A bank salvage model by impulse stochastic controls

The present paper is devoted to the study of a bank salvage model with finite time horizon and subjected to stochastic impulse controls. In our model, the bank's default time is a completely inaccessible random quantity generating its own filtration, then reflecting the unpredictability of the event itself. In this framework the main goal is to minimize the total cost of the central controller who can inject capital to save the bank from default. We address the latter task showing that the corresponding quasi-variational inequality (QVI) admits a unique viscosity solution, Lipschitz continuous in space and Holder continuous in time. Furthermore, under mild assumptions on the dynamics the smooth-fit $W^{(1,2),p}_{loc}$ property is achieved for any $1


Introduction
Mainly motivated by the recent financial credit crisis, starting from 2008-2009 credit crunch, the financial and mathematical community started investigating and generalize existing models, since previous events have shown that financial models used prior to the crisis where inadequate to describe and capture main features of financial markets. Therefore the mathematical and financial communities have focus on developing general and robust models that are able to properly describe financial markets and their main peculiarities.
From a purely mathematical perspective the above mentioned attention led, among many other research topics, for instance into the study of general stochastic optimal control problems, where instead of classical type of controls, some more realistic controls have been considered. 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 economics. In this setting, the controller can intervene on the system at some random time with a discrete type control, where in this case the control solution is represented by the couple u = (τ n , K n ) n , where τ n represents the decision time at which the ocntroller intervene and K n instead denotes the action taken by the controller. Above type of control implies that at the intervention time τ n the system jumps from the state X(τ − n ) to the new state X(τ n ) = Γ(X(τ n ), K n ), for a suitable function Γ. Therefore, as standard in optimal control theory, using the dynamic programming principle, it can be shown that stochastic impulse control problems can be associated to a quasi-variational Hamilton-Jacobi-Bellman equation (HJB) of the form 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 Introduction impulse operator that characterize HJB equation for impulse type of control. The particular form 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 the contrary V = H V and the controller intervenes. Solution to equation (1) can be formally defined so that the value function is in fact a viscosity solution, in a sense to be properly defined later on, to equation (1). It is clear that, following the above characterization of the domains for the HJB equation, particular attention must be given to the intervention boundary. In fact, particular attention is usually given in this field to proving that the boundary 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 control problem, whereas instead finite horizon problem, and in particular the terminal condition of 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 optimal stopping problems and optimal switching. The literature on the topic is wide, we refer the interested reader to [26,45], or also to [4,5,12,13,21,25,38,42,46] for other related results.
A second crucial financial aspect that emerged to be fundamental in a general financial formulation after recent crisis there is possible failures of financial entities. In fact, one of the major lack of classical financial models is that no risk of failure is considered into the general setting. Recent financial event has shown that no financial operator can be considered immune from bankruptcy. Therefore it has emerged in last decade an extensive literature that focus on credit risk modeling, assessing as main object the risk that financial entities 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 the underlyng process. Typical example are default triggered by some stopping time defined as a hitting time. Such an approach has been for instance considered in [14,15,35]. The latter instead considers a default event which is completely inaccessible for the probabilistic reference 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 impulse type, where a financial supervisor controls a system, such as financial operators or also some banks. The final goal of the controller is to prevent failures, injecting capital intro the system according to a given criterion to be maximized. The controller has no perfect information regarding the failure of the bank, so that mathematically speaking the failure cannot be foreseen by the controller. The supervisor, which can be though for instance as a central bank, can intervene with some impulse type controls over a finite horizon, so that the optimal solution is represented by both am intervention time and the quantity to inject into the system.
Our approach will be based on a intensity-based approach, so that we will assume the default event to be totally inaccessible from the reference filtration, assuming only a typical density assumption. This assumptions will allows us to rewrite the system as deterministic 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 imposed, typically a finite horizon stochastic impulse control problem is more difficult to solve than infinite horizon impulse control problems. In fact, it exists an exhaustive literature on stochastic 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 present work, has often arise in the last decade, mostly as a consequence of the 2007-2008 credit crunch. This has been for instance the case of Lehman Brothers failure, which has shown the cascade effect triggered by the default of a sufficiently large and interconnected financial institution, see, e.g., [29,30] and references therein. We stress that, particular attention has to be given not only towards the magnitude of the stressed bank's financial assets, but also to its interconnection grade. Indeed, while the exposure with few financial institutions, 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 high number of connections, hidden links and over-structured contracts.
Since above mentioned financial crisis, it has became typical, within the financial oriented stochastic optimal control theory, to model a given problem up to a random terminal time instead of considering a fixed, even infinite, horizon. From a modelling point of view, the aforementioned scenario has lead to consider the stochastic optimal control approach to model such situations by considering random terminal times, instead of considering a fixed, or infinite, horizon. Analogously, data analysts as well as mathematicians, have started to consider problems of bank bailouts, where bank's default and the consequent contagion spreading 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 pushed several central banks to establish specific economic actions to help those sectors of the banking sector of (at least) national interest, under concrete failure risks. As an example, the latter has been the case of the pro bail-in procedures followed in agreement with the Directive 2014/59/UE (approved last 1 st of January, 2016 by the European Union Parliament), and then applied, e.g., in Italy, Ukraine, etc., see, e.g. [33,44]. It is relevant to underline that such actions rely also on the following grades of freedom: the possibility, as an alternative to internal rescue, to relocate goods as well as legal links to a third party, often called bridge-bank, or to a bad bank which will collect only a part of assets aiming at maximizing its long-term value; the hierarchical order of those who are called to bear the bail-in, which means that the government can decide to put small creditors on the safe side; and the principle that no shareholder, or creditor, has to bear greater losses than would be expected if there was an administrative liquidation, namely the no worse off creditor idea.
Similar situations have been recently taken into consideration by a series the Central European 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 would like to underline that quantitative easing type procedures have been experienced also outside the 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 the US Federal Reserve not only starting from 2008, but also during the Great Depression of the 1930s, see, e.g., [48,49,50,51,52].
The main contribution of the present paper is to develop a concrete financial setting that models the evolution of a financial entity, controlled by an external supervisor who is willing to 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 general financial entities. In compete generality, we will assume that the financial entity may fail at some random time that is inaccessible to the reference filtration, which represents the controller knowledge. Also, we consider a controller that can act on a system with an impulse-type control, so that the optimal solution consists in both a random time at which injecting money into the system and the precise amount of money to inject. We characterize the value function of the above problem, showing that it must solve in a given viscosity sense a certain quasi-variational inequality (QVI). At last we will prove that above QVI admits a unique solution in a viscosity sense and also we provide a regularity results for the intervention boundary, known in literature as smooth fit principle.
The paper is so organized, Section 2 introduces the general financial and mathematical setting; then Section 3 prove some regularity results for the value function and Section 4 address the problem of existence and uniqueness of a solution. At last Section 5 is devoted to the smooth fit principle.

The general setting
We will in what follows consider a complete filtered probability space Ω, F , (F t ) t∈[0,T ] , P , (F t ) t∈[0,T ] being a filtration satisfying the usual assumptions, namely right-continuity and saturation by P-null sets. Let T < ∞ be a fixed terminal time, and let x, resp. y, denotes the total value of the investments of a given bank, resp. the total amount of deposit of the same bank. We assume that x and y evolve according to the following system of SDEs where W (t) is assumed to be a standard Brownian motion adapted to the aforementioned filtration. In particular, the first term in equation (2) accounts for the increase in X due to the fact that new deposits are made, where c 1 ∈ [0, 1] denotes the fractions of deposits which are actually invested in more or less risky financial operations. We stress that by a rescaling argument, with no less of generality c 1 = 1 it can be assumed. Moreover we define the value over liability ratio X(t) := x(t) y(t) . Then, according to eq. (2) and exploiting the Itô-Döblin formula, we have We assume the process X to be stopped at completely inaccessible random time τ , not adapted to the reference filtration (F t ) t∈[0,T ] . From a financial point of view, assuming that X represents the financial value of an agent, above assumption reflects the fact that a bank'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 := ½ {τ ≤t} . Then we define the augmented 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 surveillance rights. Such controller can inject capital in the bank, at random times τ n . Then, at that time τ n , the state process X(t) jumps, in particular we have therefore X(t) evolves according to The solution to the aforementioned system is represented by a couple u = (τ n , K n ) n≥1 , where (τ n ) n≥1 is a non-decreasing sequence of stopping times representing the intervention times, while (K n ) n≥0 is a sequence of (G t )-adapted random variables taking values in A ⊂ [0, ∞). In particular the sequence (K n ) n≥0 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 of all the impulse controls u = (τ n , K n ) n≥0 such that {τ i } i≥1 are G t adapted stopping times and increasing, i.e τ 1 < τ 2 < · · · < τ i < · · · , Remark 2.1. Equivalently, we will use a different notation, ξ t (·) to express the same space, In what follows we will denote for short so that, for any admissible control u ∈ U [t, T ], define X u t,x (s) as which is the unique strong solution of dynamics (4) with initial condition X u t,x (t) = x. We aim at solving the following stochastic control problem whose value function is defined as where J u (t, x) is the expected cost of the form where f , resp. g, represents the running cost, resp. the terminal cost, while K + κ, κ > 0, is a suitable constant defining the cost required by the capital injection. Above, we have denoted by τ the bank default time, with respect to the process X(t). We assume, as specified above, that τ is a completely inaccessible random time, and it is not adapted to the reference filtration Following the standard literature, see, e.g., [32] both the running and terminal costs are usually given in terms of suitable utility functions representing the utility gains from the bank's value. A typical example is f (x) = x p p , p ∈ (0, 1). As regards the cost K + κ, it reflects 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: (ii) the functions f , g 1 , g 2 are Lipschitz continuous, namely there exist constants L f , l g1 and L g2 > 0 such that We also assume that there exist constants C f , C g1 and C g2 > 0 such that (iii) the functions µ(t), σ(t) ∈ C([0, T ]).
The boundedness properties for the running and terminal cost can be interpreted in the following sense: since we are seeking the optimal capital injection strategy for the government over a finite time horizon, we may think that there is a healthy level U > 0 such that when the bank's capital is growing to infinity, then the utility remains flat, so that the government will have no interest in injecting more capital. As to make an example, we can take Remark 2.3. A further generalization of the above optimal control problem, consists in considering a controller having two different ways to influence the evolution of the state process x, namely (1) an impulse type control (τ n , K n ) n , hence as in equation (4) by injecting capital at random 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 where α represents the continuous control variable α(t) ∈ [0,r], for a suitable constantr, where α = 0 stands for higher returns and α =r denotes lower returns. This reflects the financial assumption that the controller, e.g. a central bank, can change the interest rate according to macroeconomic variables, as the country inflation level, the forecast of supranational interest rates, the the markets' belief about the health of the financial sector under 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 refer to the above discussion, see also, e.g., [2,3,8,9,47,37,19], for more financially oriented ideas supporting the latter setting. Accordingly, we can assume that the controller aims at maximizing a functional of the following type In what follows we assume the following density hypothesis on the random time to hold, hence requiring that the distribution of τ is absolutely continuous with respect to the Lebesgue measure: The main idea of the following procedure is to switch from the reference filtration F t , to the default free filtration G t , by mean of the following lemma, see [7, Lemma 4.1.1]. with A typical example, which will be used in what follows, consists in considering a Cox process, hence taking ρ to be an exponential function of the form for a suitable function β. In this particular case we have that so that the equation (12) reads We can thus prove the following result. Hypothesis 2.6. Let us assume that τ is a Cox process, namely it is of the form with intensity given by β. Remark 2.7. Notice that we could have assumed a more general assumption, often denoted in literature as density hypothesis, requiring that there exists a process β such that see, e.g. [7].
Theorem 2.8. Let F be a G -adapted process and let us assume τ to be a Cox process defined as in equation (13), then it holds Proof. Exploiting (2.5) together with (13) we have that and this completes the proof.
Let us then denote the impulse control for this system by where 0 ≤ τ 1 ≤ τ 2 ≤ ... are G t stopping times and K j ∈ A is G τj -measurable for all j, for any u ∈ U , then, using (2.6) together with (2.8), the corresponding functional in equation (7) can be rewritten as − t≤τn≤T ρ t (τ n ) (K n + κ) , so that the original stochastic control problem, with random terminal time, turns out to be a stochastic control problem with deterministic terminal time.
Remark 2.9. A different approach would be to consider τ to be F t -adapted, for instance of the form τ = inf{t : x(t) ≤ 0} , which implies that the hypothesis (11) is no longer satisfied and, consequently, the above mentioned techniques cannot be exploited any longer. Nevertheless, under this setting it is possible 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.
for any stopping time θ valued in [t, T ].
For simplicity, we define the following functions which will be used throughout the paper.

On the regularity of the value function
The present section is devoted to prove regularity properties of the value function. In particular the next two Lemmas prove respectively that the value function is bounded, Lipschitz continuity in space and 1 2 −Hölder continuity in time of the value function V . Lemma 3.1. Let us assume that (2.2) holds, then there exist constants C 0 , C 1 such that Proof. For simplicity, in what follows, for any fixed (t, x) ∈ [0, T ] × R and u ∈ U [t, T ], we will denote for short X u t,x (s), resp. ξ t (s) by X(s), resp. ξ(s). Then by Gronwall's inequality we have 1 + |X(s)| ≤ 1 + |x| + |ξ(s)| + s t σ(r, X(r))dW r + C On the Other hand, under (2.2), we have E|σ(r, X(r))| 2 drdr where we have exploited both the Jensen's and Hölder's inequality, several times. Hence it follows that .
Again under (2.2), we achieve that For the trivial control u 0 = ξ t (.) ≡ 0, one has that 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 1 > 0 such that Lemma 3.2. If (2.2) holds, the value function V (t, x) is Lipschitz continuous in x, and 1 2 -Hölder continuous in t, namley there exists a constant C > 0 such that, ∀ t 1 , t 2 ∈ [0, T ), Proof. Again, for simplicity, for any admissible control u ∈ U [t, T ], we denote for short X u t,x1 , resp X u t,x2 by X t,x1 , resp X t,x2 dropping the explicit dependence on the control u. Notice that, applying the Itô-Döblin formula to |X t,x1 (s) − X t,x2 (s)| 2 , and using Gronwall's lemma, we can infer that Therefore, by (2.2), for any fixed t ∈ [0, T ) and all x 1 , x 2 ∈ R and u ∈ U [t, T ], |c(t, s, X t,x1 (s)) − c(t, s, X t,x2 (s))|ds+ (20) + |g(t, X t,x1 (T )) − g(t, X t,x2 (T ))| ≤ LE T t |X t,x1 (s) − X t,x2 (s)|ds + C|X t,x1 (T ) − X t,x2 (T )| (22) By interchanging x 1 and x 2 , we get For the time regularity, first we show that For notation simplicity, we suppress the subscripts t,x for X t,x , ξ t and define where C 0 and C 1 are the constants in (19) and (18). Notice that another important corollary of (23) is that for all u ∈ U |x| [t, T ], We claim that for all |x| ≤ p, the value function V (t, x) satisfies This is due to the fact that for any u ∈ U [t, T ]\U p [t, T ], Fix x ∈ R and 0 ≤ t 1 < t 2 < T . For any u 2 ∈ U |x| [t 2 , T ), extend the control to [t 1 , T ) by setting where X t1,x (s), resp X t2,x (s) represents Xũ 1 t1,x , resp X u2 t2,x and the second last row in (25) is achieved by exploiting (23). So we obtain that On the other hand, for any ε > 0, there exists u 1 ∈ U |x| [t 1 , T ), such that Then we define the impulse controlsû 2 ,ū 2 ∈ U [t 2 , T ) bŷ Notice thatû 2 is the impulse control such that at the initial time t 2 , there is a impulse of size ξ t1 (t 2 ) andū 2 is the impulse control mimicing all the impulses in ξ t1 (·) on [t 2 , T ). By denotingx = x + ξ t1 (t 2 ), which is F t2 adapted, we have that where X t1,x , resp X t2,x represents X u1 t1,x , resp Xū 2 t2,x . Notice that in (26), we extensively use the following inequality for all s ≥ t 2 and u 1 ∈ U |x| [t 1 , T ), where the last row is achieved by (24). Since (26) holds for all ε > 0, we obtain Adding (25), we finally get the 1 2 −Hölder continuity in time, i.e.
with I being the non-local impulse operator defined as We underline that the problem (27) identifies two distinct regions: the continuation region and the impulse region or action region Let us consider the following function space.  (i) viscosity supersolution a function V ∈ PB is said to be a viscosity supersolution to the QVI (27) (ii) viscosity subsolution a function V ∈ PB is said to be a viscosity subsolution to the (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.
Divide both sides of (34) by E[τ − t 0 ] and let r → 0, we obtain Since we have already proved that V (t, x) ≥ I V (t, x), we finally conclude that , Combining (29) and (36), we have that v(t, x) is a viscosity solution of (14). it is worth to mention that the terminal condition is non trivial. In fact, it has to take into account that just right before the horizon time T , the controller might act by an impulse control. To this extent we have to specify that the terminal condition in equation (27) is to be intended as To show the boundary condition 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 large enough. Then for each ε > 0 sufficiently small, consider controls u n ∈ U [t n , T ] such that V (t n , x n ) ≤ J un (t n , x n ) + ε.
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 different definition of viscosity solution, see, e.g. [28], based on the notion of jets.
For lower-semicontinuous function V , we define We can therefore state the equivalence between the two notion of viscosity solution stated before.
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 2,+ V (s, x), resp.P 2,− V (s, x), , Theorem 4.5 (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 that For r > 0 let us definẽ From the theorem hypotheses, that is U and V are viscosity super and sub solution to equation (27), we immediately have thatṼ andŨ are viscosity super and sub solution to withĨ being the non-local impulse operator defined as Let us then assume that for x 0 ∈ R we have that and from the fact thatŨ is a viscosity super solution, resp.Ṽ is a viscosity sub solution, we have that it existsx such that U (T,x) <ĨŨ (T,x) , resp.Ṽ (T,x) >ĨṼ (T,x) .
Since we also have thatṼ (T,x) ≤ e rt g 1 (x) andŨ (T,x) ≥ e rt g 1 (x), we conclude that which contradict the assumptions. Then suppose that there exists (t,x) ∈ [0, T ) × R, such that then, analogously to what we have derived above, we have that and considering ϕ n (t, x, y) : for any n ∈ N there exist a point (t n , x n , y n ) attaining the maximum of ϕ, so that, up to a subsequence, we havẽ Moreover, sinceṼ Therefore, using the optimality of (x 0 , t 0 ), we obtain that, considering up to a subsequence it holds (t n , x n , y n ) → (t 0 , x 0 , x 0 ) and n|x n − y n | → 0.
Applying the Ishii lemma, we have that there exists (p n V , q n V , M n V ) ∈P 2,+Ṽ (t n , x n ) and (p n U , q n U , M n U ) ∈P 2,−Ũ (t n , x n ), such that with A n = ∂ xy ̺ n . Therefore from the viscosity sub-solution property ofṼ , resp. the viscosity super-solution property ofŨ , by the Lipschitz continuity of µ and σ in x and (4.3.1) we have that which gives the desired contradiction.
We are now able to state the uniqueness result for the viscosity solution.
Proof. Let V 1 and V 2 two viscosity solution to equation (27); then since V 1 is a subsolution and V 2 is a supersolution, by comparison principle (4.5) we obtain that V 2 ≤ V 1 . Since it must also holds the opposite we obtain the claim.

Smooth fit principle on the value function
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 to the smooth-fit property through the switching boundaries between action and continuation regions. This results, known as smooth-fit principle, see, e.g, [27,22,25], has already been proven to hold in the infinite horizon case. Also, we will prove W The above notations are similar to the notations used in [26].
Before proceeding to the smooth fit principle, recall that we divide the region [0, T ] × R into the following regions: and for any open set Ω ∈ R 2 , the parabolic boundary ∂ P Ω is defined as Notice that in the regularity analysis in Section 2, we already show that V (T − t, x) ∈ C 0+1/2,0+1 (Ω), so we immediately have the following lemma. (R × (0, T )) viscosity solution to the QVI (27) for any 1 < p < +∞. Furthermore, for any t ∈ [0, T ), V (t, ·) ∈ C 1,γ loc (R) for any 0 < γ < 1. Proof. Using the cost function which is independent of time and satisfies the subadditivity property, i.e.

Structure of the value function
In this subsection, we study the general property of the value function V (t, x) under further assumptions of σ(t, x), β(t), µ(t, x),f (t, x) and g 1 (x). Proof. First, we show the monotonicity of V (t, x) with respect to x. By applying the same adapted control u ∈ U [t, T ] with different initial values x 1 ≤ x 2 , the solutions satisfies X u t,x1 ≤ X u t,x2 a.s. Sincef (t, x) is increasing with respect to x, one has J u (t, x 1 ) ≤ J u (t, x 2 ) for all u ∈ U [t, T ], and thus V (t, x 1 ) ≤ V (t, x 2 ) for any x 1 ≤ x 2 .
It remains to show that there exists L ∈ [−∞, +∞) such that for any fixed t > 0 and any x 0 > L, (t, x 0 ) ∈ C . Fix any t ∈ (0, T ), suppose that there exists a sequence x 1 < x 2 < ... < x k < ... such that lim k→+∞ x k = +∞ and (t, x k ) ∈ A , ∀ k > 0, and for any k > 0 there exists ξ k ∈ Θ(t, x k ) such that However, since V (t, x) is monotone, uniformly Lipschitz continuous in x and upper bounded by C 1 according to (3.1) and (3.2), for any ε > 0 one can choose L large enough such that contradicted to (46). Notice that since such choice of L is independent of t, we conclude that there exists L ∈ [−∞, +∞) such that [0, T ] × (L, +∞) ⊆ C .
On the other hand, for any δ = 0, we have