) g ( x ) , (15)which will be used throughout the paper. The present section is devoted to prove regularity properties of the value function. Inparticular the next two Lemmas prove respectively that the value function is bounded,Lipschitz continuity in space and − H¨older continuity in time of the value function V . Lemma 3.1. Let us assume that (2.2) holds, then there exist constants C , C such that C ≥ V ( t, x ) ≥ − C (1 + | x | ) . Proof. For simplicity, in what follows, for any fixed ( t, x ) ∈ [0 , T ] × R and u ∈ U [ t, T ], wewill denote for short X ut,x ( s ), resp. ξ t ( s ) by X ( s ), resp. ξ ( s ). Then by Gronwall’s inequalitywe have 1 + | X ( s ) | ≤ | x | + | ξ ( s ) | + (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) + C Z st (1 + | X ( r ) | ) dr ≤ | x | + | ξ ( s ) | + (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) ++ C R st e C ( s − r ) (cid:0) | x | + | ξ ( r ) | + (cid:12)(cid:12)R rt σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:1) dr ≤ C (cid:20) | x | + | ξ ( s ) | + Z st | ξ ( r ) | dr ++ (cid:12)(cid:12)R st σ ( r, X ( r )) dW r (cid:12)(cid:12) + R st (cid:12)(cid:12)R rt σ ( r, X ( r )) dW r (cid:12)(cid:12) dr (cid:21) , thus E | X ( s ) | ≤ C (cid:26) E | x | + E | ξ ( s ) | + E Z st | ξ ( r ) | dr ++ E (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) + E Z st (cid:12)(cid:12)(cid:12)(cid:12)Z rt σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) dr (cid:21) . (16)n the regularity of the value function On the Other hand, under (2.2), we have E (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) + E (cid:20)Z st (cid:12)(cid:12)(cid:12)(cid:12)Z rt σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) dr (cid:21) ≤ E (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) ! / + ( s − t ) / Z st E (cid:12)(cid:12)(cid:12)(cid:12)Z rt σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) dr ! / = (cid:18)Z st E | σ ( r, X ( r )) | dr (cid:19) / + ( s − t ) / (cid:18)Z st Z rt E | σ ( r, X ( r )) | drdr (cid:19) / ≤ [1 + ( s − t )] (cid:18)Z st E | σ ( r, X ( r )) | dr (cid:19) / ≤ C ( ( s − t ) / + (cid:18)Z st E | X ( r ) | dr (cid:19) / ) ≤ C (cid:26) Z st E | X ( r ) | dr (cid:27) , (17)where we have exploited both the Jensen’s and H¨older’s inequality, several times. Hence itfollows that E | X ( s ) | ≤ C (cid:26) E | x | + E | ξ ( s ) | + E Z st | ξ ( r ) | dr + Z st E | X ( r ) | dr (cid:27) ≤ C (cid:26) E | x | + E | ξ ( s ) | + E Z st | ξ ( r ) | dr (cid:27) . . Again under (2.2), we achieve that | J u ( t, x ) | ≤ Z Tt C (1 + E | X ( s ) | ) ds + C (1 + E | X ( T ) | ) − X t ≤ r n ≤ T ρ t ( r n ) ( K n + κ ) ≤ C | x | + E | ξ ( T ) | + E Z Tt | ξ ( r ) | dr ! For the trivial control u = ξ t ( . ) ≡ 0, one has that V ( t, x ) ≥ J u ≥ − C (1 + | x | ) for all( t, x ) ∈ [0 , T ] × R. (18)which proves the lower bound of the value function.The boundedness of c ( t, s, x ) , g ( t, x ), immediately gives us that value function is bounded,i.e. there exists C > V ( t, x ) ≤ C . (19) Lemma 3.2. If (2.2) holds, the value function V ( t, x ) is Lipschitz continuous in x , and –H¨older continuous in t , namley there exists a constant C > such that, ∀ t , t ∈ [0 , T ) , x , x ∈ R , | V ( t , x ) − V ( t , x ) | ≤ C (cid:16) | x − x | + (1 + | x | + | x | ) | t − t | (cid:17) , Proof. Again, for simplicity, for any admissible control u ∈ U [ t, T ], we denote for short X ut,x , resp X ut,x by X t,x , resp X t,x dropping the explicit dependence on the control u .Notice that, applying the Itˆo-D¨oblin formula to | X t,x ( s ) − X t,x ( s ) | , and using Gronwall’slemma, we can infer that E | X t,x ( s ) − X t,x ( s ) | ≤ C | x − x | , ∀ s ∈ [ t, T ] , x , x ∈ R . Therefore, by (2.2), for any fixed t ∈ [0 , T ) and all x , x ∈ R and u ∈ U [ t, T ], | J u ( t, x ) − J u ( t, x ) | ≤ E Z Tt | c ( t, s, X t,x ( s )) − c ( t, s, X t,x ( s )) | ds + (20)+ | g ( t, X t,x ( T )) − g ( t, X t,x ( T )) | (21) ≤ L E Z Tt | X t,x ( s ) − X t,x ( s ) | ds + C | X t,x ( T ) − X t,x ( T ) | (22) ≤ C | x − x | , n the regularity of the value function which implies that V ( t, x ) ≤ J u ( t, x ) ≤ J u ( t, x ) + C | x − x | , and thus V ( t, x ) ≤ V ( t, x ) + C | x − x | . By interchanging x and x , we get | V ( t, x ) − V ( t, x ) | ≤ C | x − x | . For the time regularity, first we show that E | X t,x ( s ) − x − ξ t ( s ) | ≤ C (cid:16) (1 + | x | )( s − t ) + E (cid:18)Z st | ξ t ( s ) ds | (cid:19) (cid:17) . (23)For notation simplicity, we suppress the subscripts t,x for X t,x , ξ t and define z ( s ) = X ( s ) − x − ξ ( s ) . Then by (2.2), we have | z ( s ) | ≤ C Z st (1 + | X ( r ) | ) dr + (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ C Z st (1 + | x | + | z ( r ) | + | ξ ( r ) | ) dr + (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) . By Gronwall’s inequality, we achieve | z ( s ) | ≤ C (cid:20) (1 + | x | )( s − t ) + Z ts | ξ ( r ) | dr ++ (cid:12)(cid:12)(cid:12)(cid:12)Z st σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) + Z st (cid:12)(cid:12)(cid:12)(cid:12)Z rt σ ( r, X ( r )) dW r (cid:12)(cid:12)(cid:12)(cid:12) dr (cid:21) Using (17) and again Gronwall’s, we further get E | z ( s ) | ≤ C (cid:20) (1 + | x | )( s − t ) + Z ts E | ξ ( r ) | dr + ( s − t ) + Z st E | X ( r ) | dr (cid:21) ≤ C (cid:20) (1 + | x | )( s − t ) + Z ts E | ξ ( r ) | dr + ( s − t ) ++ Z st E (1 + | x | + | ξ ( r ) | + | z ( r ) | ) dr (cid:21) ≤ C (cid:20) (1 + | x | )( s − t ) + Z ts E | ξ ( r ) | dr (cid:21) , which proves (23).For all p ∈ [0 , ∞ ), define the control space U p [ t, T ] = u ∈ U [ t, T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E X t ≤ r i 0, there exists u ∈ U | x | [ t , T ), such that ε + V ( t , x ) ≥ J u ( t , x ) . Then we define the impulse controls ˆ u , ¯ u ∈ U [ t , T ) byˆ ξ t ( s ) = ξ t ( s ) ∀ s ≥ t , ¯ ξ t ( s ) = ξ t ( s ) − ξ t ( t ) ∀ s ≥ t . Notice that ˆ u is the impulse control such that at the initial time t , there is a impulse ofsize ξ t ( t ) and ¯ u is the impulse control mimicing all the impulses in ξ t ( · ) on [ t , T ). Bydenoting ¯ x = x + ξ t ( t ), which is F t adapted, we have that J ˆ u ( t , x ) = J ¯ u ( t , ¯ x ) − ( ξ t ( t ) + κ ) , and thus ε + V ( t , x ) ≥ J ˆ u ( t , x ) + E Z Tt ( c ( t , s, X t ,x ( s )) − c ( t , s, X t , ¯ x ( s )))+ E [ g ( t , X t ,x ( T )) − g ( t , X t , ¯ x ( T ))] ds + X t ≤ r i 0, we obtain V ( t , x ) ≥ V ( t , x ) − C (1 + | x | )( t − t ) . Adding (25), we finally get the − H¨older continuity in time, i.e. | V ( t , x ) − V ( t , x ) | ≤ C (1 + | x | ) | t − t | . An application of an ad hoc dynamic programming principle (2.10), see, e.g., [39, 40],leads to the following quasi–variational inequality (QVI). min h − ∂∂t V ( t, x ) − L V ( t, x ) − f ( x ) + β ( t )( V ( t, x ) + g ( x )) , V ( t, x ) − I V ( t, x ) i = 0 , on [0 , T ) × R ,V ( T, x ) = g ( x ) , on { T } × R , (27) with I being the non–local impulse operator defined as I V ( t, x ) := sup K ∈ A ( t,x ) [ V ( t, x + K ) − ( K + κ ))] . We underline that the problem (27) identifies two distinct regions: the continuationregion C = { ( t, x ) ∈ [0 , T ) × R : V ( t, x ) > I V ( t, x ) } , and the impulse region or action region A = { ( t, x ) ∈ [0 , T ) × R : V ( t, x ) = I V ( t, x ) } . Let us consider the following function space. Definition 4.0.1. (Space of polynomially bounded functions). PB = PB ([0 , T ] × R ) is the space of all measurable function u : [0 : T ] × R → R such that | u ( t, x ) | ≤ C u (1 + | x | p )for some constant p > C u > 0, independent of t, x .Let us introduce in what follows the definition of viscosity solution to the QVI, see eq.(27), within in the general setting (possibly not continuous). Definition 4.0.2. A function V ∈ PB is said to be a viscosity solution to the QVI (27) ifthe following two properties hold:iscosity solution to the Hamilton–Jacobi–Bellman equation a function V ∈ PB is said to be a viscosity supersolution tothe QVI (27) if ∀ (ˆ t, ˆ x ) ∈ [0 , T ] × R and φ ∈ C , ([0 , T ] × R ) with0 = ( V ∗ − φ ) (ˆ t, ˆ x ) = min ( t,x ) ∈ [0 ,T ) × R ( V ∗ − φ ) , it holds min h − ∂∂t φ ( t, x ) − L φ ( t, x ) − f ( x ) + β ( t )( φ ( t, x ) + g ( x )) , V ∗ ( t, x ) − I V ∗ ( t, x ) i ≥ , on [0 , T ) × R min [ V ∗ ( T, x ) − g ( x ) , V ∗ ( T, x ) − I V ∗ ( T, x )] ≥ , on { T } × R , ; (ii) viscosity subsolution a function V ∈ PB is said to be a viscosity subsolution to theQVI (27) if ∀ (ˆ t, ˆ x ) ∈ [0 , T ] × R and φ ∈ C , ([0 , T ] × R ) with0 = ( V ∗ − φ ) (ˆ t, ˆ x ) = max ( t,x ) ∈ [0 ,T ) × R ( V ∗ − φ ) , it holds min h − ∂∂t φ ( t, x ) − L φ ( t, x ) − f ( x ) + β ( t )( φ ( t, x ) + g ( x )) , V ∗ ( t, x ) − I V ∗ ( t, x ) i ≤ , on [0 , T ) × R min [ V ∗ ( T, x ) − g ( x ) , V ∗ ( T, x ) − I V ∗ ( T, x )] ≤ , on { T } × R , ; (iii) viscosity solution a function V ∈ PB is said to be a viscosity solution to the QVI(27) if it is both a viscosity supersolution and a viscosity subsolution .In order to prove that the value function V is the viscosity solution to equation (27), wefirst need the following Lemma 4.1. Let (2.2) holds, then we have V ( t, x ) ≥ I V ( t, x ) , for all t ∈ [0 , T ) , x ∈ R .Proof. Reasoning by contradiction, we first suppose that there exists ( t, x ) ∈ S := [0 , T ) × [0 , + ∞ ), such that V ( t, x ) < I V ( t, x ) , i.e., V ( t, x ) < sup K ∈ A V ( t, x + K ) − ( K + k ) , then there exists also ǫ > K ∈ A , such that V ( t, x ) < V ( t, x + ˆ K ) − ( ˆ K + k ) − ǫ . On the other hand, according to equation (6), there exists u ∈ U [ t, T ] such that J u ( t, x + ˆ K ) > V ( t, x + ˆ K ) − ǫ . Defining now ˆ u = ˆ ξ t ( · ) . = ˆ K + ξ t ( · ), we have V ( t, x ) ≥ J ˆ u ( t, x ) = J u ( t, x + ˆ K ) − ( ˆ K + k ) . Combining all the estimates above, we have V ( t, x + ˆ K ) − ( ˆ K + k ) − ǫ > V ( t, x ) > V ( t, x + ˆ K ) − ( ˆ K + k ) − ǫ , from which we have the desired contradiction. Remark . (4.1) implies that we are considering I V ( t, x ) as a lower obstacle, which isgiven in implicit form, since it depends on the value function V itself.iscosity solution to the Hamilton–Jacobi–Bellman equation The value function V ( t, x ) is a viscosity solution to the QVI (27) on [0 , T ] × R , in the sense of (4.0.2) .Proof. (3.2) implies that the value function is continuous. Therefore the lower–semicontinuous,resp. upper–semicontinuous, envelop of V in (4.0.2), does in fact coincide with V . Also, itis an immediate consequence of (3.1) that V ∈ PB . Let us prove that V ( t, x ) is a viscosity sub-solution of (14). By (4.1), we know that V ( t, x ) ≥ I V ( t, x ), so that in what follows we only need to show that given ( t , x ) ∈ [0 , T ) × R such that V ( t , x ) > I V ( t , x ) , (28)then for every φ ( t, x ) ∈ C , ([0 , T ] × [0 , + ∞ )) and every t , x ∈ [0 , + ∞ ), such that φ ≥ V for all ( t, x ) ∈ S ∩ B r (( t , x )) and V ( t , x ) = φ ( t , x ), we want to show that − ∂∂t φ ( t , x ) − L φ ( t , x ) − f ( x ) + β ( t ) (cid:0) φ ( x ) + g ( x ) (cid:1) ≤ . (29)In fact, if V ( t , x ) ≤ I V ( t , x ), then (29) immediately follows.Choose ǫ > u = ( τ , τ , ... ; K , K , ... ) ∈ U [ t , T ] be a ǫ - optimal control, i.e., V ( t , x ) < J u ( t , x ) + ǫ. Since τ is a stopping time, { ω, τ ( ω ) = t } is F t - measurable, thus τ ( ω ) = t a.s. or τ ( ω ) > t a.s. If τ = t a.s., X ut ,x takes a immediate jump from x to the point x + K and we have J u ( t , x ) = J u ′ ( t , x + K ) − ( K + k ) where u ′ = ( τ , τ , ... ; K , K , ... ) ∈ U [ t , T ]. Thisimplies that V ( t , x ) ≤ J u ′ ( t , x + K ) − ( K + k ) + ǫ < V ( t , x + K ) − ( K + k ) ≤ I V ( t , x ) + ǫ, which is a contradiction for ǫ < V ( t , x ) − I V ( t , x ). Thus (28) implies that τ > t a.s for all ǫ - optimal controls such that ǫ < V ( t , x ) − I V ( t , x ). For any impulse control u = ( τ , τ , ... ; K , K , ... ) ∈ U [ t , T ], defineˆ τ . = τ ∧ ( t + r ) ∧ inf n t (cid:12)(cid:12)(cid:12) t > t , | x ( t ) − x | ≥ r o . By the dynamic programming principle, for any ǫ > 0, there exists a control u such that V ( t , x ) ≤ E t ,x "Z ˆ τt n ρ t ( s ) (cid:0) f ( X ( s )) + β ( s ) g ( x ( s )) (cid:1)o ds + e − R ˆ τt β ( s ) ds V (ˆ τ , X (ˆ τ ))) + ǫ. (30)By (30) and the Dynkin formula we have that V ( t , x ) ≤≤ E t ,x hR ˆ τt n ρ t ( s ) (cid:0) f ( X ( s )) + β ( s ) g ( X ( s )) (cid:1)o ds + e − R ˆ τt β ( s ) ds φ (ˆ τ , X (ˆ τ ))) i + ǫ = E t ,x "Z ˆ τt ( ρ t ( s ) (cid:16) f ( X ( s )) + β ( s ) (cid:0) g ( X ( s )) − φ ( s, X ( s )) (cid:1) + (cid:17)) ds + E t ,x "Z ˆ τt ( ρ t ( s ) (cid:16) ∂∂t φ ( s, X ( s )) + L φ ( s, X ( s )) (cid:17)) ds + φ ( t , x ) + ǫ. (31)Using that V ( t , x ) = φ ( t , x ), we further obtain E t ,x (cid:20) Z ˆ τt ( ρ t ( s ) (cid:16) f ( X ( s )) + β ( s ) (cid:0) g ( X ( s )) − φ ( s, X ( s )) (cid:1) + (32)+ ∂∂t φ ( s, X ( s )) + L φ ( s, X ( s )) (cid:17)) ds (cid:21) + φ ( t , x ) ≥ − ǫ. iscosity solution to the Hamilton–Jacobi–Bellman equation Divide both sides of (32) by E (ˆ τ − t ) and let r → 0, we further get that f ( x ) + β ( t ) (cid:0) g ( x ) − φ ( t , x ) (cid:1) + ∂∂t φ ( t , x ) + L φ ( t , x ) ≥ − ǫ. Since ǫ > − f ( x ) + β ( t ) (cid:0) φ ( t , x ) + g ( x ) (cid:1) − ∂∂t φ ( t , x ) − L φ ( t , x ) ≤ , (33)then V ( t, x ) is a viscosity sub-solution.To prove that V ( t, x ) is also a viscosity super-solution of (14), let us consider φ ∈ C , ( S ), and any ( t , x ) ∈ S such that φ ≤ V on B r ( t , x ) and φ ( t , x ) = V ( t , x ).Taking the trivial control u = 0 ( no interventions ), calling the corresponding trajectory X ( t ) = X u ( t ) with x ( t ) = x , and defining ˆ τ = ( t + r ) ∧ inf n t (cid:12)(cid:12)(cid:12) t > t , | x ( t ) − x | > r o ,then, by the dynamic programming principle and the Dynkin formula, we have V ( t , x ) ≥ E t ,x "Z ˆ τt n ρ t ( s ) (cid:0) f ( X ( s )) + β ( s ) g ( X ( s )) (cid:1)o ds + e − R ˆ τt β ( s ) ds φ (ˆ τ , X (ˆ τ ))) = E t ,x "Z ˆ τt ( ρ t ( s ) (cid:16) f ( x ( s )) + β ( s ) (cid:0) g ( X ( s )) − φ ( s, X ( s )) (cid:1)(cid:17)) ds + E t ,x "Z ˆ τt ( ρ t ( s ) (cid:16) ∂∂t φ ( s, X ( s )) + L φ ( s, X ( s )) (cid:17)) ds + φ ( t , x ) + ǫ. Using V ( t , x ) = φ ( t , x ), we obtain that E t ,x (cid:20) Z ˆ τt ( ρ t ( s ) (cid:16) f ( X ( s )) + β ( s ) (cid:0) g ( X ( s )) − φ ( s, X ( s )) (cid:1) + (34) ∂∂t φ ( s, X ( s )) + L φ ( s, X ( s )) (cid:17)) ds (cid:21) ≤ − ǫ. Divide both sides of (34) by E [ˆ τ − t ] and let r → 0, we obtain − f ( x ) + β ( t ) (cid:0) φ ( t , x ) + g ( x ) (cid:1) − ∂∂t φ ( t , x ) − L φ ( t , x ) ≥ . (35)Since we have already proved that V ( t, x ) ≥ I V ( t, x ), we finally conclude thatmin (cid:20) − ∂∂t φ ( t , x ) − L φ ( t , x ) − f ( x ) + β ( t ) (cid:0) φ ( x ) + g ( x ) (cid:1) , (36) , V ( t , x ) − I V ( t , x ) (cid:21) ≥ . (37)Combining (29) and (36), we have that v ( t, x ) is a viscosity solution of (14). it is worth tomention that the terminal condition is non trivial. In fact, it has to take into account thatjust right before the horizon time T , the controller might act by an impulse control. To thisextent we have to specify that the terminal condition in equation (27) is to be intended as V ( T, x ) := lim ( t,x ′ ) → ( T − ,x )) V ( t, x ′ ) . Since (4.1) implies that V ( t, x ) ≥ I V ( t, x ) for all ( t, x ) ∈ [0 , T ) × R , in the limit one has V ( T, x ) ≥ I V ( T, x ) for all x ∈ R . To show the boundary conditionmin (cid:8) V ( T, x ) − g ( x ) , V ( T, x ) − I V ( T, x ) (cid:9) = 0 , (38)one first consider all the x ∈ R such that V ( T, x ) > I V ( T, x ). For any sequence ( t n , x n ) → ( T, x ) with ( t n , x n ) ∈ [0 , T ) × R , by continuity one has V ( t n , x n ) > I V ( t n , x n ) for all n iscosity solution to the Hamilton–Jacobi–Bellman equation large enough. Then for each ε > u n ∈ U [ t n , T ] suchthat V ( t n , x n ) ≤ J u n ( t n , x n ) + ε. It then suffices to show that E "Z Tt n ρ t n ( s ) (cid:16) f ( X u n t n ,x n ( s )) − β ( s ) g ( X u n t n ,x n ( s )) (cid:17) ds + (39)+ E ρ t n ( T ) g ( X u n t n ,x n ( T )) − X t n ≤ τ j ≤ T ρ t n ( τ j ) ( K j + κ ) → g ( x )as n → + ∞ . Notice that since V ( t, x ′ ) > I V ( t, x ′ ) for all ( t, x ′ ) in a neighborhood of ( T, x ),for all δ > P (cid:16) sup s ∈ [ t n ,T ] | X u n t n ,x n ( s ) − x n | < δ (cid:17) → n → ∞ . Suppose that there exists F ∈ L ( P ; R ) such that Z Tt n ρ t n ( s ) (cid:16) | f ( X u n t n ,x n ( s )) | + β ( s ) | g ( X u n t n ,x n ( s )) | (cid:17) ds + ρ t n ( T ) | g ( X u n t n ,x n ( T )) | + | X t n ≤ τ j ≤ T ρ t n ( τ j ) ( K j + κ ) | ≤ F for all n large enough, an application of dominant convergence theorem proves (39). Sowe conclude that for any ( T, x ) such that V ( T, x ) > I V ( T, x ), one has V ( T, x ) ≤ g ( x ) + ε, ∀ ε > ⇒ V ( T, x ) ≤ g ( x ) . By a similar approach, one can show that V ( T, x ) ≥ g ( x ) for all x ∈ R . This completes theproof of (38).We are now to show that the value function is the unique viscosity solution to equation(27) based on a comparison principle. In order to do that let us introduce a differentdefinition of viscosity solution, see, e.g. [28], based on the notion of jets. Definition 4.3.1. Let V : [0 , T ] ∈ PB a upper–semicontinuous function, then we define P , + V ( s, x ) = n ( p, q, M ) ∈ R × R × R : V ( s, y ) ≤ V ( t, x ) + p ( t − s ) + q ( x − y ) + 12 M ( x − y ) + o ( | t − s | + | x − y | ) o ¯ P , + V ( s, x ) = { ( p, q, M ) ∈ R × R × R : ∃ ( t n , x n ) ∈ [0 , T ] × R : ( p n , q n , M n ) ∈ P , + V ( t n , x n ) , ( t n , x n , V ( t n , x n ) , p n , q n , M n ) → ( t, x, V ( t, x ) , p, q, M ) } . For lower–semicontinuous function V , we define P , − V ( s, x ) := − P , + − V ( s, x ) , ¯ P , − V ( s, x ) := − ¯ P , + − V ( s, x ) . We can therefore state the equivalence between the two notion of viscosity solution statedbefore. Proposition 4.4. A function V ∈ PB is a viscosity sub, resp. super, solution to equation (27) if and only if ∀ ( p, q, M ) ∈ ¯ P , + V ( s, x ) , resp. ¯ P , − V ( s, x ) , min (cid:20) − p − µ ( t, x ) q − σ ( t, x ) M − f ( x ) + β ( t )( V ( t, x ) + g ( x )) , (40) , V ( t, x ) − I V ( t, x ) (cid:21) ≤ ≥ . (41)iscosity solution to the Hamilton–Jacobi–Bellman equation (Comparison principle) . Suppose that (2.2) is satisfied and that U and V are, repectively, a viscosity super solution and viscority sub solution to the equation (27) .Assume also that U and V are uniformly continuous, then V ≤ U on [0 , T ] × R .Proof. Let us prove the result by contradiction, assuming thatsup [0 ,T ] × R ( V − U ) = η > . For r > V ( t, x ) := e rt V ( x, t ) , ˜ U ( t, x ) := e rt U ( t, x ) . From the theorem hypotheses, that is U and V are viscosity super and sub solution toequation (27), we immediately have that ˜ V and ˜ U are viscosity super and sub solution to min (cid:20) ru ( t, x ) − ∂∂t u ( t, x ) − L u ( t, x ) − e rt f ( x ) + e rt β ( t )( V ( t, x ) + g ( x )) ,, u ( t, x ) − ˜ I u ( t, x ) (cid:21) = 0 , on [0 , T ) × R u ( T, x ) = e rt g ( x ) , on { T } × R , , with ˜ I being the non–local impulse operator defined as˜ I u ( t, x ) := sup K ∈ A ( t,x ) (cid:2) u ( t, x + K ) − e rt ( K + κ )) (cid:3) . Let us then assume that for x ∈ R we have that˜ V ( T, x ) − ˜ U ( T, x ) > , and from the fact that ˜ U is a viscosity super solution, resp. ˜ V is a viscosity sub solution,we have that it exists ¯ x such that˜ U ( T, ¯ x ) < ˜ I ˜ U ( T, ¯ x ) , resp. ˜ V ( T, ¯ x ) > ˜ I ˜ V ( T, ¯ x ) . Since we also have that ˜ V ( T, ¯ x ) ≤ e rt g (¯ x ) and ˜ U ( T, ¯ x ) ≥ e rt g (¯ x ), we conclude that˜ V ( T, ¯ x ) − ˜ U ( T, ¯ x ) ≤ , which contradict the assumptions.Then suppose that there exists (¯ t, ¯ x ) ∈ [0 , T ) × R , such that˜ V (¯ t, ¯ x ) − ˜ U (¯ t, ¯ x ) > , then, analogously to what we have derived above, we have that˜ U ( t, x ) < ˜ I ˜ U ( t, x ) , resp. ˜ V ( t, x ) > ˜ I ˜ V ( t, x ) , for t ∈ I δ := [¯ t − δ, ¯ t + δ ] and x ∈ B δ := [¯ x − δ, ¯ x + δ ].Therefore taking ( t , x ) ∈ I δ × B δ , such thatsup I δ × B δ ˜ V − ˜ U = ( ˜ V − ˜ U )( t , x ) > , and considering ϕ n ( t, x, y ) := ˜ V ( t, x ) − ˜ U ( t, x ) − ̺ n ( t, x, y ) , with ̺ n ( t, x, y ) = n | x − y | + | x − x | + | t − t | , for any n ∈ N there exist a point ( t n , x n , y n ) attaining the maximum of ϕ , so that, up to asubsequence, we have˜ V ( t n , x n ) − ˜ U ( t n , x n ) → ˜ V ( t , x ) − ˜ U ( t , x ) , as n → ∞ . (42)mooth fit principle on the value function Moreover, since˜ V ( t , x ) − ˜ U ( t , x ) = ϕ n ( t , x , x ) ≤ ϕ n ( t n , x n , x n ) , then ˜ V ( t , x ) − ˜ U ( t , x ) ≤ lim inf n →∞ ϕ n ( t , x , y ) ≤ lim sup n →∞ ϕ n ( t , x , y ) ≤≤ ˜ V (¯ t, ¯ x ) − ˜ U (¯ t, ¯ x ) − lim inf n →∞ n | x − y | + | x − x | + | t − t | . Therefore, using the optimality of ( x , t ), we obtain that, considering up to a subse-quence it holds ( t n , x n , y n ) → ( t , x , x ) and n | x n − y n | → p nV , q nV , M nV ) ∈ ¯ P , + ˜ V ( t n , x n ) and( p nU , q nU , M nU ) ∈ ¯ P , − ˜ U ( t n , x n ), such that p nV − p nV = 2( t n t ) ,q nV = ∂ x ̺ n , q nU = − ∂ y ̺ n , and (cid:18) M n − N m (cid:19) ≤ A n + 12 n A nn , with A n = ∂ xy ̺ n . Therefore from the viscosity sub-solution property of ˜ V , resp. theviscosity super-solution property of ˜ U , by the Lipschitz continuity of µ and σ in x and(4.3.1) we have that r (cid:16) ˜ V ( t , x ) − ˜ U ( t , x ) (cid:17) ≤ , which gives the desired contradiction.We are now able to state the uniqueness result for the viscosity solution. Corollary 4.6. Let (2.2) hold true, then there exists a unique viscosity solution to equation (27) .Proof. Let V and V two viscosity solution to equation (27); then since V is a subsolutionand V is a supersolution, by comparison principle (4.5) we obtain that V ≤ V . Since itmust also holds the opposite we obtain the claim. Under further regularity assumptions on the coefficients, to be further specified in a while,one can prove the regularity property of the value function, with particular reference tothe smooth-fit property through the switching boundaries between action and continuationregions. This results, known as smooth–fit principle , see, e.g, [27, 22, 25], has already beenproven to hold in the infinite horizon case. Also, we will prove W (1 , ,ploc regularity for thevalue function V ( t, x ) on any fixed parabolic domain Q T . = ( δ, T ] × B R (0) for any constants0 < δ < T, R > W (0 , ,p (Ω) = { u ∈ L p (Ω) : u x i ∈ L p (Ω) } ,W (1 , ,p (Ω) = { u ∈ W (0 , ,p (Ω) : u x i x j ∈ L p (Ω) } ,C α , α ( ¯Ω) = n u ∈ C ( ¯Ω) : sup ( x,t ) , ( y,s ) ∈ Ω , ( x,t ) =( y,s ) | u ( t,x ) − u ( s,y ) | ( | t − s | + | x − y | ) α/ < + ∞ o ,C α , α ( ¯Ω) = (cid:8) u ∈ C ( ¯Ω) : u t , u x i x j ∈ C α , α (cid:9) ,W (1 , ,ploc (Ω) = (cid:8) u ∈ L ploc (Ω) : u ∈ W (1 , ,p ( U ) ∀ open U with ¯ U ⊂ ¯Ω \ ∂ P Ω (cid:9) . (43).1 Structure of the value function The above notations are similar to the notations used in [26].Recall that β ( t ) is the hazard rate function defined in (13) and we make the followingassumption: Hypothesis . Let α ∈ (0 , β ( t ) ∈ C α/ ([0 , T ])and σ ( s, x ) ∈ C α, α ( ¯ Q T ) satisfying the uniform elliptic condition, i.e. σ ( s, x ) ≥ δ > δ > Q T .Before proceeding to the smooth fit principle, recall that we divide the region [0 , T ] × R into the following regions: C . = { ( t, x ) : V ( t, x ) > I V ( t, x ) } , A . = { ( t, x ) : V ( t, x ) = I V ( t, x ) } and for any open set Ω ∈ R , the parabolic boundary ∂ P Ω is defined as ∂ P Ω . = (cid:8) ( t, x ) ∈ ¯Ω | ∀ ε > , Q (( t, x ) , ε ) contains points not in Ω (cid:9) , where Q (( t , x ) , r ) . = { ( t, x ); | x − x | < r, t < t } for all ( t , x , r ) ∈ R × R + . For any( t, x ) ∈ A , define the setΘ( t, x ) = { ξ | I V ( t, x ) = V ( t, x + ξ ) − ξ − κ } . Notice that in the regularity analysis in Section 2, we already show that V ( T − t, x ) ∈ C / , (Ω), so we immediately have the following lemma. Lemma 5.2. (Theorem 4.9,5.9,5.10,and 6.33 in [34]) Under (2.2) and (5.1) , for any openset Ω ⊆ C , the linear parabolic PDE u t − L u ( t, x ) + ˜ β ( t ) u ( t, x ) = ˜ f ( t, x ) in Ω ,u ( t, x ) = V ( T − t, x ) , on ∂ P Ω . (44) admits a unique solution u ( t ) ∈ C α/ , α ( ¯Ω) ∩ C α/ , αloc (Ω) where ˜ β ( t ) = β ( T − t ) , ˜ f ( t, x ) = f ( x ) − β ( t ) g ( x ) . Theorem 5.3. (Smooth fit principle) Under (2.2) and (5.1) , the value function V ( t, x ) is a unique W (1 , ,ploc ( R × (0 , T )) viscosity solution to the QVI (27) for any < p < + ∞ .Furthermore, for any t ∈ [0 , T ) , V ( t, · ) ∈ C ,γloc ( R ) for any < γ < .Proof. Using the cost function B ( K ) . = K + κ, ∀ K > , which is independent of time and satisfies the subadditivity property, i.e. B ( K + K ) + κ = B ( K ) + B ( K ) , ∀ K , K > . (45)so that the claim follows from [26] together with (5.2). In this subsection, we study the general property of the value function V ( t, x ) underfurther assumptions of σ ( t, x ), β ( t ), µ ( t, x ), ˜ f ( t, x ) and g ( x ). Hypothesis . ˜ f ( t, x ) and g ( x ) are monotonically increasing withlim x →−∞ ˜ f ( t, x ) = lim x →−∞ g ( x ) = −∞ , lim x → + ∞ ˜ f ( t, x ) = U ( t ) > , lim x → + ∞ g ( x ) = U g < ∞ . .1 Structure of the value function Under (5.4) , for any t > the value function V ( t, x ) satisfies V ( t, x ) ≤ V ( t, x ) ∀ x ≤ x . Furthermore, there exists L ∈ [ −∞ , + ∞ ) such that [0 , T ] × ( L, + ∞ ) ⊂ C . Proof. First, we show the monotonicity of V ( t, x ) with respect to x. By applying the sameadapted control u ∈ U [ t, T ] with different initial values x ≤ x , the solutions satisfies X ut,x ≤ X ut,x a.s . Since ˜ f ( t, x ) is increasing with respect to x, one has J u ( t, x ) ≤ J u ( t, x )for all u ∈ U [ t, T ] , and thus V ( t, x ) ≤ V ( t, x ) for any x ≤ x .It remains to show that there exists L ∈ [ −∞ , + ∞ ) such that for any fixed t > x > L , ( t, x ) ∈ C . Fix any t ∈ (0 , T ), suppose that there exists a sequence x < x < ... < x k < ... such thatlim k → + ∞ x k = + ∞ and ( t, x k ) ∈ A , ∀ k > , and for any k > ξ k ∈ Θ( t, x k ) such that V ( t, x k ) = V ( t, x k + ξ k ) − ξ k − κ. (46)However, since V ( t, x ) is monotone, uniformly Lipschitz continuous in x and upper boundedby C according to (3.1) and (3.2), for any ε > L large enough such that V ( t, x + ξ ) − V ( t, x ) ≤ ε, ∀ x > L, ∀ ξ > , contradicted to (46). Notice that since such choice of L is independent of t , we concludethat there exists L ∈ [ −∞ , + ∞ ) such that [0 , T ] × ( L, + ∞ ) ⊆ C . Lemma 5.6. For any ( t , x ) ∈ A , the set Θ( t , x ) is nonempty and ( t , x + ξ ) ∈ C forany ξ ∈ Θ( t , x ) .Proof. Since V is uniformly bounded, one haslim ξ → + ∞ V ( t , x + ξ ) − B ( ξ ) = −∞ , lim ξ → + V ( t , x + ξ ) − B ( ξ ) = V ( t , x ) − κ. Then the condition V ( t , x ) = I V ( t , x ) implies that the supremum in I V ( t , x ) isachieved in the interior and thus Θ( t , x ) is nonempty.By property (45), for any ξ ∈ Θ( t , x ) one has I V ( t , x ) = sup ξ ∈ R + { V ( t , x + ξ ) − B ( ξ ) }≥ sup ξ ∈ R + { V ( t , x + ξ + ξ ) − B ( ξ + ξ ) } = sup ξ ∈ R + { V ( t , x + ξ + ξ ) − B ( ξ ) } − B ( ξ ) + κ = I V ( t , x + ξ ) − B ( ξ ) + κ. On the other hand, since I V ( t , x ) + B ( ξ ) = V ( t , x + ξ ), we have V ( t , x + ξ ) ≥ I V ( t , x + ξ ) + κ, which implies that x + ξ ∈ C . Lemma 5.7. Fix any ( t , x ) ∈ A and for any ξ ∈ Θ( t , x ) , one has V x ( t , x ) = V x ( t , x + ξ ) = 1 . .1 Structure of the value function Proof. By the definition of Θ( t, x ), ξ is a global maximum of the function ξ V ( t , x + ξ ) − B ( ξ ). 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