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Long-run risk sensitive impulse control

In this paper we consider long-run risk sensitive average cost impulse control applied to a continuous-time Feller-Markov process. Using the probabilistic approach, we show how to get a solution to a suitable continuous-time Bellman equation and link it with the impulse control problem. The optimal strategy for the underlying problem is constructed as a limit of dyadic impulse strategies by exploiting regularity properties of the linked risk sensitive optimal stopping value functions.

1. Introduction. In this paper we consider long-run risk sensitive impulse control problem where the Feller-Markov process X = (X t ) starting in x is controlled by impulse strategies V for a reward function f and a shift-cost function c. The impulse strategies V = (τ i , ξ i ) i∈N are described by the sequences (τ i ) and (ξ i ) indicating impulse times and after-impulse states of the process, respectively, and X τ − i denotes the state of X right before the impulse. Hence, up to time τ 1 the process X evolves according to its usual dynamics, then it is shifted to ξ 1 and starts its evolution again. Moreover, we assume that f and c are continuous and bounded, while shifts take values in a compact set. Expectation E (x,V ) is defined on a probability space corresponding to the controlled process; see Section 2 for details. We refer to [13,16,19] where similar framework has been studied.
The main aim of this paper is to find optimal control that minimise (1.1). The optimal strategy is constructed as a limit of dyadic impulse strategies by exploiting results from [16] and [6], i.e. by combining discrete time existence results that are based on the span-contraction approach with regularity properties of risk sensitive optimal stopping value functions. More explicitly, using change of measure technique based on the Multiplicative Poisson Equation (MPE) solution, we establish a link between impulse control Bellman equation and the optimal stopping problem considered in [6]. As expected, the strategy is characterised by relation between optimal values of the dyadic problems and the type of semigroup associated with the underlying Markov process. To get the characterisation result, we show that the finite time horizon equivalent of (1.1) could be rewritten as a limit of problems in which dyadic strategies with finite number of impulses are considered; this could be of independent interest to the reader.
The risk sensitive control could be considered as a non-linear extension of the classical risk-neutral case; see [23,13] and references therein. In particular, its application to finance have been extensively studied in the literature; see e.g. [3,9,15].
While impulse control is among the most popular forms of control, its coverage in the risk-sensitive case is limited. In particular, the application of standard methods from the risk-neutral case often leads to difficult problems involving quasi variational inequalities and differential equations techniques; see [2,10,4,1] and references therein. In this paper we follow the probabilistic approach and extend the results from [18], where the risk-neutral optimal stopping problem is linked with impulse control; see also [14,11,12,5,7]. This paper is organised as follows: In Section 2 we establish the framework and the linked notation that will be used throughout the paper. Section 3 discusses the dyadic case; this section incorporates the results from [16]. In Section 4 we consider finite time horizon version of the problem. Next, in Section 5 we solve problem (1.1); Theorem 5.1 and Theorem 5.3 might be seen as the main contribution of this paper.
2. Preliminaries. Let (Ω, F , F, P) be a continuous time filtered probability space. We assume that F := (F t ) t∈T , T := R + , F 0 is trivial, F = t∈T F t , and the usual conditions are satisfied. We use X := (X t ) to denote a continuous-time standard Markov process taking values in a locally compact separable metric space E endowed with a metric ρ and Borel σ-field E; see Definition 4 in [20,Section 1.4]. Throughout the paper, C(E) denotes the set of continuous and bounded functions on E, while C 0 (E) ⊂ C(E) denotes functions that are additionally vanishing at infinity. We assume that X satisfies the C 0 -Feller property, i.e. the transition semigroup of X transforms C 0 (E) into itself; we refer to [6, Proposition 2.1] for a discussion of properties implied by the Feller property.
As in (1.1), with slight abuse of notation, we also use X to denote a controlled process. For any V = (τ i , ξ i ) i∈N , we assume that the sequence of stopping times (τ i ) is increasing while the shifts (ξ i ) are adapted to the filtration and take values in a compact set U ⊆ E, i.e. that ξ i is F τi -measurable and ξ i ∈ U , for any i ∈ N. For brevity, we use V to denote the set all (admissible) control strategies; see [16] for details. For any x ∈ E and V ∈ V we consider controlled process probability space that is constructed following the logic from [17]. In a nutshell, we consider a countable product of canonical spaces of càdlàg functions with values in E, and with the inductively defined filtration. For t ∈ [τ i−1 , τ i ) we set X t = x i−1 t , where x i−1 t corresponds to (i − 1)th coordinate of the canonical process; see [21, Section 2] and [13] for more details. For clarity, we use P (x,V ) and E (x,V ) to denote the probability measure and expectation operator corresponding to particular choice of x ∈ E and V ∈ V. For brevity, we also use E x and P x in reference to the uncontrolled process dynamics.
Our aim is to maximise the long-run version of the risk sensitive criterion with negative (risk averse) parameter γ < 0; see [16] for details. After risk aversion parameter standarisation, our objective is to minimise the functional where x ∈ E is the starting point, f : E → R and c : E × U → R are the (normalised) reward and shift-cost functions, X τ − i is the state of the process right before the ith impulse, and V ∈ V.
Let T denote the family of almost surely finite stopping times taking values in T, and let T m ⊂ T denote the dyadic stopping times defined on the time-grid {0, δ m , 2δ m , . . .}, where δ m := (1/2) m and m ∈ N. By V m ⊂ V we denote the family of impulse control strategies with stopping times restricted to T m .
In order to solve (2.1), we show the existence of a function w ∈ C(E) and a constant λ ∈ R that satisfy Bellman equation where the operator M : Of course, one needs additional assumptions imposed on the process X and related functions in order to have a proper non-degenerate solution to (2.2) that could be linked to the original problem (2.1). Our approach to solve (2.2) is based on the dyadic approximations for the linked (discrete) stopping time problem; see [16]. Based on [16] and [19], we introduce the following set of assumptions: (A.1) (Reward function constraints.) The function f : E → R is continuous and bounded. (A.2) (Cost function constraints.) The function c : E × U → R is continuous and bounded. Moreover, we assume that the cost function is bounded away from zero by c 0 > 0 and satisfies triangle inequality, i.e.
Also, c satisfies uniform limit at infinity condition, i.e.
is the type of the semigroup P f t given by Assumption (A.1) is a classic reward function contraint and directly reflects assumption (A.1) in [16].
Assumption (A.2) imposes multiple restrictions on the cost function and partly reflects assumption (A.2) in [16]. For completeness, let us provide exemplary class of functions which satisfy all conditions from (A.2). Let h : R + → R + be bounded, continuous, non-decreasing and subadditive, i.e. h(x + y) ≤ h(x) + h(y), x, y ≥ 0. Then, the function given by where ρ is the underlying E-space metric, satisfies assumption (A.2). For example, we can set h 1 (x) = x ∧ K, h 2 (x) = x 1+x , or h 3 (x) := 1 1+e −x . Assumption (A.3) corresponds to assumption (A.4) in [16] and refers to ergodic properties of the underlying uncontrolled process X; note that in our setting assumption (A.3) from [16] is trivially satisfied, as we act in the bounded framework.
Assumption (A.4) requires the existence of a solution to the Multiplicative Poisson Equation (MPE) which is tightly linked to the problem (2.2). Exemplary sufficient condition for the existence of v include existence of density p t with respect to some probability ν t , for t > 0, such that and 0 < a t ≤ p t (x, y) ≤ A t , for 0 < a t ≤ A t < ∞ and x, y ∈ E; we refer to [22,Corollary 2]  Before we proceed, let us provide further comment about the implications of Assumption (A.4). In fact, it allows us to apply change of measure technique that substantially simplifies problem (2.2) by reducing it to the optimal stopping setting considered in [6]. Assuming (A.4) holds, for any x ∈ E we can define the measure Q x via a Radon-Nikodym derivative t 0 (f (Xs)−r(f ))ds+v(Xt) , t ≥ 0; note that it can be easily shown that Y t (x) is a martingale with E x [Y t (x)] = 1. For brevity, by analogy to E x , we use E Q x to denote the expectation with respect to Q x . By introducing measure Q x we are able to reduce the problems of the type of (2.2) to the situation, where (f (·) − λ) is replaced by a constant. This is based on identity that holds for any x ∈ E, λ ∈ R, G ∈ C(E), and τ ∈ T .
3. Dyadic impulse control. In this section we study the dyadic version of the optimal impulse control problem introduced in (2.1). Recall that for any m ∈ N and time-step δ m = 1 2 m , the corresponding family of dyadic stopping times taking values in {0, δ m , 2δ m , . . .} is denoted by T m . For any m ∈ N, the dyadic version of the Bellman equation (2.2) is given by where w m ∈ C(E), λ m ∈ R, and M is defined in (2.3). In fact, due to the dyadic nature of the problem, one could consider the associated one-step equation given by We now show that, under suitable assumptions, solution to (3.2) exists and solves the dyadic version of the optimal control problem, i.e.
where V m is the set od admissible control strategies defined on the time-grid spanned by δ m ; note we will link the solution of (3.2) to (3.1) in Proposition 3.2.
, for any m ∈ N there exists a unique (up to an additive constant) function w m ∈ C(E) and a constant λ m ∈ R satisfying (3.2). Moreover, λ m is the optimal value of the problem (3.3).
Proof. The proof relies on the results presented in [16]; it should be noted that while in [16,Equation 2.5] the maximisation problem is considered, it directly corresponds to standardised problem (3.3) due to negative sign of the risk aversion parameter in the entropic utility measure. Also, note that in our setting all assumptions listed in [16, Section 2] are satisfied. Please recall that [16, Assumption A.3] is naturally satisfied in the bounded framework, while other assumptions have a direct link to assumptions stated in this paper; see Section 2 for details.
First, from [16, Proposition 3.4] we get that there exists a unique (up to an additive constant) function w m ∈ C(E) and a constant λ m ∈ R satisfying equation Second, let us show that For brevity, we use notation From (3.4), we get Z(x) ≥ e wm(x) for any x ∈ E. Therefore, for any x ∈ E, we get On the other hand, due to (2.4) and (3.4), we get which proves (3.5). Combining (3.4) and (3.5) we know that λ m and w m are solutions to (3.2). Finally, using [16,Proposition 4.3] we can link the constant λ m from (3.2) with the optimal value of the dyadic control problem (3.3), which concludes the proof.
Next, we link the solution of (3.2) with the type of the semigroup r(f ); see (A.4) for details.
Proof. Let us fix m ∈ N. First, we show that for any x ∈ E, we get λ m ≤ r(f ). Using [22,Proposition 1], the type of the semigroup may be rewritten as Thus, for any x ∈ E and no impulse strategy V 0 ∈ V, we get Using Theorem 3.1, we know that for λ m we get (3.7) we get r(f ) = J(x, V 0 ), which implies that the no impulse strategy is optimal.
Next, let us assume that λ m < r(f ). To show that w m defined in (3.2) satisfies (3.1) we use the change of measure technique based on the Multiplicative Poisson Equation.

Finite horizon impulse control.
Before we present how to extend dyadic results presented in Section 3 to generic continuous time setting, we need to show some auxiliary results for the finite time horizon version of (2.1). Namely, in this section we consider where T ≥ 0, x, ∈ E, V ∈ V, and the remaining notation is aligned with (2.1); note that for simplicity we dropped the normalising constant 1/T in (4.1). For any n, m ∈ N, let V n ⊂ V and V n m ⊂ V m denote the families of impulse control strategies and δ m -dyadic impulse control strategies that have at most n impulses, respectively.
The main goal of this section is to show that the optimal value of (4.1) could be approximated using strategies from V n m ; see Proposition 4.2. This will be later used in the proof of Theorem 5.3.
Let us fix T ∈ N and define the family (w n ) n∈N of functions w n : [0, T ] × E → R, which are given recursively by Similarly, for any m ∈ N, we define the family (w n m ) n∈N as Before we link w n and w n m to (4.1), let us state an auxiliary result that may be seen as an extension of Proposition 4.2 and Proposition 4.3 from [6]. To ease the notation we set , and x ∈ E.
Lemma 4.1. For any n ∈ N, the map (t, x) → w n (t, x) is jointly continuous. Furthermore, if n ≥ 1, then the optimal stopping time for w n (t, x) is given by Proof. For transparency, we split the proof into two steps: (1) proof of joint continuity of (t, x) → w 0 (t, x); (2) proof of joint continuity of (t, x) → w n (t, x) and optimality of τ n t , for n ≥ 1.
Step 1. We prove that (t, x) → w 0 (t, x) is jointly continuous. Noting that | ln z − ln y| ≤ 1 min(z, y) |z − y| and |e z − e y | ≤ e max(z,y) |z − y|, for z, y > 0 and z, y ∈ R, respectively, and using boundedness of f , we get for t, s ∈ [0, T ], y ∈ E, and L := e 2T f f . By Part 4 of [6, Proposition 2.1] we get that the map x → w 0 (t, x) is continuous for any fixed t ∈ [0, T ]. Thus, for any sequence ((t k , x k )) k∈N converging to (t, x), we get as k → ∞.
Step 2. We show that (t, x) → w n (t, x) is jointly continuous and τ n t is the optimal stopping time for w n (t, x), for n ≥ 1. Let ( Ω, F , F) be a filtered product space, where Ω := [0, ∞) × Ω, the σ-field is given by F := B[0, ∞) ⊗ F , and the filtration Using standard argument one can show that X is a standard Markov process on the filtered probability space ( Ω, F , F, P) that takes values in ( E, E) and satisfies C 0 ( E)-Feller property; the measure P is given via the kernel transition functions corresponding to the family ( , and x ∈ E. Note that P (h,x) might be linked to the distribution of X starting from (h, x). We refer to [20, Section 1.4.6] for a discussion of a similar homogenisation procedure.
To show joint continuity of (t, x) → w n (t, x) we use induction argument. For n = 0, the continuity of (t, x) → w 0 (t, x) was shown in Step 1. For n ≥ 1 let us assume that that (t, x) → w n−1 (t, x) is jointly continuous. Then, we get g ∈ C( E) and G n ∈ C( E). Using Proposition 4.3, Remark 4.4, and Remark 4.5 from [6], we get that (t, h, x) → v n (t, h, x) is jointly continuous and the optimal stopping time for v n (t, h, x) is given by Also, we get v n (t, h, x) = ln v n (t, h, x) and v n (t, 0, x) = w n (t, x), which follows from definition of P (h,x) . Thus, from joint continuity of (t, h, x) → v n (t, h, x), we get joint continuity of (t, x) → w n (t, x). Also, recalling (4.5), we get τ n t (0, ω) = τ n t (ω) for any ω ∈ Ω. Noting that τ n t (0, ·) is the optimal stopping time for w n (t, x), we conclude the proof.
We are ready to state the main result of this section.  Proof. For transparency, we split the proof into four steps: (1) proof of the equalities w n (0, x) = inf V ∈V n J T (x, V ) and w n m (0, x) = inf V ∈V n m J T (x, V ) for any n, m, T ∈ N and x ∈ E; (2) proof of w n m (0, x) → w n (0, x) as m → ∞ for any n ∈ N and x ∈ E; (3) proof of lim n→∞ inf V ∈V n J T (x, V ) = inf V ∈V J T (x, V ) for any T ∈ N and x ∈ E; (4) proof of (4.7).
Step 1. Let n, m, T ∈ N and x ∈ E be fixed. We show that For brevity, we present the proof only for w n ; the proof for w n m is analogous. Clearly, for n = 0, the claim is straightforward so we can assume that n ≥ 1.
Step 2. We show that w n m (0, x) → w n (0, x) as m → ∞ for any n ∈ N and x ∈ E. In fact, due to recursive nature of w n , we prove that w n m (t, x) → w n (t, x) as m → ∞ uniformly in [0, T ] × Γ, where Γ ⊂ E is a compact set.
Let us fix compact set Γ ⊂ E and proceed by induction. The claim for n = 0 is straightforward as w 0 m ≡ w 0 for any m ∈ N. Let n ∈ N and assume that the assertion holds for w n ; we show it for w n+1 . The proof is based on the argument used in [6,Lemma 4.1].
For any m ∈ N, t ∈ [0, T ] and x ∈ E, we set Noting that min h n+1 (t, x), h n+1 m (t, x) ≥ exp(−T (n+2) f ) for any m ∈ N, t ∈ [0, T ] and x ∈ E, and using inequality | ln y − ln z| ≤ 1 min(y,z) |y − z|, for y, z > 0, we get Thus, it is enough to show that h n+1 m (t, x) → h n+1 (t, x) uniformly on [0, T ]×Γ. Before we do that, we need to introduce some auxiliary notation and results.
Consider any t ∈ [0, T ] and x ∈ Γ. Let ε > 0 and τ (ε,t,x) ≤ t be an ε-optimal stopping time for h n+1 (t, x). For any m ∈ N, we set In the following, for brevity, we write τ and τ m instead of τ (ε,t,x) and τ (ε,t,x) m , and use notation  Also, noting that C n is uniformly continuous on [0, T ] 2 × B × B and C n (s, s, y, y) = 0 for any s ∈ [0, T ] and y ∈ E, we may find r 1 > 0 and m 1 ∈ N such that for any m ≥ m 1 we get Let r := min(r 1 , 1 2 ). Using Part 3 of [6, Proposition 2.1], we may find m 2 ∈ N such that for any m ≥ m 2 we get (4.13) sup Finally, it is useful to note that w k ≤ T f and w k m ≤ T f , for k, m ∈ N; these follow from Step 1 and the fact that the cost of no impulse strategy is bounded from above by T f .
Step 3. We show that lim n→∞ inf V ∈V n J T (x, V ) = inf V ∈V J T (x, V ) for any fixed T ∈ N and x ∈ E. In fact, using monotonicity and continuity of the exponent function, it is enough to show Let ε > 0 and V ε = (τ i , ξ i ) ∞ i=1 ∈ V be an ε-optimal strategy for inf V ∈V e JT (x,V ) . For any n ∈ N, let V ε n ∈ V n denote the restriction of V ε to the first n impulses. Then, for any n ∈ N, we get For clarity, we use X and Y to denote the processes controlled by the strategies V ε and V ε n , respectively. Furthermore, we set N T : for any n ∈ N and x ∈ E. Noting that, for any n ∈ N, we get and recalling (4.18), we get Next, using boundedness of f , non-negativity of c and the fact that V ε n is the restriction of V ε , we get .
Step 4. We are finally ready to show (4.7). Combining Step 1 and Step 2 we get Next, letting n → ∞ and recalling Step 3, we conclude the proof.

Continuous time impulse control.
In this section we study the impulse control problem introduced in (2.1) and linked Bellman equation (2.2). Our approach is based on the approximation of the Bellman equation by its dyadic version discussed in Section 3. We end this section by showing the connection between a solution to (2.2) and the optimal control strategy for (2.1).
Let (λ m ), m ∈ N, be a sequence of solutions to the dyadic Bellman equations (3.1); from Theorem 3.1 we know that this sequence exists. Since (λ m ) is decreasing and λ m ≥ − f we can define the finite limit Proof. For transparency, we split the proof into two steps: (1) proof of the fact that (M w mn ) → φ uniformly to some φ ∈ C(E), for predefined subsequence m n ; (2) proof of the identity φ = M w for a suitable w. In the end we comment how to combine those steps to get (2.2) and conclude the proof.
Step 1. Let (w m ) be a sequence of functions w m ∈ C(E) given by where w m is a solution to the Bellman equation (3.2); note that w m exists due to Theorem (3.1), and (w m ) is also a solution to (3.2). Now, we show that using Arzelà-Ascoli Theorem one can choose uniformly convergent subsequence of (M w m ), where M is given in (2.3). First, we show that (M w m ) is uniformly bounded on E. For any m ∈ N and ξ ∈ U , we get w m (ξ) ≥ 0. Consequently, for x ∈ E, we get M w m (x) ≥ 0, i.e. we found the uniform lower bound for (M w m ). On the other hand, recalling (3.2) and definition (5.2), for any x ∈ E and m ∈ N, we get M w m (x) ≤ 2 c as Second, as equicontinuity of (M w m ) follows directly from the inequality and continuity of c, we can use Arzelà-Ascoli Theorem. For any N ∈ N and a compact set we can find a subsequence of (M w m ), say (M w m N n ), and φ N ∈ C(E) such that (5.4) M w m N n → φ N , n → ∞, uniformly on B(0, N ).
Third, using diagonal argument we show that the limit may be chosen independently of N . Indeed, using recursive procedure, and taking consecutive subsequences Also, for the diagonal sequence (m n ) given by m n := m n n , we get uniformly on B(0, N ), for any N ∈ N. Using (5.3), we get that φ satisfies Finally, we can show that the convergence M w mn → φ is (globally) uniform. Let ε > 0. From (2.5) we know that there exists N ε ∈ N such that Since M w mn → φ uniformly on B(N ε + 1), it is sufficient to show that Consider x / ∈ B(N ε ) and y ∈ B(N ε + 1) \ B(N ε ). Recalling (5.3) and (5.6) we get Since y ∈ B(N ε + 1), starting from some n 0 ∈ N, we get |M w mn (y) − φ(y)| ≤ ε 3 for n ≥ n 0 . As the choice of ǫ was arbitraty, this inequality, together with (5.7) and (5.9), concludes the proof of (5.8).
Step 2. For brevity, we drop the subscript n from the diagonal sequence (m n ) given in Step 1, i.e. we assume that M w m → φ uniformly. We show that where w : E → R is given by The idea is to use [6, Theorem 5.1] in order to show that M w m → M w uniformly. Then, (5.10) will follow from the uniqueness of the limit. As in the proof of Proposition 3.2 we use the change of measure technique to transform the problem (5.11) into the setting, where the assumptions of [6, Theorem 5.1] are satisfied. For any m ∈ N and x ∈ E let us define Observe that G m → G uniformly, d m ↑ d, and d > 0. Moreover, using (2.9), Proposition 3.2, and (5.11), we get Hence, using [6, Theorem 5.1], we getŵ m →ŵ uniformly on compact sets and consequently w m → w uniformly on compact sets. Moreover, from [6, Theorem 4.7], we getŵ ∈ C(E) and consequently w ∈ C(E). Finally, recalling (2.3), we get This implies uniform convergence of M w m → M w. From this we immediatelly get φ(x) = M w(x), x ∈ E. Recalling (5.11), we conclude the proof of (2.2).
Finally, we are ready to link the constant λ given by (5.1) with the optimal strategy and the optimal value of the problem (2.1); see Theorem 5.3. In case λ < r(f ), Theorem 5.1 guarantees the existence of a solution to Bellman equation (2.2) that might be used to construct the optimal impulse control strategy. For the degenerate case λ = r(f ), we show that the no impulse strategy is the optimal choice; note that in this case we additionally assumed that E = U , which allowed us to use the finite horizon results from Section 4.
Before we state the next result, let us introduce some auxiliary notation. Let us fix w ∈ C(E) that is a solution to (2.2) satisfying w ≥ 0; from Theorem 5.1 we know that such w exists if λ < r(f ). LetV := (τ i ,ξ i ) ∞ i=1 be a strategy given recursively by for i = 1, 2, . . ., whereτ 0 := 0. Note that in (5.12) a slight abuse of notation was used since, for each recursive step, X t refers to a process for which ith impulse is not yet applied. More formally, (5.12) could be rewritten asτ i =σ • θτ i−1 +τ i−1 , wherê σ := inf{t ≥ 0 : w(X t ) = M w(X t )} and θ is the Markov shift operator; see e.g. [20, Section 1.4.3] for details.
Theorem 5.3. Let λ be given by (5.1). Then, 1. If λ < r(f ), then λ = inf V ∈V J(x, V ) for any x ∈ E, and the strategy defined in (5.12) via Bellman equation (2.2) is optimal. 2. If λ = r(f ) and E = U , then λ = inf V ∈V J(x, V ) for any x ∈ E, and the no impulse strategy is optimal.
Proof. For transparency, we split the proof into two parts: (1) when λ < r(f ) and (2) when λ = r(f ) and E = U .
Proof of 1. We adapt the arguments from [19,Proposition 2.3] to the continuous time case. Given strategy V ∈ V, for any i ∈ N, we use the notation denotes the state of the process (X t ) right before the (i + 1)th impulse.
First, we show that λ = J(x,V ) for any x ∈ E. By Remark 5.2 we know that where X n τn (T ) := X n τn , τ n ≤ T, X k T , τ k ≤ T < τ k+1 . Using Fatou Lemma and boundedness of f and w, we get This implies for any T ≥ 0. Thus, recalling that by (A.2) the cost function c is bounded away from zero, we conclude thatτ n ↑ ∞. Consequently, we get X n τn (T ) → X T , where X T (ω) := X k T (ω) on {τ k (ω) ≤ T <τ k+1 (ω)}. Finally, by bounded convergence theorem applied to (5.13), we get (5.14) e w(x) = E (x,V ) e Taking logarithm of both sides, dividing by T , and recalling that 0 ≤ w(·) ≤ w , we get which concludes the proof of equality λ = J(x,V ), for any x ∈ E. Second, we show that for any x ∈ E and strategy V = (τ i , ξ i ) ∈ V we get λ ≤ J(x, V ). Clearly, we can restrict our attention to the strategies for which Using the fact that w(X n−1 τn ) ≤ M w(X n−1 τn ) ≤ c(X n−1 τn , ξ n ) + w(ξ n ), we get )+w(X n τn (T )) .
Recalling (5.15) and letting n → ∞, we get As before, we get λ ≤ J(x, V ), which concludes this part of the proof.
Proof of 2. Using Theorem 3.1 and Proposition 3.2 we get that the cost of the no impulse strategy equals r(f ). Thus, it is sufficient to show that for any x ∈ E we get inf V ∈V J(x, V ) ≥ r(f ). On the contrary, suppose that inf V ∈V J(x 0 , V ) < r(f ) for some x 0 ∈ E. Then, for some ε > 0, we get (5.16) lim sup