On the Starting and Stopping Problem: Application in reversible investments.

On the Starting and Stopping Problem: Application in reversible investments.

S. Hamad`ene

Laboratoire de Statistique et Processus,

Universit´e du Maine, 72085 Le Mans Cedex 9, France.

e-mail: hamadene@univ-lemans.fr

&

M. Jeanblanc

D´epartement de Math´ematiques Universit´e d’Evry Val d’Essonne

rue du P`ere Jarlan 91025 Evry Ceded, France.

e-mail: jeanbl@maths.univ-evry.fr

November 22, 2004

Abstract

In this work we solve completely thestarting and stopping problem when the dynamics of the system are a general adapted stochastic process. We use backward stochastic differential equations and Snell envelopes. Finally we give some numerical results.

AMS Classification subjects: 60G40 ; 93E20 ; 62P20 ; 91B99.

Keywords: Real options ; Backward SDEs ; Snell envelope; Stopping time ; Stopping and starting.

0. Introduction: First let us deal with an example in order to introduce the problem we consider in this paper.

Assume that a power station produces electricity whose selling price fluctuates and depends on many factors such as consumer demand, oil prices, weather and so on. It is well known that electricity cannot be stored and when produced it should be consumed. Now for obvious economic reasons we suppose that electricity is produced only when its profitability is satisfactory. Otherwise the power station is closed up to time when the profitability is coming back,i.e., till the time when the market selling price of electricity reaches a level which makes the production profitable again.

So for the power station there are two modes: operating and closed. At the initial time, we assume it is in its operating mode. On the other hand, like every economic unit, there are expenditures when the station is in its operating mode as well as in the closed one. In addition, switching from a mode to another is not free and generates sunk costs.

The problem we are interested in is to find the sequence of stopping times where one should make decisions to stop the production and to resume it again successively in order to maximize the profitability of the station and then to determine the maximum profit.

More precisely suppose the electricity market selling price is given by a stochastic process X = (X_t)_t≤T. As it is discussed previously, a management strategy of the power station is an increasing sequence of stopping times δ = (τ_n)_n≥1 where for n ≥ 1, τ_n ≤ τ_n+1 and τ_2n−1 (resp. τ_2n) are the times where the station is switched from the operating to the closed mode (resp. conversely). Now let J(δ) be the profit of the power station provided by the implementation of the strategyδ. Naturally it depends on the given processX. Therefore we look for a strategy δ^∗ such that J(δ^∗)≥J(δ) for any other δ.

The problem we consider in this paper is of real options type. It is usually called the re- versible investment problem. In recent years, real options area has attracted considerable interest ([BO],[BS],[DP],[DZ],...). The motivations are mainly related to decision making in the economic sphere. For more details on this subject seee.g.the book by Dixit & Pindyck [DP] and the references therein.

In the literature, our problem is also called starting and stopping (or switching). In the previous example, we have considered electricity production. However there are many real cases where this problem intervenes. Among others, we can quote the management of oil tankers, oil fields,....

¿From the economic point of view, the problem ofstarting and stopping has been already considered by A.Dixit [D] in the case whenT is infinite and X is a geometric Brownian motion. His approach is based on elliptic PDEs.

In this article we solve completely the starting and stopping problem when the dynamics of the system is a stochastic process X = (X_t)_t≤T adapted with respect to a Brownian filtration, whatever it may be and when T is finite. The main tools are the notions of reflected backward stochastic differential equation (BSDE in short) and Snell envelope. We show that our problem turns into the existence of a pair of adapted processes (Y¹, Y²) which satisfies a system expressed by means of Snell envelopes. In a second step, we show that the existence of an optimal strategy and its expression is given. At the end we discuss a method to simulate the optimal strategy and we give some numerical results.

Another interest of our work is that we bring a new point of view to tackle the starting and

stopping problem when the dynamic of the system is affected by the control. With respect to the above example, it means that the process X depends on the running implemented strategy δ. Such problems are met in the management of raw material mines like copper, gold, steel,... In [BO] and [DZ], the approach is based on PDEs, and solution are provided only under fairly stringent conditions.

We think that our approach based on BSDEs could bring new results. Though this is a task with which we deal with in a forthcoming paper.

This paper is organized as follows: Section 1 is devoted to the setting of thestarting and stopping problem. Further we show that our problem reduces to the existence of a pair of processes (Y¹, Y²) solution of a system expressed by means of Snell envelopes. Then we construct the optimal strategy.

Finally we prove the existence of (Y¹, Y²). In Section 3, we study the case where the processX is a solution of a standard stochastic differential equation and we focus on some numerical aspects of the optimal strategy. Finally we consider some specific cases.

1 Setting of the problem. Preliminary results

Throughout this paper (Ω,F, P) is a fixed probability space on which is defined a standard d- dimensional Brownian motion B = (B_t)_t≤T whose natural filtration is (F_t⁰ := σ{B_s, s ≤ t})_t≤T. Let F = (F_t)_t≤T be the completed filtration of (F_t⁰)_t≤T with the P-null sets of F, hence (F_t)_t≤T satisfies the usual conditions,i.e., it is right continuous and complete. We now introduce the following notations: let

- P be the σ-algebra on [0, T]×Ω ofF-progressively measurable sets

-M^2,k be the set ofP-measurable andIR^k-valued processesw= (w_t)_t≤T which belongs toL²(Ω× [0, T], dP ⊗dt)

- S² be the set ofP-measurable, continuous processesw= (w_t)_t≤T such thatE[sup_t≤T |w_t|²]<∞ - S_i² be the subset of S² of processes K := (K_t)_t≤T which are non-decreasing and satisfyK₀ = 0.

In particular, ifK ∈ S_i², thenE(K_T²)<∞.

For any stopping timeτ ∈[0, T], T_τ denotes the set of all stopping timesθ such thatτ ≤θ≤T. A management strategy is an increasing sequence ofF-stopping times δ:= (τ_n)_n≥1 where for any n≥1, τ_2n (resp. τ_2n−1) are the moments where the production is frozen (resp. on).

A strategy δ := (τ_n)_n≥1 is called admissible if P-a.s., lim_n→∞τ_n = T. The set of admissible strategies is denotedD_a.

Now in a short period of time dt, when the production is open, it provides a profit which is equal toψ₁(t, X_t)dt. The quantityψ₁(t, X_t) can be negative. Such a situation happens when the electricity price is low enough at point that management expenses are not recovered. On the other hand, when

the production is frozen there are sunk costs which are equal to ψ₂(t, X_t)dt. At least because one should maintain the production equipment in a good state in order to operate in due time. Finally there are also costs linked to stop the production or to start it again. For example, one can think of the fees generated by laying of the workers or engaging them again.

So the outcome of those considerations is that when an admissible management strategy δ :=

(τ_n)_n≥1 is implemented, the average global profit is given by:

J(δ) =E[

Z _T

0 Φ(s, X_s, u_s)ds−^X

n≥1

{D11_[τ2n−1<T]+a11_[τ2n<T]}]

where :

[i]X_t is the electricity market price att ; the process (X_t)_t≤T belongs to M^2,1

[ii] for anyt≤T,u_t= 1 if the production is open and 0 otherwise. Actually the processu= (u_t)_t≤T is linked to the implemented strategyδ and for anyt≤T we haveu_t= 11_[0,τ1](t) +^P_n≥111_{]τ2n,τ2n+1]}(t)

[iii] Φ(t, x,0) =ψ₂(t, x) and Φ(t, x,1) =ψ₁(t, x)

[iv] D(resp. a) stands for the sunk cost when the production is stopped (resp. starts)

[v] the functionsψ_j(t, x), (t, x)∈[0, T]×IR^dandj= 1,2, are sub-linearly growing,i.e, there exists C such that |ψ_j(t, x)| ≤C(1 +|x|) .

Now the problem we are interested in is to find an optimal strategy for the manager,i.e, a strategy δ^∗ = (τ_n^∗)_n≥1 ∈ D_a such that J(δ^∗)≥ J(δ), for any admissible strategy δ. In a second stage we deal with the numerical results of the optimal profit and strategy.

Note that here, the function Φ may depend on time in a general way, which is not the case in Dixit [D].

1.1 The Snell envelope notion

LetU = (U_t)_t≤T be anF-adaptedIR-valued c`adl`ag process without negative jumps and which belongs to the class [D], i.e., the set of random variables {U_τ, τ ∈ T₀} is uniformly integrable. Then there exists a uniqueF-adaptedIR-valued continuous processZ := (Z_t)_t≤T (see e.g. [CK], [EK], [H]), called theSnell envelope ofU, such that :

Z is the smallest super-martingale which dominates U, i.e, if ( ¯Z_t)_t≤T is another c`adl`ag super- martingale such that∀t≤T, ¯Z_t≥U_t then ¯Z_t≥Z_tfor anyt≤T.

The following properties of the processZ hold true : (i) Z can be expressed as : for any F-stopping timeγ,

Z_γ= esssup_τ∈TγE[U_τ|F_γ] (and then Z_T =U_T) (1)

(ii) let γ be anF-stopping time andτ_γ^∗= inf{s≥γ, Z_s=U_s} ∧T thenτ_γ^∗ is optimal afterγ,i.e., Z_γ =E[Z_τγ^∗|F_γ] =E[U_τγ^∗|F_γ] = esssup_τ≥γE[U_τ|F_γ]. (2) (iii) if U_n, n ≥ 0, and U are c`adl`ag and uniformly square integrable processes such that the se- quence (U_n)_n≥0 converges increasingly and pointwisely to U then (Z^Un)_n≥0 converges increasingly and pointwisely toZ^U ;Z^Un and Z^U are the Snell envelopes of respectivelyU_n and U.

The proof of (iii) is given in the appendix of [CK]. For more details on the Snell envelope notion, one can refer to [EK]¦

Letδ= (τ_n)_n≥1 be an admissible strategy. The strategyδis calledfiniteif during the time interval [0, T] it allows to the manager to make only a finite number of decisions,i.e, P(ω, τ_n(ω) < T,∀n ≥ 1) = 0. Hereafter the set of finite strategies will be denotedD. Obviously optimal strategies should be necessarily finite, otherwise the sunk costs would be infinite. Therefore we have the following result whose proof is quite easy and then is omitted.

Proposition 1 : The supremum over admissible strategies and finite strategies are the same:

sup_δ∈DaJ(δ) = sup_δ∈DJ(δ) ¦

We now focus on the optimal profit and we have the following verification theorem.

Proposition 2 : Assume there exist two IR-valued processes Y¹ = (Y_t¹)_t≤T and Y² = (Y_t²)_t≤T of S² such that∀t≤T,

Y_t¹ = esssup_τ∈TtE[

Z _τ

t ψ₁(s, X_s)ds+ (−D+Y_τ²)11_{[τ <T]}|F_t], (3) Y_t² = esssup_τ∈TtE[

Z _τ

t ψ₂(s, X_s)ds+ (−a+Y_τ¹)11_{[τ <T]}|F_t]. (4) ThenY₀¹= sup_δ∈DJ(δ). Moreover the strategyδ^∗ = (τ_n^∗)_n≥1 defined as follows:

τ₀^∗ = 0

∀n≥1, τ_2n−1^∗ = inf{s≥τ_2n−2^∗ , Y_s¹ =−D+Y_s²} ∧T τ_2n^∗ = inf{s≥τ_2n−1^∗ , Y_s² =−a+Y_s¹} ∧T is optimal.

P roof: First recall thatY¹ and Y² are continuous and verifyY_T¹ =Y_T² = 0. Therefore the jumps of ((−D+Y_t²)11_[t<T])_t≤T and ((−a+Y_t¹)11_[t<T])_t≤T at T are non-negative. Now for any t≤T, we have

Y_t¹+ Z _t

0 ψ₁(s, X_s)ds= esssup_τ≥tE[

Z _τ

0 ψ₁(s, X_s)ds+ (−D+Y_τ²)11_{[τ <T]}|F_t].

The random variableY₀¹ isF₀-measurable then it isP−a.s. a constant and thenY₀¹ =E[Y₀¹].On the other hand, according to (2),

Y₀¹=E[

Z _τ∗ 1

0 ψ₁(s, X_s)ds+ (−D+Y_τ^2∗

1)11_[τ1^∗_<T]], whereτ₁^∗ is given as in the proposition. From the properties of Snell’s envelope

Y_τ^2∗

1 = esssup_τ∈T

τ∗ 1

E[

Z _τ

τ₁^∗ψ₂(s, X_s)ds+ (−a+Y_τ¹)11_{[τ <T]}|F_τ1^∗]

= E[

Z _τ∗ 2

τ₁^∗ ψ₂(s, X_s)ds+ (−a+Y_τ^1∗

2)11_[τ^∗

2<T]|F_τ1^∗].

It implies that Y₀¹ = E[

Z _τ^∗

1

0 ψ₁(s, X_s)ds+ Z _τ^∗

2

τ₁^∗ ψ₂(s, X_s)ds−D11_[τ^∗

1<T]+E[(−a+Y_τ¹∗ 2)11_[τ^∗

2<T]|F_τ1^∗] 11_[τ^∗

1<T]]

= E[

Z _τ∗ 1

0 ψ₁(s, X_s)ds+ Z _τ∗

2

τ₁^∗ ψ₂(s, X_s)ds−D11_[τ^∗

1<T]−a11_[τ^∗

2<T]+Y_τ^1∗

211_[τ^∗

2<T]] since [τ₁^∗ < T]∈ F_τ1^∗ and [τ₂^∗ < T]⊂[τ₁^∗ < T]. Therefore

Y₀¹=E[

Z _τ^∗

2

0 Φ(s, X_s, u_s)ds−D11_[τ^∗

1<T]−a11_[τ^∗

2<T]+Y_τ^1∗

211_[τ^∗

2<T]]

since between 0 andτ₁^∗ (resp. τ₁^∗ and τ₂^∗) the production is open (resp. suspended) and then u_t= 1 (resp. u_t= 0) which implies that

Z _τ∗ 2

0 Φ(s, X_s, u_s)ds= Z _τ∗

1

0 ψ₁(s, X_s)ds+ Z _τ∗

2

τ₁^∗ ψ₂(s, X_s)ds.

Now following this reasoning as many times as necessary we obtain Y₀¹ =E[

Z _τ∗ 2n

0 Φ(s, X_s, u_s)ds− ^X

1≤k≤n

(D11_[τ^∗

2k−1<T]+a11_[τ^∗

2k<T]) +Y_τ^1∗

2n11_[τ^∗

2n<T]]. (5) But the strategyδ^∗ is finite. Indeed let A ={ω, τ_n^∗ < T,∀n≥1} and let us show that P(A) = 0. If P(A)>0 then for any n≥1,

Y₀¹ ≤ E[

Z _T

0 (|ψ₁(s, X_s)| ∨ |ψ₂(s, X_s)|)ds−( ^X

1≤k≤n

(D11_[τ^∗

2k−1<T]+a11_[τ^∗

2k<T]))11_A

−(^P_1≤k≤n(D11_[τ^∗

2k−1<T]+a11_[τ^∗

2k<T]))11A¯+ sup_s≤T |Y_s¹|11_[τ2n^∗ _<T]].

The right-hand side converges to −∞ as n→ ∞ since the process Y¹ belongs toS² and ψ_i(., X_.) ∈ M^2,1, then Y₀¹ = −∞. But this is contradictory because, once again, Y¹ ∈ S². Henceforth the strategyδ^∗ is finite. Going back to (5) and taking the limit asn→ ∞ we obtainY₀¹=J(δ^∗).

Now let us show that Y₀¹ ≥J(δ) for anyδ∈ D. Letδ = (τ_n)_n≥1 be a finite strategy. According to (2), τ₁^∗ is optimal and then

Y₀¹ ≥E[

Z _τ1

0 ψ₁(s, X_s)ds+ (−D+Y_τ²₁)11_[τ1<T]].

On the other hand

Y_τ²₁ ≥E[

Z _τ2

τ1

ψ₂(s, X_s)ds+ (−a+Y_τ¹₂)11_[τ2<T]|F_τ1] and then

Y₀¹ ≥ E[

Z _τ1

0 ψ₁(s, X_s)ds−D11_[τ1<T]+E[(−a+Y_τ¹₂)11_[τ2<T]|F_τ1]11_[τ1<T]]

≥ E[

Z _τ1

0 ψ₁(s, X_s)ds+ Z _τ2

τ1

ψ₂(s, X_s)ds−D11_[τ1<T]−a11_[τ2<T]+Y_τ¹₂11_[τ2<T]] since [τ₁< T]∈ F_τ1 and [τ₂ < T]⊂[τ₁ < T]. Therefore we have,

Y₀¹ ≥E[

Z _τ2

0 Φ(s, X_s, u_s)ds−D11_[τ1<T]−a11_[τ2<T]+Y_τ¹₂11_[τ2<T]].

Now making this reasoning as many times as necessary we obtain for anyn≥0, Y₀¹ ≥E[

Z _τ2n

0 Φ(s, X_s, u_s)ds− ^X

1≤k≤n

(D11_[τ2k−1<T]+a11_[τ2k<T]) +Y_τ¹_2n11_[τ2n<T]]. (6) As the strategyδ is finite then the right-hand side of (6) converges toJ(δ) as n→ ∞. Therefore we haveY₀¹=J(δ^∗)≥J(δ) which implies actually that the strategyδ^∗ is optimal.

Remark 1 The random variableY_t¹ (resp. Y_t²) stands for the optimal expected profit if attthe station is in its operating (resp. stopping) mode¦

2 Existence of the pair (Y

, Y

).

We now focus on the existence of the pair (Y¹, Y²). First let us recall the following result stated by El-Karoui et al. [EKal] and which is related to BSDEs with one reflecting barrier.

Let ξ ∈L²(Ω,F_T, IR;dP), (f_t)_t≤T a process ofM^2,1 and finally let S := (S_t)_t≤T be an IR-valued process ofS² such thatS_T ≤ξ. Then we have :

Theorem 1 (EKal) : There exists a triple (Y, Z, K) := (Y_t, Z_t, K_t)_t≤T of P-measurable processes, with values inIR¹×IR^d×IR¹ such that:















Y ∈ S², Z∈ M^2,d and K ∈ S_i²(K₀ = 0) Y_t=ξ+

Z _T

t f_sds+K_T −K_t− Z _T

t Z_sdB_st≤T

∀t≤T, Y_t≥S_t and Z _T

0 (Y_t−S_t)dK_t= 0.

(7)

In addition,Y can be interpreted as a Snell envelope in the following way: ∀ t≤T,

Y_t=esssup_τ≥tE[

Z _τ

t f_sds+S_τ11_{[τ <T]}+ξ11_[τ=T]|F_t]¦ (8)

We are going now to provide a solution for (3)-(4). It is based on BSDEs with two reflecting barriers studied by several authors (see e.g. [CK], [HL], [HLM], ...). Actually we have:

Theorem 2 : There exists a pair of continuous processes (Y_t¹, Y_t²)_t≤T which satisfies (3)-(4).

P roof: Since−D < athen there exists a unique quadruple ofP-measurable processes (Y_t, Z_t, K_t⁺, K_t⁻)_t≤T which satisfies :















Y ∈ S², Z∈ M^2,d and K^±∈ S_i²(K₀^±= 0) Y_t=

Z _T

t (ψ₁(s, X_s)−ψ₂(s, X_s))ds+K_T⁺−K_t⁺−(K_T⁻−K_t⁻)− Z _T

t Z_sdB_s, t≤T

∀t≤T,−D≤Y_t≤aand Z _T

0 (Y_t+D)dK_t⁺= Z _T

0 (a−Y_t)dK_t⁻ = 0.

(9)

The existence of the quadruple (Y, Z, K^±) stems from a result by ([CK], pp.2034) or ([HLM], pp.165) since the barriers −D and a are constants and then they are regular and satisfy also the so-called Mokobodzki’s condition which means the location of a difference of non-negative supermartingales between−Dand a. Actually it is enough to choose those supermartingales null identically.

Now fort≤T, let us set : Y_t¹=E[

Z _T

t ψ₁(s, X_s)ds+K_T⁺−K_t⁺|F_t] and Y_t²=E[

Z _T

t ψ₂(s, X_s)ds+K_T⁻−K_t⁻|F_t].

Therefore for anyt≤T we have Y_t=Y_t¹−Y_t². Now let γ¹ and (Z_t¹)_t≤T be respectively the constant and the process ofM^2,d such that :

Z _T

0 ψ₁(s, X_s)ds+K_T⁺ =γ¹+ Z _T

0 Z_s¹dB_s.

There is no existence problem for those items since (ψ₁(t, X_t))_t≤T belongs to M^2,1 and K_T⁺ is square integrable. Henceforth the triple (Y¹, Z¹, K⁺) satisfies :







−dY_t¹ =ψ₁(t, X_t)dt+dK_t⁺−Z_t¹dB_t, Y_T¹= 0 Y_t¹ ≥ −D+Y_t² and (Y_t¹−Y_t²+D)dK_t⁺= 0 Now by (7)-(8) we have :

Y_t¹= esssup_τ∈TtE[

Z _τ

t ψ₁(s, X_s)ds+ (−D+Y_τ²)11_{[τ <T]}|F_t], t≤T.

In the same way we can show thatY² verifies : Y_t²= esssup_τ∈TtE[

Z _τ

t ψ₂(s, X_s)ds+ (−a+Y_τ¹)11_{[τ <T]}|F_t], t≤T.

Henceforth the pair (Y¹, Y²) satisfies actually the system (3)-(4) ¦

Remark 2 Let us consider the sequences (Y^1,n)_n≥0 and (Y^2,n)_n≥1 defined recursively as follows :

∀t≤T,

Y_t^2,n=esssup_τ≥tE[

Z _τ

t ψ₂(s, X_s)ds+ (−a+Y_τ^1,n−1)11_{[τ <T]}|F_t] (10) and

Y_t^1,n =esssup_τ≥tE[

Z _τ

t ψ₁(s, X_s)ds+ (−D+Y_τ^2,n)11_{[τ <T]}|F_t] (11) where Y_t^1,0 =E[

Z _T

t ψ₁(s, X_s)ds|F_t]. Then we have the following characterization of Y_t^1,n :

∀t≤T, Y_t^1,n =esssup_δ∈Dⁿ

tE[

Z _T

t Φ(s, X_s, u_s)ds−^X

n≥1

(D11_[τ2n−1<T]+a11_[τ2n<T]|F_t] (12) where D_tⁿ is the set of admissible strategies δ = (τ_k)_k≥1 such that τ₁ ≥ t and τ_2n+1 = T, P-a.s..

Therefore we can show that the sequence (Y^1,n)_n≥0 (resp. (Y^2,n)_n≥1) converges increasingly in S² to Y¹ (resp. Y²)¦

3 Properties of the optimal strategy, numerical aspects and exam- ples

Let once again (Y_t)_t≤T be the process of (9). SinceY¹−Y² =Y then it is easily seen that the stopping times τ_n^∗ which give the optimal strategy are the ones where the process Y reaches successively the barriersaand −D,i.e,

∀n≥1, τ_2n−1^∗ = inf{s≥τ_2n−2^∗ , Y_s=−D} ∧T (τ₀^∗ = 0) andτ_2n^∗ = inf{s≥τ_2n−1^∗ , Y_s =a} ∧T.

Assume now that the processX is the unique solution of the following SDE :

dX_t=b(t, X_t)dt+σ(t, X_t)dB_t, t≤T; X₀ =x∈IR^k (13) where the functionsband σ, with appropriate dimensions, are jointly continuous and satisfy :

|b(t, x)|+|σ(t, x)| ≤k(1 +|x|) and |σ(t, x)−σ(t, x⁰)|+|b(t, x)−b(t, x⁰)| ≤k|x−x⁰|

for anyt≤T andx, x⁰ ∈IR^k. Let us recall that under those assumptions onb,σ it is well known that for anyp≥1 there exists a real constantC_p such that E[(sup_t≤T|X_t|)^p]≤C_p.

In [HH], it has been shown the existence of a continuous function u(t, x), (t, x) ∈ [0, T]×IR^k, such that Y_t =u(t, X_t), for anyt≤T. Moreover the function u is solution, in viscosity sense, of the following double obstacle partial differential inequality :







min{u(t, x) +D,max[−^∂u_∂t(t, x)−L_tu(t, x)−ψ(t, x), u(t, x)−a]}= 0, u(T, x) = 0

whereψ=ψ₁−ψ₂ and the operatorL is the generator associated withX,i.e., L_t= 1

2 Xk

i,j=1

(σσ^∗(t, x))_i,j ∂²

∂x_i∂x_j + Xk

i=1

b_i(t, x) ∂

∂x_i.

Therefore the optimal strategy can be expressed via the beginnings of some deterministic sets. Actually we have: (τ₀^∗ = 0),

∀n≥1, τ_2n−1^∗ = inf{s≥τ_2n−2^∗ , u(s, X_s) =−D} ∧T and τ_2n^∗ = inf{s≥τ_2n−1^∗ , u(s, X_s) =a} ∧T ¦ Let us now focus on some numerical aspects of the optimal strategy of the stopping and starting problem. However keep in mind that our purpose in this article is not to provide an exhaustive treatment of this issue, which of course is a very interesting subject but is a bit far from the main objective of our work.

So for n, k≥0 let us consider the following standard or reflected BSDEs : for anyt≤T, Y_t^n,k=

Z _T

t ψ(s, X_s)ds−n Z _T

t (Y_s^n,k−a)⁺ds+k Z _T

t (Y_s^n,k+D)⁻ds− Z _T

t Z_s^n,kdB_s,





 Y˜_t^k=

Z _T

t ψ(s, X_s)ds−( ˜K_T^k,−−K˜_t^k,−) +k Z _T

t ( ˜Y_s^k+D)⁻ds− Z _T

t

Z˜_s^kdB_s Y˜_t^k≤aand ( ˜Y_t^k−a)dK_t^k,− = 0,





 Y_tⁿ=

Z _T

t ψ(s, X_s)ds+ (K_T^n,+−K_t^n,+)−n Z _T

t (Y_sⁿ−a)⁺ds− Z _T

t Z_sⁿdB_s Y_tⁿ≥ −D and (Y_tⁿ+D)dK_t^n,+= 0.

It is well known that for anyn ≥0 the sequence (Y^n,k)_k≥0 converges decreasingly in S² to Yⁿ. On the other hand the sequence ( ˜Y^k)_k≥0 (resp. ((Yⁿ)_n≥0) converges increasingly (resp. decreasingly) to Y in S² (see e.g. [HLM]). So we are going to focus on the estimates ofY^n,m−Y and Yⁿ−Y. To begin with let us recall the following properties related toY^n,k, ˜Y^k and Yⁿ. The proofs can be found in [CK] or [HLM].

Proposition 3 : The following properties hold true:

(i) for any n, k we have Y˜^k≤Y^n,k≤Yⁿ (ii) ∀ k≥0 and t≤T,

Y˜_t^k=essinf_ν≥tE[

Z _ν

t {ψ(s, X_s) +k( ˜Y_s^k+D)⁻}ds+a1_[ν<T]|F_t].

In addition the infimum is reached at the stopping timeν˜_t^k= inf{s≥t,Y˜_s^k=a} ∧T

(iii) let U be the set of P-measurable processes(v_t)_t≤T with values in [0,1]. Then for anyn, k≥0 and t≤T we have,

Y_t^n,k=essinf_v∈UE[

Z _T

t e⁻ R_s

t nvsds× {ψ(s, X_s) +k(Y_s^n,k+D)⁻+nav_s}ds|F_t].

In addition the infimum is reached at(˜v_t= 1_[a<Yn,k t ])_t≤T

(iv) there exists a constant C_ψ which does depend only (ψ(t, X_t))_t≤T (and not on k, n) such that E[sup_s≤T((Y_s^n,k+D)⁻)²]≤C_ψk⁻² and E[sup_s≤T((Y_s^n,k−a)⁺)²]≤C_ψn⁻² ¦

Then we have :

Proposition 4 For any n, k≥1, it holds true that E[sup

t≤T |Y_t^n,k−Y_t|²]≤C(n⁻²+k⁻²) (14)

where C is a constant which depends only on ψ.

P roof: Let t≤T and let us focus on the difference Y_t^n,k−Y˜_t^k. Letθ_t be a stopping time such that t≤θ_t≤T, P-a.s.. On the other hand fors≤T let us setv_s= 1_[s≥θt]. Then we have :

E[ Z _T

t e⁻ R_s

t nvrdr× {ψ(s, X_s) +k(Y_s^n,k+D)⁻+nav_s}ds|F_t]−

E[

Z _θt

t {ψ(s, X_s) +k( ˜Y_s^k+D)⁻}ds+a1_[θt<T]|F_t] = E[

Z _θt

t {k(Y_s^n,k+D)⁻−k( ˜Y_s^k+D)⁻}ds−a1_[θt<T]|F_t]+

E[

Z _T

θt

e^{−n(s−θt)}{ψ(s, X_s) +k(Y_s^n,k+D)⁻+na}ds|F_t]≤ E[−a1_[θt<T]+

Z _T

θt

e^{−n(s−θt)}{ψ(s, X_s) +k(Y_s^n,k+D)⁻+na}ds|F_t]

since for any n, k,∀s≤T, (Y_s^n,k+D)⁻−( ˜Y_s^k+D)⁻ ≤0, through Prop.3-(i). Now let us deal with latter term.

Z _T

θt

e^{−n(s−θt)}{ψ(s, X_s) +k(Y_s^n,k+D)⁻}ds≤ 1 nsup

s≤T{|ψ(s, X_s)|+k(Y_s^n,k+D)⁻} and

−a1_[θt<T]+a Z _T

θt

ne^{−n(s−θt)}ds=−a1_[θt<T]+a(1−e^{−n(T−θt)})1_[θt<T]≤0 Therefore we have :

E[

Z _T

t e⁻ R_s

t nvrdr× {ψ(s, X_s) +k(Y_s^n,k+D)⁻+nav_s}ds|F_t]−

E[

Z _θt

t {ψ(s, X_s) +k( ˜Y_s^k+D)⁻}ds+a1_[θt<T]|F_t]≤ 1 nE[sup

s≤T{|ψ(s, X_s)|+k(Y_s^n,k+D)⁻}|F_t].

Taking θ_t = ˜ν_t^k (keep in mind that in the expression of ˜Y^k the infimum is reached at ˜ν_t^k) we deduce that:

Y_t^n,k ≤Y˜_t^k+ 1 nE[sup

s≤T

{|ψ(s, X_s)|+k(Y_s^n,k+D)⁻}|F_t].

Now since Y_t^n,k ≥Y˜_t^k and through the convergence of (Y^n,k)_k≥0 (resp. ( ˜Y^k)_k≥0) to Yⁿ (resp. Y) in S² and finally using Doob’s inequality we deduce that

E[sup

t≤T|Y_tⁿ−Y_t|²]≤Cn⁻². (15)