Switching problem and related system of reflected backward SDEs

Transcription

1 Sochasic Processes and heir Applicaions 20 (200) Swiching problem and relaed sysem of refleced backward SDEs Said Hamadène a, Jianfeng Zhang b, a Universié du Maine, Déparemen de Mahémaiques, Equipe Saisique e Processus, Avenue Olivier Messiaen, Le Mans, Cedex 9, France b USC Deparmen of Mahemaics, S. Vermon Ave, KAP 08, Los Angeles, CA 90089, USA Received 3 Ocober 2008; received in revised form 6 January 200; acceped 6 January 200 Available online 4 January 200 Absrac This paper sudies a sysem of backward sochasic differenial equaions wih oblique reflecions (RBSDEs for shor), moivaed by he swiching problem under Knighian uncerainy and recursive uiliies. The main feaure of our sysem is ha is componens are inerconneced hrough boh he generaors and he obsacles. We prove exisence, uniqueness, and sabiliy of he soluion of he RBSDE, and give he expression of he price and he opimal sraegy for he original swiching problem via a verificaion heorem. c 200 Elsevier B.V. All righs reserved. MSC: 60G40; 93E20; 62P20; 9B99 Keywords: Real opions; Saring and sopping problem; Swiching problem; Backward SDEs; Refleced BSDEs; Oblique reflecion; Snell envelope; Opimal sopping problem. Inroducion This paper sudies he wellposedness of a general sysem of Backward SDEs wih oblique reflecions, moivaed by our sudy on swiching problems. An earlier version of his paper was eniled The Saring and Sopping Problem under Knighian Uncerainy and Relaed Sysems of Refleced BSDEs. Corresponding auhor. Tel.: ; fax: addresses: hamadene@univ-lemans.fr (S. Hamadène), jianfenz@usc.edu (J. Zhang) /$ - see fron maer c 200 Elsevier B.V. All righs reserved. doi:0.06/j.spa

2 404 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) The sandard saring and sopping (or wo modes swiching) problem has araced a lo of ineres during he pas decades; see a long lis [ 3,6,9,7,8,2,,7,2,20,28,3,33 35], and he references herein. Assume, for example, ha a power plan produces elecriciy whose selling price flucuaes and depends on many facors such as consumer demand, oil prices, weaher and so on. I is well known ha elecriciy canno be sored (or oo expensive o sore) and once produced i should be consumed almos immediaely. Therefore elecriciy is produced only when here is enough profiabiliy in he marke. Oherwise he power plan is closed ill he ime when he profiabiliy is coming back again. Then for his plan here are wo modes: operaing and closed. Accordingly, a managemen sraegy of he plan is an increasing sequence of sopping imes δ = (τ n ) n 0 wih τ 0 = 0. A ime τn, he manager swiches he mode of he plan from is curren one o he oher. Such a swich of modes is no free and generaes expendiures. Suppose now ha we have an adaped sochasic process X which sands for eiher he marke elecriciy price or facors which deermine he price. When he plan is run under a sraegy δ, is yield is given by a quaniy denoed by J(δ), which depends also on X and many oher parameers such as uiliy funcions, expendiures, ec. Therefore he main problem is o find an opimal managemen sraegy δ = (τ n ) n such ha J(δ ) = sup δ J(δ), and consequenly, he value J(δ ) is nohing bu he fair price of he power plan in he energy marke. We noe ha his swiching problem has also been used o model indusries like copper or aluminium mines,..., where pars of he producion process are emporarily reduced or shu down when e.g. fuel, elecriciy or coal prices are oo high o be profiable from running hem. A furher area of applicaions includes Tolling Agreemens (see [4,6] for more deails). A naural exension of he wo-mode problem is he muli-mode swiching problem. This has been recenly sudied by several auhors amongs we quoe Carmona and Ludkovski [4], Djehiche e al. [0] and Porche e al. [3]. The idea of using RBSDEs in saring and sopping problems was iniiaed by Hamadène and Jeanblanc [2]. In heir model he wo-dimensional RBSDE is linear and can be ransformed ino a one-dimensional RBSDE wih double barriers. Then he wellposedness of he RBSDE is known in he lieraure. Moreover, via a verificaion heorem hey express boh he opimal sraegy δ and he plan s value J(δ ) in erms of he soluion o he RBSDE. Several oher papers have also used his ool (see e.g. [4,3]). In [4], he auhors consider a muli-mode swiching problem. However hey lef open he quesion of he exisence of he soluion of he associaed RBSDEs wih oblique reflecion. The problem is solved by Djehiche e al. in [0]. Our firs goal of he paper is o exend he work Hamadène and Jeanblanc [2] by considering Knighian uncerainy and recursive uiliy. All he works quoed above assume ha fuure uncerainy of marke condiions X is characerized by a cerain probabiliy measure P. The Knighian uncerainy inroduced by Knigh [27] assumes insead ha he marke evolves according o one of many possible probabiliies P u, u U, bu we do no know which one i is. The noion of ambiguiy follows similar idea; see, e.g. [5]. The noion of recursive uiliies was inroduced by Duffie Epsein [3,4]. These wo new feaures lead o a nonlinear RBSDE wih oblique reflecions. There are only very few resuls on hese kinds of RBSDEs in he lieraure; see e.g. [32]. We nex consider a very general muli-dimensional RBSDE of which boh he generaors and he obsacles are inerconneced. We prove he exisence of soluions by using he noion of he smalles g-supermaringales inroduced by Peng [29] and Peng and Xu [30]. This noion can be undersood as a nonlinear version of he Snell envelope. We prove he uniqueness by a verificaion heorem. However, for our general case he opimal sraegy does no exis, so one

3 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) can only obain approximaely opimal sraegies. This requires some sophisicaed esimaes and is in fac he main echnical par of he paper. As an inermediary resul we obain some sabiliy resul for high-dimensional RBSDEs, which is ineresing in is own righs. The paper closes o ours is a recen work by Hu and Tang [26], which we learned afer we finished he firs version of his paper. They sudy RBSDEs wih generaor aking he form f (, y, z) and barrier h(, y) = y c where c is a consan. Our model is more general in wo aspecs: (i) he generaor akes he form f (, y,..., y m, z), ha is, he uiliies under differen modes are inerconneced in he generaor; (ii) he barrier h(, y) is nonlinear and random. We hink such a generalizaion is ineresing from heoreical poin of view. From applied poin of view, he dependence of f on all he y-componens can be inerpreed as a nonzero-sum game problem, where he players uiliies affec each oher and consequenly he generaors are inerconneced. The general coninuous ime nonzero-sum game problem is much more difficul. We have some resul on he exisence of equilibrium in a differen framework; see [24]. The general barrier h allows one o consider more general swiching cos. In fac, even if he original cos akes he form h(, y) = y c, in some applicaions (especially in he case of risk-sensiive payoffs) one needs o ake he sandard exponenial ransformaion and hen he barrier becomes h(, ỹ) = e c ỹ; see, e.g. [23]. Hu Tang [26] ake he penalizaion approach o prove he exisence. While heir esimaes are nice, heir approach relies heavily on applying he Iô Tanaka formula o [ seems difficul o obain similar esimaes for [ Y i c j,i ] +. I h j,i (, Y i )]+ when general h is considered. We also noe ha, in [2,26], he opimal sraegy can be consruced explicily and hen he uniqueness follows immediaely via he verificaion heorem; see Theorem 2.. However, in our general case he opimal sraegy may no exis, and hus some echnical esimaes have o be involved in order o prove he uniqueness of soluions. The res of he paper is organized as follows. In he nex secion we inroduce he swiching problem, review he works of Hamadène Jeanblanc [2] and Hu Tang [26], and inroduce our general RBSDE. In Secion 3 we prove he exisence of soluions, and finally in Secion 4 we prove he uniqueness. 2. The swiching problem and refleced BSDEs In his secion we review some resuls in he lieraure and inroduce our general RBSDE. 2.. The swiching problem and some exising resuls We sar wih reviewing he work Hamadène Jeanblanc [2]. Le (Ω, F, P) denoe a fixed complee probabiliy space on which is defined a sandard d-dimensional Brownian moion B = (B ) 0 T, and F = (F ) 0 T be he filraion generaed by B and augmened by all he P-null ses. Throughou his paper we assume all he processes are progressively measurable and F-adaped. Furhermore, we le: - H be he space of processes η such ha E[ 0 η s 2 ds] < ; - S be he space ofcàdlàg processes η such ha E[sup 0 T η 2 ] < ; - S c be he subspace of S wih coninuous elemens; - A be he space of càdlàg and non-decreasing scalar processes η wih η 0 = 0 and E[ηT 2 ] < ; - A c be he subspace of A wih coninuous elemens;

4 406 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Le us now fix he daa of he problem. - Le X S c wih dimension k which sands for he facors deermining he marke elecriciy price. - Le ψ i : [0, T ] R k R, i =, 2, be Borelean funcions wih linear growh in x which represen he rae of uiliies for he power plan when i is in is operaing and closed modes, respecively. - Le c (resp. c 2 ) be a posiive consan which represens he sunk cos when he plan is swiched from he operaing (resp. closed) mode o he closed (resp. operaing) one. Le D denoe he se of all admissible sraegies δ = (τ n ) n 0 such ha τ n s are an increasing sequence of F-sopping imes wih τ 0 = 0 and lim n τ n = T, P-a.s. Assume for convenience ha he power plan is in is operaing mode a he iniial ime = 0. Then τ 2n+ (resp. τ 2n ) are he imes where he plan is swiched from he operaing (resp. closed) mode o he closed (resp. operaing) one. Under sraegy δ D, he mean yield of he power plan is given by: { J(δ) = E ψ δ (, X )d A δ T, 0 where ψ δ (, x) = [ ] ψ (, x) [τ2n,τ 2n+ )() + ψ 2 (, x) [τ2n+,τ 2n+2 )() ; n 0 A δ = [ ] c {τ2n+ < + c 2 {τ2n+2 <. n 0 (2.) Therefore he price of he power plan in he energy marke is jus sup δ D J(δ). As showed in [2], he above problem is closed relaed o he following wo-dimensional RBSDEs wih linear generaor and oblique reflecions: Y, Y 2 S c, Z, Z 2 H and K, K 2 A c, T Y i = ψ i (s, X s )ds Zs i db s + KT i K i, i =, 2; Y Y 2 c ; [Y Y 2 + c ]dk = 0; Y 2 Y c 2 ; [Y 2 Y + c 2 ]dk 2 = 0. Here Y (resp. Y 2 ) sands for he opimal uiliy a ime if he mode a ha ime is operaing (resp. closed). We noe ha, Y = Y Y 2, Z = Z Z 2 saisfy he following RBSDE wih double reflecions, which has been sudied by many auhors (see, e.g. [9,22,30]): Y = T [ ] ψ (s, X s ) ψ 2 (s, X s ) ds Z s db s + (KT K ) (K T 2 K 2 ); c Y c 2, [ Y + c ]dk = [ Y c 2 ]dk 2 = 0. Then he wellposedness of (2.2) follows immediaely. Moreover, [2] obains he following imporan verificaion heorem. (2.2) (2.3)

5 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Theorem 2.. sup δ D J(δ) = Y 0 and he opimal sraegy δ D 2 is given by τ 0 = 0 and, τ2n+ τ2n+2 = inf{ τ2n : Y = Y 2 c T ; = inf{ τ2n+ : Y 2 = Y c 2 T ; n = 0,,... The above work can be naurally exended o swiching problems under Knighian uncerainy and recursive uiliies. We refer o Knigh [27] and Chen Epsein [5] for Knigh uncerainy and Duffie Epsein [3,4] for recursive uiliies. We noe ha such an exension has been carried ou independenly by a recen work Hu Tang [26], which we learned afer we finished he firs version of his paper. Mahemaically, his amouns o solving he following BSDE wih some nonlinear generaor H and oblique reflecions: Y, Y 2 S c, Z, Z 2 H and K, K 2 A c, Y i = T [ψ i (s, X s ) + H (s, X., Y i s, Z i s )]ds Zs i db s + KT i K i, i =, 2; Y Y 2 c ; [Y Y 2 + c ]dk = 0; Y 2 Y c 2 ; [Y 2 Y + c 2 ]dk 2 = 0. Noe ha one canno ransform (2.4) ino a one-dimensional RBSDE wih double barriers. Hu Tang [26] esablishes he wellposedness of he following RBSDE (afer some obvious ransformaion) in higher-dimensional case: S c, Z j H and K j A c, = ξ j + max i j (Y i f j (s, ω, s, Z j s )ds c j,i ); [ Z j s db s + K j T K j ; max (Y i c j,i )]dk j = 0. i j (2.4) j =,..., m. (2.5) Here ξ j L 2 (F T ), f j saisfies he sandard measurabiliy and Lipschiz condiions, and c j,i are consans saisfying c j,i 0 and c k, j + c j,i > c k,i. (2.6) They prove he exisence of he soluion by penalizaion approach, and he uniqueness by a verificaion heorem which in he meanime provides he opimal sraegy δ, in he spiri of Theorem The general BSDEs wih oblique reflecion In his paper we exend he RBSDEs (2.2) and (2.5) o he following general m-dimensional RBSDEs wih oblique reflecions for some m 2: for j =,..., m, S c, Z j H and K j A c, = ξ j + max i A j h j,i (, Y i f j (s, Y s,..., Y m s, Z j s )ds ); [ max h j,i (, Y i )]dk j = 0. i A j Z j s db s + K j T K j ; (2.7)

6 408 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Here ξ j are F T -measurable, he coefficiens f j, h j,i can depend on ω, and A j {,..., m { j. For simpliciy we denoe Y = (Y,..., Y m ), and similarly for oher vecors. The consrain A j means ha from mode j he plan can only be swiched o hose modes in A j. We emphasize ha A j can be empy and if so we ake he convenion ha he maximum over he empy se, denoed as, is. Then in his case has no lower barrier and hen we ake K j = 0. Consequenly, saisfies he following BSDE wihou reflecion: = ξ j + Also, for any j we define h j, j (, y) = y. f j (s, Y s, Z j s )ds Then a soluion of (2.7) always saisfies max i A j { j h j,i(, Y i ). Z j s db s, 0 T. We noe ha our work is done independenly of Hu Tang [26]. Our RBSDE (2.7) is no he same as he one of (2.5) in wo aspecs: (i) he componens Y,..., Y m are inerconneced in he generaors f j ; (ii) he barriers h j,i can be random and nonlinear. We hink such a generalizaion is ineresing from heoreical poin of view. From applied poin of view, he dependence of f j on Y can be inerpreed as a nonzero-sum game problem, where he players uiliies affec each oher and consequenly he generaors are inerconneced. The general barrier h j,i allows one o consider more general swiching cos. In fac, even if he original cos akes he form h j,i (, y) = y c j,i, in he risk-sensiive swiching problem (see [23]) one needs o ake he sandard exponenial ransformaion and hen he barrier becomes h j,i (, ỹ) = e c j,i ỹ. 3. Exisence To prove he exisence of soluions, we use he noion of he smalles g-supermaringales inroduced by Peng [29] and Peng and Xu [30], which can be undersood as a nonlinear version of he Snell envelope (see e.g. [5]). Throughou his secion we shall adop he following assumpions. Assumpion 3.. For any j =,..., m, i holds ha: { (i) E 0 sup y :y j =0 f j(, y, 0) 2 d + ξ j 2 <. (ii) f j (, y, z) is uniformly Lipschiz coninuous in (y j, z) and is coninuous in y i for any i j; and h j,i (, y) is coninuous in (, y) for i A j. (iii) f j (, y, z) is increasing in y i for i j, and h j,i (, y) is increasing in y for i A j. (iv) For i A j, h j,i (, y) y. Moreover, here is no sequence j 2 A j,..., j k A jk, j (2.8) (2.9) A jk, and (y,..., y k ) such ha y = h j, j 2 (, y 2 ), y 2 = h j2, j 3 (, y 3 ),..., y k = h jk, j k (, y k ), y k = h jk, j (, y ). (v) For any j =,..., m, ξ j max i A j h j,i (T, ξ i ). We noe ha (i), (ii) and (v) are sandard; and (iii) implies he m players are parners. The assumpion (iv) means ha i is no free o make a circle of insananeous swichings. This is saisfied, for example, in [26] under condiion (2.6).

7 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Our main resul of his secion is: Theorem 3.2. Assume Assumpion 3. holds. Then RBSDE (2.7) has a soluion. Proof. We shall use Picard ieraion, and proceed in five seps. Sep. We firs consruc he Picard ieraions. Denoe: f j (, y, z) = inf f j (, y, z) and f j (, y, z) = sup f j (, y, z). y :y j =y y :y j =y By Assumpion 3.(i) and (ii), f j, f j are uniformly Lipschiz coninuous in (y, z) and { E [ f j (, 0, 0) 2 + f j (, 0, 0) 2 ]d <. 0 Le (,0, Z j,0 ) be he soluion o he following BSDE wihou reflecion:,0 = ξ j + f j (s, Ys j,0, Zs j,0 )ds Z j,0 s db s, j =,..., m. (3.) For j =,..., m and n =, 2,..., recursively define,n via he following RBSDEs whose soluion exiss hanks o he resul by El-Karoui e al. [6]:,n = ξ j Zs j,n db s + K j,n T K j,n T + f j (s, Ys,n,..., Ys j,n, Ys j,n, Ys j+,n,..., Ys m,n, Zs j,n )ds; (3.2),n max h j,i (, Y i,n ); [,n max h j,i (, Y i,n )]dk j,n = 0. i A j i A j Noe ha, given Y i,n, i =,..., m, for each j (3.2) is a one-dimensional BSDE (when A j = φ) or RBSDE. Under Assumpion 3., (3.2) has a unique soluion. Moreover, by comparison heorem (see e.g. [6], Theorem 4.) i is obvious ha,,0. Then by inducion one can easily show ha,n is increasing as n increases. Sep 2. We show ha { E sup,n T j= 0 Z j,n 2 d + K j,n T 2 To his end, denoe: m ξ = ξ j and m f (, y, z) = f j (, y, z), j= and le ( Y, Z) be he soluion o he following BSDE: Y = ξ + Denoe, for j =,..., m, Ȳ j = Y, Z j f (s, Y s, Z s )ds = Z, K j = 0. Z s db s. C, j, n. (3.3)

8 40 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Obviously,0 Ȳ j. By Assumpion 3.(iv) we know ha (Ȳ j, Z j, K j ) saisfies Ȳ j = ξ + f (s, Ȳs j, Z s j ) Z s j db s + K j T K j ; Ȳ j max h j,i (, Ȳ i ); [Ȳ j max h j,i (, Ȳ i )]d K j = 0. i A j i A j Once more apply he comparison heorem repeaedly, we ge,n Y, n. Recall ha,n,0. Then { m E sup,n 2 C <, 0 T n. (3.4) j= Moreover, { E sup [max h j,i (, Y i,n )] + 2 E 0 T i A j { sup [max Y i,n ] + 2 C. 0 T i A j This, ogeher wih (3.4) and applying he resuls in [6], proves (3.3). Sep 3. Now le denoe he limi of,n. By Peng s monoonic limi heorem [29] or [30], we know ha is an càdlàg process, and following similar argumens here one can easily show ha here exis Z j H and K j A such ha = ξ j + max i A j h j,i (, Y i f j (s, Y s, Z j s )ds ). Z j s db s + K j T K j ; Consider now he following RBSDEs whose soluion exiss hanks o he resul by Hamadène [8] or Peng and Xu [30]: Ỹ j S, Z j H and K j A; Ỹ j Ỹ j + = ξ j Z j s db s + K j T K j f j (s, Ys,..., Y s j, Ỹs j, Ys j+,..., Ys m, Z s j )ds; max i A j h j,i (, Y i ); [Ỹ j max i A j h j,i (, Y i )]d K j = 0. We noe ha (3.5) and (3.6) have he same lower barrier. Since Ỹ j is he smalles f j - supermaringale wih lower barrier max i A j h j,i (, Y i j ), we have Ỹ (see [30], Theorem 2.). On he oher hand, since Y i,n max h j,i (, Y i,n ) max h j,i (, Y i ). i A j i A j (3.5) (3.6) Y i for any (i, n ), by he monooniciy of h j,i we ge Then once more by comparison heorem for RBSDEs we have,n Ỹ j. Therefore, Ỹ j =. This furher implies ha Z j Ỹ j, which implies ha = K j for = Z j, d dp-a.s., K j

9 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) any 0 T, P-a.s., and ha = ξ j + f j (s, Y s, Zs j )ds Zs j db s + K j T K j ; max h j,i (, Y i ), [ max h j,i (, Y i )]dk j = 0. i A j i A j (3.7) Sep 4. We show ha is coninuous. This obviously implies ha K j is also coninuous and hus ( Y, Z, K ) is a soluion o (2.7). We firs noe ha, by (3.7), = K j 0, and if K j 0, hen = max i A j h j,i (, Y i ). I is obvious ha is coninuous when A j =. We now assume 0 for some j and. Then A j and < 0. Noe ha in his case K j > 0, which furher implies ha = max i A j h j,i(, Y i ). Le j 2 A j be he opimal index, hen h j, j 2 (, 2 ) = > max h j,i(, Y i ) h j, j 2 (, 2 ). i A j Thus 2 < 0, and herefore A j2. Repea he argumens we obain j k A jk and k < 0 for any k. Since each j k can ake only values,..., m, we may assume, wihou loss of generaliy ha j = j k+ for some k 2 (noe again ha j A j and hus j 2 j ). Then we have = h j, j 2 (, 2 ),..., k = h j k, j k (, k ), k = h j k, j (, ). This conradics wih Assumpion 3.(iv). Therefore, all processes are coninuous. Sep 5. Finally, as a by-produc we show ha, for j =,..., m, { lim E n sup [,n 2 + K j,n K j 2 ] + 0 T 0 Z j,n Z j 2 d In fac, since is coninuous and,n, by Dini s Theorem we know ha lim sup n 0 T,n = 0, a.s. = 0. (3.8) Applying Dominaed Convergence Theorem we prove he convergence of,n in (3.8). Now by sandard argumens, see e.g. [6], one can prove (3.8). By applying comparison heorem repeaedly, he following wo resuls are direc consequences of Theorem 3.2, and heir proofs are omied. Corollary 3.3. The soluion Y consruced in Theorem 3.2 is he minimum soluion of (2.7). Tha is, if Ỹ is anoher soluion of (2.7), hen Ỹ j, j =,..., m. Corollary 3.4. Assume ( ξ j, f j ) also saisfy Assumpion 3., and f j f j, ξ j ξ j.

10 42 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Le Y and Ỹ denoe he soluion of (2.7) consruced in Theorem 3.2, wih coefficiens (ξ j, f j, h j,i ) and ( ξ j, f j, h j,i ), respecively. Then Ỹ j, j =,..., m. We also have he convergence of he penalized BSDEs, which is obained by Hu and Tang [26] in heir case using a differen approach. Theorem 3.5. Assume Assumpion 3. holds, and ( Y, Z, K ) denoe he soluion of (2.7) consruced in Theorem 3.2. Le ( Y n, Z n ) denoe he soluions of he following penalized BSDEs wihou reflecion: Y n, j = ξ j + f j (s, Ys n, Z s n, j )ds + n Z n, j s db s. Then Y n, j is increasing in n and { where lim E n K n, j = n [Y n, j s sup [ Y n, j 2 + K n, j K j 2 ] + 0 T 0 [Y n, j s max h j,i (s, Ys n,i )] ds. i A j max h j,i (s, Ys n,i )] ds i A j 0 Z n, j Z j 2 d (3.9) = 0, (3.0) Proof. The proof is similar o Theorem 3.2, we hus only inroduce he main idea and leave he deails o he ineresed readers. n, j,0 Firs, i is obvious ha he BSDEs (3.9) have a unique soluion. Define Y =,0, and for k = 0,,..., recursively define n, j,k+ Y = ξ j + + n f j (s, Y n,,k s,..., Y n, j,k+ [Ys max n, j,k s, Y h j,i (s, Ys n,i,k i A j n, j,k+ s, Y )] ds By sandard argumens in BSDE heory one can easily see ha { lim E k sup 0 T n, j,k Y Y n, j n, j,k Z Z n, j 2 = 0. n, j+,k s,..., Y n,m,k s n, j,k+ Zs db s., Zs n, j )ds Moreover, by comparison heorem we have Y n, j,k is increasing in n. Thus Y n, j is increasing in n. We noe ha one can also use he comparison heorem for high-dimensional BSDEs, see [25], o prove he monooniciy of Y n, j. Le Ỹ j denoe he limi of Y n, j as n. By inducion one can show ha Y n, j,k,k for any (n, j, k). Then Y n, j and hus Ỹ j. Now apply he resuls in [30] and he argumens in Theorem 3.2, we can prove Ỹ j = and (3.0). Anoher by-produc of Theorem 3.2 is he exisence of a soluion of he sysem (2.7) considered beween wo sopping imes. This resul is in paricular useful o show uniqueness of (2.7) in he nex secion.

11 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) To be precise, le λ and be wo sopping imes such ha P-a.s., 0 λ T and le us consider he following RBSDE over [λ, ]: for j =,..., m, P-a.s., ( ) [λ, ] coninuous, { (K j ) [λ, ] coninuous and non-decreasing, K j λ = 0, and E λ2 = ξ j + max h j,i (, Y i i A j Then we have: sup 2 + [λ, ] f j (s, Y s, Z j s )ds ) and [Y j λ2 λ λ2 Z j s 2 ds + (K j ) 2 < ; Z j s db s + K j K j, [λ, ]; max h j,i (, Y i )]dk j = 0, [λ, ]. i A j (3.) Theorem 3.6. Assume Assumpion 3. holds and ha, for j =,..., m, ξ j F λ2 and saisfies: E{ ξ j 2 < and ξ j max i A j h j,i (, ξ i ). (3.2) Then he RBSDE (3.) has a soluion. 4. Uniqueness We now focus on uniqueness of he soluion of RBSDE (3.), hence ha of RBSDE (2.7). To do ha we need a sronger assumpion. Assumpion 4.. (i) f j is uniformly Lipschiz coninuous in all y i. (ii) If i A j, k A i, hen k A j { j. Moreover, (iii) For any i A j, h j,i (, h i,k (, y)) < h j,k (, y). (4.) h j,i (, y ) h j,i (, y 2 ) y y 2. (4.2) Noe again ha hese assumpions are saisfied if A j = {,..., m { j for any j =,..., m and h j,i (ω,, y) = y c j,i under condiion (2.6), as in [26]. Our main resul of his secion is he following heorem. Theorem 4.2 (Uniqueness). (i) Assume Assumpions 3. and 4. are in force, and ξ j saisfies (3.2). Then he soluion of RBSDE (3.) is unique. (ii) Moreover, assume for j =,..., m, f j saisfies Assumpions 3. and 4., and ξ j saisfies (3.2). Le (Ỹ j, Z j ) be he soluion o RBSDE (3.) corresponding o ( f j, ξ j ). For j =,..., m, denoe, = f = Ỹ j m j=, ξ j = ξ j ξ j, esssup [ f j f j ](, y, z). ( y,z) Then here exiss a consan C, which is independen of λ,, such ha: (4.3)

12 44 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) { max λ j m 2 E λ e C( λ ) max ξ j j m 2 + C Here E λ { denoes he condiional expecaion E{ F λ. λ2 λ f 2 d. (4.4) We noe ha he sabiliy resul (4.4) is no only ineresing in is own righ, we need i o prove he uniqueness in (i). The main idea is o prove a verificaion heorem in he spiri of Theorem 2.. However, he proof here is much more involved because for our general RBSDEs he opimal sraegy like he δ in Theorem 2. does no exis. We can only consruc some approximaely opimal sraegy, and hen we need some precise esimaes of he errors, which will be obained by using (4.4). The res of his secion is organized as follows. In Secion 4. we discuss heurisically how o find he approximaely opimal sraegies, which will lead o he definiion of admissible sraegies. In Secion 4.2 we define rigorously he admissible sraegy δ and he corresponding value funcion Y δ. In Secion 4.3 we esimae he error beween and he given soluion, which leads o he verificaion heorem. Finally in Secion 4.4 we prove Theorem Heurisic discussion We wan o exend he argumens for Theorem 2. o his case. For an arbirary soluion, he idea is o express Y0 as he supremum of Y 0 δ for some appropriaely defined Y δ. The sraegy δ we can use here is much more suble and in fac he argumens are very echnical. To explain he difference and o moivae our definiion of admissible sraegies, le us firs consider he following wo-dimensional RBSDEs: = ξ j + Y Y 2 f j (s, Y s, Y 2 s, Z j s )ds h (, Y 2 ); [Y h (, Y 2 = 0; h 2 (, Y ); 2 [Y h 2 (, Y 2 = 0. Z j s db s + K j T K j, j =, 2; Assume (Y, Y 2 ) is an arbirary soluion of (4.5). As in Theorem 2. we wan o express Y 0 as Y δ 0 for some δ and appropriaely defined Y δ. Very naurally we wan o define τ (4.5) = inf{ 0 : Y = h (, Y 2 ) T. (4.6) When f does no depend on Y 2 Y = ξ {τ =T + h (τ, Y τ 2 ) {τ <T +, as in (2.2) or (2.5), we have τ τ f (s, Ys, Z s )ds Zs db s. This is a BSDE wihou reflecion and is well posed. Therefore, once we can deermine Yτ 2, Y is unique on [0, τ ]. Nex we can define τ 2 by using Y 2 and express Yτ 2 in erms of Yτ 2. Repea he argumens we can mimic he proof of Theorem 2.. However, in our case, we have o consider he following RBSDE over [0, τ ], τ = ξ {τ =T + h (τ, Y τ 2 τ ) {τ <T + f (s, Ys, Y s 2, Z s )ds τ = Yτ 2 τ + f 2 (s, Ys, Y s 2, Z s 2 )ds Zs 2 db s + KT 2 K 2 ; h 2 (, Y 2 ); [Y h 2 (, Y )]dk 2 = 0. Y Y 2 Y 2 Z s db s; (4.7)

13 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) This iself akes he form of (3.), whose wellposedness needs o be proved. We will come back o his idea laer. There is anoher naive approach. Define τ = inf{ > 0 : Y = h (, Y 2 ) or Y 2 = h 2 (, Y ) T. Then we have Y = Yτ Y 2 = Yτ 2 + τ + τ τ f (s, Ys, Y s 2, Z s )ds Zs db s; τ f 2 (s, Ys, Y s 2, Z s 2 )ds Zs 2 db s; 0 τ. This sysem is well posed once he erminal condiions are given. However, in his approach we will have o define τ 2 = inf{ > τ : Y = h (, Y 2 ) or Y 2 = h 2 (, Y ) T. I is very likely ha τ2 = τ, and hen we have rouble o move forward. We now come back o he firs approach. Tha is, we consider (4.6) and (4.7). One key observaion is ha, alhough we do no know is uniqueness ye, RBSDE (4.7) has only one reflecion while he original RBSDE (4.5) has wo reflecions. Therefore, by doing his we reduce he number of reflecions, and hus by repeaing he procedure we can ransform he sysem o BSDEs wihou reflecion which is well posed. There is anoher difficuly o prove he verificaion heorem for RBSDEs in he form of (4.7). To illusrae he idea le us consider he following RBSDE insead of (4.7): Y = ξ + f (s, Ys, Y s 2, Z s )ds Zs db s + KT K ; T Y 2 = ξ 2 + f 2 (s, Ys, Y s 2, Z s 2 )ds Zs 2 db (4.8) s; Y h (, Y 2 ); [Y h (, Y 2 )]dk = 0. Again we define τ by (4.6). Then over [0, τ ] we have τ Y = ξ {τ =T + h (τ, Y τ 2 τ ) {τ <T + f (s, Ys, Y s 2, Z s )ds Zs db s; Y 2 = Yτ 2 + τ τ f 2 (s, Ys, Y s 2, Z s 2 )ds Zs 2 db s. This is well posed. However, Y 2 has no reflecion, hus we canno define τ2 as in Theorem 2.. Our second key observaion is ha, when τ < T, Y τ = h (τ, Y τ 2 ). Noe ha Y, Y 2, h are all coninuous. This implies ha if τ2 is close o τ, hen Y h (, Y 2) for [τ, τ 2 ], and herefore, Y 2 Yτ τ 2 f 2 (s, h (, Y 2 ), Y 2 s, Z 2 s )ds τ 2 Z 2 s db s. (4.9) Ignoring he approximaion, his is a BSDE wihou reflecion and is well posed. We should, of course, esimae he error due o his approximaion. We now summarize he above idea and discuss heurisically how o find he approximaely opimal sraegy for he m-dimensional RBSDE (3.). Le µ denoe he number of nonempy

14 46 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) ses A j in (3.), ha is, he number of reflecions in (3.). We proceed by inducion on µ. Firs, when µ = 0, (3.) becomes an m-dimensional BSDE wihou reflecion. By sandard argumens one can easily show ha Theorem 4.2 holds. Now assume Theorem 4.2 is rue for µ = m for some m m. For µ = m, le (, Z j, K j ) be an arbirary soluion of (3.). Le τ0 = λ, and wihou loss of generaliy assume A. Se τ = inf{ τ0 : Y = max h,i (, Y i ). i A When τ <, we have Y τ = max h,i (τ, Y i A τ i ). Tha is, here exiss an index, denoed as η A, such ha Y τ = h,η (τ, Y η τ ). So, besides he sopping ime τ, we need o keep rack of he opimal index η. We noe ha η is random and is F τ holds ha: Y η 0 τ = τ + max h j,k (, Y k k A j = Y η 0 + τ measurable. A his poin, le us denoe η 0 =. Noe ha, over [τ 0, τ ], i τ f j (s, τ Y s, Zs j )ds Zs j db s + K j τ K j, j η 0 ; ); [ max h j,k (, Y k )]dk j = 0, j η 0 ; k A j f η0 (s, Y s, Z η 0 s )ds τ Z η 0 s db s. (4.0) This is a sysem wih only m reflecions, and hus is well posed by our inducion assumpion. Now assume τ <. To define (τ 2, η 2), we need o consider wo differen cases. Case. A η. Denoe τ 2 = inf{ τ : Y η = max h η,i(, Y i ), i A η and, when τ 2 <, le η 2 A η such ha Y η τ 2 = h η,η 2 (τ2, Y η 2 τ2 ). Then Y saisfies a sysem wih m reflecions over [τ, τ 2 ], where he η h equaion has no reflecion. Case 2. A η =. In his case, he η h equaion has no reflecion. Noe ha Y η 0 τ h η0,η (τ, Y η τ ). As in (4.9), choose τ 2 close o τ, hen for any [τ, τ 2 ], we have Y η 0 h η0,η (τ, Y η ). On he oher hand, by (4.) and (2.9) one can see ha τ for any j such ha η 0 A j. Since τ2 is close o τ j, le us assume Y [τ, τ 2 ]. So approximaely, over [τ, τ 2 ], { j η0 saisfy τ2 τ 2 τ 2 + Zs j db s + K j τ K j ; 2 max h j,k(, Y k ); k A j {η 0 f j (s, h,η (τ, Y η s ), Ys 2,..., Y s m, Z s j )ds [ max h j,k(, Y k )]dk j = 0. k A j {η 0 = > h j,η0 (τ, Y η 0 τ ) > h j,η0 (τ, Y η 0 ) for (4.)

15 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) This is a sysem of m equaions wih m reflecions, where we remove he equaion for Y η 0 compleely. In order o move forward, we need o define η 2 so ha A η2. I urns ou ha he bes way is o se η 2 = η0. Now we can coninue he procedure and define a sequence of (τ n, η n) Consrucion of Y δ The argumens in Secion 4. is only heurisic. We now make everyhing rigorous. Firs, le us inroduce he following definiion: Definiion 4.3. δ = (τ 0,..., τ n ; η 0,..., η n ) is called an admissible sraegy if (i) λ = τ 0 τ n is a sequence of sopping imes; (ii) η 0,..., η n are random index aking values in {,..., m such ha η i F τi ; (iii) A η0 ; (iv) If A ηi, hen η i+ A ηi ; (v) If A ηi =, hen η i+ = ηi. We noe ha, unlike in Secion 2, here δ mus be a finie sequence. Remark 4.4. By Definiion 4.3(iii), A ηi = implies ha i. Then he (v) above makes sense. Moreover, by inducion we see in his case A ηi+ = A ηi. We assume Theorem 4.2 holds for µ = m and for any m m. Now assume µ = m. For an admissible sraegy δ, we consruc (, Z δ, j ) as follows. Firs, for [τ n, ] and j =,..., m, se = Y 0, j, Z δ, j = Z 0, j, (4.2) where (Y 0, j, Z 0, j ) is he soluion o (3.) consruced in Secion 2. Then in paricular we have τ n max i A j h j,i (τ n, Y δ,i τ n ), j =,..., m. (4.3) For i = n,..., 0, assume we have consruced for j =,..., m, which we will do laer. Noe ha may be disconinuous a. Corresponding o Case and Case 2 when we defined (τ2, η 2) in Secion 4., we define (, Z δ, j ) over [τ i, ) in wo cases. Case. A ηi. Assume our consruced saisfies max k A j h j,k (, Y δ,k ), j η i. (4.4) Recall (4.0). We consider he following RBSDE by removing he consrain of he η i h equaion: Y δ,η i τi+ = + τi+ f j (s, Y δ s, Z s δ, j )ds Zs δ, j db s + Kτ δ, j i+ K δ, j, j η i ; max h j,k (, Y δ,k ); [ k A j = Y δ,η i + τi+ max h j,k (, Y δ,k k A j f ηi (s, Y δ s, Z δ,η i s )ds τi+ )]dk δ, j = 0, j η i ; Z δ,η i s db s. (4.5)

16 48 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) I is obvious ha he f j, h j,i, A j here saisfy Assumpions 3. and 4., and (4.4) implies ha he erminal condiions of (4.5) saisfy (3.2). Since (4.5) has only m reflecions, by inducion assumpion i has he unique soluion (, Z δ, j ), j =,..., m over [τ i, ). Case 2. A ηi =. By Remark 4.4 we have i and A ηi. Assume our consruced saisfies max h j,k(, Yτ δ,k k A j {η i i+ ), j η i. (4.6) Recall (4.). We omi he η i h equaion and consider he following m -dimensional RBSDE wih a mos m reflecions: for j η i, τi+ = Zs δ, j db s + Kτ δ, j i+ K δ, j Here: + τi+ f j (s, Y δ,,..., Y δ,η i, Y δ,η i+,..., Y δ,m, Zs δ, j )ds; s s s max h j,k(, Y δ,k ), [ max h j,k(, Y δ,k )]dk δ, j = 0. k A j {η i k A j {η i f j (, y,..., y ηi, y ηi +,..., y n, z) s (4.7) = f j (, y,..., y ηi, h ηi,η i (τ i, y ηi ), y ηi +,..., y n, z). (4.8) One can easily check ha f j, h j,i, A j {η i here saisfy Assumpions 3. and 4., and (4.6) implies ha he erminal condiions of (4.7) saisfy (3.2). Since RBSDE (4.7) has a mos m reflecions, by inducion assumpion i has he unique soluion (, Z δ, j ), j η i, over [τ i, ). We emphasize ha Y δ,η i is no involved in his case. I remains o consruc saisfying (4.4) or (4.6). Firs, se = Yτ 0, j n, if i + = n; = ξ j, if =. (4.9) By (4.2) and (3.2) we know ha boh (4.4) and (4.6) hold. Now assume i < n and <. Assume we have solved eiher (4.5) or (4.7) over [, τ i+2 ). Again we consruc in wo cases. Case 2. A ηi =. In his case we need o consruc only for j η i and o check (4.6). By Remark 4.4 we know ha i, η i+ = η i, and A ηi+. Then Yτ δ, j i+ were obained from (4.5) over [, τ i+2 ) and hus saisfy: Define max k A j h j,k (, Y δ,k ), j η i+ = η i. (4.20) = Yτ δ, j i+, j η i. (4.2) Then (4.6) follows immediaely from (4.20). Case. A ηi. In his case we need o consruc in wo cases. for all j and o check (4.4). We do i

17 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Case.. A ηi+ saisfy =. Then, j η i were obained from (4.7) over [, τ i+2 ) and hus Define Yτ δ, j i+ max h j,k(, Yτ δ,k k A j {η i i+ ), j η i. (4.22) = Yτ δ, j i+, j η i ; Y δ,η i = h ηi,η i+ (, Y δ,η i+ ). (4.23) By (4.22), o prove (4.4) i suffices o show ha h j,ηi (, h ηi,η i+ (, Y δ,η i+ )), if η i A j. (4.24) Assume η i A j. By Definiion 4.3(iv) and Assumpion 4.(ii), we have η i+ [A j {η i ] { j, and h j,ηi (, h ηi,η i+ (, Y δ,η i+ )) < h j,ηi+ (, Y δ,η i+ ). (4.25) If η i+ A j {η i, hen (4.24) follows from (4.22) and (4.25). If η i+ = j, hen (4.24) follows from (2.8) and (4.25). So in boh cases (4.24) holds, hen so does (4.4). Case.2. A ηi+. Then Yτ δ, j i+ were obained from (4.5) over [, τ i+2 ) and hus saisfy: Define max k A j h j,k (, Y δ,k ), j η i+. (4.26) = Yτ δ, j i+, j η i, η i+ ; Y δ,η i+ = Y δ,η i+ max h η i+,k(, Yτ δ,k k A ηi+ {η i i+ ); Y δ,η i = h ηi,η i+ (, Y δ,η i+ ). We now check (4.4) for j η i. Firs, for j = η i+, by (4.27), Y δ,η i+ max h η i+,k(, Yτ δ,k k A ηi+ {η i i+ ). Moreover, if η i A ηi+, by (4.) and (2.8) we have h ηi+,η i (, Y δ,η i ) = h η i+,η i (, h ηi,η i+ (, Y δ,η i+ )) < Y δ,η i+. So (4.4) holds for j = η i+. I remains o check (4.4) for j η i, η i+. By (4.26) and he firs line in (4.27) we have (4.27) max h j,k(, Yτ δ,k k A j {η i,η i+ i+ ). (4.28) If η i+ A j, recall he definiion of Y δ,η i+ h j,ηi+ (, Y δ,η i+ ) Yτ δ, j i+ =. in (4.27). Firs, by (4.26) we have Second, for any k A ηi+ {η i, similar o (4.24) one can easily prove h j,ηi+ (, h ηi+,k(, Y δ,k )) =.

18 420 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Thus, by Assumpion 3.(iii) we have h j,ηi+ (, Y δ,η i+ ). (4.29) Finally, if η i A j, by Definiion 4.3(iv), Assumpion 4.(ii), and (4.29), we have η i+ A j { j and h j,ηi (, Y δ,η i ) = h j,η i (, h ηi,η i+ (, Y δ,η i+ )) < h j,ηi+ (, Y δ,η i+ ). This, ogeher wih (4.28) and (4.29), proves (4.4) for j η i, η i+. Now repea he argumens backward in ime, we see in each [τ i, ), eiher (4.5) or (4.7) is well defined and is well posed. Thus we obain over he whole inerval [λ, ], wih he excepion of Y δ,η i for [τ i, ) when A ηi = φ. By applying Corollary 3.4 and comparison heorem repeaedly, one can easily show ha: Lemma 4.5. Assume Assumpions 3. and 4. hold, and ha Theorem 4.2 is rue for µ = m. Then for µ = m and for any admissible sraegy δ and any j, we have whenever is well defined Verificaion heorem We now prove he verificaion heorem. Theorem 4.6. Assume Assumpions 3. and 4. hold, and ha Theorem 4.2 is rue for µ = m. Then for µ = m and for any soluion of RBSDE (3.), we have λ = esssup δ λ for all j, where he esssup is aken over all admissible sraegies δ. Proof. We prove he heorem in several seps. Sep. Fix ε > 0 and le D ε = {iε : i = 0,,.... We consruc an approximaely opimal admissible sraegy as follows. Firs, se τ 0 = λ and choose η 0 such ha A η0. For i = 0,,..., we define (, η i+ ) in wo cases. Case. A ηi. Se = inf{ τi : Y η i = max k A ηi h ηi,k(, Y k ). If <, se η i+ A ηi be he smalles index such ha Y η i = h ηi,η i+ (, Y η i+ ). (4.30) Oherwise choose arbirary η i+ A ηi. Case 2. A ηi =. Since A η0, we have i. Se η i+ = ηi. If τ i =, define = λ2. Now assume τ i <. I is more involved o define in his case. By he definiion of η i, one can check ha in his case we mus have A ηi, and hus by Case, η i A ηi and Y η i τ i = h ηi,η i (τ i, Y η i τ i ). Moreover, by Assumpions 3.(iii) and 4.(ii), one can easily see Yτ j i > h j,ηi (τ i, Y η i ) for any j such ha η i A j. We now define τ i = τ i+ τ 2 i+ ;

19 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) where τ i+ is he smalles number in D ε such ha τ i+ > τ i; and τi+ 2 = inf{ > τ i : j s.. η i A j, = h j,ηi (, Y η i ). Now se δ = δ n,ε = (τ 0,..., τ n ; η 0,..., η n ). Recall Definiion 4.3. One can easily check ha δ is an admissible sraegy. Sep 2. We esimae he errors backward in ime. Recall Secion 4.2 and denoe =. Firs, by (4.9) i is obvious ha τ n τ n = τ n. Now assume i < n. Case. A ηi φ. We claim ha (4.3) max Y τ j j m i 2 E τi {e C( τ i ) max Yτ j j η i+ 2. (4.32) i In fac, in his case (, Z j, K j ) saisfies τi+ = Yτ j i+ + f j (s, Y s, Zs j )ds Y η i max h j,k (, Y k ); k A j = Y η i + τi+ [ τi+ Z j s db s + K j K j, j η i ; max h j,k (, Y k )]dk k = 0, j η i ; k A j f ηi (s, τi+ Y s, Z η i s )ds Z η i s db s. (4.33) Compare (4.33) and (4.5). They have only m reflecions, hus by inducion assumpion we can apply Theorem 4.2 (ii) and obain max Y τ j j m i 2 E τi {e C( τ i ) So o prove (4.32) i suffices o show ha max Y τ j j m i+ 2. max Y τ j j m i+ max Yτ j j η i+. (4.34) i If =, hen by (4.9), we have = ξ j ξ j = 0, j. Thus (4.34) holds. Now assume <. Noe ha case, by (4.2) we have max Yτ j j η i+ = max Yτ j i j η i+ ; i is defined by eiher (4.23) or (4.27). In he former Y η i Y δ,η i = h η i,η i+ (, Y η i+ ) h ηi,η i+ (, Y δ,η i+ ) Y η i+. Since η i+ A ηi and hus η i+ η i. We prove (4.34) in his case.

20 422 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) In he laer case, ha is, is defined by (4.27), we firs have max Yτ j j η i,η i+ = i+ max Yτ j j η i,η i+. i+ Nex, for j = η i+, by Lemma 4.5 and Assumpion 3.(iii) we have Y η i+ Y δ,η i+ and Y η i+ max h ηi+,k(, Yτ k k A i+ ) max h ηi+,k(, Yτ δ,k ηi+ k A i+ ). ηi+ Then Y η i+ Y δ,η i+. Therefore, Y η i+ Y δ,η i+ = Y η i+ Y δ,η i+ Y η i+ Y δ,η i+ = Y η i+. Finally, for j = η i, by Assumpion 4.(iii) we have Y η i Y δ,η i = h η i,η i+ (, Y η i+ ) h ηi,η i+ (, Y δ,η i+ ) Thus (4.34) also holds. Y η i+ Y δ,η i+ Y η i+. Case 2. A ηi = φ. In his case (, Z j, K j ), j η i saisfies τi+ = Yτ j i+ Zs j db s + Kτ j i+ K j τi+ + fˆ j (s, Ys,..., Y η i s, Y η i + s,..., Ys m, Z s j )ds; max h j,k(, Y k ); k A j {η i where, recalling (4.8), ˆ f j (, y,..., y ηi, y ηi +,..., y n, z) I j [ max k A j {η i h j,k(, Y k )]dk k = 0; (4.35) = f j (, y,..., y ηi, y ηi +,..., y n, z) + I j ; (4.36) = f j (, Y, Z j ) f j (, Y,..., Y η i, h ηi,η i (τ i, Y η i ), Y η i +,..., Y n, Z j ). (4.37) We noe ha here I j is considered as a random coefficien. Compare (4.35) and (4.7). Recalling (4.2), by inducion assumpion again we ge max Yτ j j η i 2 E τi i ec( τ i ) max Yτ j j η i+ 2 + C τi+ I j d i. (4.38) Noe ha Y η i τ i I j C Y η i [ C Y η i = h ηi,η i (τ i, Y η i τ i ). Then [ C Y η i h ηi,η i (τ i, Y η i ) 2 j η i Y η i τ i 2 + h ηi,η i (τ i, Y η i τ i ) h ηi,η i (τ i, Y η i m Y η i τ i 2 + Y η i τ i Y η i 2] C k= τ i ) 2] Y k Y k τ i 2.

21 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Noe ha in his case τ i ε. Then I j C m Thus (4.38) implies sup k= λ < 2 : 2 ε max Yτ j j η i 2 E τi {e C( τ i ) i Y k Y k 2 2 = I ε. (4.39) max Y τ j j m i+ 2 + I ε [ τ i ]. (4.40) Sep 3. We claim ha, for a.s. ω, τ i = for i large enough. We prove i by conradicion. Assume ω is in he se ha all Y j (ω) and h j,i (, ω, y) are coninuous and τ i (ω) < for all i. Denoe = limi τ i. Firs, i is obvious ha here can be only finiely many i such ha A ηi = φ and = τi+. Second, assume here is an infinie sequence i k such ha A ηik = φ and τ ik + = τi 2 k +. Noe ha in his case η ik A and here exiss ˆη ηik i k + such ha η ik A. Then ˆηik + Y η i k τ ik = h ηik,η ik (τ ik, Y η i k τ ik ); Y ˆη i k + τ ik + = h ˆη ik +,η (τ ik i k +, Y η i k τ ik + ). (4.4) The vecor ( ˆη ik +, η ik, η ik ) can ake only finiely many values, hen here exis ( j, j 2, j 3 ) and an infinie subsequence of i k, wihou loss of generaliy we assume i is he whole sequence i k, such ha j 2 A j, j 3 A j2 and ˆη ik + = j, η ik = j 2, η ik = j 3, k. By (4.4) we ge τ ik + = h j, j 2 (τ ik +, 2 τ ik + ), 2 τ ik = h j2, j 3 (τ ik, 3 τ ik ), k. Send k, we have = h j, j 2 (, 2 ), 2 = h j2, j 3 (, 3 ). Then, by Assumpion 4.(ii), j 3 A j { j and = h j, j 2 (, h j2, j 3 (, 3 )) < h j, j 3 (, 3 ). This conradics wih (2.9). Therefore, here are only finiely many i such ha A ηi = φ. Finally, by he above resuls we mus have some n 0 such ha A ηi φ for all i n 0. Then η i+ A ηi and (4.30) holds for all i n 0. We say (η i, η i+,..., η i+l ) is a loop if hey are all differen and η i+l = η i. Since each η i akes only values,..., m, here are in oal finiely many possible loops. Thus here exis ( j,..., j l ) and an infinie sequence i k such ha (η ik,..., η ik +l, η ik +l) = ( j,..., j l, j ). Therefore, by (4.30), we have τ ik + = h j, j 2 (τ ik +, 2 τ ik + ),..., l τ ik +l = h j l, j l (τ ik +l, l τ ik +l ), and l τ ik +l = h jl, j (τ ik +l, τ ik +l ). Send k, we ge = h j, j 2 (, 2 ),..., l = h jl, j l (, l ), l = h jl, j (, ). This conradics wih Assumpion 3.(iv). Therefore, we prove he claim.

22 424 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Sep 4. We are now ready o complee he proof. Given A ηi, if A ηi+ =, by (4.32) and (4.40) we have max Y τ j j m i 2 E τi {e C(τ i+2 τ i ) max Y τ j j m i I ε [τ i+2 ]. (4.42) By Definiion 4.3(v), we have A ηi+2. Therefore, if A ηi, hen eiher A ηi+ and (4.32) holds, or A ηi+2 and (4.42) holds. Since A η0, one ges immediaely ha max Y τ j j m 0 2 C E τ0 { max Y τ j j m n 2 0, j + I ε = C E λ { max Yτ j m n Yτ j n 2 + I ε. Firs send n. Since τ n Convergence Theorem we have lim max n j m λ 2 C E λ {I ε., we ge Y 0, j τ n ξ j and τ n ξ j. By Dominaing Now send ε 0. Since is coninuous, by Dominaing Convergence Theorem again we ge lim lim max ε 0 n λ j m 2 = 0. This, ogeher wih Lemma 4.5, proves he heorem Proof of Theorem 4.2 As menioned before, we prove he heorem by inducion on µ. When µ = 0, (3.) is an m-dimensional BSDE wihou reflecions. Then (i) holds, and by sandard argumens one can easily prove (ii). Assume Theorem 4.2 holds for µ = m. Now assume µ = m. (i) By Theorem 4.6, λ is unique. Similarly is unique for any [λ, ]. By he uniqueness of he Doob Meyer decomposiion we ge Z j is unique, which furher implies he uniqueness of K j immediaely. (ii) For any admissible sraegy δ, define Ỹ δ, j similarly and denoe = Ỹ δ, j. If A ηi, recalling (4.5), (4.23) and (4.27), by inducion we have: δ, j max Yτ j m i 2 E τi {e C( τ i ) max Yτ δ, j j η i+ 2 + C i If A ηi =, recalling (4.7) and (4.2), by inducion we have: { max Yτ δ, j j η i 2 E τi e C( τ i ) δ, j max Y 2 + C i j m τi+ τ i λ τi+ τ i f 2 d. f 2 d. Noe ha A η0. Applying he above esimaes repeaedly we ge: { λ2 δ, j max Yλ j m 2 E λ e C( λ ) max ξ j λ j m C f 2 d, δ. Then (ii) follows from Theorem 4.6 immediaely.

23 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) Acknowledgemens The second auhor s research was suppored in par by NSF grans DMS and DMS Par of he work was done while his auhor was visiing Universié du Maine, whose hospialiy is grealy appreciaed. References [] K.A. Brekke, B. Oksendal, The high conac principle as a sufficiency condiion for opimal sopping, in: D. Lund, B. Oksendal (Eds.), Sochasic Models and Opion Values, Norh-Holland, Amserdam, 99, pp [2] K.A. Brekke, B. Oksendal, Opimal swiching in an economic aciviy under uncerainy, SIAM J. Conrol Opim. 32 (994) [3] M.J. Brennan, E.S. Schwarz, Evaluaing naural resource invesmens, J. Business 58 (985) [4] R. Carmona, M. Ludkovski, Opimal swiching wih applicaions o energy olling agreemens, Ph.D. Thesis, Princeon Universiy, USA, [5] Z. Chen, L. Epsein, Ambiguiy, risk, and asse reurns in coninuous ime, Economerica 70 (4) (2002) [6] S.J. Deng, Z. Xia, Pricing and hedging elecric supply conracs: A case wih olling agreemens, 2005, Preprin. [7] A. Dixi, Enry and exi decisions under uncerainy, J. Poliical Economy 97 (989) [8] A. Dixi, R.S. Pindyck, Invesmen Under Uncerainy, Princeon Univ. Press, 994. [9] B. Djehiche, S. Hamadène, On a finie horizon saring and sopping problem wih risk of abandonmen, In. J. Theor. Appl. Finance 2 (4) (2009) [0] B. Djehiche, S. Hamadène, A. Popier, A finie horizon opimal muliple swiching problem, SIAM J. Conrol Opim. 48 (4) (2009) [] K. Duckworh, M. Zervos, A model for invesmen decisions wih swiching coss, Ann. Appl. Probab. () (200) [2] K. Duckworh, M. Zervos, A problem of socahsic impulse conrol wih discreionary sopping, in: Proceedings of he 39h IEEE Conference on Decision and Conrol, IEEE Conrol Sysems Sociey, Piscaaway, NJ, 2000, pp [3] D. Duffie, L. Epsein, Sochasic differenial uiliy, Economerica 60 (992) [4] D. Duffie, L. Epsein, Asse pricing wih sochasic differenial uiliies, Rev. Financial Sudies 5 (992) [5] N. El Karoui, Les aspecs probabilises du conrôle sochasique, in: Ecole d éé de probabiliés de Sain-Flour, in: Lec. Noes in Mah., vol. 876, Springer Verlag, 980. [6] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.C. Quenez, Refleced soluions of backward SDEs and relaed obsacle problems for PDEs, Ann. Probab. 25 (2) (997) [7] X. Guo, H. Pham, Opimal parially reversible invesmen wih enry decision and general producion funcion, Soch. Proc. Appl. 5 (2005) [8] S. Hamadène, Refleced BSDEs wih disconinuous barriers, Soch. Soch. Rep. 74 (3 4) (2002) [9] S. Hamadène, M. Hassani, BSDEs wih wo reflecing barriers: The general resul, Probab. Theory Relaed Fields 32 (2005) [20] S. Hamadène, I. Hdhiri, On he saring and sopping problem wih Brownian and independan Poisson noise, Ph.D. Thesis, Univ. du Maine (Le Mans, Fr.), [2] S. Hamadène, M. Jeanblanc, On he saring and sopping problem: Applicaion in reversible invesmens, Mah. Oper. Res. 32 () (2007) [22] S. Hamadène, J.-P. Lepelier, Refleced BSDEs and mixed game problem, Sochasic Process. Appl. 85 (2000) [23] S. Hamadène, H. Wang, Opimal risk-sensiive swiching problem under knighian uncerainy and is relaed sysems of refleced BSDEs, Ph.D. Thesis, Univ. du Maine, Le Mans (Fr.), [24] S. Hamadène, J. Zhang, The coninuous ime nonzero-sum Dynkin game problem and applicaion in game opions, SIAM J. Compu. (200) (in press). [25] Y. Hu, S. Peng, On he comparison heorem for mulidimensional BSDEs, C. R. Mah. Acad. Sci. Paris 343 (2) (2006) [26] Y. Hu, S. Tang, Muli-dimensional BSDE wih oblique reflecion and opimal swiching, Probab. Theory Relaed Fields (2009) doi:0.007/s [27] F.H. Knigh, Risk, Uncerainy, and Profi, Boson, MA, 92. [28] T.S. Knudsen, B. Meiser, M. Zervos, Valuaion of invesmens in real asses wih implicaions for he sock prices, SIAM J. Conrol Opim. 36 (998)

24 426 S. Hamadène, J. Zhang / Sochasic Processes and heir Applicaions 20 (200) [29] S. Peng, Monoonic limi heory of BSDE and nonlinear decomposiion heorem of Doob Meyer s ype, Probab. Theory Relaed Fields 3 (999) [30] S. Peng, M. Xu, The smalles g-supermaringale and refleced BSDE wih single and double L 2 -obsacles, Ann. Ins. H. Poincaré Probab. Sais. 4 (2005) [3] A. Porche, N. Touzi, X. Warin, Valuaion of a power plan under producion consrains and marke incompleeness, Mah. Mehods Oper. Res. (2008) doi:0.007/s [32] S. Ramasubramanian, Refleced backward sochasic differenial equaions in an orhan, Proc. Indian Acad. Sci. 2 (2) (2002) [33] L. Trigeorgis, Real Opions: Managerial Flexibiliy and Saregy in Resource Allocaion, MIT Press, 996. [34] L. Trigeorgis, Real opions and ineracions wih financial flexibiliy, Financ. Manage. 22 (993) [35] M. Zervos, A problem of sequenial enry and exi decisions combined wih discreionary sopping, SIAM J. Conrol Opim. 42 (2) (2003)