Switching problems with controlled randomisation and associated obliquely reflected BSDEs

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Switching problems with controlled randomisation and associated obliquely reflected BSDEs

Cyril Bénézet, Jean-François Chassagneux, Adrien Richou

To cite this version:

Cyril Bénézet, Jean-François Chassagneux, Adrien Richou. Switching problems with controlled ran-

domisation and associated obliquely reflected BSDEs. Stochastic Processes and their Applications,

Elsevier, 2022, pp.23-71. �10.1016/j.spa.2021.10.010�. �hal-02463626v1�

Switching problems with controlled randomisation and associated obliquely reected BSDEs

Cyril Bénézet

^∗

, Jean-François Chassagneux

^†

, Adrien Richou

^‡

January 30, 2020

Abstract

We introduce and study a new class of optimal switching problems, namely switching problem with controlled randomisation, where some extra-randomness impacts the choice of switching modes and associated costs. We show that the optimal value of the switching problem is related to a new class of multidimen- sional obliquely reected BSDEs. These BSDEs allow as well to construct an optimal strategy and thus to solve completely the initial problem. The other main contribution of our work is to prove new existence and uniqueness results for these obliquely reected BSDEs. This is achieved by a careful study of the domain of re- ection and the construction of an appropriate oblique reection operator in order to invoke results from [7].

1 Introduction

In this work, we introduce and study a new class of optimal switching problems in stochastic control theory. The interest in switching problems comes mainly from their connections to nancial and economic problems, like the pricing of real options [4]. In a celebrated article [14], Hamadène and Jeanblanc study the fair valuation of a company producing electricity. In their work, the company management can choose between two modes of production for their power plant operating or close and a time of switch- ing from one state to another, in order to maximise its expected return. Typically, the company will buy electricity on the market if the power station is not operating.

The company receives a prot for delivering electricity in each regime. The main point here is that a xed cost penalizes the prot upon switching. This switching problem has been generalized to more than two modes of production [10]. Let us now discuss this switching problem with d ¥ 2 modes in more details. The costs to switch from

∗Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris- Saclay, Route de Saclay, 91128 Palaiseau Cedex, France (cyril.benezet@polytechnique.edu)

†UFR de Mathématiques & LPSM, Université de Paris, Bâtiment Sophie Germain, 8 place Aurélie Nemours, 75013 Paris, France (chassagneux@lpsm.paris)

‡UMR 5251 & IMB, Université de Bordeaux, F-33400 Talence, France (adrien.richou@math.u-bordeaux.fr)

one state to another are given by a matrix p c

_i,j

q

1¤i,j¤d

. The management optimises the expected company prots by choosing switching strategies which are sequences of stopping times p τ

n

q

n¥0

and modes p ζ

n

q

n¥0

. The current state of the strategy is given by a

_t

°

₈

k0

ζ

_k

1

_{rτk,τk 1q}

p t q , t P r 0, T s , where T is a terminal time. To formalise the problem, we assume that we are working on a complete probability space pΩ, A, Pq supporting a Brownian Motion W . The stopping times are dened with respect to the ltration p F

_t

q

t¥0

generated by this Brownian motion. Denoting by f p t, i q the instan- taneous prot received at time t in mode i, the time cumulated prot associated to a switching strategy is given by ³

_T

0

f p t, a

t

q dt °

₈

k0

c

_{ζk,ζk 1}

1

_{tτk 1¤t^Tu}

. The management solves then at the initial time the following control problem

V

₀

sup

aPA

E

»

T 0

f p t, a

t

q dt ¸

⁸

k0

c

_{ζk,ζk 1}

1

_{tτk 1¤Tu}

, (1.1)

where A is a set of admissible strategies that will be precisely described below (see Section 2.1). We shall refer to problems of the form (1.1) under the name of classical switching problems. These problems have received a lot of interest and are now quite well understood [14, 10, 17, 5]. In our work, we introduce a new kind of switching problem, to model more realistic situations, by taking into account uncertainties that are encountered in practice. Coming back to the simple but enlightning example of an electricity producer described in [14], we introduce some extra-randomness in the production process. Namely, when switching to the operating mode, it may happen with hopefully a small probability that the station will have some dysfunction. This can be represented by a new mode of production with a greater switching cost than the business as usual one. To capture this phenomenon in our mathematical model, we introduce a randomisation procedure: the management decides the time of switching but the mode is chosen randomly according to some extra noise source. We shall refer to this kind of problem by randomised switching problem. However, we do not limit our study to this framework. Indeed, we allow some control by the agent on this randomisation.

Namely, the agent can chose optimally a probability distributions P

^u

on the modes space given some parameter u P C , in the control space. The new mode ζ

_{k 1}

is then drawn, independently of everything up to now, according to this distribution P

^u

and a specic switching cost c

^u_ζ

k,ζk 1

is applied. The management strategy is thus given now by the sequence p τ

_k

, u

_k

q

k¥0

of switching times and controls. The maximisation problem is still given by (1.1). Let us observe however that E

c

^u_ζ^k

k,ζk 1

E °

1¤j¤d

P

_ζ^uk

k,j

c

^u_ζ^k

k,j

, thanks to the tower property of conditional expectation. In particular, we will only work with the mean switching costs c ¯

^u_i^k

: °

1¤j¤d

P

_i,j^uk

c

^u_i,j^k

in (1.1). We name this kind of control problem switching problem with controlled randomisation. Although their apparent modeling power, this kind of control problem has not been considered in the literature before, to the best of our knowledge. In particular, we will show that the classical or randomised switching problem are just special instances of this more generic problem. The switching problem with controlled randomisation is introduced rigorously in Section 2.1 below.

A key point in our work is to relate the control problem under consideration to a

new class of obliquely reected Backward Stochastic Dierential Equations (BSDEs).

In the rst part, following the approach of [14, 10, 17], we completely solve the switching problem with controlled randomisation by providing an optimal strategy. The optimal strategy is built by using the solution to a well chosen obliquely reected BSDE. Al- though this approach is not new, the link between the obliquely reected BSDE and the switching problem is more subtle than in the classical case due to the state uncertainty.

In particular, some care must be taken when dening the adaptedness property of the strategy and associated quantities. Indeed, a tailor-made ltration, studied in details in Appendix A.2, is associated to each admissible strategy. The state and cumulative cost processes are adapted to this ltration, and the associated reward process is dened as the Y -component of the solution to some switched BSDE in this ltration. The classical estimates used to identify an optimal strategy have to be adapted to take into account the extra orthogonal martingale arising when solving this switched BSDE in a non Brownian ltration.

In the second part of our work, we study the auxiliary obliquely reected BSDE, which is written in the Brownian ltration and represents the optimal value in all the possible starting modes. Reected BSDEs were rst considered by Gegout-Petit and Pardoux [13], in a multidimensional setting of normal reections. In one dimension, they have also been studied in [11] in the so called simply reected case, and in [8] in the doubly reected case. The multidimensional RBSDE associated to the classical switching problem is reected in a specic convex domain and involves oblique directions of reection. Due to the controlled randomisation, the domain in which the Y -component of the auxiliary RBSDE is constrained is dierent from the classical switching problem domain and its shape varies a lot from one model specication to another. The existence of a solution to the obliquely reected BSDE has thus to be studied carefully. We do so by relying on the article [7], that studies, in a generic way, the obliquely reected BSDE in a xed convex domain in both Markovian and non-Markovian setting. The main step for us here is to exhibit an oblique reection operator, with the good properties to use the results in [7]. We are able to obtain new existence results for this class of obliquely reected BSDEs. Because we are primarily interested in solving the control problem, we derive the uniqueness of the obliquely reected BSDEs in the Hu and Tang specication for the driver [17], namely f

ⁱ

p t, y, z q : f

ⁱ

p t, y

ⁱ

, z

ⁱ

q for i P t 1, . . . , d u. But our results can be easily generalized to the specication f

ⁱ

pt, y, zq : f

ⁱ

pt, y, z

ⁱ

q by using similar arguments as in [6].

The rest of the paper is organised as follows. In Section 2, we introduce the switching

problem with controlled randomisation. We prove that, if there exists a solution to

the associated BSDE with oblique reections, then its Y -component coincides with

the value of the switching problem. A verication argument allows then to deduce

uniqueness of the solution of the obliquely reected BSDE. In Section 3, we show that

there exists indeed a solution to the obliquely reected BSDE under some conditions

on the parameters of the switching problem and its randomisation. We also prove

uniqueness of the solution under some structural condition on the driver f . Finally, we

gather in the Appendix section some technical results.

Notations If n ¥ 1 , we let B

ⁿ

be the Borelian sigma-algebra on R

ⁿ

.

For any ltered probability space pΩ, G, F, Pq and constants T ¡ 0 and p ¥ 1, we dene the following spaces:

• L

^pn

p G q is the set of G -measurable random variables X valued in R

ⁿ

satisfying E r| X |

^p

s   8,

• P pFq is the predictable sigma-algebra on Ω r 0, T s,

• H

^pn

pFq is the set of predictable processes φ valued in R

ⁿ

such that E

»

_T

0

| φ

_t

|

^p

dt

  8 , (1.2)

• S

^pn

pFq is the set of càdlàg processes φ valued in R

ⁿ

such that E

sup

0¤t¤T

| φ

_t

|

^p

  8 , (1.3)

• A

^pn

pFq is the set of continuous processes φ valued in R

ⁿ

such that φ

_T

P L

^pn

p F

_T

q and φ

ⁱ

is nondecreasing for all i 1, . . . , n .

If n 1 , we omit the subscript n in previous notations.

For d ¥ 1 , we denote by p e

_i

q

^d_i1

the canonical basis of R

^d

and S

_d

pRq the set of symmetric matrices of size d d with real coecients.

If D is a convex subset of R

^d

(d ¥ 1) and y P D , we denote by C pyq the outward normal cone at y , dened by

C p y q : t v P R

^d

: v

^J

p z y q ¤ 0 for all z P D u . (1.4) We also set n p y q : C p y q X t v P R

^d

: | v | 1 u.

If X is a matrix of size n m , I € t 1, . . . , n u and J € t 1, . . . , m u, we set X

^pI,Jq

the matrix of size p n | I |q p m | J |q obtained from X by deleting rows with index i P I and columns with index j P J . If I t i u we set X

^pi,Jq

: X

^pI,Jq

, and similarly if J t j u.

If v is a vector of size n and 1 ¤ i ¤ n , we set v

^piq

the vector of size n 1 obtained from v by deleting coecient i.

For p i, j q P t 1, . . . , d u, we dene i

^pjq

: i 1

_ti¡ju

P t 1, . . . , d 1 u , for d ¥ 2 .

We denote by ¥ the component by component partial ordering relation on vectors and

matrices.

2 Switching problem with controlled randomisation

We introduce here a new kind of stochastic control problem that we name switching problem with controlled randomisation. In contrast with the usual switching problems [14, 16, 17], the agent cannot choose directly the new state, but chooses a probability distribution under which the new state will be determined. In this section, we assume the existence of a solution to some auxiliary obliquely reected BSDE to characterize the value process and an optimal strategy for the problem, see Assumption 2.2 below.

Let p Ω, G, Pq be a probability space. We x a nite time horizon T ¡ 0 and κ ¥ 1, d ¥ 2 two integers. We assume that there exists a κ-dimensional Brownian motion W and a sequence p U

_n

q

n¥1

of independent random variables, independent of W , uniformly distributed on r 0, 1 s. We also assume that G is generated by the Brownian motion W and the family p U

_n

q

n¥1

. We dene F

⁰

p F

_t⁰

q

t¥0

as the augmented Brownian ltration, which satises the usual conditions.

Let C be an ordered compact metric space and F : C t 1, . . . , d u r 0, 1 s Ñ t 1, . . . , d u a measurable map. To each u P C is associated a transition probability function on the state space t 1, . . . , d u, given by P

_i,j^u

: Pp F p u, i, U q j q for U uniformly distributed on r 0, 1 s. We assume that for all p i, j q P t 1, . . . , d u

²

, the map u ÞÑ P

_i,j^u

is continuous.

Let ¯ c : t 1, . . . , d u C Ñ R , p i, u q ÞÑ ¯ c

^u_i

a map such that u ÞÑ c ¯

^u_i

is continuous for all i 1, . . . , d. We denote sup

iPt1,...,du,uPC

¯ c

^u_i

: c ˇ and inf

_{iPt1,...,du,uP}C

¯ c

^u_i

: ˆ c.

Let ξ p ξ

¹

, . . . , ξ

^d

q P L

²_d

p F

_T⁰

q and f : Ω r 0, T s R

^d

R

^dκ

Ñ R

^d

a map satisfying

• f is P pF

⁰

q b B

^d

b B

^dκ

-measurable and f p , 0, 0 q P H

²_d

pF

⁰

q.

• There exists L ¥ 0 such that, for all p t, y, y

¹

, z, z

¹

q P r 0, T sR

^d

R

^d

R

^dκ

R

^dκ

,

| f p t, y, z q f p t, y

¹

, z

¹

q| ¤ L p| y y

¹

| | z z

¹

|q .

These assumptions will be in force throughout our work. We shall also use, in this section only, the following additional assumptions.

Assumption 2.1. i) Switching costs are assumed to be positive, i.e. ˆ c ¡ 0 . ii) For all p t, y, z q P r 0, T s R

^d

R

^dκ

, it holds almost surely,

f p t, y, z q p f

ⁱ

p t, y

ⁱ

, z

ⁱ

qq

1¤i¤d

. (2.1) Remark 2.1. i) It is usual to assume positive costs in the litterature on switching prob- lem. In particular, the cumulative cost process, see (2.2), is non decreasing. Introducing signed costs adds extra technical diculties in the proof of the representation theorem (see e.g. [19] and references therein). We postpone the adaptation of our results in this more general framework to future works.

ii) Assumption (2.1) is also classical since it allows to get a comparison result for BS-

DEs which is key to obtain the representation theorem. Note however than our results

can be generalized to the case f

ⁱ

p t, y, z q f

ⁱ

p t, y, z

ⁱ

q for i P t 1, ..., d u by using similar

arguments as in [5].

2.1 Solving the control problem using obliquely reected BSDEs We dene in this section the stochastic optimal control problem. We rst introduce the strategies available to the agent and related processes. The denition of the strategy is more involved than in the usual switching problem setting since its adaptiveness property is understood with respect to a ltration built recursively.

A strategy is thus given by φ p ζ

0

, p τ

n

q

n¥0

, p α

n

q

n¥1

q where ζ

0

P t 1, . . . , d u, p τ

n

q

n¥0

is a nondecreasing sequence of random times and p α

_n

q

n¥1

is a sequence of C -valued random variables, which satisfy:

• τ

0

P r 0, T s and ζ

0

P t 1, . . . , d u are deterministic.

• For all n ¥ 0 , τ

n 1

is a F

ⁿ

-stopping time and α

n 1

is F

_τⁿ

n 1

-measurable (recall that F

⁰

is the augmented Brownian ltration). We then set F

^{n 1}

p F

_t^{n 1}

q

t¥0

with F

_t^{n 1}

: F

_tⁿ

_ σ p U

_{n 1}

1

_{tτn 1¤tu}

q.

Lastly, we dene F

⁸

p F

_t⁸

q

t¥0

with F

_t⁸

: š

n¥0

F

_tⁿ

, t ¥ 0 . For a strategy φ p ζ

0

, p τ

n

q

n¥0

, p α

n

q

n¥1

q, we set, for n ¥ 0 ,

ζ

_{n 1}

: F p α

_{n 1}

, ζ

_n

, U

_{n 1}

q and a

_t

: ¸

⁸

k0

ζ

_k

1

_{rτk,τk 1q}

p t q , t ¥ 0 ,

which represents the state after a switch and the state process respectively. We also introduce two processes, for t ¥ 0 ,

A

^φ_t

¸

⁸

k0

¯ c

^α_ζ^{k 1}

k

1

_{tτk 1¤tu}

and N

_t^φ

: ¸

k¥0

1

_{tτk 1¤tu}

. (2.2) The random variable A

^φ_t

is the cumulative cost up to time t and N

_t^φ

is the number of switches before time t . Notice that the processes p a, A

^φ

, N

^φ

q are adapted to F

⁸

and that A

^φ

is a non decreasing process.

We say that a strategy φ p ζ

0

, p τ

n

q

n¥0

, p α

n

q

n¥1

q is an admissible strategy if the cumu- lative cost process satises

A

^φ_T

A

^φ_τ0

P L

²

pF

⁸T

q and E

A

^φ_τ0 ²

F

_τ⁰

0

  8 a.s. (2.3)

We denote by A the set of admissible strategies, and for t P r 0, T s and i P t 1, . . . , d u, we denote by A

tⁱ

the subset of admissible strategies satisfying ζ

₀

i and τ

₀

t . Remark 2.2. i) The denition of an admissible strategy is slightly weaker than usual

[17], which requires the stronger property A

^φ_T

P L

²

p F

_T⁸

q. But, importantly, the

above denition is enough to dene the switched BSDE associated to an admissible

control, see below. Moreover, we observe in the next section that optimal strategies

are admissible with respect to our denition, but not necessarily with the usual one,

due to possible simultaneous jumps at the initial time.

ii) For technical reasons involving possible simultaneous jumps, we cannot consider the generated ltration associated to a, which is contained in F

⁸

.

We are now in position to introduce the reward associated to an admissible strategy. If φ p ζ

₀

, p τ

_n

q

n¥0

, p α

_n

q

n¥1

q P A , the reward is dened as the value E

U

_τ^φ₀

A

^φ_τ0

F

_τ⁰

0

, where p U

^φ

, V

^φ

, M

^φ

q P S

²

pF

⁸

q H

²κ

pF

⁸

q H

²

pF

⁸

q is the solution of the following switched BSDE [17] on the ltered probability space p Ω, G, F

⁸

, Pq:

U

t

ξ

^aT

»

_T

t

f

^as

p s, U

s

, V

s

q ds

»

_T

t

V

s

dW

s

»

_T

t

dM

s

»

_T

t

dA

^φ_s

, t P r τ

0

, T s . (2.4) Remark 2.3. This switched BSDE rewrites as a classical BSDE in F

⁸

, and since A

^φ

A

^φ_t

P S

²

pF

⁸

q, the terminal condition and the driver are standard parameters, there exists a unique solution to (2.4) for all φ P A . We refer to Section A.2.2 for more details.

For t P r0, T s and i P t1, . . . , du, the agent aims thus to solve the following maximisation problem:

V

_tⁱ

ess sup

φPAtⁱ

E

U

_t^φ

A

^φ_t

F

_t⁰

. (2.5)

We rst remark that this control problem corresponds to (1.1) as soon as f does not depend on y and z. Moreover, the term E

A

^φ_t

F

_t⁰

is non zero if and only if we have at least one instantaneous switch at initial time t .

The main result of this section is the next theorem that relates the value process V to the solution of an obliquely reected BSDEs, that is introduced in the following assumption:

Assumption 2.2. i) There exists a solution p Y, Z, K q P S

²_d

pF

⁰

q H

²_dκ

pF

⁰

q A

²_d

pF

⁰

q to the following obliquely reected BSDE:

Y

_tⁱ

ξ

»

_T

t

f

ⁱ

ps, Y

_sⁱ

, Z

_sⁱ

qds

»

_T

t

Z

_sⁱ

dW

s

»

_T

t

dK

_sⁱ

, t P r0, T s, i P I, (2.6)

Y

t

P D, t P r0, T s, (2.7)

»

_T

0

Y

_tⁱ

sup

uPC

#

_d

¸

j1

P

_i,j^u

Y

_t^j

c ¯

^u_i

+

dK

_tⁱ

0, i P I, (2.8)

where I : t 1, . . . , d u and D is the following convex subset of R

^d

: D :

#

y P R

^d

: y

i

¥ sup

uPC

#

_d

¸

j1

P

_i,j^u

y

j

¯ c

^u_i

+

, i P I +

. (2.9)

ii) For all u P C and i P t 1, . . . , d u, we have P

_i,i^u

1 .

Let us observe that the positive costs assumption implies that D has a non-empty interior. Except for Section 2.2, this is the main setting for this part, recall Remark 2.1.

In Section 3, the system (2.6)-(2.7)-(2.8) is studied in details in a general costs setting.

An important step is then to understand when D has non-empty interior.

Theorem 2.1. Assume that Assumptions 2.1 and 2.2 are satised.

1. For all i P t 1, . . . , d u, t P r 0, T s and φ P A

tⁱ

, we have Y

_tⁱ

¥ E

U

_t^φ

A

^φ_t

F

_t⁰

. 2. We have Y

_tⁱ

E

U

_t^φ

A

^φ_t

F

_t⁰

, where φ

p i, p τ

_n

q

n¥0

, p α

_n

q

n¥1

q P A

tⁱ

is dened in (2.27)-(2.28).

The proof is given at the end of Section 2.3. It will use several lemmata that we introduce below. We rst remark, that as an immediate consequence, we obtain the uniqueness of the BSDE used to characterize the value process of the control problem.

Corollary 2.1. Under Assumptions 2.1 and 2.2, there exists a unique solution p Y, Z, K q P S

²_d

pF

⁰

q H

²_dκ

pF

⁰

q A

²_d

pF

⁰

q to the obliquely reected BSDE (2.6)-(2.7)-(2.8).

Remark 2.4. The classical switching problem is an example of switching problem with controlled randomisation. Indeed, we just have to consider C t 1, ..., d 1 u,

P

_i,j^u

#

1 if j i u mod d,

0 otherwise. , @ u P C, 1 ¤ i, j ¤ d and

c

i,j

$ '

&

' %

¯

c

^j_iⁱ

if j ¡ i

¯

c

^j_i^{i d}

if j   i 0 if j i.

@ u P C, 1 ¤ i, j ¤ d.

We observe that, in this specic case, there is no extra-randomness introduced at each switching time and so there is no need to consider an enlarged ltration. In this setting, Theorem 2.1 is already known and Assumption 2.2 is fullled, see e.g. [16, 17].

2.2 Uniqueness of solutions to reected BSDEs with general costs In this section, we extend the uniqueness result of Corollary 2.1. Namely, we consider the case where inf

_1¤i¤d,uPC

¯ c

^u_i

ˆ c can be nonpositive, meaning that only Assumptions 2.1-ii) and 2.2 hold here. Assuming that D has a non empty interior, we are then able to show uniqueness to (2.6)-(2.7)-(2.8) in Proposition 2.1 below.

Fix y

⁰

in the interior of D . It is clear that for all 1 ¤ i ¤ d , y

_i⁰

¡ sup

uPC

#

_d

¸

j1

P

_i,j^u

y

⁰_j

c ¯

^u_i

+

.

We set, for all 1 ¤ i ¤ d and u P C ,

˜

c

^u_i

: y

⁰_i

¸

d j1

P

_i,j^u

y

_j⁰

¯ c

^u_i

¡ 0,

so that ˆ ˜ c : inf

_1¤i¤d,uPC

c ˜

^u_i

¡ 0 by compactness of C . We also consider the following set D ˜ :

#

˜

y P R

^d

: ˜ y

_i

¥ sup

uPC

#

_d

¸

j1

P

_i,j^u

y ˜

_j

˜ c

^u_i

+

, 1 ¤ i ¤ d +

.

Lemma 2.1. Assume that D has a non empty interior. Then, D ˜ y y

⁰

: y P D (

.

Proof. If y P D , let y ˜ : y y

⁰

. For 1 ¤ i ¤ d and u P C , we have

˜

y

i

y

i

y

_i⁰

¥

¸

d j1

P

_i,j^u

y

j

¯ c

^u_i

y

_i⁰

¸

d j1

P

_i,j^u

py

j

y

_j⁰

q p¯ c

^u_i

y

_i⁰

¸

d j1

P

_i,j^u

y

⁰_j

q

¸

d j1

P

_i,j^u

y ˜

j

˜ c

^u_i

, hence y ˜ P D ˜ .

Conversely, let y ˜ P D ˜ and let y : y ˜ y

⁰

. We can show by the same kind of calculation

that y P D . l

Proposition 2.1. Assume that D has a non empty interior. Under Assumptions 2.1-ii) and 2.2-ii), there exists at most one solution to (2.6)-(2.7)-(2.8) in S

²_d

pF

⁰

q H

²_dκ

pF

⁰

q A

²_d

pF

⁰

q.

Proof. Let us assume that pY

¹

, Z

¹

, K

¹

q and pY

²

, Z

²

, K

²

q are two solutions to (2.6)- (2.7)-(2.8). We set Y ˜

¹

: Y

¹

y

0

and Y ˜

²

: Y

²

y

0

. Then one checks easily that p Y ˜

¹

, Z

¹

, K

¹

q and p Y ˜

²

, Z

²

, K

²

q are solutions to (2.6)-(2.7)-(2.8) with terminal condition ξ ˜ ξ y

0

, driver f ˜ given by

f ˜

ⁱ

p t, y ˜

_i

, z

_i

q : f

ⁱ

p t, y ˜

_i

y

_i⁰

, z

_i

q , 1 ¤ i ¤ d, t P r 0, T s , y ˜ P R

^d

, z P R

^dκ

,

and domain D ˜ . This domain is associated to a randomised switching problem with ˆ ˜

c ¡ 0 , hence Corollary 2.1 gives that p Y ˜

¹

, Z

¹

, K

¹

q p Y ˜

²

, Z

²

, K

²

q which implies the

uniqueness. l

2.3 Proof of the representation result

We prove here our main result for this part, namely Theorem 2.1. It is divided in several steps.

2.3.1 Preliminary estimates

We rst introduce auxiliary processes associated to an admissible strategy and prove some key integrability properties.

Let i P t 1, . . . , d u and t P r 0, T s. We set, for φ P A

tⁱ

and t ¤ s ¤ T , Y

_s^φ

: ¸

k¥0

Y

_s^ζk

1

_{rτk,τk 1q}

p s q , (2.10) Z

_s^φ

: ¸

k¥0

Z

_s^ζk

1

_{rτk,τk 1q}

p s q , (2.11) K

^φ_s

: ¸

k¥0

K

_s^ζk

1

_{rτk,τk 1q}

psq, (2.12) M

^φ_s

: ¸

k¥0

Y

_τ^ζ_k^k₁¹

E

Y

_τ^ζ_k^k₁¹

F

_τ^k

k 1

1

_{tt τk 1¤su}

, (2.13) A

^φ_s

¸

k¥0

Y

_τ^ζ_k^k₁

E

Y

τ^ζ_k^k₁¹

F

_τ^k

k 1

¯ c

^α_ζ^{k 1}

k

1

_{tt τk 1¤su}

. (2.14) Remark 2.5. For all k ¥ 0 , since α

_{k 1}

P F

_τ^k

k 1

, we have E

Y

_τ^ζ_k^k₁¹

F

_τ^k

k 1

¸

d j1

P

ζ

_{k 1}

j | F

_τ^k

k 1

Y

_τ^j

k 1

¸

d j1

P

_ζ^αk

k,j

Y

_τ^j

k 1

. (2.15) Lemma 2.2. Assume that assumption (2.2) is satised. For any admissible strategy φ P A

tⁱ

, M

^φ

is a square integrable martingale with M

^φ_t

0 . Moreover, A

^φ

is increasing and satises A

^φ_T

P L

²

p F

_T⁸

q. In addition,

E

¸

k¥0

Y

τ^ζk^k1¹

E

Y

τ^ζk^k1¹

F

_τ^k

k 1

1

_{tτk 1¤tu}

2

F

_t⁰

  8 a.s. (2.16)

Proof. Let φ P A

tⁱ

. Using (2.14) and (2.15), we have, for all s P r t, T s, A

^φ_s

¸

k¥0

Y

_τ^ζ_k^k₁

¸

d j1

P

_ζ^{αk 1}

k,j

Y

_τ^j_{k 1}

¯ c

^α_ζ^{k 1}

k

1

_{tt τk 1¤su}

, (2.17) which is increasing since each summand is positive as Y P D .

We have, for t ¤ s ¤ T , Y

_s^φ

Y

_t^φ

¸

k¥0

Y

_τ^ζ_k^k_1^s

Y

_τ^ζ_k^k_^s

¸

k¥0

Y

τ^ζ_k^k₁¹

Y

_τ^ζ_k^k₁

1

_{tt τk 1¤su}

. (2.18)

Using (2.6), we get, for all k ¥ 0 , Y

_τ^ζ_k^k_1^s

Y

_τ^ζ_k^k_^s

»

_{τk 1^s}

τk^s

f

^ζk

p u, Y

_u^ζk

, Z

_u^ζk

q du

»

_{τk 1^s}

τk^s

Z

_u^ζk

dW

_u

»

_{τk 1^s}

τk^s

dK

_u^ζk

,

Recalling ζ

_k

is F

_τk

-measurable. We also have, using (2.15), for all k ¥ 0 , Y

τ^ζk^k1¹

Y

_τ^ζ_k^k₁

Y

τ^ζk^k1¹

E

Y

τ^ζk^k1¹

F

_τ^k

k 1

Y

_τ^ζ_k^k₁

¸

d j1

P

_ζ^{αk 1}

k,j

Y

_τ^j_{k 1}

c ¯

^α_ζ^{k 1}

k

¯ c

^α_ζ^{k 1}

k

.

Plugging the two previous equalities into (2.18), we get:

Y

_s^φ

Y

_t^φ

¸

k¥0

»

_{τk 1^s}

τk^s

f

^ζk

p u, Y

_u^ζk

, Z

_u^ζk

q du

»

_{τk 1^s}

τk^s

Z

_u^ζk

dW

u

»

_{τk 1^s}

τk^s

dK

_u^ζk

¸

k¥0

Y

τ^ζk^k1¹

E

Y

τ^ζk^k1¹

F

_τ^k

k 1

1

_{tt τk 1¤su}

A

^φ_s

A

^φ_t

¸

k¥0

Y

_τ^ζk

k 1

¸

d j1

P

_ζ^{αk 1}

k,j

Y

_τ^j

k 1

c ¯

^α_ζ^{k 1}

k

1

_{tt τk 1¤su}

. (2.19) By denition of Y

^φ

, Z

^φ

, K

^φ

, M

^φ

, A

^φ

, we obtain, for all s P r t, T s,

Y

_s^φ

ξ

^aT

»

_T

s

f

^au

pu, Y

_u^φ

, Z

_u^φ

qdu

»

_T

s

Z

_u^φ

dW

u

»

_T

s

dM

^φ_u

»

_T

s

dA

^φ_u

A

^φ_T

K

^φ_T

A

^φ_s

K

_s^φ

. (2.20)

For any n ¥ 1 , we consider the admissible strategy φ

n

p ζ

0

, p τ

_kⁿ

q

k¥0

, p α

ⁿ_k

q

k¥1

q dened by ζ

₀ⁿ

i ζ

₀

, τ

_kⁿ

τ

_k

, α

ⁿ_k

α

_k

for k ¤ n , and τ

_kⁿ

T 1 for all k ¡ n . We set Y

ⁿ

: Y

^φn

, Z

ⁿ

v : Z

^φn

, and so on.

By (2.20) applied to the strategy φ

ⁿ

, we get, recalling that A

ⁿ_t

0 , A

ⁿ_τ

n^T

Y

_tⁿ

Y

_τⁿ

n^T

»

_τn^T

t

f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q ds

»

_τn^T

t

Z

_sⁿ

dW

_s

»

_τn^T

t

dM

ⁿ_s

»

_τn^T

t

dA

ⁿ_s

»

_τn^T

t

dK

ⁿ_s

. (2.21)

We obtain, for a constant Λ ¡ 0 , E

| A

ⁿ_τn^T

|

²

¤ Λ

E

| Y

_tⁿ

|

²

| Y

_τⁿ_n^T

|

²

»

_τn^T

t

| f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q|

²

ds (2.22)

»

_τn^T

t

| Z

_sⁿ

|

²

ds

»

_τn^T

t

dr M

ⁿ

s

s

pA

^φ_T

A

^φ_t

q

²

p K

ⁿ_T

q

²

.

We have

E

| Y

_rⁿ

|

²

¤

¸

d j1

E

| Y

_r^j

|

²

E

| Y

_r

|

²

¤ E

sup

t¤r¤T

| Y

_r

|

²

} Y }

²_S²

dpF⁰q

, E

»

_τn^T

t

| Z

_sⁿ

|

²

ds

¤ } Z }

_H²_dκpF⁰_q

, E

»

_τn^T

t

| f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q|

²

ds

¤ 4L

²

T } Y }

²_S²

dpF⁰q

4L

²

} Z }

_H²

dpF⁰q

2 } f p , 0, 0 q}

²_H²

dpF⁰q

and

E

p K

ⁿ_T

q

²

¤ E

| K

_T

|

²

.

Thus, by these estimates and the fact that A

^φ_T

A

^φ_t

P L

²

p F

_T⁸

q as φ is admissible, there exists a constant Λ

₁

¡ 0 such that

E | A

ⁿ_τ

n^T

|

²

¤ Λ

1

Λ E

»

τn^T t

d r M

ⁿ

s

s

. (2.23)

Using (2.20) applied to φ

ⁿ

and Itô's formula between t and τ

_n

^ T , since M

ⁿ

is a square integrable martingale orthogonal to W and A

ⁿ

, A

ⁿ

, K

ⁿ

are nondecreasing and nonnegative, we get

E

| Y

_tⁿ

|

²

»

_τn^T

t

| Z

_sⁿ

|

²

ds

»

_τn^T

t

d r M

ⁿ

s

s

E

| Y

_τⁿ_n^T

|

²

2

»

_τn^T

t

Y

_sⁿ

f

^ans

ps, Y

_sⁿ

, Z

_sⁿ

qds 2

»

_τn^T

t

Y

_sⁿ

dA

ⁿ_s

2

»

_τn^T

t

Y

_sⁿ

dA

ⁿ_s

2

»

_τn^T

t

Y

_sⁿ

dK

ⁿ_s

¤ E

| Y

_τⁿ

n^T

|

²

2 E

»

_τn^T

t

| Y

_sⁿ

f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q| ds

2 E

»

_τn^T

t

| Y

_sⁿ

| dA

ⁿ_s

2 E

»

_τn^T

t

| Y

_sⁿ

| dA

ⁿ_s

2 E

»

_τn^T

t

| Y

_sⁿ

| dK

ⁿ_s

. (2.24)

We have, using Young's inequality, for some ¡ 0 , and (2.23), E

»

_τn^T

t

| Y

_sⁿ

f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q| ds

¤ 1 2 E

»

_τn^T

t

| Y

_sⁿ

|

²

ds

1

2 E

»

_τn^T

t

| f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q|

²

ds

¤ T p 1

2 2L

²

q} Y }

²_S²

dpF⁰q

2L

²

} Z }

_H²

dpF⁰q

} f p , 0, 0 q}

²_H²

dpF⁰q

, E

»

_τn^T

t

| Y

_sⁿ

| dA

ⁿ_s

¤ 1 2 } Y }

²_S²

dpF⁰q

1 2 E

p A

^φ_T

A

^φ_t

q

²

,

E

»

_τn^T

t

| Y

_sⁿ

| dK

ⁿ_s

¤ 1 2 } Y }

²_S²

dpF⁰q

1 2 E

| K

_T

|

²

,

and

E

»

_τn^T

t

| Y

_sⁿ

| dA

ⁿ_s

¤ 1 2 } Y }

²_S²

dpF⁰q

2 E

p A

ⁿ_τ

n^T

q

²

¤ 1 2 } Y }

²_S²

dpF⁰q

2

Λ

₁

Λ E

»

_τn^T

t

d r M

ⁿ

s

s

.

Using these estimates together with (2.24) gives, for a constant C

¡ 0 independent of n ,

p 1 Λ q E

»

_τn^T

t

d r M

ⁿ

s

s

¤ C

} Y }

²_S²

dpF⁰q

} Z }

_H²_dκpF⁰_q

} f p , 0, 0 q}

²_H²

dpF⁰q

E

| K

T

|

²

, (2.25) and chosing

_2Λ¹

gives that E ³

_τn^T

t

d r M

ⁿ

s

s

is upper bounded independently of n . We also get an upper bound independent of n for E

p A

ⁿ_τ

n^T

q

²

by (2.23).

Since ³

_τn^T

t

d r M

ⁿ

s

s

(resp. | A

ⁿ_τ

n^T

|

²

) is nondecreasing to ³

_T

t

d r M

^φ

s

s

(resp. to | A

^φ_T

|

²

), we obtain by monotone convergence the rst part of Lemma (2.2). It is also clear that M

^φ

is a martingale satisfying M

^φ_t

0 .

We now prove (2.16). Using that E

pN

_t^φ

q

²

F

_t⁰

is almost-surely nite as φ is admissible and ˆ c ¡ 0 , we compute,

E

¸

k¥0

Y

τ^ζk^k1¹

E

Y

τ^ζk^k1¹

F

_τ^k

k 1

1

_{tτk 1¤tu}

2

F

_t⁰

¤ E

¸

k¥0

Y

τ^ζk^k1¹

¸

d j1

p

^α_ζ^{k 1}

k,j

Y

_t^j

1

_{tτk 1¤tu}

2

F

_t⁰

¤ 4 | Y

_t

|

²

E

p N

_t^φ

q

²

F

_t⁰

  8 a.s. (2.26)

l 2.3.2 An optimal strategy

We now introduce a strategy, which turns out to be optimal for the control problem.

This strategy is the natural extension to our setting of the optimal one for classical switching problem, see e.g. [17]. The rst key step is to prove that this strategy is admissible, which is more involved than in the classical case due to the randomisation.

Let φ

p ζ

₀

, p τ

_n

q

n¥0

, p α

_n

q

n¥1

q dened by τ

₀

t and ζ

₀

i and inductively by:

τ

_{k 1}

inf

#

τ

_k

¤ s ¤ T : Y

s^ζk

max

_uPC

#

_d

¸

j1

P

_ζ^u

k,j

Y

_s^j

c ¯

^u_ζ k

++

^ p T 1 q , (2.27)

α

_{k 1}

min

#

α P arg max

uPC

#

_d

¸

j1

P

_ζ^u

k,j

Y

_τ^j k 1

¯ c

^u_ζ

k

++

, (2.28)

recall that C is ordered.

In the following lemma, we show that, since D has non-empty interior, the number of switch (hence the cost) required to leave any point on the boundary of D is square integrable, following the strategy φ

. This result will be used to prove that the cost associated to φ

satises E

A

^φ_t ²

F

_t⁰

is almost surely nite.

Lemma 2.3. Let Assumption 2.2-ii) hold. For y P D , we dene S p y q

#

1 ¤ i ¤ d : y

_i

max

uPC

#

_d

¸

j1

P

_i,j^u

y

_j

c ¯

^u_i

++

, (2.29)

and p u

i

q

iPSpyq

the family of elements of C given by

u

i

min arg max

uPC

#

_d

¸

j1

P

_i,j^u

y

j

¯ c

^u_i

+

.

Consider the homogeneous Markov Chain X on S p y q Y t 0 u dened by, for k ¥ 0 and i, j P S p y q

²

,

Pp X

_{k 1}

j | X

_k

i q P

_i,j^ui

, Pp X

_{k 1}

0 | X

_k

i q 1 ¸

jPSpyq

P

_i,j^ui

,

Pp X

_{k 1}

0 | X

_k

0 q 1, Pp X

_{k 1}

i | X

_k

0 q 0.

Then 0 is accessible from every i P S p y q, meaning that X is an absorbing Markov Chain.

Moreover, let N p y q inf t n ¥ 0 : X

_n

0 u. Then N p y q P L

²

pP

ⁱ

q for all i P S p y q, where P

ⁱ

is the probability satisfying P

ⁱ

p X

₀

i q 1 .

Proof. Assume that there exists i P S p y q from which 0 is not accessible. Then every communicating class accessible from i is included in Spyq. In particular, there exists a recurrent class S

¹

€ S p y q. For all i P S

¹

, we have P

_i,j^ui

0 if j R S

¹

since S

¹

is recurrent.

Moreover, since S

¹

€ S p y q, we obtain, for all i P S

¹

, by denition of S p y q, y

i

¸

jPS¹

P

_i,j^ui

y

j

c ¯

^u_iⁱ

. (2.30) Since S

¹

is a recurrent class, the matrix P ˜ p P

_i,j^ui

q

i,jPS¹

is stochastic and irreducible.

By denition of D , we have

D € R

^d|S1|

$ &

% z P R

^|S1|

| z

i

¥ ¸

jPS¹

P

_i,j^ui

z

j

c ¯

^u_iⁱ

, i P S

¹

, .

- R

^d|S1|

D

¹

.

With a slight abuse of notation, we do not renumber coordinates of vectors in D

¹

. Let i

0

P S

¹

and let us restrict ourself to the domain D

¹

. According to Lemma 3.1, D

¹

is invariant by translation along the vector p 1, ..., 1 q of R

^|S1|

. Moreover, Assumption 3.1 is fullled since P ˜ is irreducible and controls p u

_i

q

iPSpyq

are set. So, Proposition 3.1 gives us that D

¹

X t z P R

^|S1|

| z

_i0

0 u is a compact convexe polytope. Recalling (2.30), we see that py

i

y

i0

q

iPS¹

is a point of D

¹

X tz P R

^|S1|

|z

i0

0u that saturates all the inequalities.

So, p y

i

y

i0

q

iPS¹

is an extreme points of D

¹

X t z P R

^|S1|

| z

i0

0 u and all extreme points are given by

E :

$ &

% z P R

^|S1|

| z

_i

¸

jPS¹

P

_i,j^ui

z

_j

¯ c

^u_iⁱ

, i P S

¹

, z

_i0

0 , . - .

Recalling that D

¹

is compact, E is a nonempty bounded ane subspace of R

^|S1|

, so it is a singleton. Since D

¹

X t z

_i0

0 u is a compact convex polytope, it is the convex hull of E and so it is also a singleton. Hence D

¹

is a line in R

^|S1|

. Moreover, |S

¹

| ¥ 2 as P

_i,i^u

1 for all u P C and i P t 1, . . . , d u. Thus D € R

^d|S1|

D

¹

gives a contradiction with the fact that D has non-empty interior and the rst part of the lemma is proved.

Finally, we have N p y q P L

²

pP

ⁱ

q for all i P S p y q thanks to Theorem 3.3.5 in [18]. l Lemma 2.4. Assume that assumption (2.2) is satised. The strategy φ

is admissible.

Proof. For n ¥ 1 , we consider the admissible strategy φ

_n

p ζ

₀

, p τ

_kⁿ

q

k¥0

, p α

ⁿ_k

q

k¥1

q dened by ζ

₀ⁿ

i ζ

₀

, τ

_kⁿ

τ

_k

, α

ⁿ_k

α

_k

for k ¤ n, and τ

_kⁿ

T 1 for all k ¡ n. We set Y

_sⁿ

: Y

_s^φn

, Z

_sⁿ

: Z

_s^φn

and so on, for all s P r t, T s.

By denition of τ

, α

, recall (2.27)-(2.28), it is clear that A

ⁿ_s^τ

n

0 and that ³

τ_{k 1}^s

τ_k^s

dK

_u^ζk

0 for all k   n and s P rt, T s. The identity (2.20) for the admissible strategy φ

ⁿ

gives

Y

_tⁿ

Y

_τⁿ n^T

»

_τ n^T t

f

^ans

p s, Y

_sⁿ

, Z

_sⁿ

q ds

»

_τ n^T t

Z

_sⁿ

dW

s

»

_τ n^T t

dM

ⁿ_s

»

_τ n^T t

dA

ⁿ_s

. Using similar arguments and estimates as in the precedent proof, we get

E | A

ⁿ_τ

n^T

A

ⁿ_t

|

²

¤ Λ

₁

Λ E

»

_τn^T

t

d r M

^u

s

s

, (2.31)

and, for ¡ 0 , p 1 Λ q E

»

_τn^T

t

d r M

ⁿ

s

s

¤ C

} Y }

²_S²

dpF⁰q

} Z }

_H²

dκpF⁰q

} f p , 0, 0 q}

²_H²

dpF⁰q

. (2.32) Choosing

_2Λ¹

gives that E

³

_τ n^T

t

d r M

ⁿ

s

s

and E | A

ⁿ_τ

n^T

A

ⁿ_t

|

²

are upper bounded uniformly in n , hence by monotone convergence, we get that A

^φ_T

A

^φ_t

P L

²

p F

_T⁸

q.

It remains to prove that A

^φ_t

P L

²

p F

_t⁰

q. We have A

^φ_t

¤ ˇ cN

_t^φ

, and E

p N

_t^φ

q

²

F

_t⁰

  8 a.s. is immediate from Lemma 2.3, since E

p N

_t^φ

q

²

F

_t⁰

Ψ p Y

t

q with Ψ p y q E

ⁱ

p N p y qq

²

, y P D , where E

ⁱ

is the expectation under the probability P

ⁱ

dened in

Lemma 2.3. l