Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

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1 Sochasic conrol under progressive enlargemen of filraions and applicaions o muliple defauls risk managemen Huyên PHAM Laboraoire de Probabiliés e Modèles Aléaoires CNRS, UMR 7599 Universié Paris 7 pham@mah.jussieu.fr and Insiu Universiaire de France May 11, 21 Absrac We formulae and invesigae a general sochasic conrol problem under a progressive enlargemen of filraion. The global informaion is enlarged from a reference filraion and he knowledge of muliple random imes ogeher wih associaed marks when hey occur. By working under a densiy hypohesis on he condiional join disribuion of he random imes and marks, we prove a decomposiion of he original sochasic conrol problem under he global filraion ino classical sochasic conrol problems under he reference filraion, which are deermined in a finie backward inducion. Our mehod revisis and exends in paricular sochasic conrol of diffusion processes wih finie number of jumps. This sudy is moivaed by opimizaion problems arising in defaul risk managemen, and we provide applicaions of our decomposiion resul for he indifference pricing of defaulable claims, and he opimal invesmen under bilaeral counerpary risk. The soluions are expressed in erms of BSDs involving only Brownian filraion, and remarkably wihou jump erms coming from he defaul imes and marks in he global filraion. Key words: sochasic conrol, progressive enlargemen of filraions, decomposiion in he reference filraion, muliple defaul imes, risk managemen. MSC Classificaion 2): 6J75, 932, 6H2. I would like o hank N. l Karoui, Y. Jiao and I. Kharroubi for useful commens. 1

2 1 Inroducion The field of sochasic conrol has known imporan developmens over hese las years, inspired especially by various problems in economics and finance arising in risk managemen, opion hedging, opimal invesmen, porfolio selecion or real opions valuaion. A vas lieraure on his opic and is applicaions has grown wih differen approaches ranging from dynamic programming mehod, Hamilon-Jacobi-Bellman Parial Differenial quaions PDs) and Backward Sochasic Differenial quaions BSDs) o convex maringale dualiy mehods. We refer o he monographs [7, [25, [17 or [18 for recen updaes on his subjec. In paricular, he heory of BSDs has emerged as a major research opic wih original and significan conribuions relaed o sochasic conrol and is financial applicaions, see a recen overview in [5. On he oher hand, he field of enlargemen of filraions is a radiional subjec in probabiliy heory iniiaed by fundamenal works of he French school in he 8s, see e.g. [11, [9, [12, and he recen lecure noes [16. I knows a renewed ineres due o is naural applicaion in credi risk research where i appears as a powerful ool for modelling defaul evens. For an overview, we refer o he books [3, [4, [22 or he lecure noes [2. The sandard approach of credi even is based on he enlargemen of a reference filraion F he defaul-free informaion srucure) by he knowledge of a defaul ime when i occurs, leading o he global filraion G, and called progressive enlargemen of filraions. Moreover, i assumes ha he credi even should arrive by surprise, i.e. i is a oally inacessible random ime for he reference filraion. Hence, he main approaches consis in modelling he inensiy of he random ime usually referred o as he reduced-form approach), or more generally in he modelling of he condiional law of his random ime, and referred o as densiy hypohesis, see [6. The sabiliy of he class of semimaringale, usually called H ) hypohesis, and meaning ha any F-semimaringale remains a G-semimaringale, is a fundamenal propery boh in probabiliy and finance where i is closely relaed o he absence of arbirage. I holds rue under he densiy hypohesis, and he relaed canonical decomposiion in he enlarged filraion can be explicily expressed, as shown in [1. A sronger assumpion han H ) hypohesis is he so-called immersion propery or H) hypohesis, denoing he fac ha F-maringales remain G-maringales. The purpose of his paper is o combine boh feaures of sochasic conrol and progressive enlargemen of filraions in view of applicaions in finance, in paricular for defauls risk managemen. We formulae and sudy he general srucure for such conrol problems by considering a progressive enlargemen wih muliple random imes and associaed marks. These marks represen for example in credi evens jump sizes of asse values, which may arrive several imes by surprise and canno be prediced from he pas observaion of asse processes. We work under he densiy hypohesis on he condiional join disribuion of he random imes and marks. Our new approach consiss in decomposing he iniial conrol problem in he G-filraion ino a finie sequence of conrol problems formulaed in he F-filraion, and which are deermined recursively. This is based on an enlighening represenaion of any G-predicable or opional process ha we spli ino indexed F-predicable or opional processes beween each random ime. This poin of view allows 2

3 us o change of regimes in he sae process, and o modify he conrol se and he gain funcions beween random imes. This flexibiliy in he formulaion of he sochasic conrol problem appears also quie useful and relevan for financial inerpreaion. Our mehod consis basically in projecing G-processes ino he reference F-filraion beween wo random imes, and feaures some similariies wih filering approach. This conrass wih he sandard approach in progressive enlargemen of filraion focusing on he represenaion of conrolled sae process in he G-filraion where he conrol se has o be fixed a he iniial ime. Moreover, in his global approach, one usually assumes ha H) hypohesis holds in order o ge a maringale represenaion in he G-filraion. In his case, he soluion is hen characerized from dynamic programming mehod in he G-filraion via PDs wih inegrodifferenial erms or BSDs wih jumps. By means of our F-decomposiion resul under he densiy hypohesis and wihou assuming H) hypohesis), we can solve each sochasic conrol problem by dynamic programming in he F-filraion, which leads ypically o PDs or BSDs relaed only o Brownian moion, hus simpler a priori han Inegro-PDs and BSDs wih jumps. Our decomposiion mehod revisis and more imporanly exends sochasic conrol of diffusion processes wih finie number of jumps, and gives some new insigh for sudying Inegro-PDs and BSDs wih jumps. We illusrae our mehodology wih wo financial applicaions in defaul risk managemen. The firs one considers he problem of indifference pricing of defaulable claims, and he second applicaion deals wih an opimal invesmen problem under bilaeral conagion risk wih wo nonordered defaul imes. The soluions are explicily expressed in erms of BSDs involving only Brownian moion. The paper is organized as follows. The nex secion presens he general framework of progressive enlargemen of filraion wih successive random imes and marks. We sae he decomposiion resul for a G-predicable and opional process, and as a consequence we derive under he densiy hypohesis he compuaion of expecaion funcionals of G- opional processes in erms of F-expecaions. In Secion 3, we formulae he absrac sochasic conrol problem in his conex and connec i in paricular o diffusion processes wih jumps. Secion 4 conains he main F-decomposiion resul of he iniial sochasic conrol problem. The case of enlargemen of filraion wih muliple and no necessarily successive) random imes is considered in Secion 5, and we show how o derive he resuls from he case of successive random imes wih auxiliary marks. Finally, Secion 6 is devoed o some applicaions in risk managemen, where we presen he resuls and pospone he deailed proofs and more examples in a forhcoming paper [13. 2 Progressive enlargemen of filraion wih successive random imes We fix a probabiliy space Ω, G, P), and we sar wih a reference filraion F = F ) saisfying he usual condiions F conains he null ses of P and F is righ coninuous: F = F := s> F s ). We consider a vecor of n random imes τ 1,..., τ n i.e. nonnegaive G-random variables) and a vecor of n G-measurable random variables ζ 1,..., ζ n valued in some Borel subse of R m. The defaul informaion is he knowledge of hese defaul 3

4 imes τ k when hey occur, ogeher wih he associaed marks ζ k. For each k = 1,..., n, i is defined in mahemaical erms as he smalles righ-coninuous filraion D k = D k ) such ha τ k is a D k -sopping ime, and ζ k is Dτ k k -measurable. In oher words, D k = D k, where D k = σζ k 1 τk s, s ) σ1 τk s, s ). The global marke informaion is hen defined by he progressive enlargemen of filraion G = F D 1... D n. The filraion G = G ) is he smalles filraion conaining F, and such ha for any k = 1,..., n, τ k is a G-sopping ime, and ζ k is G τk -measurable. Wih respec o he classical framework of progressive enlargemen of filraion wih a single random ime exensively sudied in he lieraure, we consider here muliple random imes ogeher wih marks. For simpliciy of presenaion, we firs consider he case where he random imes are ordered, i.e. τ 1... τ n, and so valued in n on {τ n < }, where { } k =,..., ) R ) k :...,, k = 1,..., n. This means acually ha he observaions of ineres are he ranked defaul imes ogeher wih he marks). We shall indicae in Secion 5 how o adap he resuls in he case of muliple random imes no necessarily ordered. We inroduce some noaions used hroughou he paper. - PF) resp. PG)) is he σ-algebra of F resp. G)-predicable measurable subses on R Ω, i.e. he σ-algebra generaed by he lef-coninuous F-adaped resp. G-adaped) processes. We also le P F resp. P G ) denoe he se of processes ha are F-predicable resp. G-predicable), i.e. PF)-measurable resp. PG)-measurable). - OF) resp. OG)) is he σ-algebra of F resp. G)-opional measurable subses on R Ω, i.e. he σ-algebra generaed by he righ-coninuous F-adaped resp. G-adaped) processes. We also le O F resp. O G ) denoe he se of processes ha are F-opional resp. G-opional), i.e. OF)-measurable resp. OG)-measurable). - For k = 1,..., n, we denoe by PF k k, k ) resp. OF k k, k )) he se of of indexed processes Y k.) such ha he map, ω,,...,, e 1,..., e k ) Y k ω,,...,, e 1,..., e k ) is PF) B k ) B k )-measurable resp. OF) B k ) B k )-measurable). - For θ =,..., θ n ) n, e = e 1,..., e n ) n, we denoe by θ k) =,..., ), e k) = e 1,..., e k ), k = 1,..., n. The following resul provides he key decomposiion of predicable and opional processes wih respec o his progressive enlargemen of filraion. This exends a classical resul, see e.g. Lemma 4.4 in [11 or Chaper 6 in [2, saed for a progressive enlargemen of filraion wih a single random ime. Lemma 2.1 Any G-predicable process Y = Y ) is represened as n 1 Y = Y 1 τ1 Y k τ 1,..., τ k, ζ 1,..., ζ k )1 τk < τ k1 k=1 Y n τ 1,..., τ n, ζ 1,..., ζ n )1 τn<,, 2.1) 4

5 where Y P F, and Y k P k F k, k ), for k = 1,..., n. Any G-opional process Y = Y ) is represened as n 1 Y = Y 1 <τ1 Y k τ 1,..., τ k, ζ 1,..., ζ k )1 τk <τ k1 k=1 Y n τ 1,..., τ n, ζ 1,..., ζ n )1 τn,, 2.2) where Y O F, and Y k O k F k, k ), for k = 1,..., n. Proof. We prove he decomposiion resul for predicable processes by inducion on n. We denoe by G n = F D 1... D n. Sep 1. Suppose firs ha n = 1, so ha G = F D 1. Le us consider generaors of PG), which are processes in he form Y = f s gζ 1 1 τ1 s)hτ 1 s)1 >s,, wih s, f s F s -measurable, g measurable defined on {}, and h measurable defined on R. By aking Y = f s g)hs)1 >s, and Y 1, e) = f s ge1 θ1 s)h s)1 >s, we see ha he decomposiion 2.1) holds for generaors of PG). We hen exend his decomposiion for any PG)-measurable processes, by he monoone class heorem. Sep 2. Suppose ha he resul holds for n, and consider he case wih n 1 ranked defaul imes, so ha G = G n D n1, D n1 n1 n1 = D, where D = σζ n1 1 τn1 s, s ) σ1 τn1 s, s ). By he same argumens of enlargemen of filraion wih one defaul ime as in Sep 1, we derive ha any PG)-measurable process Y is represened as Y = Y,n) 1 τn1 Y 1,n) τ n1, ζ n1 )1 τn1 <, 2.3) where Y,n) is PG n )-measurable, and, ω, θ n1, e n1 ) Y 1,n) ω, θ n1, e n1 ) is PG n ) BR ) B)-measurable. Now, from he inducion hypohesis for G n, we have Y,n) = Y,,n) n 1 1 τ1 Y k,,n) τ 1,..., τ k, ζ 1,..., ζ k )1 τk < τ k1 k=1 Y n,,n) τ 1,..., τ n, ζ 1,..., ζ n )1 τn<,, where Y,,n) P F, and Y k,,n) P k F k, k ), for k = 1,..., n. Similarly, we have Y 1,n) θ n1, e n1 ) = Y,1,n) θ n1, e n1 )1 τ1 n 1 Y k,1,n) τ 1,..., τ k, ζ 1,..., ζ k, θ n1, e n1 )1 τk < τ k1 k=1 Y n,1,n) τ 1,..., τ n, ζ 1,..., ζ n, θ n1, e n1 )1 τn<,, 5

6 where Y,1,n) PF 1R, ), Y k,1,n) P k1 F k R, k1 ), k = 1,..., n. Finally, plugging hese wo decomposiions wih respec o PG n ) ino relaion 2.3), and recalling ha τ 1... τ n τ n1, we ge he required decomposiion a level n 1 for G: Y = Y,,n) 1 τ1 n k=1 Y k,,n) τ 1,..., τ k, ζ 1,..., ζ k )1 τk < τ k1 Y n1 τ 1,..., τ n1, ζ 1,..., ζ n1 )1 τn1 <,, where we noice ha he indexed process Y n1 defined by Y n1,..., θ n1, e 1,..., e n1 ) := Y n,1,n),..., θ n, e 1,..., e n, θ n1, e n1 ), lies in P n1 F n1, n1 ). The decomposiion resul for G-opional processes is proved similarly by inducion and considering generaors of OG 1 ), which are processes in he form Y = f s gζ 1 1 τ1 s)hτ 1 s)1 s,, wih s, f s F s -measurable, g measurable defined on {}, and h measurable defined on R. Obviously, any process in he form 2.1) resp. 2.2)) is G-predicable resp. G- opional). Lemma 2.1 saes he converse propery. Therefore, we can idenify any Y P G resp. O G ) wih an n 1-uple Y,..., Y n ) P F... P n F n, n ) resp. O F... O n F n, n )) arising from is decomposiion 2.1) resp. 2.2)). We now require a densiy hypohesis on he random imes and heir associaed jumps by assuming ha for any, he condiional disribuion of τ 1,..., τ n, ζ 1,..., ζ n ) given F is absoluely coninuous wih respec o a posiive measure λdθ)ηde) on B n ) B n ), wih λ he Lebesgue measure λdθ) = d... dθ n, and η a produc measure ηde) = η 1 de 1 )... η 1 de n ) on B)... B). More precisely, we assume ha here exiss γ OF n n, n ) such ha DH) P [ τ 1,..., τ n, ζ 1,..., ζ n ) dθde F = γ,..., θ n, e 1,..., e n ) d... dθ n η 1 de 1 )... η 1 de n ), Remark 2.1 In he paricular case where γ is in he form γ θ, e) = ϕ θ)ψ e), he condiion DH) means ha he random imes τ 1,..., τ n ) and he jump sizes ζ 1,..., ζ n ) are independen given F, for all, and [ [ P τ 1,..., τ n ) dθ F = ϕ θ)λdθ), P ζ 1,..., ζ n ) de F = ψ e)ηde), a.s. This condiion exends he usual densiy hypohesis for a random ime in he heory of iniial or progressive enlargemen of filraion, see [9 or [1. An imporan resul in he heory of enlargemen of filraion under he densiy hypohesis is he semimaringale invariance propery, also called H ) hypohesis, i.e. any F-semimaringale remains a G- semimaringale. This resul is relaed in finance o no-arbirage condiions, and is hus also a desirable propery from a economic viewpoin. Random imes saisfying he densiy hypohesis are very well suiable for he analysis of credi risk evens, as shown recenly in [6. We also refer o his paper for a discussion on he relaion beween he densiy hypohesis and he reduced-form or inensiy) approach in credi risk modelling. 6 a.s.

7 In he sequel, i is useful o inroduce he following noaions. We denoe by γ he F-opional process defined by γ = P [ τ 1 > F =... n θ n 1 γ,..., θ n, e 1,..., e n )dθ n... d η 1 de 1 )... η n de n ), and we denoe by γ k, k = 1,..., n 1, he indexed process in O k F k, k ) defined by 2.4) = γ k,...,, e 1,..., e k )... γ,..., θ n, e 1,..., e n )dθ n... d1 η 1 de k1 )... η 1 de n ), n k 1 θ n 1 so ha for k = 1,..., n 1, P [ τ k1 > F = γ k,...,, e 1,..., e k )d... d η 1 de 1 )... η 1 de k ).2.5) k k Noice ha he family of measurable maps γ k, k =,..., n can be also wrien in backward inducion by γ k,...,, e 1,..., e k ) = γ k1,..., 1, e 1,..., e k1 )d1 η 1 de k1 ), for k =,..., n 1, saring from γ n = γ. In view of 2.4)-2.5), he process γ k may be inerpreed as he survival densiy process of τ k1, k =,..., n 1. The nex resul provides he compuaion for he opional projecion of a OG)-measurable process on he reference filraion F. Lemma 2.2 Le Y = Y,..., Y n ) be a nonnegaive or bounded) G-opional process. Then for any, we have Ŷ F := [ Y F = Y γ n k=1 k... Y k,...,, e 1,..., e k ) 1 γ k,...,, e 1,..., e k )d... d η 1 de 1 )... η 1 de k ), where we used he convenion ha 1 = for k = 1 in he above inegral. quivalenly, we have he backward inducion formula for Ŷ F = Ŷ,F, where he Ŷ k,f are given for any, by Ŷ n,f θ, e) = Y n θ, e)γ θ, e) Ŷ k,f θ k), e k) ) = Y k θ k), e k) )γ k θ k), e k) ) Ŷ k1,f θ k), 1, e k), e k1 )η 1 de k1 )d1, for θ =,..., θ n ) n [, n, e = e 1,..., e n ) n. 7

8 Proof. Le Y = Y,..., Y n ) be a nonnegaive or bounded) G-opional process, decomposed as in 2.2) so ha: [ Y F [ = Y F 1 <τ1 n k=1 [ Y k F τ 1,..., τ k, ζ 1,..., ζ k )1 τk <τ k1, 2.6) wih he convenion ha τ n1 =. Now, for any k = 1,..., n, we have under he densiy hypohesis DH) [ Y k F τ 1,..., τ k, ζ 1,..., ζ k )1 τk <τ k1 = Y k,...,, e 1,..., e k )1 θk <1 γ,..., θ n, e 1,..., e n )λdθ)ηde) n n = k... Y k,...,, e 1,..., e k )γ k,...,, e 1,..., e k )d... d 1 η 1 de 1 )... η 1 de k ), where he second inequaliy follows from Fubini s heorem and he definiion of γ k. We also have [ Y F 1 <τ1 = Y P[τ 1 > F = Y γ. We hen ge he required resul by plugging hese wo las relaions ino 2.6). Finally, he backward formula for he F-opional projecion of Y is obained by a sraighforward inducion. As a consequence of he above backward inducion formula for he opional projecion, we derive a backward formula for he compuaion of expecaion funcionals of G-opional processes, which involves only F-expecaions. Proposiion 2.1 Le Y = Y,..., Y n ) and Z = Z,..., Z n ) be wo nonnegaive or bounded) G-opional processes, and fix T, ). The expecaion [ T Y d Z T can be compued in a backward inducion as where he J k, k =,..., n are given by [ T J n θ, e) = θ n [ T J k θ k), e k) ) = T [ T Y d Z T Y n = J γ θ, e)d ZT n γ T θ, e) F θn Y k γ k θ k), e k) )d Z k T γ k T θ k), e k) ) J k1 θ k), 1, e k), e k1 )η 1 de k1 )d1 Fθk, for θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n, wih he convenion θ =. 8

9 Proof. For any θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n, le us define [ T J k θ k), e k) ) = Ŷ k,f θ k), e k) )d Ẑk,F T θk), e k) ) Fθk, where he Ŷ k,f and Ẑk,F are defined in Lemma 2.2, associaed respecively o Y and Z. Then J = [ T Y d Z T, and we see from he backward inducion for Ŷ k,f and Ẑk,F ha he J k, k =,..., n, saisfy [ T J n θ, e) = θ n [ T J k θ k), e k) ) = γ θ, e)d ZT n θ, e)γ T θ, e) F θn Y n Y k γ k θ k), e k) )d Z k T γ k T θ k), e k) ) T T [ T = T T [ T = T Ŷ k1,f θ k), 1, e k), e k1 )η 1 de k1 )d1 d Ẑ k1,f T θ k), 1, e k), e k1 )η 1 de k1 )d1 Fθk Y k γ k θ k), e k) )d Z k T γ k T θ k), e k) ) T Ŷ k1,f θ k), 1, e k), e k1 )d η 1 de k1 )d1 1 Ẑ k1,f T θ k), 1, e k), e k1 )η 1 de k1 )d1 Fθk Y k γ k θ k), e k) )d Z k T γ k T θ k), e k) ) J k1 θ k), 1, e k), e k1 ) η 1 de k1 )d1 Fθk, where we used Fubini s heorem in he second equaliy for J k, and he law of ieraed condional expecaions for he las equaliy. This proves he required inducion formula for J k, k =,..., n. 3 Absrac sochasic conrol problem In his secion, we formulae he general sochasic conrol problem in he conex of progressively enlargemen of filraion wih successive random imes and marks. 3.1 Conrols and sae process A conrol is a G-predicable process α = α,..., α n ) P F... P n F n, n ), where he α k, k =,..., n, are valued in some given Borel se A k of an uclidian space. We denoe by P F A ) resp. P k F k, k ; A k ), k = 1,..., n), he se of elemens in P F resp. P k F, k ), k = 1,..., n) valued in A resp. A k, k = 1,..., n). We se A = A... A n, and denoe by A G he se of admissible conrols as he produc A F... An F, where A F resp. Ak F, k 9

10 = 1,..., n) is some separable meric space of P F A ) resp. P k F k, k ; A k ), k = 1,..., n). The separabiliy condiion is required for measurabiliy selecion issue. The descripion of he conrolled sae process is formulaed as follows: Conrolled sae process beween defaul imes: we are given a collecion of measurable mappings: x, α ) R d A F X,x,α O F 3.1) x, α k ) R d A k F X k,x,αk O k F k, k ), k = 1,..., n, 3.2) such ha we have he iniial daa: X,x,α = x, x R d, X k,ξ,αk,...,, e 1,..., e k ) = ξ, ξ F θk measurable, k = 1,..., n. Jumps of he conrolled sae process: we are given a collecion of maps Γ k on R Ω R d A k 1, for k = 1,..., n, such ha, ω, x, a, e) Γ k ω, x, a, e) is PF) BR d ) BA k 1 ) B) measurable. Global conrolled sae process: he conrolled sae process is hen given by he mapping where X x,α is he process equal o x, α = α,..., α n )) R d A G X x,α O G, X x,α = X n 1 1 <τ1 k=1 wih X,..., X n ) O F... O n F n, n ) given by X k τ 1,..., τ k, ζ 1,..., ζ k )1 τk <τ k1 X n τ 1,..., τ n, ζ 1,..., ζ n )1 τn,, 3.3) X = X,x,α X k,...,, e 1,..., e k ) = X k,γk k 1 X θ,α k 1 k θ,e k ),α k k,...,, e 1,..., e k ), for k = 1,..., n. The inerpreaion is he following. Beween he ime inerval τ k = and τ k1 = 1, k =,..., n 1 wih he convenion θ = ), he sae process X = X k is conrolled by α k, which is based on he basic informaion F, and he knowledge of he pas jump imes and marks,...,, e 1,..., e k ). Then, a ime 1, here is a jump on he sae process deermined by he map Γ k1, which depends on he curren sae value, conrol and informaion, bu also on a nonpredicable mark ζ k1 = e k1 a ime 1 : X τk1 = Γ k1 τ k1 X τ, ατ k k1 k1, ζ k1 ). 1

11 3.2 Typical conrolled sae process In ypical applicaions, he dynamics of X = X,x,α, X k = X k,x,αk, k = 1,..., n, are governed by diffusion processes: dx = b X, α )d σ X, α )dw, 3.4) dx k = b k X k, α k,,...,, e 1,..., e k )d 3.5) σ k X k, α k,,...,, e 1,..., e k )dw,. Here, W is a sandard m-dimensional P, F)-Brownian moion, and, ω, x, a) b ω, x, a), σ ω, x, a) are PF) BR d ) BA )-measurable maps valued respecively in R d and R d m, for k = 1,..., n, he maps, ω, x, a,,...,, e 1,..., e k ) b k ω, x, a,,...,, e 1,..., e k ), σ k ω, x, u,,...,, e 1,..., e k ) are PF) BR d ) BA k ) B k ) B k )-measurable valued respecively in R d and R d m. To alleviae noaions, we omied in 3.5) he dependence of X k, α k in,...,, e 1,..., e k ). We make he linear growh and Lipschiz assumpions on he funcions x b k x,.), σ k x,.), k =,..., n, in order o ensure for all,...,, e 1,..., e k ) k k, he exisence and uniqueness of a soluion X k,...,, e 1,..., e k ) o he sde 3.4), 3.5), given he conrols and he iniial condiions, and his indexed process X k lies in OF k k, k ). The dependence of he coefficiens b k, σ k on he pas jump imes,...,, and marks e 1,..., e k, corresponds o change of regimes afer each jump ime, and may be inerpreed in finance as raing upgrades or downgrades. Also, ypical choice for he se of admissible conrols A k F is subse of indexed F-predicable processes in L p, p [1, ), and he separabiliy of A k F follows from he separabiliy of Lp, see he discussion in [23. Connecion wih conrolled jump-diffusion processes. Consider he paricular case where he ses of conrols A k are idenical, equal o A, and le us define he mappings b and σ on R Ω R d A by: n 1 b x, a) = b x, a)1 τ1 b k x, a, τ 1,..., τ k, ζ 1,..., ζ k )1 τk < τ k1 k=1 b n x, a, τ 1,..., τ n, ζ 1,..., ζ n )1 >τn, n 1 σ x, a) = σ x, a)1 τ1 σ k x, a, τ 1,..., τ k, ζ 1,..., ζ k )1 τk < τ k1 k=1 σ n x, a, τ 1,..., τ n, ζ 1,..., ζ n )1 >τn, and noice ha he maps, ω, x, a) b ω, x, a), σ ω, x, a) are PG) BR d ) BA)- measurable. Denoe also by δ he mapping on R Ω R d A : δ x, a, e) = n 1 k= Γ k1 ) x, a, e) x 1 τk < τ k1 which is PG) BR d ) BA) B)-measurable. Le us denoe by µd, de) he inegervalued random measure associaed o he imes τ k and he marks ζ k, k = 1,..., n, which 11

12 is hen given by µ[, B) = k 1 1 τn 1 B ζ k ),, B B). The progressive enlarged filraion G can hen be wrien also as: G = F F µ where F µ is he righ-coninuous filraion generaed by he ineger-valued random measure µ. Now, since he semimaringale propery is preserved under he densiy hypohesis for his progressive enlargemen of filraion, see [1), he process W remains a semimaringale under P, G) wih a canonical decomposiion, which can be explicily expressed in erms of he densiy). Then, we can wrie he dynamics of he sae process X = X x,α in 3.3) as a conrolled jump-diffusion process under P, G): dx = b X, α )d σ X, α )dw δ X, α, e)µd, de). However, noice ha in he above G-formulaion, he process W is no in general a Brownian moion under P, G), unless he so-called H) immersion propery is saisfied, i.e. he maringale propery is preserved from F o G, which corresponds o he paricular case where he densiy saisfies: γ θ, e) = γ θ θ, e) for θ. In he classical formulaion by conrolled jump-diffusion processes, one has o fix a conrol se A, which is invarian during he ime horizon. Here, he more general formulaion 3.3) allows us o consider differen conrol ses A k beween wo defaul imes, and his may be relevan in pracical applicaions. Moreover, we have a suiable decomposiion of he coefficiens and conrolled sae process beween random imes, which provides a naural inerpreaion in economics and finance. 3.3 Sochasic conrol problem In he general framework for he conrolled process in 3.3), le us formulae he objecive funcion for he sochasic conrol problem on a finie horizon T. The erminal gain funcion is given by a nonnegaive map G T on Ω R d such ha ω, x) G T ω, x) is G T BR d )- measurable, and which may be represened as n 1 G T x) = G T x)1 T <τ1 G k T x, τ 1,..., τ k, ζ 1,..., ζ k )1 τk T <τ k1 k=1 G n T x, τ 1,..., τ n, ζ 1,..., ζ n )1 τn T, where G T is F T BR d )-measurable, and G k T is F T BR d ) B k ) B k )-measurable, for k = 1,..., n. The running gain funcion is given by a nonnegaive map f on Ω R d A such ha, ω, x, a) f ω, x, a) is OG) BR d ) BA)-measurable, and which may be decomposed as n 1 f x, a) = f x, a )1 <τ1 f k x, a k, τ 1,..., τ k, ζ 1,..., ζ k )1 τk <τ k1 k=1 f n x, a n, τ 1,..., τ n, ζ 1,..., ζ n )1 τn, 12

13 for a = a,..., a n ) A = A... A n, where f is OF) BR d ) BA )-measurable, and f k is OF) BR d ) BA k ) B k ) B k )-measurable, for k = 1,..., n. In oher words, here is a change of regimes in he running and erminal gain afer each defaul ime. The value funcion for he sochasic conrol problem is hen defined by: [ T V x) = sup α A G f X x,α, α )d G T X x,α T, ) x R d. 3.6) 4 F-decomposiion of he sochasic conrol problem In his secion, we provide a decomposiion of he value funcion for he sochasic conrol problem in he G-filraion, defined in 3.6), ha we formulae in a backward inducion for value funcions of sochasic conrol in he F-filraion. To alleviae noaions, we shall ofen omi in 3.2) he dependence of X k,x on α k and,...,, e 1,..., e k ) when here is no ambiguiy. Theorem 4.1 The value funcion V is obained from he backward inducion formula: [ T V n x, θ, e) = ess sup α n A n F θ n f n X n,x, α n, θ, e)γ θ, e)d G n T X n,x T, θ, e)γ T θ, e) F θn [ V k x, θ k), e k) T ) = ess sup f k X k,x, α k, θ k), e k) )γ k θ k), e k) )d α k A k θ F k G k T X k,x T, θk), e k) )γt k θ k), e k) ) T V k1 Γ k1 1 X k,x 1, α k 1, e k1 ), θ k), 1, e k), e k1 ) 4.1) η 1 de k1 )d1 Fθk, k =,..., n 1, 4.2) for all θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n, x R d. Remark 4.1 ach sep in he backward inducion for he deerminaion of he original value funcion V leads o he formulaion of a family of value funcions associaed o sandard sochasic conrol problem in he F-filraion. Indeed, a sep n, V n x,.) is a family of value funcions paramerized by,..., θ n ) n, e 1,..., e n ) n, and corresponding o he sochasic conrol problem afer he las defaul a ime θ n, wih a running gain funcion f n and erminal gain funcion G n T on he conrolled sae process Xn in he F- filraion, and weighed by he OF)-measurable process γ. Now, suppose ha a sep k 1, we have deermined he family of value funcions V k1 x,.),,..., 1 ) k1, e 1,..., e k1 ) k1, and denoe by ˆV k1 he map on Ω R d A k k1 k : = ˆV k1 x, ak, θ k), 1, e k)) V k1 Γ k1 1 x, a k, e k1 ), θ k), 1, e k) ), e k1 η1 de k1 ). 13

14 Then, he family of value funcions a sep k, represening he value for he sochasic conrol problem afer k defauls, is compued from he sochasic conrol problem in he F-filraion wih he running gain funcion f k and erminal gain funcion G k T weighed by he OF)-measurable random variable γ k, and wih he running gain funcion ˆV k1 : [ T V k x) = ess sup f k X k,x, α k )γ k d G k T X k,x )γk T α k A k θ F k T ˆVk1 X k,x 1, α k 1, 1 )d1 Fθk. 4.3) Here, we omied he dependence in θ k) =,..., ), e k) = e 1,..., e k ) o alleviae noaions. The wo firs erms in he rhs of 4.3) represen he gain funcional when here is no more defaul afer he k-h one, while he las erm represens he gain in he case when a k 1-h defaul would occur beween he las one a ime τ k = and he finie horizon T. Finally, he decomposiion in Theorem 4.1 also shows ha an opimal conrol for he global problem in he G-filraion is obained by a concaenaion of opimal conrols for each subproblems V k in he F-filraion. Proof of Theorem 4.1. Fix x R d, α = α,..., α n ) A G, and consider he conrolled sae process X x,α. By definiion of X x,α in 3.3), G T.) and f.), observe ha he G T -measurable random variable G T X x,α T ) is decomposed according o he n 1-uple G T X T ),..., Gn T X T n )), and he G- opional process f X x,α, α ) is decomposed as f X, α ),..., f n X n, α n )). Le us now define by backward inducion he maps J k, k =,..., n by [ T J n x, θ, e, α) = f n X n,x, α n, θ, e)γ θ, e)d G n T X n,x T, θ, e)γ T θ, e) F θn θ n [ T J k x, θ k), e k), α) = f k X k,x, α k, θ k), e k) )γ k θ k), e k) )d G k T X k,x T, θk), e k) )γt k θ k), e k) ) J k1 Γ k1 1 X k,x 1, α k1, e k1 ), θ k), 1, e k), e k1, α ) T η 1 de k1 )d1 Fθk, 4.4) for any x R d, θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n, and α = α,..., α n ) A F... An F. Le us denoe by J k θ k), e k) ) = J k X k, θ k), e k), α), k =,..., n, and observe by definiion of X x,α and X k in 3.3) ha J k saisfy he backward inducion formula: [ T J n θ, e) = f n X n, α n, θ, e)γ θ, e)d G n T X T n, θ, e)γ T θ, e) F θn θ n [ J k θ k), e k) T ) = f k X k, α k, θ k), e k) )γ k θ k), e k) )d G k T X T k, θ k), e k) )γt k θ k), e k) ) J k1 θ k), 1, e k) Fθk, e k1 )η 1 de k1 )d1. T 14 T

15 Therefore, from Proposiion 2.1, we have he equaliy: [ T fx x,α, α )d G T X x,α T ) Le us now define he value funcion processes: = J = J x, α). 4.5) V k x, θ k), e k) ) := ess sup α A G J k x, θ k), e k), α), 4.6) for k =,..., n, x R d, and θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n. Firs, observe ha his definiion for k = is consisen wih he definiion of he value funcion V of he sochasic conrol problem 3.6) from he relaion 4.5). Then, i remains o prove ha he value funcions V k defined in 4.6) saisfy he backward inducion formula in he asserion of he heorem. For k = n, and since J n x, θ, e, α) depends on α only hrough is las componen α n, he relaion 4.1) holds rue. Nex, from he backward inducion 4.4) for J k, and he definiion of V k1, we have for all α = α,..., α n ) A G : [ T J k x, θ k), e k), α) f k X k,x, α k, θ k), e k) )γ k θ k), e k) )d G k T X k,x T, θk), e k) )γt k θ k), e k) ) V k1 Γ k1 1 X k,x 1, α k1, e k1 ), θ k), 1, e k) ), e k1 T η 1 de k1 )d1 Fθk V k x, θ k), e k) ), 4.7) where V k is defined by he rhs of 4.2). By aking he supremum over α in he inequaliy 4.7), his shows ha V k V k. Conversely, fix x R d, θ =,..., θ n ) n [, T n, e = e 1,..., e n ) n, and le us prove ha V k x, θ k), e k) ) V k x, θ k), e k) ). Fix an arbirary α k A k F, and he associaed conrolled process Xk,x. By definiion of V k1, for any ω Ω, ε >, here exiss α ω,ε A G, which is an ε-opimal conrol for V k1., θ k), e k) ) a ω, Γ k1 1 X k,x 1, α k1, e k1 )). Recalling ha he se of admissible conrols is a separable meric space, one can use a measurable selecion resul see e.g. [24) o find α ε A G s.. α ε ω) = α ω,ε ω), d dp a.e., and so V k1 Γ k1 1 X k,x 1, α k 1, e k1 ), θ k), 1, e k), e k1 ) ε J k1 Γ k1 1 X k,x 1, α k 1, e k1 ), θ k), 1, e k), e k1, α ε), a.s. Denoe by α ε,,..., α ε,n ) he n 1-uple associaed o α ε A G, and le us consider he admissible conrol α ε = α ε,,..., α k, α ε,k1,..., α ε,n ) A G consising in subsiuing he k-h componen of α ε by α k A k F. Since J k1x, θ, e, α) depends on α only hrough is las 15

16 componens α k1,..., α n ), we have from 4.4) V k x, θ k), e k) ) J k x, θ k), e k), α ε ) [ T = f k X k,x, α k, θ k), e k) )γ k θ k), e k) )d G k T X k,x T, θk), e k) )γt k θ k), e k) ) J k1 Γ k1 1 X k,x 1, α k1, e k1 ), θ k), 1, e k), e k1, α ε) T [ T η 1 de k1 )d1 Fθk f k X k,x, α k, θ k), e k) )γ k θ k), e k) )d G k T X k,x T, θk), e k) )γt k θ k), e k) ) V k1 Γ k1 1 X k,x 1, α k1, e k1 ), θ k), 1, e k) ), e k1 T η 1 de k1 )d1 Fθk ε. From he arbirariness of α k A k F and ε >, we obain he required inequaliy: V kx, θ k), e k) ) V k x, θ k), e k) ), and he proof is complee. 5 The case of enlarged filraion wih muliple random imes In his secion, we consider he case where he random imes are no assumed o be ordered. In oher words, his means ha one has access o he defaul imes hemselves wih heir indexes, and no only o he ranked defaul imes. This general case can acually be derived from he case of successive random imes associaed wih suiable auxiliary marks. Le us consider he progressive enlargemen of filraion from F o G wih muliple random imes τ 1,..., τ n ) associaed wih he marks ζ 1,..., ζ n ). Denoe by ˆτ 1... ˆτ n he corresponding ranked imes, and by ι i he index mark valued in {1,..., n}) of he i-h order saisic of τ 1,..., τ n ) for i = 1,..., n, so ha ˆτ 1,..., ˆτ n ) = τ ι1,..., τ ιn ). Then, i is clear ha he progressive enlargemen of filraion of F wih he successive random imes ˆτ 1,..., ˆτ n ) ogeher wih he marks ι 1, ζ ι1,..., ι n, ζ ιn ) leads o he filraion G, so ha one can apply he resuls of he previous secions. For simpliciy of noaions, we shall focus on he case of wo random imes τ 1 and τ 2, associaed o he marks ζ 1 and ζ 2 valued in Borel space of R m. The decomposiion of opional and predicable process wih respec o his progressive enlargemen of filraion is given by he following lemma, which is derived from Lemma 2.1, wih he specific feaure ha we have also o ake ino accoun he index of he order saisic in τ 1, τ 2 ). Lemma 5.1 Any G-opional resp. predicable) process Y = Y ) is represened as Y = Y 1 <ˆτ1 Y 1,1 τ 1, ζ 1 )1 τ1 <τ 2 Y 1,2 τ 2, ζ 2 )1 τ2 <τ 1 Y 2 τ 1, τ 2, ζ 1, ζ 2 )1 ˆτ2, resp. = Y 1 ˆτ1 Y 1,1 τ 1, ζ 1 )1 τ1 < τ 2 Y 1,2 τ 2, ζ 2 )1 τ2 < τ 1 Y 2 τ 1, τ 2, ζ 1, ζ 2 )1 >ˆτ2 ), 16

17 for all, where Y O F resp. P F ), Y 1,1, Y 1,2 O 1 F R, ) resp. P 1 F R, )), and Y 2 O 2 F R2, 2 ) resp. P 2 F R2, 2 )). Any Y O G resp. P G ) can hen be idenified wih a quadruple Y, Y 1,1, Y 1,2, Y 2 ) O F O 1 F R, ) O 1 F R, ) O 2 F R2, 2 ) resp. P F P 1 F R, ) P 1 F R, ) P 2 F R2, 2 )). Similarly as in Secion 1, we now make a densiy hypohesis on he condiional disribuion of τ 1, τ 2, ζ 1, ζ 2 ) given he reference informaion. We assume ha here exiss a OF) BR 2 ) B 2 )-measurable map, ω,,, e 1, e 2 ) γ ω,,, e 1, e 2 ) such ha DH) P [ τ 1, τ 2, ζ 1, ζ 2 ) dθde F = γ,, e 1, e 2 )d d ηde 1 )ηde 2 ), a.s. where η is a nonnegaive measure on B). We nex inroduce some useful noaions. We denoe by γ he F-opional process defined by γ = P[τ 1 >, τ 2 > F = γ,, e 1, e 2 )d d ηde 1 )ηde 2 ), 2 [, ) 2 and we denoe by, ω,, e 1 ) γ 1,1, e 1 ), and, ω,, e 2 ) γ 1,2, e 2 ),, he OF) BR ) B)-measurable maps defined by γ 1,1, e 1 ) = γ,, e 1, e 2 )d ηde 2 ), γ 1,2, e 2 ) = γ,, e 1, e 2 )d ηde 1 ), so ha P[τ 2 > F = γ 1,1, e 1 )d ηde 1 ), P[τ 1 > F = γ 1,2, e 2 )d ηde 2 ). Hence, γ 1,1, e 1 ) is inerpreed as h probabiliy for {τ 2 > } condiioned on F, and {τ 1, ζ 1 ) =, e 1 )}, and a similar inerpreaion holds for γ 1,2. The nex resul, which is analog o Proposiion 2.1, provides a backward inducion formula involving F-expecaions for he compuaion of expecaion funcionals of G-opional processes. Proposiion 5.1 Le Y = Y, Y 1,1, Y 1,2, Y 2 ) and Z = Z, Z 1,1, Z 1,2, Z 2 ) be wo nonnegaive or bounded) G-opional processes, and fix T, ). The expecaion [ T Y d Z T can be compued in a backward inducion as [ T Y d Z T = J 17

18 where he J, J 1,1, J 1,2, J 2 ) are given by [ T J 2,, e 1, e 2 ) = γ,, e 1, e 2 )d ZT 2 γ T,, e 1, e 2 ) F θ1 Y 2 [ T J 1,1, e 1 ) = Y 1,1 γ 1,1, e 1 )d Z 1,1 T γ1,1 T, e 1 ) T J 2,, e 1, e 2 )d ηde 2 ) F θ1 [ T J 1,2, e 2 ) = Y 1,2 γ 1,2, e 2 )d Z 1,2 T γ1,2 T, e 2 ) T J 2,, e 1, e 2 )d ηde 2 ) F θ2 [ T J = Y γ d ZT γt T T J 1,1, e 1 )d ηde 1 ) J 1,2, e 2 )d ηde 2 ). Le us now formulae he general sochasic conrol problem in his framework. A conrol is a G-predicable process α = α, α 1,1, α 1,2, α 2 ) P F PF 1R, ) PF 1R, ) PF 2R2, 2 ), where α, α 1,1, α 1,2 and α 2 are valued respecively in A, A 1,1, A 1,2 and A 2, Borel ses of some uclidian space. We denoe by A = A A 1,1 A 1,2 A 2, and by A G he se of admissible conrol processes, which is a produc space A F A1,1 F A1,2 F A2 F, where A F, A1,1 F, A1,2 F and A 2 F are some separable meric spaces respecively in P FA ), P 1 F R, ; A 1,1 ), P 1 F R, ; A 1,2 ) and P 2 F R2, 2 ; A 2 ). We are nex given a collecion of measurable mappings: such ha we have he iniial daa x, α ) R d A F X,x,α O F x, α 1,1 ) R d A 1,1 F X 1,1,x,α1,1 OF 1 R, ) x, α 1,2 ) R d A 1,2 F X 1,2,x,α1,2 OF 1 R, ) x, α 2 ) R d A 2 F X 2,x,α2 OF 2 R2, 2 ), X,x,α = x, x R d, X 1,1,ξ,α1,1, e 1 ) = ξ, ξ F θ1 measurable, X 1,2,ξ,α1,2, e 2 ) = ξ, ξ F θ2 measurable, X 2,ξ,α2,, e 1, e 2 ) = ξ, ξ F θ1 measurable. We are also given a collecion of maps Γ 1,1, Γ 1,2, on R Ω R d A, Γ 2,1 on R Ω R d A 1,1 and Γ 2,2 on R Ω R d A 1,2 such ha, ω, x, a, e) Γ 1,1 ω, x, a, e), Γ 1,2 ω, x, a, e) are PF) BR d ) BA ) B) measurable, ω, x, a, e) Γ 2,1 ω, x, a, e) is PF) BR d ) BA 1,1 ) B) measurable, ω, x, a, e) Γ 2,2 ω, x, a, e) is PF) BR d ) BA 1,2 ) B) measurable 18

19 The conrolled sae process is hen given by he mapping x, α) R d A G X x,α O G, where for α = α, α 1,1, α 1,2, α 2 ), X x,α is he process equal o X x,α = X 1 <ˆτ1 X 1,1 τ 1, ζ 1 )1 τ1 <τ 2 X 1,2 τ 2, ζ 2 )1 τ2 <τ 1 X 2 τ 1, τ 2, ζ 1, ζ 2 )1 ˆτ2, wih X, X 1,1, X 1,2, X 2 ) O F O 1 F R, ) O 1 F R, ) O 2 F R2, 2 ) given by X = X,x,α X 1,1, e 1 ) = X 1,1,Γ1,1 X,α,e 1 ),α 1,1, e 1 ) X 1,2, e 2 ) = X 1,2,Γ1,2 θ X 2 θ,α 2 θ,e 2 ),α 1,2 2, e 2 ) { X 2,, e 1, e 2 ) = X 2,Γ2,2 1,1 θ X 2 θ,α 1,1 2 θ,e 2 ),α 2 2,, e 1, e 2 ) if X 2,Γ2,1 X 1,2,α 1,2,e 1 ),α 2,, e 1, e 2 ) if <. The inerpreaion is he following: X is he conrolled sae process before any defaul, X 1,1 resp. X 1,2 ) is he conrolled sae process beween τ 1 and τ 2 resp. beween τ 2 and τ 1 ) if he defaul of index 1 resp. index 2) occurs firs, and X 2 is he conrolled sae process afer boh defauls. Moreover, Γ 1,1 resp. Γ 1,2 ) represens he jump of X a τ 1 resp. τ 2 ) if he defaul of index 1 resp. index 2) occurs firs, and Γ 2,2 resp. Γ 2,1 ) represens he jump of X 1,1 resp. X 1,2 ) a τ 2 resp. τ 1 ) when he defaul of index 2 resp. index 1) occurs in second afer index 1 resp. index 2). For a fixed finie horizon T <, we are given a nonnegaive map G T on Ω R d such ha ω, x) G T ω, x) is G T BR d )-measurable, hus in he form G T x) = G T x)1 T <ˆτ1 G 1,1 T x, τ 1, ζ 1 )1 τ1 T <τ 2 G 1,2 T x, τ 2, ζ 2 )1 τ2 T <τ 1 G 2 T x, τ 1, τ 2, ζ 1, ζ 2 )1ˆτ2 T, where G T is F T BR d )-measurable, G 1,1 T, G1,2 T are F T BR d ) BR ) B)-measurable, and G 2 T is F T BR d ) BR 2 ) B 2 )-measurable. The running gain funcion is given by a nonnegaive map f on Ω R d A such ha, ω, x, a) f ω, x, a) is OG) BR d ) BA)- measurable, and which may be decomposed as f x, a) = f x, a )1 <ˆτ1 f 1,1 x, a 1,1, τ 1, ζ 1 )1 τ1 <τ 2 f 1,2 x, a 1,2, τ 2, ζ 2 )1 τ2 <τ 1 f 2 x, a 2, τ 1, τ 2, ζ 1, ζ 2 )1ˆτ2 T, for a = a, a 1,1, a 1,2, a 2 ) A = A A 1,1 A 1,2 A 2, where f is OF) BR d ) BA )- measurable, and f 1,1 is OF) BR d ) BA 1,1 ) BR ) B)-measurable, f 1,2 is OF) BR d ) BA 1,2 ) BR ) B)-measurable and f 2 is OF) BR d ) BA 2 ) BR 2 ) B 2 )- measurable. The value funcion for he sochasic conrol problem is hen defined by [ T V x) = sup α A G f X x,α, α )d G T X x,α T, ) x R d. 19

20 The main resul of his secion provides a decomposiion of he value funcion in he reference filraion, which is analog o he decomposiion in Theorem 4.1. To alleviae he noaions, we omi he dependence of he sae process in he conrols and in he parameers θ, e, when here is no ambiguiy. Theorem 5.1 The value funcion V is obained from he backward inducion formula [ T V 2 x,,, e 1, e 2 ) = ess sup f 2 X 2,x, α 2,,, e 1, e 2 )γ,, e 1, e 2 )d α 2 A 2 θ F 1 G 2 T X 2,x T, θ 1,, e 1, e 2 )γ T,, e 1, e 2 ) F θ1 [ T V 1,1 x,, e 1 ) = ess sup X 1,1,x, e 1 )d α 1,1 A 1,1 F T [ T V 1,2 x,, e 2 ) = ess sup α 1,2 A 1,2 F T [ T V x) = sup α A F for all, ) [, T 2, e 1, e 2 ) 2. f 1,1, α 1,1,, e 1 )γ 1,1 G 1,1 T X1,1,x T,, e 1 )γ 1,1 T, e 1 ) V 2 Γ 2,2 X 1,1,x, α 1,1 ) Fθ1, e 2 ),,, e 1, e 2 ηde2 )d f 1,2 X 1,2,x, α 1,2,, e 2 )γ 1,2, e 2 )d G 1,2 T X1,2,x T,, e 2 )γ 1,2 T, e 2 ) V 2 Γ 2,1 X 1,2,x, α 1,2 ) Fθ2, e 1 ),,, e 1, e 2 ηde1 )d f X,x, α )γ d G T X,x )γ T T T T V 1,1 Γ 1,1 X,x, α, e 1 ),, e 1 ) ηde1 )d V 1,2 Γ 1,2 X,x, α, e 2 ),, e 2 ) ηde2 )d, Remark 5.1 As menioned in Remark 4.1, he value funcions V 2, V 1,1 and V 1,2 correspond o sandard sochasic conrol problem in he F-filraion. This is also he case for V in he decomposiion formula of Theorem 5.1. Indeed, denoe by V 1 he map on Ω [, T R d A : V 1 x, θ, a ) = V 1,1 Γ 1,1 θ x, a, e), θ, e) V 1,2 Γ 1,2 x, a, e), θ, e) ηde). Then, V is compued from he sochasic conrol problem in he F-filraion wih he erminal gain funcion G T weighed by he F T -measurable random variable γt, and wih he running gain funcions f γ and V 1 : [ T V x) = sup G T X,x T )γ T f X,x α A F θ, α )γ V 1 X,x,, α )d. 2

21 6 Applicaions in mahemaical finance 6.1 Indifference pricing of defaulable claims We consider a sock subjec o a single counerpary defaul a a random ime τ, which induces a jump of random relaive size ζ valued in 1, ). The price process of he sock is described by where S is governed by S = S 1 <τ S 1 τ, ζ)1 τ, ds = S b d σ dw ), and he indexed process S 1 θ, e), θ, e) R is given by ds 1 θ, e) = S 1 θ, e) b 1 θ, e)d σ 1 θ, e)dw ), θ, S θ, e) = S e). Here W is a P, F)-Brownian moion, b, σ > are F-adaped processes, b 1, σ 1 > OF 1R, ). The marke informaion is represened by he progressive enlarged filraion G = F D, wih D = D ), D = ε> {σζ1 τ s, s ε) σ1 τ s, s ε)}. Denoing by b, σ he G-adaped processes: b = b 1 <τ b 1 τ, ζ)1 τ, σ = σ 1 <τ σ 1 τ, ζ)1 τ, and by µd, de) he random measure associaed o τ, ζ), we can wrie he dynamics of he sock price under G as: ds = S b d σ dw eµd, de) ), where W is a P, G)-semimaringale under he densiy hypohesis. By Girsanov s heorem and under suiable inegrabiliy condiions on he model coefficiens, one can find a probabiliy measure Q P such ha S is a Q, G)-local maringale, so ha his model is arbirage-free see he discussion in Remark 2.3 in [14 for more deails). An invesor can rade in a riskless bond wih zero ineres rae, and in he defaulable sock. Her rading sraegy is a G-predicable process α = α, α 1 ) P F PF 1R, ) represening he amoun raded in he sock. We allow consrains on rading sraegy by considering closed ses A and A 1 in which he conrols α and α 1 ake values. Noice also ha A and A 1 may differ. The conrolled wealh process of he invesor is hen given by X = X 1 <τ X 1 τ, ζ)1 τ, 6.1) where X is he wealh process before he defaul, and governed by dx = α ds S = α b d σ dw ), and X 1 θ, e) is he wealh indexed process afer-defaul, governed by dx 1 θ, e) = α 1 θ, e) ds1 θ, e) S 1 θ, e) = α 1 θ, e) b 1 θ, e)d σ 1 θ, e)dw ), θ X 1 θ θ, e) = X θ α θ e. 21

22 Le us now consider a defaulable coningen claim wih payoff a mauriy T given by H T = H T 1 T <τ H 1 T τ, ζ)1 τ T, where HT is a bounded F T -measurable random variable, and HT 1, ) is a bounded F T BR ) B)-measurable map. We use he popular indifference pricing crierion for valuing his defaulable claim. We are hen given an exponenial uiliy funcion U on R, i.e. Ux) = exp px), x R, for some p >, and we consider he opimal invesmen problem for an agen delivering he defaulable claim a mauriy T : V H x) = sup α A G [ UX x,α T H T ). 6.2) Here X x,α is he wealh process in 6.1) conrolled by he rading sraegy α, and saring from x a ime. We denoe by V he value funcion for he opimal invesmen problem wihou he defaulable claim, i.e. when H T = in 6.2), and he indifference price for H T is he amoun of iniial capial such ha he invesor is indifferen beween holding or no he defaulable claim. I is hen defined as he unique number π such ha V H x π) = V x). A similar problem wihou unpredicable mark ζ) was recenly considered in [15 and [1 by using a global G-filraion approach under H) hypohesis, see also [19. The paper [14 sudied an opimal invesmen problem wih power uiliy funcions under a single counerpary defaul by using a densiy approach for decomposing he problem in he F-filraion. We follow his mehodology and solve he sochasic conrol problem 6.2) by applying he F-decomposiion mehod. From Theorem 4.1, he value funcion V H is obained in wo seps via he resoluion of he afer-defaul problem [ V1 H x, θ, e) = ess sup U X 1,x T θ, e) H1 T θ, e) ) γ T θ, e) F θ, 6.3) α 1 A 1 F and hen via he resoluion of he before-defaul problem [ T V H x) = sup UX,x T HT )γt V1 H X,x θ αθ. e, θ, e)ηde)dθ 6.4) α A F Soluion o he afer-defaul problem. For fixed θ, e) [, T, problem 6.3) is a classical uiliy maximizaion problem wih random endowmen in he complee marke model afer defaul described by he indexed price process S 1 θ, e). Indeed, noice ha we can remove he posiive erm γ T θ, e) in 6.3) by defining he modified claim H 1 T θ, e) = H1 T θ, e) 1 p ln γ T θ, e) so ha V H 1 x, θ, e) = ess sup α 1 A 1 F [ U X 1,x T θ, e) H T 1 θ, e) ) Fθ. 6.5) 22

23 This problem was addressed by several mehods in he lieraure, and we know from dynamic programming and BSD mehods see [21 or [8)) ha V H 1 x, θ, e) = U x Y 1,H θ θ, e) ) where Y 1,H θ, e) is he unique bounded soluion o he BSD Y 1,H θ, e) = HT 1 θ, e) 1 T p ln γ T θ, e) T f 1 r, Zr 1,H, θ, e)dr Zr 1,H dw r and he generaor f 1 is he PF) BR) BR ) B)-measurable map defined by f 1, z, θ, e) = b1 θ, e) 1 b 1 σ 1 θ, e) ) 2 p z θ, e)z 2p σ 1θ, e) 2 inf 1 a A 1 p b 1 θ, e) ) σ 1θ, e) aσ 1 θ, e) 2. Global soluion The global soluion is finally obained from he resoluion of he before-defaul problem, which is hen reduced o [ V H x) = sup UX,x T HT )γt α A F T UX,x θ αθ e Y 1,H θ θ, e))ηde)dθ. From he addiive dependence of he wealh process X,x in funcion of x, and he exponenial form of he uiliy funcion U, we know ha he value funcion V H is in he form V H x) = Ux Y,H ), for some quaniy Y,H independen of x, and which may be characerized by dynamic programming mehods in he F-filraion. This can be achieved eiher via PD mehods in a Markovian seing, or via BSD mehods in he general case. The BSD associaed o Y,H is Y,H = HT 1 T p ln γ T f,h r, Y,H r, Zr,H T )dr Zr,H dw r, 6.6) where he generaor f,h is he OF) BR) BR)-measurable map defined by f,h, y, z) = b σ z 1 b ) 2 2p σ 6.7) p z 2 inf 1 b ) aσ 2 a A p p Uy) U ae Y 1,H, e) ) ηde) 2. σ The soluion o he opimal invesmen problem wihou defaulable claim is obained similarly as for he case wih claim, by considering H =. We hus have V x) = Ux Y ), where he BSD associaed o Y is given by Y = 1 T T p ln γ T f r, Yr, Zr )dr Zr dw r, 23

24 wih a generaor f as in 6.7) for H =, i.e. Y 1,H replaced by Y 1 soluion o he BSD Y 1 θ, e) = 1 T T p ln γ T θ, e) f 1 r, Zr 1, θ, e)dr Zr 1 dw r. Finally, he indifference price is given by π = Y,H Y. Remark 6.1 Noice ha he quadraic generaor f,h in 6.7) of he BSD 6.6) is no sandard due o he addiional erm arising from he inegral gain involving Y 1,H. However, one can prove exisence and uniqueness of his BSD and obain a verificaion heorem relaing he soluion of his BSD o he original value funcion by choosing a suiable se of admissible conrols A G = A F A1 F. The deails are provided in he companion paper [13. Acually, in his relaed paper, we consider a muli-dimensional exension of he above model wih asses subjec o successive counerpary defaul imes, and we apply he F- decomposiion mehod for solving he indifference pricing of defaulable claims, including credi derivaives such as CDOs. 6.2 Opimal invesmen under bilaeral counerpary risk We consider a porfolio wih wo names, each one subjec o an exernal counerpary defaul, bu also o he defaul of he oher one due o a conagion effec. We denoe by S 1 and S 2 he value process of hese wo names, by τ 1 and τ 2 heir defaul imes, no necessarily ordered, and by ˆτ 1 = minτ 1, τ 2 ), ˆτ 2 = maxτ 1, τ 2 ). Once he name i defauls a random ime τ i, meaning ha he value of S i drops o zero, i also incurs a jump drop or gain) on he oher value process S j, i, j {1, 2}, i j. The reference filraion F is he filraion generaed by a wo-dimensional Brownian moion W = W 1, W 2 ), driving he evoluion of he names in absence of defauls, and he global marke informaion is represened by G = F D 1 D 2, wih D i = D i ), D i = ε> σ1 τi s, s ε), i = 1, 2. The G-adaped value processes S i of names i = 1, 2, are given by S i = S i, 1 <ˆτ1 S i,j τ j )1 τj <τ i,, i, j = 1, 2, i j, where S = S 1,, S 2, ) is he vecor price process of he wo names in absence of any defaul, governed by ds = diags ) b d σ dw ), b = b 1,, b 2, ) is F-adaped, σ is he 2 2-diagonal F-adaped marix wih diagonal diffusion coefficiens σ 1, >, σ 2, >, and he indexed process S i,j θ j ), θ j R, represening he price process of name i afer he defaul of name j a ime θ j, is given by ds i,j θ j ) = ds i,j θ j ) b i,j θ j )d σ i,j θ j )dw i ), θj, S i,j θ j θ j ) = S i, θ j.1 e i,j ), 24

25 where e i,j represens he proporional jump induced by he defaul of name j on name i, and assumed consan for simpliciy and valued in 1, ). The coefficiens b i,, σ i, > are F-adaped processes, and b i,j, σ i,j > are in O 1 F R ). As in he model of Secion 6.1., each asse price process is a P, G)-semimaringale wih nondegenerae diffusion erm as long as i can be raded, and so he wo-asses model is arbirage-free. The rading sraegy of he invesor is a G-predicable measurable process α represening he fracion of wealh invesed in he wo names. I is hen decomposed in four componens: he firs componen α is a pair of F-predicable processes represening he fracion invesed in he wo names before any defaul, he second componen α 1,1 is an indexed F-predicable process represening he fracion invesed in he name 2 when he name 1 defauls, he hird componen α 1,2 is an indexed F-predicable process represening he fracion invesed in he name 1 when he name 2 defauls, and he fourh componen is zero when boh names defaul. The wealh process of he invesor is hen given by X = X 1 <ˆτ1 X 1,1 τ 1 )1 τ1 <τ 2 X 1,2 τ 2 )1 τ2 <τ 1 X 2 τ 1, τ 2 )1 ˆτ2, where X is he wealh process before any defaul, governed by dx = X α ) diags ) 1 ds = X α.b d α ) σ ) dw, X 1,1 ) is he wealh indexed process afer defaul of name 1, governed by dx 1,1 ) = X 1,1 )α 1,1 ) ds2,1 ) X 1,1 ) = X.1 α. 1, e 2,1 )), S 2,1 ), X 1,2 ) is he wealh indexed process afer defaul of name 2, governed by dx 1,2 ) = X 1,2 )α 1,2 ) ds1,2 ) X 1,2 ) = X. 1 α.e 1,2, 1) ), S 1,2 ), and X 2, ) is he wealh indexed process afer boh defauls, hence consan afer, and hen given by { X 2 X 1,1, ) = ). 1 α 1,1 ) ), X 1,2 ). 1 α 1,2 ) ), < In order o ensure ha he wealh process is sricly posiive, we assume ha α is valued in a closed subse A {a R 2 : 1 a. 1, e 2,1 ) >, and 1 a.e 1,2, 1) > }, and α 1,1, α 1,2 are valued respecively in closed subses A 1,1, A 1,2, 1). We are nex given a uiliy funcion U on R, over a finie horizon T, and we consider he opimal invesmen problem V x) = sup α A G [ UX x,α T ). 6.8) 25