Two-sided Poisson control of linear diffusions

We study a class of two-sided optimal control problems of general linear diffusions under a so-called Poisson constraint: the controlling is only allowed at the arrival times of an independent Poisson signal processes. We give a weak and easily verifiable set of sufficient conditions under which we derive a quasi-explicit unique solution to the problem in terms of the minimal r-excessive mappings of the diffusion. We also investigate limiting properties of the solutions with respect to the signal intensity of the Poisson process. Lastly, we illustrate our results with an explicit example.


Introduction
In a classical stochastic singular control problems the objective is to maximise the expected discounted cumulative yield, given by a function of a stochastic process, and the maximisation takes place over controls that can be continuously applied.In these problems, the controlling can be allowed only downwards or upwards (one-sided problems) or both (two-sided problems) depending on the application.Both one-sided and two-sided singular problems have been studied extensively due to their mathematical attractiveness and applicability in various fields.These include, for example, a reversible or irreversible investment problems (cf.[29,7,10]), where the controls can be interpreted as investor purchasing capital and possibly selling it, and rational harvesting (cf.[23,5]), where the controls can be seen as harvesting and replanting.
In Poisson optimal control problems, also called constrained control problems (these terminologies are from [15] and [27]), the potential control opportunities are restricted to jump times of an independent Poisson process.The main motivation for introducing Poisson control problem is that in applications it is not often possible to apply the control continuously.For example, in [30] and [24] due to liquidity effects when trading financial assets and in [18] the possibility of modelling imperfect flow of information is considered.Nowadays the literature on Poisson type control problems is quite extensive.Some examples include optimal stopping [9,17], stopping games [21,22], ergodic control [31,19], optimal switching [20], extensions to inhomogeneous Poisson processes [12,13] and more general signal processes [27,26,28].
In the classical singular control problem when the controlling is costless the optimal policy is often a local time type reflecting barrier policy.The effect of introducing the Poisson restriction to control opportunities has usually similar effect on the optimal strategy, although for different reasons as introducing a constant transaction cost to the model.Indeed, when the controlling is costly the optimal strategy is a sequential impulse control, where the decision maker chooses a sequence of stopping times tτ 1 , τ 2 , . ..u to exert the control and corresponding impulse sizes tζ 1 , ζ 2 , . ..u, see [4,3,11].A similar strategy is optimal in the Poisson control problems, but the possible intervention times are restricted exogenously by the signal process [31,18,19].
We extend the Poisson control framework in the following way.Our problem setting is closely related to [31], [25] and [18].Similarly to [25] and [18], we assume that the underlying dynamics follow a general one-dimensional diffusion.In [31], the Poisson control problem was first introduced in the literature by considering controlling of standard Brownian motion with quadratic payoff at the jump times of independent Poisson process.These results were then extended to a more general underlying and payoff structure by [18], where one-sided problem is considered.In [25], a two-sided singular control problem is solved using the techniques from classical theory of linear diffusions and r-excessive mappings, which lead to quasi-explicit solution.Our study extends the findings of [18] by considering two-sided Poisson control policies using similar, rather easily verifiable set of assumptions as in [25].Further, our results can also be seen as a generalization to those of [25], since the results coincide when the Poisson arrival rate approaches infinity.To the best of our knowledge these results are new.
The remainder of the study is organized as follows.In Section 2 we set up the underlying dynamics and formulate the two-sided Poisson control problem.Main assumptions and auxiliary calculations are done in Section 3, whereas in Section 4 we derive a candidate solution for the problem and verify its optimality.Asymptotic results connecting the problem to the singular problem are proved in Section 5.The study is concluded by considering explicit example in Section 6.

Underlying dynamics and problem setting
Let pΩ, F, tF t u tě0 , Pq be a filtered probability space that satisfies the usual conditions.We consider an uncontrolled process X t defined on pΩ, F, tF t u tě0 , Pq, which lives in R `, and is given as a strong solution to a regular Itô diffusion where W t is the Wiener process and the functions µ : R `Ñ R and σ : R `Ñ R `are sufficiently smooth (see e.g.[14] chapter 5).The boundaries of the state space R `are assumed to be natural.Even though we consider the case where the process evolves in R `, we remark that the results would remain the same with obvious changes even if the state space would be replaced with any interval.
As usual, we define the second-order linear differential operator A which represents the infinitesimal generator of the diffusion X as and for a given r ą 0 we respectively denote the increasing and decreasing solutions to the differential equation pA ´rqf " 0 by ψ r ą 0 and ϕ r ą 0. These solutions are called fundamental solutions or minimal excessive mappings ([6] p.19, p.33).
For r ą 0, we denote by L r 1 the set of functions f on R `, which satisfy the integrability condition For any f P L r 1 , we define the functional pR r f q : R `Ñ R by This functional, called the resolvent, is the inverse of the differential operator r ´A.Also define the scale density of the diffusion by which is the derivative of the monotonic (and non-constant) solution to the differential equation AS " 0. Given the scale density and the fundamental solutions, the resolvent pR r f qpxq can be re-expressed as ( [6] p. 19) is the constant Wronskian (does not depend on x) and is the density of the speed measure.We also recall the resolvent equation ( [6] where q ą r ą 0.
Having setup the underlying dynamics we next describe the control problem.We study a maximization problem of the expected value of the cumulative payoff when the controlling of X is allowed only at the jump times of a signal process N .Assumption 1.The process N is assumed to be Poisson process with parameter λ, that is independent of X. Further, the filtration tF t u tě0 is augmented such that it is rich enough to carry the Poisson process.
We call a control policy pζ d t , ζ u t q admissible if both processes are non-negative, nondecreasing, right-continuous and can be represented as where η d and η u are tF t u-predictable processes.Thus, the controlled process X ζ t is given by Jpx, ζq, where the supremum is taken over all admissible controls and γ d and ´γu are constants, called the unit price and unit cost, respectively.The aim is also to characterize semiexplicitly the optimal control policy pζ d , ζ ů q that realizes the supremum in (2).We end this section by stating useful bounds on the value function that hold in given regions of the state space.The lemma follows directly from corollary 2.4 of [18].

Lemma 1.
piq If there exists an interval pb, 8q of the state space R `such that it is always suboptimal to use the control ζ u t , the value function satisfies V pxq ď pR r`λ π γ d qpxq `λ r sup xPR `pR r`λ θ d qpxq when x ą b. piiq If there exists an interval p0, aq of the state space R `, such that it is always suboptimal to use the control ζ d t , the value satisfies V pxq ě pR r`λ π γu qpxq `λ r inf xPR `pR r`λ θ u qpxq when x ă a.

Auxiliary results
To set up the framework further, we denote n P td, uu, and define the functions π γn : R `Ñ R and g n : R `Ñ R as g n pxq " γ n x ´pR r πqpxq, π γn pxq " λγ n x `πpxq.(4) Also, define the functions θ n : R `Ñ R as θ n pxq " πpxq `γn pµpxq ´rxq.
In the literature, the function θ n pxq is known as the net convenience yield of holding inventories cf.[1,8] and is often in a key role when determining the optimal policies [1,4,18].
The following lemma gives convenient relationships between the defined auxiliary functions.It can be proved by using the resolvent equation ( 1) and the harmonicity properties of pR r πq.Lemma 2. Let r ą 0 and g n , π γn , θ n P L r 1 .Then g n pxq " ´pR r θ n qpxq pR r`λ π γ qpxq " λpR r`λ gqpxq `pR r πqpxq λpR r`λ gqpxq " pR r`λ θ r qpxq `gpxq.
We now collect the main assumptions that are needed to prove that the solution to the control problem is well-defined and unique.
Assumption 2. Assume that: (i) γ d ă γ u , (ii) the functions θ n and id: x Þ Ñ x are in L r 1 , (iii) the payoff π is positive, continuous and non-decreasing, (iv) µ 1 pxq ă r, (v) there is a unique state x n ě 0 such that θ 1 n pxq ą " ă 0 when x ă " ą x n, (vi) θ n satisfies the limiting conditions lim xÑ0`θn pxq ą 0 and lim xÑ8 θ n pxq " ´8.Some remarks on these assumptions are in order.The converse of the assumption (i) would easily lead to infinite value functions.Item (ii) guarantees sufficient integrability so that the resolvents of the defined functions exist.The assumptions (iii), (v) and (vi) are in key role when proving the existence of the solution.These type of assumptions are quite standard in stochastic control problems, where explicit solutions are desired, see e.g.[1,25,18].The assumption (iv) can be seen as an upper bound for the Lipschitz constant for the coefficient µ, but it also guarantees together with assumption (ii) that the fundamental solutions ψ r and ϕ r are convex, see Corollary 1 in [2].
Remark 1.It follows from the assumptions 2 (i), (iv) and (v) that x ů ă x d .To see this, note that for x ą x d θ 1 u pxq " π 1 pxq `γu pµ 1 pxq ´rq ď π 1 pxq `γd pµ 1 pxq ´rq " θ 1 d pxq ď 0 Having stated the main assumptions, we define the functionals L r f : R `Ñ R and K r f : R `Ñ R for any f P L 1 r as and prove some auxiliary results.
Proof.The representation of L r`λ f is proven in lemma 2 of Lempa 2014 and the representation of K r`λ f is proven analogously.
The next two lemmas provide us with the monotonicity properties of functionals related to L r`λ f and K r`λ f , when f " θ n .The first part of the first lemma is an analogue of the proof of lemma 3 in [18] and the second part of lemma 3.1 in [4].
Lemma 5.The monotonicity of the functions r`λ pxq is determined by the signs of L r`λ θ d and K r`λ θu .
Proof.A straight differentiation and the usage of harmonicity properties of pR r πq and ϕ r give g 1 d pxqL r`λ ϕ pxq 1 ´ϕ1 r pxqL r`λ g d pxq 1 " 0. Thus, r`λ pxqq.Using Lemma 2 and resolvent equation ( 1) we see that Therefore, by the lemma 3, we arrive at 1 2 λσ 2 pxqS 1 pxqL r`λ ´gd `λ´1 θ pxq " g 2 d pxqϕ 1 r`λ pxq ´g1 d pxqϕ 2 r`λ pxq and also 1 2 λσ 2 pxqS 1 pxqL r`λ ϕ " ϕ 1 r pxqϕ 2 r`λ pxq ´ϕ2 r pxqϕ 1 r`λ pxq.Hence, the derivative reads as This implies that the monotonicity is determined only by the sign of L r`λ θ d as the sign of all the other terms are known: all the other terms are positive, except L r`λ ϕ is negative.Similarly, we can calculate that The following lemma about the minimal excessive functions ψ r and ϕ r is useful in further analysis of Q and H. Lemma 6.The minimal excessive functions ψ r pxq and ϕ r pxq satisfy the following inequalities for z ă x ă y Proof.We observe that for all s ą 0 and z ď x we have (see [6] pp. 18) Moreover, utilizing that pA ´rqψ r`λ " pA ´pr `λqqψ r`λ `λψ r`λ " λψ r`λ with the Corollary 3.2 of [3], we have ψ 1 r pxqψ r`λ pxq ´ψ1 r`λ pxqψ r pxq " ´λS 1 pxq ż x 0 ψ r pyqψ r`λ pyqm 1 pyq ď 0.
Reorganizing the above we get (6) Combining ( 5) and ( 6) yields inequalities for ψ r and the inequalities for ϕ r are proven similarly.pA ´pr `λqqF pxq " ´πpxq ´λpγ u px ´aq `F paqq,

The solution
x ď a (9) We first consider the equation (7).This equation gives where B 1 and B 2 are constants.Using the first order conditions (10) and (11), we find after some algebraic manipulation that .
Because the boundaries of the state space are natural, we find from the differential equations ( 8) and ( 9) that particular solutions to these are pR r`λ π γ d qpxq `λ λ `r pF pbq ´γd bq, x ą b pR r`λ π γu qpxq `λ λ `r pF paq ´γu aq, x ă a.Thus, by growth conditions given by lemma 1, continuity over the boundaries and first order conditions (10) and (11) we see that the general solutions can be written as where the functions π γ d , π γu are defined in (4) and the constants C and D are To solve the boundary points a and b, we first use the C 2 condition at the upper boundary point b.This reads as (12) B 1 ϕ 2 r pbq `B2 ψ 2 r pbq `pR r πq 2 pbq " Cϕ 2 r`λ pbq `pR r`λ π γ d q 2 pbq.To simplify this, we recall the definition of the functions g n (3) and lemma 2. These yield for the right-hand side of ( 12) Here, by lemma 3, the second term is an integral operator To deal with the left-hand side of ( 12), a straight forward algebraic manipulation yields `´γ d ϕ 1 r paq `γu ϕ 1 r pbq ´ϕ1 r pbqpR r πq 1 paq `ϕ1 r paqpR r πq 1 pbq ϕ 1 r pbqψ 1 r paq ´ϕ1 r paqψ 1 r pbq r pbqψ 1 r paq ´ϕ1 r paqψ 1 r pbq .
We note that ´2rS 1 pbqσ ´2pbqB r " ψ 2 r pbqϕ 1 r pbq ´ϕ2 r pbqψ 1 r pbq.Next we combine two terms from above, one from left side and one from right side of (12), yielding Again by lemma 3, we find that ϕ 1 r`λ pbqϕ 2 r pbq ´ϕ2 r`λ pbqϕ 1 r pbq " ´2λS 1 pbqσ ´2pbqL ϕ pbq and ϕ 2 r`λ pbqψ 1 r pbq ´ϕ1 r`λ pbqψ 2 r pbq " 2λS 1 pbqσ ´2pbqL ψ pbq.Finally, using the above calculations the C 2 condition (12) reads as Reorganizing the above we have By entirely similar arguments we arrive at the lower boundary to the equation Since the function F is r-harmonic in the interval pa, bq, we find by uniqueness, that the optimal thresholds should satisfy the pair of equations Using the notation defined in lemma 5 the pair can be written as # Hpψ r , g u ; aq " Qpg d , ψ r ; bq, Hpϕ r , g u ; aq " Qpg d , ϕ r ; bq.

Uniqueness and existence.
In this section we prove that an unique solution pa ˚, b ˚q to the pair of equations ( 13) exists.To this end, we use a slight generalisation of the method that was first used in [16] and [4] in a analogous setting and notice that in the solution pa ˚, b ˚q the point a ˚must necessarily be a fixed point of the function Kpxq " H ´1 u,ϕ pQ d,ϕ pQ ´1 d,ψ pH u,ψ pxqqqq, where slightly shorter notation H u,ϕ pxq " Hpϕ r , g u ; xq, H u,ψ pxq " Hpψ r , g u ; xq, Q d,ψ pxq " Qpg d , ψ r ; xq, Q d,ϕ pxq " Qpg d , ϕ r ; xq, is introduced.Thus, in order to prove that the solution exists we must first ensure that K is well-defined and then study its fixed points.
The following is the main result on the uniqueness of the solution.
has a solution pa ˚, b ˚q.Then the solution is unique.
Proof.Let K : p0, xs Ñ p0, xs be defined as (14) Kpxq " Ȟ´1 u,ϕ p Qd,ϕ p Q´1 d,ψ p Ȟu,ψ pxqqqq, where ˆand ˇare restrictions to domains rx, 8q and p0, xs respectively.We notice that if a solution pa ˚, b ˚q to the pair exists, then a ˚must be fixed point of K.Because the functions H and Q are monotonic in their domains we get ( 15) u,ψ pxq ą 0, and hence K is increasing in its domain p0, xi s.We observe using the fixed point property that Consequently, whenever K intersects the diagonal of R `, the intersection must be from above.
To prove the existence of the solution we need to study the function K in more detail.First, we analyze the limiting properties of the functions Q and H.We notice that by lemma 2 and 3 we have Carrying out similar calculations for Hpψ r , g u ; aq, Qpg d , ψ r ; bq and Hpϕ r , g u ; aq the pair of equations reads as ( 16) Assuming that x ą maxpx, x 0 d q (here x 0 d is the unique root of θ d pxq) we find using mean value theorem that And again, similarly Hpϕ r , g u ; xq ă 0. We summarize these findings: Hpψ r , g u ; 0`q " 0 Hpϕ r , g u ; 0`q ą 0, Hpϕ r , g u ; xq ă 0, Unfortunately, the analysis so far is not enough for the existence of the solution.To guarantee that the fixed point of the function K exists, the inequalities respectively, we note by ( 16) that the first inequality is equivalent to K r θ d pxq ´Kr θu pxq ď 0. By lemma 4 we have that x ă x d and x ą x ů.Thus, by our assumption x ă x we have that x ů ă x ă x ă x d .Hence, we find that where the first two inequalities follow from part pvq of assumptions 2 and last from parts piq and pivq of assumptions 2.
The assumption x ă x may seem restricting, but based on numerical calculations it cannot easily be relaxed, because there exists cases under the main assumptions 2, where x ą x and the solution does not exist.Furthermore, because the points x and x are known to be the unique roots of the functionals K r`λ θu and L r`λ θ d , it is straightforward to calculate them, at least numerically, and verify the assumption.Proposition 2. Let the assumptions 1 and 2 hold.Assume further that x ă x, then the pair of equations # Hpψ r , g u ; aq " Qpg d , ψ r ; bq, Hpϕ r , g u ; aq " Qpg d , ϕ r ; bq.
has a unique solution pa ˚, b ˚q.
Proof.Define the function K : p0, xs Ñ p0, xs as in (14).Then by lemma 7, remark 2 and limiting properties (17) the function K is well-defined.Further, by (15) K is monotonic mapping.Thus, K is a monotonic mapping from a set to its open subset, and hence it must have at least one fixed point, which we denote by a ˚.Then the pair pa ˚, b ˚q, where b ˚" Q ´1 d,ψ pH u,ψ pa ˚qq, is a solution to the equations (13).The uniqueness follow from proposition 1.

4.3.
Verification.We begin by stating the verification theorem.
Proposition 3. Let assumptions 2 hold and denote by pa ˚, b ˚q the unique solution to the necessary conditions # Hpψ r , g u ; aq " Qpg d , ψ r ; bq, Hpϕ r , g u ; aq " Qpg d , ϕ r ; bq.
Then the optimal policy is as follows.If the controlled process X ζ is not inside the interval pa ˚, b ˚q at a jump time T i of N , i.e.X ζ T i´R pa ˚, b ˚q for any i, the optimal policy is to take the controlled process X ζ to the closest boundary of the interval pa ˚, b ˚q.Moreover, the optimal value function V pxq reads as (18) V pxq " where Before proving the proposition, we show few properties of the candidate value function F .

Lemma 8.
(i) F 2 pxq ď 0 for all x P pa, bq (ii) The function x Þ Ñ F pxq ´γd x has an unique global maximum at b. Similarly, the function x Þ Ñ F pxq ´γu x has an unique global maximum at a.
Proof.The item piq is same as part (A) of lemma 4.3 in [25].
To prove the item piiq, we find by straight differentiation in (18) that when x ą b where we have used that b ą x and lemma 4. Similarly, when x ă a we find that F 1 pxq γu ą 0. Furthermore, as F 1 pa`q " γ u ą γ d " F 1 pb´q we find by item piq that we must have γ u ď F 1 pxq ď γ d , when x P pa, bq.Hence, the item piq follows by F 1 pxq ą γ u ą γ d when x ă a and F 1 pxq ă γ d ă γ d when x ą b.
Proof of proposition 3. The proof is a slight modification of the proof of theorem 3.6 in [18].Define the almost surely finite stopping times τ :" ρ ^τρ , where τ ρ " inftt ě 0 : X ζ t ě ρu and let x P R `. Applying generalised Ito's lemma to the stopped process e ´rpt^τ q F pX ζ t^τ q we get ( 19) To prove that the candidate value is attainable with admissible policy we show that F pxq ď Jpx, ζ ˚q.We first note that as the integrand in M t^τ is continuous and the stopped process X ζ t^τ is bounded and thus M t^τ is a martingale.Furthermore, because Φ b pxq ě ´F pxq.Since π γ d pxq and ϕ r`λ pxq are in L r 1 we observe by resolvent equation ( 1) that ( 21) is bounded uniformly from above by integrable random variable and consequently is a submartingale.Treating the other integral term in Z t^τ similarly, we see that Z t^τ is a submartingale.As the inequality (20) is equality for the proposed optimal control, we get by taking expectations that By lemma 8 we find that E x re ´rpt^τ pρqq F pX ζ t^τ pρq qs ď E x re ´rpt^τ pρqq pF pb ˚q `γpX ζ t^τ pρq ´b˚q qs.
Hence, we must have lim inf τ,ρÑ8 E x re ´rpt^τ q F pX ζ t^τ qs " 0, as otherwise we would contradict the assumption that id P L r 1 .Consequently, V pxq " Jpx, ζ ˚q.Remark 3. We note that the above verification theorem and uniqueness of the optimal pair in proposition 1, do not require most of our assumptions, and these are only needed to prove the existence of the solution.Thus, other type of assumptions could be allowed as long as one can be sure that the solution exists.Unfortunately, it seems very hard to find general conditions that cover interesting cases.A hint to this direction is given in p. 252 of [25], where the shape of the functions θ n are somewhat relaxed, but then more restrictive assumptions about the boundary behaviour of the diffusion are needed.

Note on Asymptotics
A similar control problem, where controlling is not restricted by a signal process is studied in [25].In this singular control case, we know that under the assumptions 2, the optimal control thresholds pa s , b s q are the unique solution to the pair of equations L r θu pa s q " L r θ d pb s q, K r θu pa s q " K r θ d pb s q.
Intuitively, this solution should coincide with ours when the signal rate λ tend to infinity, because then the controlling opportunities are more and more frequent.The next proposition verifies this intuition.
Proposition 4. Let K λ pxq be as in ( 14) and define a function k : p0, a s s Ñ p0, a s s as (see [25] p. 248) kpxq " Ľr ´1 θu p Lr θ d p Kr ´1 θ d p Kr θu pxqqqq, where ˆand ˇare restrictions to domains rx d , 8q and p0, x ůs.Then the unique fixed point a ˚of K λ converges to the unique fixed point a s of k as λ tends to infinity.
Hence, we observe from the representation of the pair of equations ( 16) and monotonicity that lim Also, a rather straightforward calculation of the limit λ Ñ 0 in p18q yields V pxq " pR r πqpxq for all x P R `.This result corresponds to the case where the signal process does not jump at all and thus there is no opportunities to control the underlying process.Hence, the reward that the controller gets is the resolvent pR r πqpxq, i.e. the expected cumulative present value of the instantaneous payoff π.

Illustration: geometric Brownian motion
We assume that the underlying diffusion is a standard geometric Brownian motion and thus the infinitesimal generator reads as where µ P R `and σ P R `are given constants.Furthermore, the scale density and the density of the speed measure read as Assume that µ ă r and denote Then the minimal r-excessive functions for X read as ψ r pxq " x β 0 , ϕ r pxq " x α 0 , ψ r`λ pxq " x β λ , ϕ r`λ pxq " x α λ .
Define the instantaneous payoff by πpxq " x δ , 0 ă δ ă 1.Then the net convenience yield is given by θ n pxq " x δ ´γn pr ´µqx, where n P tu, du and γ d ă γ u .We readily verify that our assumptions 2 hold in this case.The resolvent reads as pR r πqpxq " x δ r ´δµ ´1 2 σ 2 δpδ ´1q .
that on one hand increased uncertainty (in terms of decreasing signal rate λ) shrinks the inactivity region by hastening the usage of control policies, but on the other hand increased uncertainty (in terms of increasing volatility σ) expands the inactivity region.Finally, the value function of the problem is shown in Figure 2.

4. 1 .
The associated free boundary problem.Similar to[9,18,19], we can use straightforward heuristic arguments to formulate a free boundary problem for the candidate value function of the control problem.Denoting the positive twice continuously differentiable candidate value function by F and two constant boundaries by a and b, where a ă b, the heuristics give us the free boundary problem pA ´rqF pxq " ´πpxq, a ă x ă b (7) pA ´pr `λqqF pxq " ´πpxq ´λpγ d px ´bq `F pbqq,x ě b (8)

Lemma 7 .
Hpψ r , g u ; xq ď Qpg d , ψ r ; xq, Hpϕ r , g u ; xq ď Qpg d , ϕ r ; xq have to hold.Our assumptions are not enough for these inequalities to hold in general, but the next lemma gives an easily verifiable sufficient condition.Assume that x ă x.Then Hpψ r , g u ; xq ď Qpg d , ψ r ; xq, Hpϕ r , g u ; xq ď Qpg d , ϕ r ; xq Proof.We prove the first inequality as the second is shown similarly.Because x and x are zeros of K r`λ θu and L r`λ θ d
This proves the claim.Remark 2. Lemma 4 and 5 together show that there exists two points x ă x d and x ą x Qpg d , ψ r ; xq is increasing on p0, xq and decreasing on px, 8q, (ii) Hpψ r , g u ; xq is increasing on p0, xq and decreasing on px, 8q, (iii) Qpg d , ϕ r ; xq is decreasing on p0, xq and increasing on px, 8q, (iv) Hpϕ r , g u ; xq is decreasing on p0, xq and increasing on px, 8q.
Define the functions Φ b pxq " 1 txąbu ppF pxq ´γd xq ´pF pbq ´γd bqq, Φ a pxq " 1 txăau ppF pxq ´γu xq ´pF paq ´γu aqq.Because the control can jump only if the Poisson process jumps we have by lemma 8 that