Finite-Horizon Optimal Multiple Switching with Signed Switching Costs

Transcription

1 Finie-Horizon Opimal Muliple Swiching wih Signed Swiching Coss Randall Maryr This version: 24 November 215 Firs version: 13 November 214 Research Repor No. 15, 214, Probabiliy and Saisics Group School of Mahemaics, The Universiy of Mancheser

2 Finie-horizon opimal muliple swiching wih signed swiching coss Randall Maryr November 24, 215 Absrac This paper is concerned wih opimal swiching over muliple modes in coninuous ime and on a finie horizon. The performance index includes a running reward, erminal reward and swiching coss ha can belong o a large class of sochasic processes. Paricularly, he swiching coss are modelled by righ-coninuous wih lef-limis processes ha are quasi-lefconinuous and can ake boh posiive and negaive values. We provide sufficien condiions leading o a well known probabilisic represenaion of he value funcion for he swiching problem in erms of inerconneced Snell envelopes. We also prove he exisence of an opimal sraegy wihin a suiable class of admissible conrols, defined ieraively in erms of he Snell envelope processes. MSC21 Classificaion: 93E2, 6G4, 91B99, 62P2. Key words: opimal swiching, real opions, sopping imes, opimal sopping problems, Snell envelope. 1 Inroducion. The recen paper by Guo and Tomecek [8 showed a connecion beween Dynkin games and opimal swiching problems wih signed posiive and negaive swiching coss. The resuls were obained for a model in which he cos/reward processes were merely required o be adaped and saisfy mild inegrabiliy condiions. However, here are few heoreical resuls on he exisence of opimal swiching conrol policies under such general condiions. Opimal swiching for models driven by disconinuous sochasic processes has been sudied previously in papers such as [1, 2. The paper [2 used opimal sopping heory o sudy he opimal swiching problem on an infinie ime horizon wih muliple modes. The model described in [2 has bounded and non-negaive running rewards which are driven by righconinuous processes, and swiching coss ha are sricly posiive and consan. The paper [1 sudied he finie ime horizon opimal swiching problem wih wo modes. The model has running rewards adaped o a filraion generaed by a Brownian moion and an independen Poisson random measure, bu excludes erminal daa and assumes swiching coss ha are sricly posiive and consan. The more recen paper [4 has a model similar o [1 wih swiching coss assumed o be coninuous sochasic processes and filraion generaed by a Brownian moion. Neverheless, he auhors saed [4, p ha heir resuls can be adaped o a more general seup, possibly by using he same approach as in [1. This research was parially suppored by EPSRC gran EP/K557X/1. School of Mahemaics, The Universiy of Mancheser, Oxford Road, Mancheser M13 9PL, Unied Kingdom. randall.maryr@posgrad.mancheser.ac.uk 1

3 Finie-horizon opimal muliple swiching wih signed swiching coss 2 Mos of he lieraure on opimal swiching assumes non-negaive swiching coss. However, signed swiching coss are imporan in models where he conroller can parially recover is invesmen, or receive a subsidy/gran for invesing in a new echnology such as renewable green energy producion [8, 14, 17. The preprin [5 sough o generalise he resuls of [4 by permiing signed swiching coss, bu a he expense of limiing he oal number of swiches incurring negaive coss. This limiaion is absen in papers such as [2, 15 where he opimal swiching problem was sudied wihin a Markovian seing. There are, however, oher srucural condiions and hypoheses made in [2, 15. For example, here is an assumpion in [15, p ha he erminal reward is he same for all modes which also implies he erminal values of he swiching coss are non-negaive. The Snell envelope approach, also known as he mehod of essenial supremum [23, is a general approach o opimal sopping problems which does no require Markovian assumpions on he daa. I was used in he aforemenioned papers [4, 1 and he paper [1 on opimal swiching problems for one-dimensional diffusions albei in a slighly differen manner. In his paper we use he heory of Snell envelopes o exend Theorems 1 verificaion and 2 exisence of [4. Our model allows for non-zero erminal daa and swiching coss which are real-valued sochasic processes wih pahs ha are righ-coninuous wih lef limis and quasilef-coninuous. In conras o [5, our resuls do no presuppose a limi on he oal number of swiches incurring negaive coss. We do, however, require ha a cerain maringale hypohesis M on he swiching coss be saisfied see Secion 6.2 below. This hypohesis can be verified in many cases of ineres, including he case of wo modes, and does no require he erminal reward o be he same for all modes. We also assume ha he filraion, in addiion o saisfying he usual condiions of righ-coninuiy and compleeness, is quasi-lef-coninuous. This propery, which generalises he assumpion made in [1, is saisfied in many applicaions. For example, i holds when he filraion is he naural compleed filraion of a Lévy process. Such models have wide ranging applicaions in finance, insurance and conrol heory [22. The layou of paper is as follows. Secion 2 inroduces he probabilisic model and opimal swiching problem. Preliminary conceps from he general heory of sochasic processes and opimal sopping are recalled in Secion 3. The modelling assumpions for he opimal swiching problem are given in Secion 4. A verificaion heorem esablishing he relaionship beween he opimal swiching problem and ieraive opimal sopping is given in Secion 5. Sufficien condiions for validaing he verificaion heorem s hypoheses are discussed in Secion 6. The conclusion, appendix, acknowledgemens and references hen follow. 2 Definiions. 2.1 Probabilisic seup. We work on a ime horizon [, T, where < T <. I is assumed ha a complee filered probabiliy space, Ω, F, F, P, has been given and he filraion F = F T saisfies he usual condiions of righ-coninuiy and augmenaion by he P-null ses. Le E denoe he corresponding expecaion operaor. We use 1 A o represen he indicaor funcion of a se even A. The shorhand noaion a.s. means almos surely. Le T denoe he se of F- sopping imes ν which saisfy ν T P-a.s. For a given S T, wrie T S = {ν T : ν S P a.s.}. Unless oherwise saed, a sopping ime is assumed o be defined wih respec o F. For noaional convenience he dependence on ω Ω is ofen suppressed.

4 Finie-horizon opimal muliple swiching wih signed swiching coss Problem definiion. The conroller in an opimal swiching problem influences a dynamical sysem over he horizon [, T by choosing operaing modes from a finie se I = {1,..., m} wih m 2. The insananeous profi in mode i I is a mapping ψ i : Ω [, T R. There is a cos for swiching from mode i o j which is given by γ i,j : Ω [, T R. There is also a reward for being in mode i I a ime T, denoed by Γ i, which is a real-valued random variable. The assumpions on hese coss / rewards are discussed below in Secion 4. Definiion 2.1 Admissible Swiching Conrol Sraegies. Le [, T and i I be given. An admissible swiching conrol sraegy saring from, i is a double sequence α = τ n, ι n n of sopping imes τ n T and mode indicaors ι n such ha: 1. τ = and he sequence {τ n } n is non-decreasing; 2. Each ι n : Ω I is F τn -measurable; ι = i and ι n ι n+1 for n ; 3. Only a finie number of swiching decisions can be made before he erminal ime T : P {τ n < T, n } = The family of random variables {Cn α } n 1, where Cn α is he oal cos of he firs n 1 swiches n Cn α := γ ιk 1,ι k τ k 1 {τk <T } saisfies E [ sup n C α n <. 2.2 Le A,i denoe he se of admissible swiching conrol sraegies henceforh, jus sraegies. We wrie A i when = and drop he subscrip i if i is no imporan for he discussion. Remark 2.2. Processes or funcions wih supersub-scrips in erms of he random mode indicaors ι n are inerpreed in he following way: Noe ha he summaions are finie. Y ιn = 1 {ιn=j}y j, n j I γ ιn 1,ι n = 1 {ιn 1 =j}1 {ιn=k}γ j,k, n 1. j I k I We shall frequenly use he noaion Nα o denoe he random number of swiches before T under sraegy α: Nα = n 1 1 {τn<t }, α A. 2.3 Associaed wih each sraegy α A is a mode indicaor funcion u: Ω [, T I ha gives he acive mode a each ime [4, 8: u := ι 1 [τ,τ 1 + n 1 ι n 1 τn,τ n+1, [, T. 2.4

5 Finie-horizon opimal muliple swiching wih signed swiching coss 4 For a fixed ime [, T and given mode i I, he performance index for he opimal swiching problem saring a in mode i is given by: T Jα;, i = E ψ us sds + Γ ut γ ιn 1,ι n τ n 1 {τn<t } F, α A,i. 2.5 n 1 The goal is o find a sraegy α A,i ha maximises he performance index: Jα ;, i = ess sup α A,i J α;, i =: V, i. 2.6 The random funcion V, i is called he value funcion for he opimal swiching problem. Remark 2.3. For α A, define C α o be he oal swiching cos under α: C α := n 1 γ ιn 1,ι n τ n 1 {τn<t } By he finieness condiion 2.1, we have α A: C α = lim n Cα n P a.s. Furhermore, by using condiion 2.2, we have he following dominaed convergence propery: α A : lim E n [Cα n B = E [C α B a.s. for every σ-algebra B F Preliminaries. 3.1 Some resuls from he general heory of sochasic processes. We need o recall a few resuls from he general heory of sochasic processes ha are essenial o his paper. For more deails he reader is kindly referred o he references [7, 13, Righ-coninuous wih lef-limis processes. An adaped process X = X T is said o be càdlàg if i is righ-coninuous and admis lef limis. The lef-limis process associaed wih a càdlàg process X is denoed by X = X < T here we follow he convenion of [25. We also define he process X by X := X X and le X := X X denoe he size of he jump in X a, T Predicable random imes. A random ime S is an F-measurable mapping S : Ω [, T. For wo random imes ρ and τ, he sochasic inerval [ρ, τ is defined as: [ρ, τ = {ω, Ω [, T : ρω τω}. Sochasic inervals ρ, τ, [ρ, τ, ρ, τ are defined analogously. A random ime S > is said o be predicable if he sochasic inerval [, S is measurable wih respec o he predicable σ-algebra he σ-algebra on Ω, T generaed by he adaped processes wih pahs ha are lef-coninuous wih righ-limis on, T. Noe ha every predicable ime is a sopping ime [13, p. 17. By Meyer s previsibiliy predicabiliy heorem [25, Theorem VI.12.6, a sopping ime S > is predicable if and only if i is announceable in he following sense: here exiss a sequence of sopping imes {S n } n saisfying S n ω S n+1 ω < Sω for all n and lim n S n ω = Sω.

6 Finie-horizon opimal muliple swiching wih signed swiching coss Quasi-lef-coninuous processes and filraions. A càdlàg process X is called quasi-lef-coninuous if S X = a.s. for every predicable ime S Definiion I.2.25 of [13. The sric pre-s σ-algebra associaed wih a random ime S >, F S, is defined as [25, p. 345: F S = σ {A {S > u}: u T, A F u }. According o [25, p. 346, a filraion F = F T which saisfies he usual condiions is said o be quasi-lef-coninuous if F S = F S for every predicable ime S. We have he following equivalence resul for quasi-lef-coninuous filraions see [25, Theorem VI.18.1 and [7, Theorem Proposiion 3.1 Characerisaion of quasi-lef-coninuous filraions. The following saemens are equivalen: 1. F saisfies he usual condiions righ-coninuous and P-complee and is quasi-lef-coninuous; 2. For every bounded and hen for every uniformly inegrable càdlàg maringale M and every predicable ime S, we have S M = a.s.; 3. For every increasing sequence of sopping imes {S n } wih limi, lim n S n = S, we have F S = n F Sn. 3.2 Some noaion. Le us now define some noaion ha is frequenly used below. 1. For 1 p <, we wrie L p o denoe he se of random variables Z saisfying E [ Z p <. 2. Le Q denoe he se of adaped, càdlàg processes which are quasi-lef-coninuous. 3. Le M 2 denoe he se of progressively measurable processes X = X T saisfying, [ T E X 2 d <. 4. Le S 2 denoe he se of adaped, càdlàg processes X saisfying: E [ sup T X 2 <. 3.3 Properies of Snell envelopes. The following properies of Snell envelopes are also essenial for our resuls. Recall ha a progressively measurable process X is said o belong o class [D if he se of random variables {X τ, τ T } is uniformly inegrable. Proposiion 3.2. Le U = U T be an adaped, R-valued, càdlàg process ha belongs o class [D. Then here exiss a unique up o indisinguishabiliy, adaped R-valued càdlàg process Z = Z T such ha Z is he smalles supermaringale which dominaes U. The process Z is called he Snell envelope of U and i enjoys he following properies.

7 Finie-horizon opimal muliple swiching wih signed swiching coss 6 1. For any sopping ime θ we have: Z θ = ess sup E [U τ F θ, and herefore Z T = U T. 3.1 τ T θ 2. Meyer decomposiion: There exis a uniformly inegrable righ-coninuous maringale M and wo non-decreasing, adaped, predicable and inegrable processes A and B, wih A coninuous and B purely disconinuous, such ha for all T, Z = M A B, A = B =. 3.2 Furhermore, he jumps of B saisfy { B > } {Z = U }. 3. Le a sopping ime θ be given and le {τ n } n be an increasing sequence of sopping imes ending o a limi τ such ha each τ n T θ and saisfies E [ U τ n <. Suppose he following condiion is saisfied for any such sequence, lim sup U τn U τ 3.3 n Then he sopping ime τ θ defined by is opimal afer θ in he sense ha: Z θ = E [Z τ F θ θ τ θ = inf{ θ : Z = U } T 3.4 = E [U τ F θ θ = ess sup E [U τ F θ. 3.5 τ T θ 4. For every θ T, if τθ is he sopping ime defined in equaion 3.4, hen he sopped process Z τ is a uniformly inegrable càdlàg maringale. θ θ T 5. Le {U n } n and U be adaped, càdlàg and of class [D and le Z U n and Z denoe he Snell envelopes of U n and U respecively. If he sequence {U n } n is increasing and converges poinwise o U, hen he sequence {Z U n } n is also increasing and converges poinwise o Z. Furhermore, if U S 2 hen Z S 2. References for hese properies can be found in he appendix of [9 and oher references such as [6, 19, 23. Proof of he fifh propery can be found in Proposiion 2 of [4. We also have he following resul concerning inegrabiliy of he componens in he Doob-Meyer decomposiion. Proposiion 3.3. For T, le Z = M A where 1. he process Z = Z T is in S 2 ; 2. he process M = M T is a càdlàg, quasi-lef-coninuous maringale wih respec o F; 3. he process A = A T is an F-adaped càdlàg increasing process. Then A and herefore M is also in S 2. Proof. The proof essenially uses an inegraion by pars formula on A T 2 and he decomposiion Z = M A in he hypohesis. See Proposiion A.5 of [9 for furher deails, noing ha he same proof works for quasi-lef-coninuous M.

8 Finie-horizon opimal muliple swiching wih signed swiching coss 7 4 Assumpions Assumpion 4.1. The filraion F saisfies he usual condiions of righ-coninuiy and P- compleeness and is also quasi-lef-coninuous. Assumpion 4.2. For every i, j I we suppose: 1. he insananeous profi saisfies ψ i M 2 ; 2. he swiching cos saisfies γ i,j Q S he erminal daa Γ i L 2 and is F T -measurable. Assumpion 4.3. For every i, j, k I and [, T, we have a.s.: γ i,i = γ i,k < γ i,j + γ j,k, if i j and j k, Γ i max {Γ j γ i,j T }. j i Remark 4.4. The firs line in condiion 4.1 shows here is no cos for saying in he same mode. The oher wo rule ou possible arbirage opporuniies also see [8, 11. In paricular, we can always resric our aenion o hose sraegies α = τ n, ι n n A such ha P {τ n = τ n+1, τ n < T } = for n 1. Indeed, if H n := {τ n = τ n+1, τ n < T } saisfies PH n > for some n 1, hen by he second line in condiion 4.1 we ge γιn 1,ι n τ n + γ ιn,ι n+1 τ n+1 1 Hn = γ ιn 1,ι n τ n + γ ιn,ι n+1 τ n 1 Hn 4.1 > γ ιn 1,ι n+1 τ n 1 Hn which shows i is subopimal o swich wice a he same ime. 5 A verificaion heorem. Throughou his secion, we suppose ha here exis processes Y 1,..., Y m in Q S 2 defined by [ τ Y i { = ess sup E ψ i sds + Γ i 1 {τ=t } + max Y j τ γ i,j τ } 1 {τ<t } F, τ T j i 5.1 YT i = Γ i. Sufficien condiions ensuring he exisence of Y 1,..., Y m wih hese properies are given in Secion 6. Theorem 5.2 below verifies ha he soluion o he opimal swiching problem 2.6 can be wrien in erms of hese m sochasic processes. In preparaion of his verificaion heorem, we need a few preliminary resuls. Le U i = U i, i I, be a càdlàg process T defined by: { } U i := Γ i1 {=T } + max Y j γ i,j 1 {<T }, T. 5.2 j i Recall ha for every i, j I we have γ i,j, Y i Q S 2, Γ i L 2 by assumpion. Hence he process U i S 2 and is herefore of class [D. Recalling Proposiion 3.2 and rewriing equaion 5.1 for Y i as follows, [ τ Y i = ess sup E ψ i sds + U i τ F τ T [ τ = ess sup E ψ i sds + Uτ i F ψ i sds, P a.s. we can verify ha τ T Y i + ψ isds T is he Snell envelope of U i + ψ isds T.

9 Finie-horizon opimal muliple swiching wih signed swiching coss 8 Lemma 5.1. Suppose ha Y 1,..., Y m defined in 5.1 are in Q S 2. For each i I, le U i be defined as in equaion 5.2. Then for every τ n T and F τn -measurable ι n : Ω I, we have [ τ Y ιn = ess sup E ψ ιn sds + Uτ ιn F, P a.s. τ n T. 5.3 τ T Furhermore, here exis a uniformly inegrable càdlàg maringale M ιn predicable, coninuous, increasing process A ιn = A ιn τ n T such ha Y ιn + ψ ιn sds = M ιn = M ιn τn T and a A ιn, P a.s. τ n T. 5.4 Proof. The claim 5.3 is esablished in he same way as he firs few lines of Theorem 1 in [4 so he proof is skeched. We need o show ha Y ιn + ψ ι n sds is he Snell envelope of U ιn + ψ ι n sds for τ n T. Our previous discussion esablished under he curren hypoheses ha, for every i I, Y i + ψ isds is he Snell envelope of U i + ψ isds on [, T. Since 1 {ιn=i} is non-negaive and F τn -measurable, we can show ha Y i + ψ isds 1 {ιn=i} is he smalles càdlàg supermaringale dominaing U i + ψ isds 1 {ιn=i} on [τ n, T. By summing over i I recall I is finie, we have Y ιn + ψ ι n sds is he smalles càdlàg supermaringale dominaing U ιn + ψ ι n sds for τ n T. In paricular, we have [ τ Y ιn + ψ ιn sds = ess sup E ψ ιn sds + U ιn τ F, P a.s. τ n T, τ T and equaion 5.3 follows by F -measurabiliy of he inegral erm for τ n. For he second par of he claim, we use he unique Meyer decomposiion of he Snell envelope propery 2 of Proposiion 3.2 o show ha for every i I, Y i + ψ i sds = M i A i for [, T, where M i = M i T is a càdlàg uniformly inegrable maringale and Ai = A i T is a predicable, increasing process. The Snell envelope Y i + ψ isds is in Q T S2 since Y i Q S 2 and ψ i M 2. This means he Snell envelope is a regular supermaringale of class [D and Theorem VII.1 of [3 assers ha is compensaor, A i, is coninuous. Using he Meyer decomposiion, we see ha Y ιn + ψ ιn sds := Y i + ψ i sds 1 {ιn=i} = M i A i 1{ιn=i}. 5.5 i I i I Now, using 1 {ιn=i} is non-negaive and F -measurable for τ n, we see ha M ιn defined on [τ n, T by M ιnω ω := M i ω1 {ιn=i}ω, ω, [τ n, T 5.6 i I is a uniformly inegrable càdlàg maringale P-a.s. for every τ n T. Likewise, A ιn defined on [τ n, T by A ιnω ω := A i ω1 {ιn=i}ω, ω, [τ n, T 5.7 i I is a coninuous, predicable increasing process P-a.s. for every τ n T. By equaion 5.5, M ιn and A ιn provide he unique Meyer decomposiion of Y ιn P-a.s. for every τ n T.

10 Finie-horizon opimal muliple swiching wih signed swiching coss 9 Theorem 5.2 Verificaion. Suppose here exis m unique processes Y 1,..., Y m in Q S 2 which saisfy equaion 5.1. Define a sequence of imes {τ n} n and mode indicaors {ι n} n as follows: τ =, ι = i, 5.8 { τn = inf s τn 1 : Y ι n 1 s = max Y j ι s j γ ι n 1,j s } T, n 1 ι n = j1 ι j I F n 1 j F ι n 1 j := for n 1, where F ι n 1 j is he even : { Y j τ n γ ι n 1,j τ n = max k ι n 1 Then he sequence α = τ n, ι n n A,i and saisfies Y k τ n γ ι n 1,k τ n }. Y i = Jα ;, i = ess sup α A,i Jα;, i P a.s. 5.9 Proof. Sandard argumens can be used o verify ha τn is a sopping ime and each ι n is F τ n - measurable. The appendix confirms ha α A,i. As for he claim 5.9, i holds rivially for = T since YT i = Γ i = V, i a.s. for every i I. Henceforh, we assume ha [, T. Recall he process U i = U i defined in equaion 5.2. By our assumpions on T Y i, ψ i, Γ i and γ i,j for every i, j I, we have U i S 2 and we asser ha Y i + ψ isds T is he Snell envelope of U i + ψ isds. For i, j I, using Y j T = Γ j P-a.s., quasi-lefconinuiy of Y j and γ i,j, and Assumpion 4.3 on he erminal condiion for he swiching coss, T we have { lim max Y j γ i,j } = max {Γ j γ i,j T } Γ i P a.s. T j i j i Therefore, U i is quasi-lef-coninuous on [, T and lim T U i UT i P-a.s. Combining his wih he coninuiy of he inegral, we see ha U i + ψ isds saisfies he hypoheses of T propery 3 in Proposiion 3.2. Le τn, ι n n be he pair of random imes and mode indicaors in he saemen of he heorem and u be he associaed mode indicaor funcion. In conjuncion wih Lemma 5.1, {τn} defines a sequence of sopping imes where, for n 1, τn is opimal for an appropriaely defined opimal sopping problem. The remaining argumens, which are similar o hose esablishing Theorem 1 in [4, are only skeched here. The main idea is as follows: saring from an iniial mode i I a ime [, T, ieraively solve he opimal sopping problem on he righ-hand-side of 5.1 using he heory of Snell envelopes and Lemma 5.1. The minimal opimal sopping imes characerise he swiching imes whils he maximising modes are paired wih hem o give he swiching sraegy. This characerisaion will evenually lead o: [ τ N 1, Y i N N N = E ψ u s sds + Γ ι n 1 1 {τ n 1 <T }1 {τ n =T } γ ι n 1,ι τ n n 1 {τ n <T } F [ + E Y ι N τ 1 {τ N N <T } F 5.1

11 Finie-horizon opimal muliple swiching wih signed swiching coss 1 By Lemma A.1 and Theorem A.4 in he appendix respecively, he imes {τn} n saisfy he finieness condiion 2.1 and E [ sup n Cn α < holds for he cumulaive swiching coss. Appealing also o he condiional dominaed convergence heorem cf. 2.7, we may ake he limi as N in equaion 5.1 and use he definiion of u o ge: T Y i = E ψ u s sds + Γ u T γ ι n 1,ι τ n n 1 {τ n <T } F = Jα ;, i. n 1 Now, ake any arbirary admissible sraegy α = τ n, ι n n A,i. Since he sequence τ n, ι n n 1, does no necessarily achieve he essenial suprema / maxima in he ieraed opimal sopping problems, we have for all N 1: [ τn N N Y i E ψ us sds + Γ ιn 1 1 {τn 1 <T }1 {τn=t } γ ιn 1,ι n τ n 1 {τn<t } F [ + E Y ι N τn 1 {τn <T } F Passing o he limi N and using he condiional dominaed convergence heorem, we obain T Jα ;, i = Y i E ψ us sds + Γ ut γ ιn 1,ι n τ n 1 {τn<t } F = Jα;, i. n 1 Since α A,i was arbirary we have jus proved Exisence of he candidae opimal processes. The exisence of he processes Y 1,..., Y m which saisfy Theorem 5.2 is proved in his secion following he argumens of [4. The ineresed reader may also compare he proof o ha of Lemma 2.1 and Corollary 2.1 in [ The case of a mos n swiches. For each n, define process Y 1,n,..., Y m,n recursively as follows: for i I and for any T, firs se [ T Y i, = E ψ i sds + Γ i F, 6.1 and for n 1, Y i,n [ τ = ess sup E τ T { ψ i sds + Γ i 1 {τ=t } + max Y j,n 1 τ γ i,j τ } 1 {τ<t } F. 6.2 j i Define anoher process Û i,n = Û i,n T by: Û i,n := { } ψ i sds + Γ i 1 {=T } + max Y j,n 1 γ i,j 1 {<T } j i If Û i,n is of class [D, hen by Proposiion 3.2 is Snell envelope exiss and is defined by ess sup τ T E [ Û i,n F = Y i,n τ + ψ i sds.

12 Finie-horizon opimal muliple swiching wih signed swiching coss 11 Some properies of Y i,n which verify his are proved in he following lemma. In order o simplify some expressions in he proof, inroduce a new process Ŷ i,n = Ŷ i,n T which is defined by: Ŷ i,n := Y i,n + ψ i sds. Lemma 6.1. For all n, he processes Y 1,n,..., Y m,n defined by 6.1 and 6.2 are in Q S 2. Proof. The proof is similar o he one in [4. By F -measurabiliy of he inegral erm, we have Ŷ i, := Y i, + [ T ψ i sds = E ψ i sds + Γ i F. Since ψ i M 2 and Γ i L 2, he condiional expecaion is well-defined and Ŷ i, is a uniformly inegrable maringale which we can ake o be càdlàg Secion II.67 of [24. By Doob s maximal inequaliy i follows ha Ŷ i, S 2 and herefore Y i,. Since he filraion is assumed o be quasi-lef-coninuous, Proposiion 3.1 verifies ha Ŷ i, Q and herefore Y i, Q. Therefore, Y i,n Q S 2 for every i I when n =. Now, suppose by an inducion hypohesis on n ha for all i I, Y i,n Q S 2. We firs show ha Y i,n+1 S 2. By he inducion hypohesis on Y i,n and since γ i,j Q S 2 and ψ i M 2, we verify ha Û i,n+1 S 2. Therefore, by Proposiion 3.2, Ŷ i,n+1 is he Snell envelope of Û i,n+1. I is hen no difficul o show ha Û i,n+1 S 2 = Ŷ i,n+1 S 2 also propery 5 of Proposiion 3.2. Since ψ i M 2 we conclude ha Y i,n+1 S 2. We now show ha Y i,n+1 Q by arguing similarly as in he proof of Proposiion 1.4a in [12. Firs, recall ha Y i,n is in Q S 2 for every i I by { he inducion hypohesis, } and γ ij Q S 2 for every i, j I. This means he process max j i γ i,j + Y j,n is also in Q T S2. Using Y j,n T = Γ j, P-a.s. and Assumpion 4.3 on he swiching coss, we also have { lim max Y j,n γ i,j } = max {Γ j γ i,j T } Γ i. T j i j i Thus Û i,n+1 is quasi-lef-coninuous on [, T and has a possible posiive jump a ime T. Nex, by Proposiion 3.2, Ŷ i,n+1 has a unique Meyer decomposiion: Ŷ i,n+1 = M A B, where M is a righ-coninuous, uniformly inegrable maringale, and A and B are predicable, non-decreasing processes which are coninuous and purely disconinuous respecively. Le τ T be any predicable ime. The process A is coninuous so A τ = A τ holds almos surely. Moreover, he maringale M also saisfies M τ = M τ a.s. since, by Proposiion 3.1, i is quasilef-coninuous. Predicable jumps in Ŷ i,n+1 herefore come from B, and we need only consider he wo evens { τ B > } and { τ B = } since B is non-decreasing. By propery 2 of Proposiion 3.2, we have { τ B > } {Ŷ i,n+1 τ = Û i,n+1 τ } and, using he dominaing propery of Ŷ i,n+1 and non-negaiviy of he predicable jumps of Û i,n+1, his gives i,n+1 i,n+1 i,n+1 i,n+1 E [Ŷ 1 τ { τ B>} = E [Û 1 τ { τ B>} E [Û τ 1 { τ B>} E [Ŷ τ 1 { τ B>} 6.3

13 Finie-horizon opimal muliple swiching wih signed swiching coss 12 On he oher hand, he Meyer decomposiion of Ŷ i,n+1 and he almos sure coninuiy of M and A a τ yield he following: i,n+1 E [Ŷ 1 τ { τ B=} = E [ M τ A τ B τ 1 { τ B=} = E [ M τ A τ B τ 1 { τ B=} i,n+1 = E [Ŷ τ 1 { τ B=} 6.4 From 6.3 and 6.4 we ge he inequaliy, E [ Ŷ i,n+1 [Ŷ τ E i,n+1 [Ŷ τ. However, E i,n+1 τ E [ Ŷτ i,n+1 since Ŷ i,n+1 is a righ-coninuous supermaringale in S 2 and τ is predicable Theorem VI.14 of [3. Thus E [ Ŷ i,n+1 [Ŷ τ = E i,n+1 τ for every predicable ime τ. This means Y i,n+1 is a regular supermaringale of class [D and, by Theorem VII.1 of [3, he predicable nondecreasing componen of he Meyer decomposiion of Y i,n+1 mus be coninuous. Therefore, B and Y i,n+1 Q since he only jumps i experiences are hose from he quasi-lefconinuous maringale M. Lemma 6.2. For every i I, he process Y i,n solves he opimal swiching problem wih a mos n swiches: T n Y i,n = ess sup E ψ us sds + Γ ut γ ιj 1,ι j τ j 1 {τj <T } F, [, T. 6.5 α A n,i j=1 Moreover, he sequence { Y i,n} is increasing and converges poinwise P-a.s. for any T n o a càdlàg process Ỹ i saisfying: [, T, Ỹ i = ess sup α A,i J α;, i =: V, i a.s. 6.6 Proof. Le [, T, i I be given and for n define A n,i as he subse of admissible sraegies wih a mos n swiches: Define a double sequence ˆα n = ˆτ k, ˆι k n+1 k= A n,i = {α A,i : τ n+1 = T, P a.s.} as follows ˆτ =, ˆι = i, { ˆτ k = inf s ˆτ k 1 : Y ˆι k 1,n k 1 s = max Ys j,n k γˆιk 1,j s } T, j ˆι k 1 ˆι k = j1 ˆι j I F k 1 j for k = 1,..., n where F ˆι k 1 j is he even : 6.7 { F ˆι k 1 j := Y j,n k ˆτ k γˆιk 1,j ˆτ k = max Y l,n k l ˆι ˆτ k γˆιk 1,l ˆτ k }, k 1 and se ˆτ n+1 = T, ˆι n+1 ω = j I wih j ˆι n ω. Since Y i,n Q S 2, one verifies ha ˆα n A n i,n,i and, using he argumens of Theorem 5.2, ha Y = Jˆα n ;, i and has he represenaion 6.5. Furhermore, since A n,i An+1,i A,i, i follows ha Y i,n is non-decreasing in n for all [, T and Y i,n Y i,n+1 V, i almos surely. Recalling also he processes Û i,n and Ŷ i,n from Lemma 6.1 and ha Ŷ i,n is he Snell envelope of Û i,n for each n, we deduce {Ŷ i,n } n

14 Finie-horizon opimal muliple swiching wih signed swiching coss 13 is an increasing sequence of càdlàg supermaringales. Theorem VI.18 of [3 shows ha his sequence converges o a limi Ŷ i defined poinwise on [, T by Ŷ i := sup Ŷ i,n = sup Y i,n + ψ i sds. n n This random funcion Ŷ i = Ŷ i T is indisinguishable from a càdlàg process, bu is no necessarily a supermaringale since we have no esablished is inegrabiliy. Neverheless, he sequence {Y i,n } n converges poinwise on [, T o a limi Ỹ i which, modulo indisinguishabiliy, is a càdlàg process given by Ỹ i = sup n Y i,n = Ŷ i ψ i sds. 6.8 Nex, le α = τ k, ι k k A,i be arbirary. By Remark 4.4, we can resric our aenion o hose sraegies such ha P {τ k = τ k+1, τ k < T } = for k 1. Define α n = τ n k, ιn k k o be he sraegy obained from α when only he firs n swiches are kep: { τ n k, ιn k = τ k, ι k, k n, τ n k = T, k > n. The difference beween he performance indices under α and α n is: [ T Jα;, i Jα n ;, i = E ψus s ψ ι n s ds + Γ ut Γ ι n γ ιk 1,ι k τ k 1 {τk <T } F τ n k>n [ T = E ψus s ψ ι n s ds + Γ ut Γ ι n C α Cn α F τ n where u is he mode indicaor funcion associaed wih α and ι n n = ι n Nα is he las mode swiched o before T under α n. Since α A,i, ψ i M 2 and Γ i L 2 for every i I, he condiional expecaion above is well-defined for every n 1. This also leads o an inegrable upper bound for Jα;, i, [ T Jα;, i E ψ us s ψ ι n s ds + Γ ut Γ ι n + C α Cn α 1 {Nα>n} F τ n Jα n ;, i Using hese inegrabiliy condiions again ogeher wih he observaion ha Nα < P- a.s. and {τ k } is sricly increasing owards T, we may pass o he limi n in equaion 6.9 o ge, Jα;, i lim n Jαn ;, i a.s. 6.1 However, as α n A n,i for each n, from 6.1 and 6.8 we ge for every [, T : Jα;, i lim n Jαn ;, i lim Y i,n n = Ỹ i a.s. Since α A,i was arbirary, we have jus shown for every [, T V, i := ess sup J α;, i Ỹ i α A,i a.s. The reverse inequaliy holds since Y i,n = Jˆα n ;, i V, i almos surely for n cf. 6.1 and Ỹ i is he poinwise supremum of he sequence {Y i,n } n.

15 Finie-horizon opimal muliple swiching wih signed swiching coss The case of an arbirary number of swiches. This secion gives sufficien condiions under which he limiing processes Ỹ 1,..., Ỹ m saisfy he verificaion heorem 5.2. The main difficuly is in proving ha Ỹ i S 2, and in order o achieve his we make he following hypohesis. M There exiss a family of maringales {M ij = M ij T : i, j I} such ha for every i, j, k I: i. M i,j S 2 ii. γ i,j M i,j, P a.s. if i j iii. M i,j + M j,k M i,k, P a.s. if i j and j k. This hypohesis can be verified in he following cases: The swiching coss are maringales since we can se M i,j = γ i,j wih sric inequaliy in propery iii. above. This includes he case γ i,j = γ i,j, [, T, wih γ i,j L 2 and F -measurable; The swiching coss are non-negaive since we can se M i,j for i, j I; There are wo modes I = {, 1} as per convenion. For i {, 1} and j = 1 i, le Z i,j denoe he Snell envelope of γ i,j T, which exiss and is in S 2 since γ i,j S 2 see Proposiion 3.2. We may hen ake M i,j o be he maringale componen in he Doob-Meyer decomposiion of Z i,j, and se M, = M 1,1 = M,1 +M 1,. This case includes many examples of Dynkin games see [18. Lemma 6.3. Assume Hypohesis M, hen α = τ n, ι n n A,i,, i [, T I: N 1, E [ N γ ιn 1,ι n τ n F [ E max M j 1,j 2 T j 1,j 2 I F P a.s Proof. Le α = τ n, ι n n A,i be arbirary. For n 1 and i, j I we have τ n T, 1 {ιn 1 =i}1 {ιn=j} is non-negaive and F τn -measurable, M ij S 2 is a maringale wih γ i,j M i,j for i j. We can herefore show for N 1: E [ [ N N [ γ ιn 1,ι n τ n F N E M ιn 1,ι n τ n F = E M ιn 1,ι n T F The proof can be compleed by showing N 1, N M ιn 1,ι n T max j 1,j 2 I M j 1,j 2 T P a.s and concluding by arbirariness of α. The inequaliy 6.12 shall be proved via inducion similarly o [16, p Firs noe ha 6.12 is rue for N = 1. Now, suppose ha 6.12 is saisfied for N 1. Since M ιn 1,ι N T + M ιn,ι N+1 T M ιn 1,ι N+1 T a.s. we have N+1 M ιn 1,ι n T N 1 M ιn 1,ι n T + M ιn 1,ι N+1 T P a.s.

16 Finie-horizon opimal muliple swiching wih signed swiching coss 15 Define a new sraegy α = τ n, ι n n A,i by τ n, ι n = τ n, ι n for n = 1,..., N 1 and τ n, ι n = τ n+1, ι n+1 for n N. Then, using he inducion hypohesis on α, one ges N+1 M ιn 1,ι n T N M ιn 1, ι n T max j 1,j 2 I M j 1,j 2 T P a.s. Theorem 6.4 Exisence. Suppose Hypohesis M. Then he limi processes Ỹ 1,..., Ỹ m of Lemma 6.2 saisfy he following: for i I, 1. Ỹ i Q S For any T, [ τ Ỹ i = ess sup E Ỹ i T = Γ i. τ } j ψ i sds + Γ i 1 {τ=t } + max {Ỹ τ γ i,j τ 1 {τ<t } F, j i 6.13 In paricular, Ỹ 1,..., Ỹ m are unique and saisfy he verificaion heorem. Proof. Recall he limi processes Ŷ 1,..., Ŷ m and Ỹ 1,..., Ỹ m from Lemma 6.1, equaion 6.8. Under Hypohesis M one verifies direcly using Lemma 6.3 and he argumens in Lemma 6.2 ha he F-maringale ζ = ζ T defined by [ T ζ := E max ψ js ds + max Γ j + max M j 1,j 2 T j I j I j 1,j 2 I F 6.14 saisfies ζ S 2 and n, Ŷ i,n ζ P-a.s. for every [, T. Moreover, since Ŷ i is he poinwise supremum of {Ŷ i,n } n we also have Ŷ i ζ for each [, T. These observaions give ζ Ŷ i ζ P a.s. [, T. Since ζ S 2, i follows ha Ŷ i S 2 and also Ỹ i S 2 since ψ i M 2. Now define a process Û i = Û i T for i = 1,..., m similarly o Û i,n used in Lemma 6.1: Û i := } j ψ i sds + Γ i 1 {=T } + max {Ỹ γ i,j 1 {<T } j i The S 2 processes Ŷ i and Û i are he respecive limis of he increasing sequences of càdlàg S 2 processes {Ŷ i,n } n and {Û i,n } n. Since Ŷ i,n is also he Snell envelope of Û i,n, propery 5 of Proposiion 3.2 verifies ha Ŷ i is he Snell envelope of Û i. This leads o equaion 6.13 for Ỹ i and he uniqueness claim. The final par is o show ha Ỹ i Q. Le τ T be any predicable ime. Since Ŷ i is he Snell envelope of Û i, i has a Meyer decomposiion cf. Proposiion 3.2 Ŷ i = M A B, where M is a uniformly inegrable càdlàg maringale and A resp. B is non-decreasing, predicable and coninuous resp. disconinuous. Remember ha M is also quasi-lef-coninuous due o Assumpion 4.1 and Proposiion 3.1. We herefore have τ Ŷ i = M τ A τ B τ M τ A τ B τ = τ B a.s. By propery 2 of Proposiion 3.2 concerning he jumps of Ŷ i and herefore Ỹ i, we have { τ B > } {Ŷ i τ = Û i τ }

17 Finie-horizon opimal muliple swiching wih signed swiching coss 16 and by using he definiions of Ŷ i and Û i we ge: Ỹ i τ < Ỹ i τ = max j i } j {Ỹ γ τ i,j τ on { τ B > } Since I is finie, 6.15 implies ha here exiss an I-valued random variable j, j i, such ha } Ỹτ i = Ỹ j γ τ i,j τ j = max {Ỹ γ j i τ i,j τ on { τ B > } However, 6.15 also implies ha he process j max j i {Ỹ γ i,j } T jumps a ime τ since i is dominaed by Ỹ i. As he swiching coss are quasi-lef-coninuous, we conclude ha Ỹ j jumps a ime τ. Using he Meyer decomposiion of Ŷ j and he properies of he jumps as before, his leads o } Ỹ j l = max {Ỹ τ l j τ γ j,lτ on { τ B > } and here exiss an I-valued random variable l, l j, such ha } Ỹ j = τ Ỹ l τ γ j,l τ l = max {Ỹ l j τ γ j,lτ on { τ B > } 6.17 Puing 6.16 and 6.17 ogeher, hen using he quasi-lef-coninuiy of he swiching coss and Assumpion 4.3, he following almos sure inequaliy and conradicion o he opimaliy of j is obained: Ỹτ i = γ i,j τ + Ỹ j = γ τ i,j τ γ j,l τ + Ỹ τ l = γ i,j τ γ j l,lτ + Ỹ τ < γ i,l τ + Ỹ l τ < γ i,l τ + Ỹ l τ on { τ B > }. This means τ B = a.s. for every predicable ime τ, and Ỹ i Q for every i I. 7 Conclusion. This paper exended he sudy of he muliple modes opimal swiching problem in [4 o accoun for 1. non-zero, possibly differen erminal rewards; 2. signed swiching coss modelled by càdlàg, quasi-lef-coninuous processes; 3. filraions which are only assumed o saisfy he usual condiions and quasi-lef-coninuiy. Jus as in Theorem 1 of [4, i was shown ha he value funcion of he opimal swiching problem can be defined sochasically in erms of inerconneced Snell envelope-like processes. The exisence of hese processes was proved in a manner similar o Theorem 2 of [4, by a limiing argumen for sequences of processes solving he opimal swiching problem wih a mos n swiches. The limis of hese sequences are righ-coninuous processes, bu may no saisfy he inegrabiliy assumpions of he Snell envelope represenaion in general. Sufficien condiions for his represenaion were obained by furher hypohesizing he exisence of a family of maringales saisfying paricular relaions among hemselves and he swiching coss. We explained ha his maringale hypohesis can be verified quie easily in he following cases:

18 Finie-horizon opimal muliple swiching wih signed swiching coss 17 he swiching coss are maringales; he swiching coss are non-negaive; he case of wo modes saring and sopping problem. A Admissibiliy of he candidae opimal sraegy. Le α = τn, ι n n be he sequence of imes and random mode indicaors defined in equaion 5.8 of Theorem 5.2. In his secion we prove ha α A,i cf. Definiion 2.1. One readily verifies by righ-coninuiy ha {τn} n T is non-decreasing wih τ =, and each ι n is an F τ n -measurable I-valued random variable wih ι = i and ι n ι n+1 for n. The remaining properies are esablished in a number of seps, beginning wih he following lemma on he swiching imes. Lemma A.1. Le {τ n} n be he swiching imes defined in equaion 5.8 of Theorem 5.2. Then hese imes saisfy i P{τ n = τ n+1, τ n < T } =, n 1 ii P {τ n < T, n } = A.1 Proof. Condiion A.1-i can be proved via conradicion using Assumpion 4.3 recall Remark 4.4. Condiion A.1-ii can also be proved by conradicion using Assumpion 4.3 and he same argumens of [11, pp since he swiching coss are quasi-lef-coninuous. The deails are herefore omied. The res of his secion is devoed o verifying condiion 2.2 for he sraegy α. Recall ha he cumulaive cos of swiching n 1 imes is given by, C α n = n γ ι k 1,ι τ k k 1 {τk <T } Since he swiching coss saisfy γ i,j S 2 for every i, j in he finie se I, Cn α n 1. We define a sequence n := 1 {τ k <T }, n = 1, 2,... which we use o rewrie he expression for C α n C α n = N n as follows: γ ι k 1,ι τ k k. L 2 for every A.2 The following proposiion gives an alernaive represenaion of Cn α in erms of he processes Y 1,..., Y m and heir Meyer decomposiion wih random superscrips cf. Lemma 5.1. Proposiion A.2. Le α = τn, ι n n A,i be he swiching conrol sraegy defined in equaion 5.8 of Theorem 5.2 and le u be he associaed mode indicaor funcion. Then Cn α, he cumulaive cos of swiching n 1 imes under α, saisfies C α n = Y ι N n τ N n Y ι τ τ + τ N n ψ u s sds M ι k 1 τ k M ι k 1 τk 1 P a.s. A.3 where M ι k, k, is he maringale componen of he Meyer decomposiion 5.4 in Lemma 5.1.

19 Finie-horizon opimal muliple swiching wih signed swiching coss 18 Proof. By definiion of he sraegy α cf. 5.8, opimaliy of he ime τn and he definiion of ι n, for n 1 he cos of swiching a τn is, γ ι n 1,ι τ n n1 {τ n <T } = Y ι n τn Y ι n 1 τn 1 {τ n <T } P a.s. A.4 Therefore, from equaion A.2 and A.4 he cos of he firs n swiches can be rewrien as, C α n = N n Y ι k τ k Y ι k 1 τk P a.s. A.5 Now, Lemma 5.1 proved ha he following Meyer decomposiion holds for k cf. equaion 5.4: Y ι k + ψ ι k sds = M ι k A ι k, P a.s. τ k T. A.6 where, on [τ k, T, M ι k is a uniformly inegrable càdlàg maringale and A ι k is a predicable, coninuous and increasing process. The Meyer decomposiion is used o rewrie equaion A.5 for he cumulaive swiching coss as follows: P-a.s., C α n = N n M ι k τ k M ι k 1 τk N n τ k N n A ι k τ A ι k 1 k τk τ k ψ ι k sds ψ ι k 1 sds. A.7 The firs summaion erm in equaion A.7 can be rewrien as: N n M ι k τ k M ι k 1 τk = M ι τ M ι τ N n M ι k 1 τ k M ι k 1 τk 1 A.8 For every k, by he definiion of τk+1 and propery 4 of Proposiion 3.2, we know ha Y ι k + ψ ι sds is a maringale P-a.s. for every τ k k τ k+1. By using he Meyer decomposiion A.6, we herefore observe ha k, A ι k is consan P-a.s. τk τ k+1. The summaion erm in A.7 wih respec o A ι k 1 can hen be simplified as follows, N n A ι k τ A ι k 1 k τk = N n A ι k τ A ι k 1 k τk 1 = A ι τ A ι τ P a.s. A.9 By wriing ou he erms and using he definiion of he mode indicaor funcion u, he hird summaion erm in A.7 is simplified as follows: P-a.s., = = N n τ k τ 1 τ 1 ψ ι sds + ψ ι sds + τ k ψ ι k sds ψ ι k 1 sds N n 1 τ k+1 τk τ τ 1 ψ u s sds τ ψ ι k sds ψ ι sds N n τ N n ψ ι sds A.1

20 Finie-horizon opimal muliple swiching wih signed swiching coss 19 Subsiue equaions A.8, A.9, and A.1 ino equaion A.7 for he cumulaive swiching cos, hen use he Meyer decomposiion A.6 and he definiion of u o ge, C α n = M ι τ M ι τ N n τ 1 + ψ ι sds + = Y ι N n τ N n N n = Y ι N n τ N n Y ι τ M ι k 1 τ k Y ι τ M ι k 1 τ k τ N n τ 1 τ + ψ ι sds M ι k 1 τk 1 τ + τ M ι k 1 τk 1 A ι τ + A ι τ τ ψ u s sds ψ u s sds ψ ι sds τ 1 + ψ ι sds + N n M ι k 1 τ k τ N n τ 1 M ι k 1 τk 1 ψ u s sds P a.s. A.1 Convergence of he family of cumulaive swiching coss. A.1.1 A discree-parameer maringale. For k, define an F τ k -measurable random variable ξ k by, M ι k 1 τ ξ k := M ι k 1 k τ on k 1 and {τk < T }, k 1 oherwise. A.11 Noe ha he limi ξ is a well-defined F T -measurable random variable which saisfies {, on {Nα < }, ξ := lim ξ k = k, a.s. on {Nα = }. where he second line holds since M i, i I, is quasi-lef-coninuous, and he swiching imes {τk } k 1 announce T on {Nα = } cf. Lemma A.1. In his case se ι := u T. Since M i S 2 for i I cf. Proposiion 3.3 and he se I is finie, he sequence {ξ k } k is in L 2. Properies of square-inegrable maringales and condiional expecaions can be used o show: [ n [ n n 1, E ξ k 2 = E m M ι k 1 τk n i=1 M ι k 1 τ k {τ k <T } [ E Mτ i k 2 2 Mτ i k 1 E[ Mτ i k m E [ sup s T Ms i 2 i=1 Fτ k 1 + M i τ k m max i I E [ M i T 2 A.12 Finally, almos surely for 1 k, E [ [ ξ k ι F τ k 1 = E M k 1 τk M ι k 1 τk 1 F τ k 1 = 1 {ι k 1 =i}e [ Mτ i k M τ i k 1 i I F τ k 1 =

21 Finie-horizon opimal muliple swiching wih signed swiching coss 2 and leing n shows ha E [ ξ k Fτ k 1 = for k 1. Now define an increasing family of sub-σ-algebras of F, G = G n n, by G n := F τ n. Applying Lemma A.1 and Proposiion 3.1 shows ha G := G n = F τ n = F T. n n The sequence X n, G n n wih X n defined by X n := n k= ξ k A.13 is a discree-parameer maringale in L 2. The probabiliy space Ω, F, P wih filraion G = G n n will be used o discuss convergence and inegrabiliy properies of X n n. A.1.2 Convergence of he discree-parameer maringale. As discussed previously, he G-maringale X n n is in L 2. I is no hard o verify, by he condiional Jensen inequaliy for insance, ha he sequence Xn 2 is a posiive G-submaringale. n By Doob s Decomposiion Proposiion VII-1-2 of [21, Xn 2 can be decomposed uniquely n as Xn 2 = Q n + R n A.14 where Q n n is an inegrable G-maringale and R n n is an increasing process saring from wih respec o G. Convergence of X n n depends on he properies of he compensaor R n n, and his is made more precise by he following proposiion. Proposiion A.3 [21, Proposiion VII-2-3. Le X n n be a square-inegrable G-maringale such ha wihou loss of generaliy X =, and R n n denoe he increasing process associaed wih he G-submaringale Xn 2 n by he Doob decomposiion A.14. Then if E[R <, he maringale X n n converges in L 2 ; furhermore, E[sup n X n 2 4E[R. We can now prove he main resul. Theorem A.4 Square-inegrable cumulaive swiching coss. The sequence {Cn α } n 1 converges in L 2 and also saisfies E [ sup n Cn α 2 <. Proof. Proposiion A.2 gave he following represenaion for he swiching cos sum: C α n = Y ι N n τ N n = Y ι N n τ N n Y ι τ Y ι τ τ + τ + τ N n τ N n ψ u s sds M ι k 1 τ k M ι k 1 τk 1 ψ u s sds X N n P a.s. A.15 Since Nα < almos surely, he sequences {τn n } n 1 and {ι } n 1 converge almos surely o τnα T and ι Nα = u T respecively. Noing ha Y i S 2 and ψ i M 2 for every i I, we can prove he claim by showing ha he maringale X n n converges in L 2 and E[sup n X n 2 <. For his i suffices o prove he hypohesis of Proposiion A.3. Towards his end, we apply Faou s Lemma o he increasing process R n n associaed wih he G-submaringale Xn 2 n o ge E [R lim inf n E [R n. A.16

22 Finie-horizon opimal muliple swiching wih signed swiching coss 21 For n 1, he random variable R n can be decomposed as follows [21, p. 148: n 1 n 1 R n = R k+1 R k = k= k= E [X k+1 X k 2 n 1 G k = k= E [ξ k+1 2 G k. A.17 Using equaion A.17 in A.16 and applying he ower propery of condiional expecaions leads o [ n 1 E [R lim inf E ξ k+1 2. A.18 n The inequaliies leading up o A.12 above show ha he righ-hand side of A.18 is finie and we conclude by applying Proposiion A.3. Acknowledgmens This research was parially suppored by EPSRC gran EP/K557X/1. The auhor would like o hank his PhD supervisor J. Moriary and colleague T. De Angelis for heir feedback which led o an improved draf of he paper. The auhor also expresses his graiude o ohers who commened on a previous version of he manuscrip. References [1 Erhan Bayrakar and Masahiko Egami. On he One-Dimensional Opimal Swiching Problem. Mahemaics of Operaions Research, 351:14 159, 21. [2 Bruno Bouchard. A sochasic arge formulaion for opimal swiching problems in finie horizon. Sochasics An Inernaional Journal of Probabiliy and Sochasic Processes, 812: , 29. [3 Claude Dellacherie and Paul-André Meyer. Probabiliies and Poenial B - Theory of Maringales, volume 72 of Norh-Holland Mahemaics Sudies. Elsevier, Amserdam, [4 Boualem Djehiche, Said Hamadène, and Alexandre Popier. A Finie Horizon Opimal Muliple Swiching Problem. SIAM Journal on Conrol and Opimizaion, 484: , 29. [5 Brahim El Asri and Imade Fakhouri. Opimal Muli-Modes Swiching wih he Swiching Cos no necessarily Posiive, 212, arxiv: [6 Nicole El Karoui. Les aspecs probabilises du conrôle sochasique. Ecole d Eé de Probabiliés de Sain-Flour IX-1979, [7 Rober J. Ellio. Sochasic Calculus and Applicaions Applicaions of Mahemaics 18. Springer-Verlag, New York, [8 Xin Guo and Pascal Tomecek. Connecions beween Singular Conrol and Opimal Swiching. SIAM Journal on Conrol and Opimizaion, 471: , 28. [9 Said Hamadène. Refleced BSDE s wih disconinuous barrier and applicaion. Sochasics An Inernaional Journal of Probabiliy and Sochasic Processes, 743: , 22. [1 Said Hamadène and I Hdhiri. The sopping and saring problem in he model wih jumps. PAMM, 71: , 27. k=

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