Academic Editor: Mogens Steffensen Received: 24 September 2017; Accepted: 16 October 2017; Published: 23 October 2017

Transcription

1 risks Aricle Opional Defaulable Markes Mohamed N. Abdelghani 1, *, and Alexander V. Melnikov 2, 1 Machine Learning, Morgan Sanley, New York Ciy, NY 119, USA 2 Mahemaical and Saisical Sciences, Universiy of Albera, Edmonon, AB 6G 2R3, Canada; melnikov@ualbera.ca * Correspondence: mnabdelghani@gmail.com he research is suppored by he NSERC discovery gran 591. Academic Edior: Mogens Seffensen Received: 24 Sepember 217; Acceped: 16 Ocober 217; Published: 23 Ocober 217 Absrac: he paper deals wih defaulable markes, one of he main research areas of mahemaical finance. I proposes a new approach o he heory of such markes using echniques from he calculus of opional sochasic processes on unusual probabiliy spaces, which was no presened before. he paper is a foundaion paper and conains a number of fundamenal resuls on modeling of defaulable markes, pricing and hedging of defaulable claims and resuls on he probabiliy of defaul under such condiions. Moreover, several imporan examples are presened: a new pricing formula for a defaulable bond and a new pricing formula for credi defaul swap. Furhermore, some resuls on he absence of arbirage for markes on unusual probabiliy spaces and markes wih defaul are also provided. Keywords: defaulable claims; hazard process; maringale deflaors; opional processes; hedging 1. Inroducion Under he usual condiions, a probabiliy space Ω, F, P is equipped wih a non-decreasing family of σ-algebras F F such ha F is complee and righ coninuous for all. A wide class of processes adaped o F are righ coninuous wih lef limis pahs RCLL semimaringales wih a well defined sochasic calculus. Many fundamenal resuls in probabiliy heory and mahemaical finance were proved wih he use of he usual condiions and RCLL semimaringales. However, i is no difficul o give examples showing he exisence of a sochasic basis wihou he usual condiions see, for insance, Fleming and Harringon 211, p. 24. herefore, in he middle of 197, he famous exper of sochasic analysis Dellacherie 1975 iniiaed sudies of sochasic processes wihou his convenien echnical assumpion. Dellacherie called his case he unusual nonsandard condiions. Furher developmens were done by many mahemaicians: Horowiz 1978; Lenglar 198; Lepingle 1977; Merens 1972 bu mosly by Galchuk 198, 1982, In hese publicaions, a heory of sochasic analysis was consruced for processes on un usual probabiliy spaces. I urns ou ha many processes on unusual spaces are no necessarily lef or righ coninuous bu have lef and righ limis. For a review of opional calculus and is applicaions o finance, see Abdelghani 216; Abdelghani and Melnikov 216, 217a, 217b. A shor summary is given in he Appendix A. Defaulable markes are markes wih he possibiliy of defaul evens occurring. Research on defaul risk is concerned wih modeling of defaul ime, defaulable claims and recovery rules and wih pricing and hedging sraegies. he wo main approaches o modelling defaul are he srucural and he reduced form models. In srucural models, he value of he firm deermines if a defaul even occur. his approach was founded by Black and Scholes 1973 and Meron 1974, 1976 and exended by many see Cox and Ross 1976; Geske 1977, 1979; Geske and Johnson 1984; Leland 1994; Risks 217, 5, 56; doi:1.339/risks5456

2 Risks 217, 5, 56 2 of 21 Leland and of 1996; Longsaff and Schwarz he advanage of srucural models is ha one sees how he corporae condiions affec defaul. However, mos ofen he firm s value is no in iself a radable asse, bu he resul of an accouning of all of he firm s asses and liabiliies where some are visible o he marke and ohers are no. Hence, he parameers of srucural models are difficul o esimae in pracice. Moreover, in srucural models where a boundary condiion is used o demarcae he onse of defaul, i is difficul o define wha he boundary condiion should be. In conras o srucural models, in reduced form models, he firm s value plays only an auxiliary role and defauls arrive as a oal-surprise o all counerparies. herefore, defaul is he resul of exogenous facors. In his case, he random ime of defaul is defined as a oally inaccessible sopping ime on an enlarged filraion ha encompass all marke informaion, asses and defauls. o compue pricing rules and hedging porfolios, a fundamenal problem in a reduced form approach is he compuaion of he condiional probabiliy of defaul given available marke informaion. o compue he condiional probabiliy of defaul, several approaches arose depending on wheher or no he informaion abou defaul-free marke is available. If defaul-free marke informaion is no given, hen one mus suppose he exisence of an inensiy process associaed wih he probabiliy of defaul o be able o price a defaulable claim Duffie and Singleon his approach is known as he inensiy based approach. he main problem wih his approach is ha he pricing rule is hard o compue Jeanblanc and Yann 28. On he oher hand, in he hazard process approach, he reference filraion of he defaul-free marke is enlarged by he progressive knowledge of defaul evens. I is more convenien o derive probabiliy of defaul in his framework; however, one needs o assume knowledge of he defaul-free marke informaion. Much of reduced form pricing heory has been elaboraed by many mahemaicians see Arzner and Delbaen 1995; Duffie and Singleon 1997, 1999; Belanger e al. 21; Ellio e al. 2; Lando 1998 and Bielecki and Rukowski 22, ElKaroui e al. 21, Jeanblanc 21, for a good review. he basic mahemaics of srucural and reduced form models is given here as a review and, o compare hese models o models, we will presen in his paper. Le us consider a usual probabiliy space Ω, G, G = G, P, where he marke evolves: asse, X, and defaul, τ. Associaed wih defaul ime is he defaul process of H = 1 τ. Le F be he defaul-free σ-algebra generaed by X and H be he σ-algebra generaed by H wih F = F and H = H. Boh F and H are sub-σ-algebras of G for all. In srucural form models, he process X is a deerminan facor of τ. In oher words, he filraions F and H may have common elemens for any ime. In addiion, in srucural models where defaul ime τ is predicable, H F. On he oher hand, in reduced form models, defaul is no measurable in he reference-filraion F, i.e., τ / F for any ime. A way o analyze reduced form models is o consider τ o be an inaccessible sopping ime wih respec o he marke filraion G. Wih his consideraion of τ, one can proceed o compue he condiional probabiliy of defaul P τ G given G using inensiy-based echniques. If one is able o define F, he defaul-free filraion, hen a new filraion can be consruced by joining he filraion H = H generaed by H = 1 τ o F, i.e., G = σ F H G and G = G, hen compue P τ G using hazard process based echniques. However, his enlargemen of he filraion F by H leads o several problems in rying o esablish a pricing heory based on no-arbirage principle. Enlargemen of filraion changes he properies of maringales and semimaringale. For example, if X is a maringale under F, i migh no be a maringale under G or G. o deal wih hese effecs and o esablish exisence of local maringale measures for pricing, one has o invoke wo invariance principles known as he H and H hypoheses. H hypohesis saes ha every local maringale in he smaller filraion F is a local maringale in he larger filraion G, whereas he H hypohesis saes ha a semimaringale under he smaller filraion F remains a semimaringale under he larger filraion G. While here are sill many open and ineresing problems o consider in he classical approaches o credi risk, we choose o ake anoher approach o he problem. Our approach is based on he calculus of processes on unusual probabiliy spaces. Even hough we are going o use a nonsandard calculus o approach he problem of defaul, i is possible o view our models as exension or enrichmen

3 Risks 217, 5, 56 3 of 21 of reduced form models by mehods from he calculus of processes on unusual probabiliy spaces. hus, our goal wih his paper is o develop iniial resuls and presen applicaions of he calculus of opional processes o defaulable markes. However, before we delve ino our approach, le us moivae i by few examples where we ake a closer look a he mechanics of defaul and how i affecs he value of an asse X. Le us place ourselves on he usual probabiliy space Ω, F, F = F, P, where he marke evolves. In addiion, suppose he asse, X, remains in exisence afer defaul while is value changes. Now, le us fix an insance of ime ; if defaul ime is predicable or a sopping ime in F, hen τ F informaion abou defaul is incorporaed in F. On he oher hand, if τ / F, so ha H = 1 τ is no F measurable, hen he defaul ime τ is a random ime ha is a resul of exernal facors. However, afer defaul akes place, he surprising informaion abou i ges incorporaed in fuure-values of he asse, X; if X is RCLL and F = F +, hen obviously τ / F + and, loosely speaking, we may wan o say τ F s for any s > such ha F s F + o mean ha he informaion abou defaul ime τ a is par of a fuure-filraion of he marke. In reduced form modeling, o be precise, an enlarged filraion F H is consruced, such ha he view ha informaion abou defaul is incorporaed in fuure values of he asse is subsumed in he definiion of he enlarged marke filraion F H. However, we are going o show in his paper ha, wih he use of he calculus of processes on unusual probabiliy spaces, one needs no consider a filraion enlargemen o incorporae defaul informaion in he dynamic changes of he value of he asse. Le us consider defaul from a differen viewpoin. Suppose ha he even A F such ha PA >. Ideally, F conains all evens including defauls, for example, for he defaul ime τ, σ τ F. he expeced value a = E 1 A F a some fixed ime,, is implicily affeced by a fuure defaul even τ. For example, consider for some fixed ime u such ha 2 < u < and τ = u where i is possible ha A τ = u =. However, a ime, F conains absoluely no informaion abou he defaul even τ = u, bu he naural filraion F a generaed by he process, a, is no defaul-free. Essenially, in a defaulable marke processes ha are affeced by defaul a some poin over heir lifeime, heir curren value is implicily affeced by defaul even-hough curren informaion does no announce he exisence of fuure defaul evens. herefore, he consrucion of a defaulable marke filraion G from a defaul-free filraion F and a filraion associaed wih he defaul process H is hard o realize in real defaulable markes. his issue can be circumvened by considering defaulable markes on unusual probabiliy spaces. Le us consider anoher example. Suppose he asse X ha evolves according o X 1 dx = µd + σdw, where W is a Wiener process and µ and σ are unknown consans o be esimaed from marke-raded values of X. In his marke, even hough defaul evens migh be fuure evens beyond he curren marke ime are implici in he psyche of he marke, e.g., corporae bonds are riskier han US/Canadian Governmen bonds, and can weakly influence he curren raded values of X. As a resul, he esimaed values of µ and σ from he raded values of X are inadverenly conaminaed by defaul evens. herefore, in defaulable markes, asses and defauls are siched ogeher and are difficul o segregae. o avoid difficulies of previous approaches, he reconsrucion of enlarged marke filraions, he requiring of invariance properies o be saisfied as a way o ge decomposiion resuls, he iniial compleion of filraion by all null ses from he final filraion, and he requiremen ha he filraion be righ coninuous. Wihou all of hese consrucs, we propose a differen and more naural approach based on he sochasic calculus of opional processes on unusual probabiliy spaces. he paper is organized as follows: Secion 2 presens he essence of our approach o defaulable markes on unusual sochasic basis. Secion 3 deals wih defaulable claims and cash-flows. Secion 4 is devoed o he condiional probabiliy of defaul. In Secion 5, we consider valuaion of defaulable coningen claims and illusraive examples. Finally, in Secion 6, we discuss he imporan opic of no-arbirage condiion in markes on unusual spaces.

4 Risks 217, 5, 56 4 of Defaulable Markes on Unusual Spaces o illusrae our approach, we begin wih an example of a simple marke ha experiences defaul. Even hough simple, he ideas we presen here are general and can be carried over o oher more delicae marke srucures. Le us consider now he unusual sochasic basis Ω, F, F = F, P and an asse whose value is given by he process Y on his space. Assume ha he asse final value, η, realized a he end of ime is a sricly posiive random variable measurable wih respec o F. Le F = F represen he hisory of he marke, he good, he bad and defauls. However bad, we assume ha he asse never vanquishes, in oher words, Y > for all ime. All is well wih he marke excep for a single defaul even τ F +, i.e., F + = F +, ha happens a some ime. Noice ha we have defined defaul ime, τ F +, as an inaccessible sopping ime in he broad sense. Furhermore, we assume ha he value of he asse before defaul τ follows X and immediaely afer defaul evolves according o x. Boh x and X are opional semimaringales adaped o F; however, X is observable only up o defaul and x is only observable afer defaul. In addiion, we define he defaul process H = 1 τ< as a lef coninuous opional finie variaion process wih respec o F. hen, Y is given by Y = 1 H X + H x. 1 Noe ha he filraion generaed by Y is he fusion by defaul of he filraions generaed by X and x, which is F Y = σ σ X τ σ x τ σh. In he inegral represenaion of Y, we have, Y = X H s d X s x s + X s x s dh s + + H sd X s x s + H s d X s+ x s+ X s x s dh s+ + H s d X s+ x s+. See Appendix A Equaions A1 and A2 for he definiion of sochasic inegrals wih respec o opional semimaringales on unusual probabiliy space. However, since H has a single lef jump bu is oherwise coninuous, hen H = and Y reduces o Y = X H s d X s x s + + X s x s H s ds + + H s d X s+ x s+ X s x s H s+ ds + X τ+ x τ+ H. o highligh he effecs of defaul τ on he value of Y, which will manifes iself as a lef jump on he value of Y a he ime of defaul, we are going o assume ha x and X are coninuous; hence, Y = X H X x + X τ+ x τ+ H + X s x s ds. τ o summarize, he basic enans o our approach are: 1 he marke evolves in he unusual probabiliy space Ω, F, F = F, P ; 2 F + is he smalles righ coninuous enlargemen of F; 3 he value of an asse or a defaulable cash-flow is an F opional semimaringale; and 4 finally, bu mos imporanly, is ha defaul ime, τ, is defined as τ is a oally inaccessible sopping ime in he broad sense, i.e., τ F +, and he associaed defaul process is H = 1 τ<, which is an opional lef coninuous process adaped o F. he definiion of defaul ime as an inaccessible sopping ime in he broad sense makes defaul evens come as a surprise o he marke given up o ime informaion, F. However, as he marke evolves o F +, he surprising informaion abou defaul has been incorporaed in F +. In his way, F + in unusual probabiliy space models of defaulable marke akes he place of filraion enlargemen in reduced form models. Nex, we consider defaulable cash-flow pricing and hedging in he conex of unusual sochasic basis.

5 Risks 217, 5, 56 5 of Defaulable Cash-Flow Again, le he marke evolve in he unusual probabiliy space Ω, F, F = F, P. P is he real-world probabiliy, as opposed o he local opional maringale measure Q, for which we assume ha a leas one exiss in his marke or choose one if many exiss Jacod and Proer 21. In addiion, we have F + he smalles righ coninuous enlargemen of F. he filraion F suppors he following objecs. he claim, Λ, is he payoff received by he owner of he claim a mauriy ime,, if here was no defaul prior o or a. he process, A, wih A =, is he promised dividends if here was no defaul prior o or a. he recovery claim ρ represens he recovery payoff received a, if defaul occurs prior o he claims mauriy dae. he recovery process, R, which specifies he recovery payoff a ime of defaul, if i occurs prior o. Finally, defaul ime, τ, is a oally inaccessible sopping ime in he broad sense, i.e., τ F +. In addiion, we define he associaed defaul process by, H = 1 τ<, which is opional and a lef coninuous process wih respec o F. Furhermore, we assume ha he processes X, R, and A are progressively measurable wih respec o he filraion F. In addiion, he random variable Λ is F -measurable and ρ is a leas F + -measurable. Furhermore, we assume wihou menioning ha all random objecs inroduced above are a leas RLL and saisfy suiable inegrabiliy condiions ha are needed for evaluaing inegrals, sochasic or oherwise. his brings us o he recovery rules; if defaul occurs afer ime, hen he promised claim Λ is paid in full a ime. Oherwise, defaul ime τ and depending on he agreed upon recovery rules eiher he amoun R τ is paid a he ime of defaul τ, or he amoun ρ is paid a he mauriy dae. herefore, in is more general seing, we consider simulaneously boh kinds of recovery payoff and hus define a defaulable claim formally as a quinuple, DC = A, Λ, ρ, R, H. Noice ha he dae, he informaion srucures F and F + and he real-world probabiliy P are inrinsic componens of he definiion of a defaulable claim see Bielecki and Rukowski 22 for he usual case. For DC = A, Λ, ρ, R, H, he dividend process is defined as Definiion 1. he dividend process, D, of a defaulable claim DC = A, Λ, ρ, R, H equals D = X1 + 1 H A + R H, 2 where X = Λ 1 H + ρh. he process D is opional and F-measurable. Lemma 1. he process D is finie variaion over finie ime segmens including,. Proof. By using Galchouk Io lemma and properies of he componens of D. he risk neural value of a defaulable claim in his marke is he discouned value of he dividend process D. Definiion 2. he ex-dividend price process X, of a defaulable claim DC = A, Λ, ρ, R, H, which seles a ime is given by X, = B E Q D D F = B E Q u dd u + Bu 1 dd u+ F,,, + B 1 where B is he discouning process, an opional sricly posiive RLL semimaringale. Under Q, he ex-dividend process, B 1 X,, is a local opional maringale, X, M loc F, Q, i.e., Bs 1 Xs, = E Q X, F s for all s. Expression 3 is referred o as he risk-neural valuaion formula of a defaulable claim see Musiela and Rukowski 1997; Duffie and Singleon 1999; Ellio and Kopp 1999 for is definiion in usual 3

6 Risks 217, 5, 56 6 of 21 probabiliy spaces. For breviy, wrie X = X, Combine Definiions 1 and 2 and he fac ha H is lef coninuous, we obain X = B E Q B 1 X + + B 1 u 1 H u da u + Bu 1 1 H u da u+ + Bu 1 R u dh u+ F. Remark 1 RCLL Cash-Flow. We have considered a general model of defaulable cash-flow, namely ha he componens of he cash-flow are opional RLL processes affeced by defaul. However, we should no hink ha we are limied by his generalizaion of defaulable cash-flow. We can consider ha he underlying componens of defaulable cash-flow be RCLL processes affeced by defaul, which induces a lef coninuous opional jump on he resulan defaulable cash-flow, and he heory of defaulable markes on opional space remains he same. From now on, we are going o work in he deflaed probabiliy space Ω, F, F = F, Q and le he expecaion operaor E o mean E Q expeced values under Q. Nex, we provide a jusificaion of definiion Porfolio wih Defaul Consider a porfolio of hree primary securiies S 2 = B, he value process of a money marke accoun, S 1 a defaul free non-dividend-paying asses, and S a dividend paying asse, D wih D =, wih he possibiliy of defaul. Inroduce he discouned price processes S i by seing S i = Si /S2. he marke lifespan is he ime inerval, and φ = φ, φ 1, φ 2 is an F-opional self-financing rading sraegy on S, S 1, S 2 see Abdelghani and Melnikov 216 for he definiion of opional self-financing rading sraegies. I is sraighforward o generalize he hree-asse porfolio o any number of asses. o begin wih, le us examine a simple rading sraegy: suppose ha a ime = we purchase one uni of he h asse a he iniial price S and holds i unil ime ; hen, we inves all he proceeds from dividends in he money marke accoun. More specifically, we consider a buy-and-hold sraegy φ = 1,, φ 2. he associaed wealh process U equals wih iniial wealh U = S + φ2 B. Since φ is self-financing, hen U = S + φ 2 B,, 4 U U = S S + φ2 B + D 5 and B φ 2 + φ 2, B =, where D is he dividend paid by S. Now, le us divide U S by B, and use he produc rule, B 1 U S = = + U S U S + φ2 B + D + = φ 2 B + D + φ 2 B + U S +, U S φ 2 B +, φ 2 B, φ 2 B where we have used Equaions 4 and 5. However, since B + B = B 2 B, B,, φ 2 B = = φ 2 hen, we are able o find he simple relaion, φ 2 B + B φ 2 + φ 2, B B 2 B + B 3 B, B, B = φ 2,, B = B 2 φ 2 B, B. B 1 U S = φ 2 B 1 B + φ 2 B φ 2 B 2 B, B + D = D.

7 Risks 217, 5, 56 7 of 21 Considering he difference beween he value of a defaulable porfolio a ime, i.e., U, and is value a mauriy, U, we arrive a he following difference equaion: B 1 U S U U = B 1 U S = D D, S B 1 S + D D. Now, we are ready o derive he risk-neural valuaion formula for he ex-dividend price S. o his end, we have assumed ha our model admis a local opional maringale measure Q equivalen o P such ha he discouned wealh process U φ of any admissible self-financing rading sraegy φ follow local opional maringales under Q wih respec o he filraion F. Moreover, we make an assumpion ha he marke value a ime of he h securiy comes exclusively from he fuure dividends sream. his means ha we have o posulae ha S =. his posulae makes sense because he value of a defaulable cash-flow a mauriy, which has paid all is value in dividends, is essenially nohing. We shall refer o S as he ex-dividend price of he h asse he defaulable claim. Given ha E U U F =, E S F = S, S =, and Equaion 6, we arrive a he definiion of he value of defaulable claim, E U U F = E S = E B 1 S B 1 S + D D F = D D F. Hence, B 1 S is an F local opional maringale under Q and B 1 S = E D D F. 7 Le us now examine rading wih a general self-financing rading sraegy φ = φ, φ 1, φ 2. he associaed wealh process is U φ = 2 i= φi Si. Since φ is self-financing, hen i mus be ha U φ = U φ + G φ for every, where he gains process Gφ is G φ := φ D + 2 i= φ i S i = φ S + D + φ 1 S 1 + φ 2 S 2. As before, S 2 = B, he value of our money marke accoun, S = X, is he dividend, D, paying asse and S 1 is he defaul-free, non-dividend paying insrumen. he erm φ S + D of he gain process is he gain acquired as a resul of rading φ of he h asse having curren value S and paid dividend D. heorem 1. For any self-financing rading sraegy, φ, and S i = S i, i =, 1, 2 are F, Q local opional maringales and he discouned wealh process B 1 U φ follows a local opional maringale under F, Q. 6

8 Risks 217, 5, 56 8 of 21 Proof. Given S i = S i = S i + S i +, S i, hen, by he produc rule we ge, U φ = B 1 U φ + U φ +, Uφ = φ D + = φ D + = φ D + = φ D + 2 i= 2 i= 2 i= 2 i= φ i S i + φ i S i + 2 i= 2 i= φ i S i + φ i S i φ i S i + φ i S i φ i S i = φ Ŝ i= + φ i 2 i=1 φ i S i,, 2 i= φ i S i φ i, S i, S i where he process Ŝ is given by Ŝ := S + B 1 D. he process Ŝ is he discouned cumulaive dividend price a ime of he defaulable dividend D. he processes S i, i =, 1, 2 are local opional maringales in F, Q. o finalize his proof, observe ha, using Equaion 7, he process Ŝ saisfies Ŝ = E D F, and hus follows a local maringale under Q. Hence, B 1 U φ is a local opional maringale in F, Q. 4. Probabiliy of Defaul he essenial componen of he defaulable cash-flows, D, Equaion 2 is he defaul process H u = 1 τ<u, where u, is a ime-horizon. he ex-dividend price process Equaion 3 relies implicily on he condiional expeced value of 1 τ<u given all known up o ime marke informaion F. hus, our goal for his secion is o undersand he condiional probabiliy of defaul F, u = E H u F, also known as he hazard process and is properies. Associaed wih he hazard process F is he survival process G, u := 1 F, u, for which we will discuss some of is properies oo. his secion is exposiory, where we will simply lis a few ineresing relaed resuls abou F and G. We begin by he lemma. Lemma 2. he process H is a submaringale. Proof. Esablished by condiioning on F s for any s. Lemma 3. F and G are posiive. Furhermore, for a fixed ime horizon, boh are opional maringales. However, for a fixed ime and variable ime horizon, F is a submaringale and G a supermaringale. Proof. F and G are posiive, and one can wrie F, u F, u = E H u F = Q τ < u F = 1 u> E H + 1 τ<u F + 1 u H u = 1 u> H + E 1 H H u F + 1 u H u. For a fixed ime horizon u and any s < u, F, u = E E H u F F s = E H u F s = Fs, u is a posiive opional maringale. In addiion, for u s, F, u = Fs, u = H u a consan. However, for s u, F, u = E E H u F F s = E H u F s = Fs, u, again, an opional maringale. Essenially for any fixed u, Fs, u is an opional maringale. he same is rue for G. For a fixed,, and any ime horizons, v u, where u and v,. hen, τ < v τ < u and F, u = E H u F = E H v F + E 1 v τ<u F = F, v + E 1 v τ<u F F, v.

9 Risks 217, 5, 56 9 of 21 Hence, F, u is increasing along is ime horizon. Moreover, for a fixed and any v u, EF, u F v = EE H u F F v = 1 v EE H v + 1 v τ<u F = 1 v E F v + 1 v< EE H + 1 τ<u F F v H v + 1 v τ<u F + 1 v< EH + 1 τ<u F v 1 v F, v + 1 v< Fv,. herefore, if v, hen F, u is a submaringale; on he oher hand, if v <, hen we re-label v as and as v o ge EFv, u F Fv,, which is also a submaringale. Hence, F is a submaringale in eiher case. I follows ha G = 1 F is a supermaringale. On a relaed noe, Galchuk 1982 showed ha for he process ξ1 τ<, where ξ is F τ+ -measurable and inegrable random variable and τ is a sopping ime in he broad sense, i can be decomposed o an opional maringale ha is F measurable and an F measurable coninuous finie variaion process. herefore, Corollary 1. he defaul process H has he following decomposiion, H = 1 τ< = M + µ, where M MF, Q is an opional maringale and µ is he F-compensaor of H ha is an F-measurable coninuous finie variaion process. Proof. A consequence of heorem 3.5 in Galchuk Using he above corollary, le us consider evaluaing E H F ; for a ime horizon >, Corollary 2. F, = H + E µ µ F and Q τ < F = E µ F µ. Proof. F, = E 1 τ< F ; herefore, F, = E 1 τ< F = E M + µ F = M + E µ F = H + E µ µ F. Consequenly, one can wrie, Q τ < F = 1 H H F = E H H F = E H F H = E µ µ F = E µ F µ. Le us now undersand he relaion beween he process 1 τ, for a sopping ime in he broad sense τ, and he defaul process H = 1 τ< as regards he filraion F. Remark 2. Recall ha, under he usual condiions, 1 τ is known as he defaul process; however, in he unusual condiions, i is replaced wih he OF process H = 1 τ<. his choice of 1 τ< as he defaul process was moivaed by he fac ha he known marke informaion up o ime is F on which 1 τ< is adaped and opional while 1 τ is no. Furhermore, he forward derivaive + H = 1 τ= marks he occurrence of defaul, which is in OF +. Proposiion 1. Le τ be a sopping ime in he broad sense and H = H + = 1 τ, where τ = τ < +, is RCLL and F + measurable or OF +. In addiion, H = H or τ = τ <. H and H are of righ-limis and lef coninuous pahs and are OF. Moreover, τ = τ < τ = or ha H = H + 1 τ=.

10 Risks 217, 5, 56 1 of 21 Proof. For every ω τ, τ ω < + so τ τ < +. On he oher hand, τ ω < + = ɛ τ < + ɛ τ. hus, H + = 1 τ<+ = 1 τ or ha τ = τ < +. By definiion, 1 τ is RCLL and OF +. Furhermore, i can be easily seen ha he he process 1 τ can be decomposed as, 1 τ, = 1 τ, + 1 τ=. In addiion, i is obvious ha H = 1 τ = 1 τ< = H. Finally, for τ a sopping ime in he broad sense, by definiion, τ F + for any ime. Moreover, since τ 1/n F 1/n+ F for all n, hen i mus be ha τ < F. Le us sudy he properies of he process F, + = E H + F = Q τ F. Se F := E H + F and consider he following lemma. Lemma 4. he ex-hazard process F can be decomposed o he defaul process H and he jump hazard process δ. Boh F and δ are OF +, whereas H is OF. Proof. he opional projecion E H + F can be reduced o E H + F = E1 τ F = E1 τ< + 1 τ= F = H + E1 τ= = H + δ, where δ can be derived by formula, δ = lim h 1 h Q τ < + h F. Corollary 3. F, + = F, + E δ F = M + E µ + δ F. Proof. he process F, + = E H + F is evaluaed as F, + = EH F + E1 τ= F = H + E µ µ F + E δ F = F, + E δ F or F, + = H µ + E µ + δ F = M + E µ + δ F. Nex, we discuss he absence of arbirage in markes on unusual spaces. 5. Valuaion of Defaulable Cash-Flow and Examples Our nex goal is o esablish a convenien represenaion of he value of a defaulable cash-flow. he ex-dividend value of defaulable cash-flow is B 1 X, = 1 τ E D D F, and, in erms of D = Λ 1 H + ρh H A + R H, is X = E B 1 + ρh F +E + B 1 u 1 H u da u + +E Bu 1 R u dh u+ F. u 1 H u da u+ F We begin valuaion of X, wih he value of E B 1 Λ 1 H F. Le λ = B 1 Λ and λ = E λ F be a maringale. λ can be hough of as he defaul-free coningen claim price a ime. Le G, u = E 1 H u F. G can be hough of as he survival process, whereas F, u = E H u F is he associaed defaul process on he inerval, u. Lemma 5. he value of Λ = E B 1 Λ 1 H F a ime is given by where λ u = E Λ = E Λ F 1 H + E Λ F u and G u+ = Gu, u+ = E 1 H u+ F u. λu + + λ u dgu+ F, 8

11 Risks 217, 5, of 21 Proof. Using he produc rule on λ 1 H, we find ha λ 1 H = λ 1 H + λ 1 H + G λ + λ, 1 H = λ 1 H + λ 1 H + G λ + λ, 1 H + λ u d 1 H u + + λ u d 1 H u+ + 1 H u dλ u + 1 H u dλ u λ u d 1 H u+ = λ 1 H + λ u d 1 H u + 1 H u dλ u + + λ u d 1 H u+. Since H is a lef coninuous finie variaion process, he quadraic variaion λ, H = + λ u + H u+ = + H u dλ u+ = + λ u dh u+. u< We have chosen he definiion λ, H = + λ H. hus, he condiional expeced value of λ 1 H is E λ 1 H F = E B 1 Λ 1 H F and E λ 1 H F = E λ 1 H + 1 H u dλ u + λ u + + λ u dg u+ F, = λ 1 H + E λ u d 1 H u+ + + λ u d 1 H u+ F where we have used he fac ha E G u dλ u F = since λ is a local maringale, G u+ = Gu, u+ and E λ 1 H F = λ 1 H. hus, we arrive a he resul E B 1 Λ 1 H F = E Λ F 1 H + E E Λ F u + + E Λ F u dg u+ F. Remark 3. We have replaced he inegral over 1 H wih an inegral over he process G as a resul of he following saemen: E α u dh u+ F where F u+ = Fu, u+ = E H u+ F u. = n E α i E H i+1 H i F i F i= n E n α i F i+1 F i F = E α i F i+1 F i i= i= Remark 4. How do we evaluae dg u+? Here is how: Gu, u+ = E 1 H u+ F u and dg u+ = + G u = Gu, u+ Gu, u. herefore, dg u+ = E 1 H u+ F u E 1 H u F u = E H u H u+ F u, where H u+ = 1 τ u = H u + 1 τ=u. Hence, dg u+ = E 1 τ=u F u = Qτ = u F u. Noe ha he even τ = u is F u+ measurable so dg u+ is he projecion of τ = u on F u. For he second erm, E B 1 ρh F : le ϱ = B 1 ρ hus ϱ = E ϱ F is a local maringale. In addiion, we have he following lemma. F,

12 Risks 217, 5, of 21 Lemma 6. he value ρ = E ϱ H F is given by where ϱ u = E ρ F u. ρ = ϱ H E ϱu + + ϱ u dgu+ F, Proof. Applying a similar argumen o he one used in he above lemma, we arrive a he resul ρ = E ϱ H F = E ϱ H + = ϱ H + E = ϱ H E ϱu + + ϱ u dfu+ F ϱu + + ϱ u dgu+ F, where E H u dϱ u F = since ϱ is a local maringale. H u dϱ u + ϱ u dh u + + ϱ u dh u+ F As for he value of he defaulable cash-flow recovery-sream, we have he following lemma. Lemma 7. he value of he defaulable cash-flow recovery sream is E Bu 1 R u dh u+ F = E Bu 1 R u dg u+ F. Proof. Using he sum approximaion of he lef opional inegral, he ex-dividend value of defaulable claim is derived as follows: E Bu 1 R u dh u+ F = E E Bu 1 R u dh u+ F u F = E Bu 1 R u df u+ F = E Bu 1 R u dg u+ F. Noe ha Bu 1 R u is F u measurable. Finally, we look a he process E 1 H A 1 H A F. Le  = A ; hen, E 1 H A 1 H A F = E 1 H  1 H  F = E 1 H u dâ u F. Now, le us apply he produc rule o 1 H  from,, 1 H  = 1 H  + Âd1 H u + 1 Hd u + +  u d1 H. Using he above equaion, we can wrie 1 Hd u = 1 H  1 H   u d1 H u +  u d1 H u+. Wihou loss of generaliy, we are going o assume ha A = because any dividend paymen made a ime is going o be par of he claim Λ paid a mauriy ime. Hence, he equaion above reduces o

13 Risks 217, 5, of 21 1 Hd u = 1 H   u d1 H u+ +  u d1 H u+. 9 We are going o use his equaion o prove he nex lemma. Lemma 8. he value of dividend payou sream is E 1 H A 1 H A F = 1 H A E A u + + A u dgu+ F. Proof. Using Equaion 9, he sum approximaion of he lef opional inegral and he fac ha 1 H A is F measurable, we ge E 1 H A 1 H A F where  = A. = E 1 Hd u F = E 1 H   u d1 H u+ F E +  u d1 H u+ F = 1 H  E Âu + +  u d1 Hu+ F = 1 H A E A u + + A u dgu+ F, Consequenly, we arrive a he following heorem for he ex-dividend price process. heorem 2. he ex-dividend price process X, ha seles a ime is X, = λ 1 H + E λu + + λ u dgu+ F +ϱ H E ϱu + + ϱ u dgu+ F 1 H  E Âu + +  u dgu+ F E ˆR u dg u+ F where λ = E Λ F = λ  1 H + ϱ H +E λu ϱ u  u ˆR u + + λ u ϱ u  u dgu+ F,, ϱ = E Proof. Follows from he lemmas above. ρ F,  = A and ˆR = R. In he special case where componens of he dividends process are RCLL processes, we ge he following corollary. Corollary 4. Suppose we are given a defaulable cash-flow A, and R and B are RCLL. hen, he ex-dividend price process X, ha seles a ime is X, = B λ  1 H + B ϱ H +B E λu ϱ u  u ˆR u + + λ u ϱ u dg u+ F.

14 Risks 217, 5, of 21 Furhermore, if λ = E + E Λ F B 1 Λ F and ϱ = E = and + E ρ F = for all and ρ F have RCLL modificaions, hen X, = B λ  1 H + B ϱ H +B E λu ϱ u  u ˆR u dgu+ F. Proof. Follows from he definiion of he ex-dividend price process. We were able o esablish a convenien represenaion of he value of a defaulable claim in erms of he probabiliy of defaul. Now, we provide some examples. Example 1 Zero-Coupon Defaulable Bond. he price of a zero-coupon bond wih face-value $1 a mauriy dae is B E B 1 F a ime. On he oher hand, he price of a zero-coupon bond ha may experience defaul is B E B 1 1 τ F. Lemma 5 ells us how o compue he price B E B 1 1 τ F a ime. Le λ = B 1 ; hus, λ = E λ F and X, = E λ 1 H F = λ 1 H + E λu + + λ u dgu+ F = λ 1 τ + E λu + + λ u dgu+ F. Suppose ha B = e r, where B, = B B 1 = e r, wih a consan ineres rae r and ha he survival process admis a consan inensiy γ such ha dg u+ = E H u H u+ F u = E 1 τ=u F u = δ u τ γe γu du, where δ u τ is Dirac dela funcion a a paricular value of defaul ime τ. hen, X, = 1 τ e r e r E δ u τ γe γu du F = e r 1 τ 1 τ< e γ 1 e γ. A =, X, = e r 1 1 τ< 1 e γ, and, a =, X, = 1 τ. If τ <, hen X, = e r+γ. 1 herefore, for a bond of face-value of $1 a mauriy dae, defaul decreases he presen value of he bond by a facor e γ. Hence, X, is a reasonable pricing formula. In addiion, noe ha, in his simple case, he price formula we presened here maches he price formula derived wih classical mehods in he usual case see he following references for comparison Duffie and Singleon 212, p. 12; Bingham and Kiesel 213, p. 396; Marellini e al. 23, p Example 2 Credi Defaul Swap. A credi defaul swap CDS is a conrac in which he holder of a defaulable asse buys an insurance agains defaul. he elemens of a CDS are: he mauriy ime, he fee rae funcion K and he recovery funcion R. he buyer of defaul proecion pays a fee a a rae K up o defaul

15 Risks 217, 5, of 21 ime τ or o mauriy ime and receives he amoun Rτ a defaul from he seller of he proecion. he price of he CDS a ime is given by he difference in price beween he value of he proecive leg and premium leg: where he proecion price is and he premium price is CDS, = Pro, Prem,, Pro, = E Rτ1 τ< F = E R u dh u+ F = E R u df u+ F, τ Prem, = E 1 τ Kudu F = E 1 H u K u du F. Suppose K and R are consans and defaul admis a consan inensiy γ such ha df u+ = E H u+ H u F u = E 1 τ=u F u = δ u τ γe γu du, a a paricular value of defaul ime τ. hen, he legs prices are Pro, = E R u df u+ F = R δ u τ γe γu du = 1 τ< Re γ 1 e γ, and Prem, = E 1 H u K u du F = KE 1 F u du F = KE δ u τ e γu du F = 1 τ< K γ e γ 1 e γ. herefore, CDS, = R γ 1 K 1 e γ e γ 1 τ<. A =, CDS, = 6. Absence of Arbirage on Unusual Spaces R γ 1 K 1 e γ 1 τ<. 11 he no-arbirage argumens culminaed in he fundamenal heorem of asse pricing: under he usual condiions, for a real valued semimaringale X, here exiss a probabiliy measure Q equivalen o P under which X is a σ-maringale if and only if X does no permi a free lunch wih vanishing risk. Given ha K = {φ X : φ admissible and φ X = lim φ X exiss a.s.} and C = {g L P : g f f K}, hen X is said o saisfy no free lunch wih vanishing risk NFLVR if C L +P = {}, where C is he closure of C in he norm opology of L +P Delbaen and Schachermayer 26. Q is known as he equivalen local maringale measure ELMM. However, in some models of financial markes, ELMM may fail o exis. An alernaive was developed, an equivalen local maringale deflaor ELMD, which is a sricly posiive local maringale ha ransforms he semimaringale X o a local maringale. I was shown ha he exisence of a sricly posiive local maringale deflaor is equivalen o he no arbirage condiion of he firs kind NA1. An F measurable random variable ζ

16 Risks 217, 5, of 21 is called an arbirage of he firs kind if P ζ = 1, P ζ > > and for any iniial wealh x > if here exiss an admissible φ such ha x + φ X ζ Kardaras 212. NA1 is a weaker condiion han NFLVR. While exending NFLVR and NA1 and he equivalence relaions ELMM and ELMD, respecively, o unusual sochasic basis is possible and is of imporance, i is definiely ou of scope here. However, here we will presen an argumen o show he viabiliy of financial markes and markes wih defauls on unusual probabiliy spaces. Again, consider a marke on unusual probabiliy space Ω, F, F = F, P. Inroduce he filraion F + = F + and is compleion under P, F = F+ P. In addiion, le Y OF be a real valued opional semimaringale ha is a leas of RLL pahs and Ỹ = Y + is O F. Ỹ is RCLL semimaringale and he sochasic basis Ω, F, F, P saisfies he usual condiions. Suppose Ỹ saisfied he condiions NFLVR wih φ P F admissible porfolio and φ Y exis a.s.; hen, here is ELMM Q P such ha φ Ỹ is a local maringale, i.e., le F s = Fs+ P F; hen, φ Ỹ s = E Q φ Ỹ F s. Knowing ha φ Ỹ s is a local maringale under F, Q where φ P F is admissible, hen how do we recover Y? and wha is he porfolio φ in F, Q? Knowing ha Ỹ = Y +, we are going o employ opional projecion of he space Ω, F, F, Q on Ω, F, F, Q o answer hese quesions. We begin by showing ha φ Y s = E Q φ Ỹ F s a.s. Q, 12 where is he sochasic inegral wih respec o RCLL semimaringale Ỹ and a predicable inegrand and is he sochasic inegral wih respec o RLL opional semimaringales Y wih an opional inegrand. Again, we are going o idenify E Q wih E for breviy. heorem 3. Le φ Ỹ be a Q-local maringale. hen, φ Y s = Eφ Ỹ F s a.s. Q. Proof. Le R k be a sequence of sopping imes in F where φ Ỹ Rk is a maringale for all k and E φ Ỹ Rk <. Noe ha Ỹ = Y + and + Y = Y + Y. hen, + Y = Y + Y = Ỹ Y = Ỹ Ỹ = Ỹ. In addiion, dỹ = dy + Ỹ = dy + + Y and E φ Ỹ φ Ỹ s F s = E E φ Ỹ φ Ỹ s F s Fs = since F s F s for all s. Moreover, if Y u is evolving in he inerval s,, hen Ỹ u is evolving in he inerval s,. Hence, E φ Ỹ φ Ỹ s F s = E E φ Ỹ φ Ỹ s F s Fs = E φ u dỹ u φ u dỹ u F s,,s = E φ u dỹ u + φ u dỹ u φ u dỹ u F s,s s,,s = E φ u dỹ u F s = E φ u dyu + Ỹ u Fs s, s, = E φ u dy u F s + E φ u Ỹ u F s s, s, = E φ u dy u F s + E φ v + Y v F s s, s, = E φ u dy u + φ v + Y v F s s, s, = E φ u dy u + φ vdy v+ F s s, s, = E φ udy u F s = E φ udy u F s + E φ udy u F s s,,,s = E φ Y φ Y s F s = E φ Y F s φ Y s = E φ Y F s = φ Y s.

17 Risks 217, 5, of 21 Alernaively, one can use approximaion mehods of sochasic inegrals by sums o arrive o he same resul. hus, in he limi of a pariion Π n of, where is some ime horizon and for any,, φ φ + φ i 1 1 i 1, i+1, where we have chosen he inervals, i 1, i+1, for which φ i 1 = φ i. hen, φ Ỹ φ i 1 Ỹi+1 Ỹ i 1 = φ i 1 Ỹi+1 Ỹ i + Ỹ i Ỹ i 1 = φ i 1 Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1 = φ i Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1. using φ i = φ i 1 Noe ha he sochasic inegral is defined in he usual sense wih respec o Ω, F, F, Q and is valid under any refinemen of he inerval,. his allows using he inervals i 1, i+1 where, in he limi, he inegral is defined. Consider a pariion of s,. hen, he projecion of he inegral s φ u dỹ u on F s is E φ u dỹ u F s = E lim φ i 1 Ỹi+1 Ỹ i 1 Fs s = lim E φ i 1 Ỹi+1 Ỹ i 1 Fs, by localizaion and inegrabiliy = lim E φ i 1 Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1 Fs = lim E φ i Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1 Fs, by φi 1 = φ i. By properies of condiional expecaions and knowing ha for any i, F i F s and φ is F -measurable, we ge E φ Ỹ φ Ỹ s F s = E φ u dỹ u F s = lim E φ i Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1 Fs s by localizaion and inegrabiliy, pass he limi hrough he expecaion, = E lim φ i Ỹi+1 Ỹ i + φi 1 Ỹi Ỹ i 1 Fs = E lim E φ i Ỹ i+1 F i+1 E φi Ỹ i F i + E φi 1 Ỹ i F i E φi 1 Ỹ i 1 F i 1 Fs = E lim φ i E Ỹ i+1 F i+1 φi E Ỹ i F i + φi 1 E Ỹ i F i φi 1 E Ỹ i 1 F i 1 Fs by Ỹ u = Y u+ = Y u + + Y u = E lim φ i 1 Yi Y i 1 + φi Yi+1 Y i + φi 1 + Y i + Y i 1 + φi + Y i+1 + Y i Fs = E lim φ i 1 Yi Y i 1 + φi Yi+1 Y i + φi 1 + Y i + Y i 1 + φi + Y i+1 + Y i Fs = E φ u dyu r + φ u dy g u+ F s s+ s s s = E φ u dyu r + φ u dy g u+ φ u dyu r φ u dy g u+ F s + + s s = E φ u dỹ u φ u dyu r φ u dy g u+ F s + = E φ Ỹ φ Ys r φ Y g s+ F s = E φ Ỹ φ Y s F s = E φ Ỹ F s φ Ys =. herefore, E φ Ỹ F s = φ Ys = φ Ys r + φ Y g s+ + Y i + Y i 1 = + Y i+1 + Y i =. for any s. Noe ha, in he limi, Lemma 9. φ Y is a local opional maringale under Q.

18 Risks 217, 5, of 21 Proof. By he resul of he above heorem, for any u s, E φ Y s F u = E E φ Ỹ F s Fu = E φ Ỹ F u = φ Yu. By he same mehod, we have esablished NFLVR in he unusual condiions, i is possible o see ha if NA1 is saisfied for Ỹ on Ω, F, F, Q, hen i is saisfied for Y on Ω, F, F, Q. herefore, one can res assured ha if NFLVR or NA1 are saisfied for RCLL semimaringales on Ω, F, F, P, hen hey mus be saisfied for heir opional version by opional projecion on Ω, F, F, P. Hence, opional markes are free of arbirage. he same is rue for opional defaulable markes as hey are a special case of opional markes, where defaulable asses and cash-flows are opional semimaringales. In pracice, i may be of ineres o know how o ransform a defaulable process, Y, o a local opional maringale wih a sricly posiive local opional maringale deflaor, Z. Here, we will demonsrae, wih a simple example, a mehod o derive Z; Example 3. Consider Y = X + HX x as in Equaion 1. However, suppose ha X and x are coninuous local maringales and ha H, X = H, x = see Remark 5. We are going o find a sricly posiive local opional maringale deflaor Z for Y. Given ha H = M + µ and knowing ha he sum of local opional maringales is a local opional maringale and ha he producs of Z wih coninuous local maringales X and X x are local opional maringales, hen i suffices o consider finding Z such ha ZH is a local opional maringale. o do so, we choose Z = E θ M > and find he appropriae condiions ha θ mus saisfy so ha ZH is a local maringale. We do his as follows: E θ M H = HE θ M θ M + E θ M M + E θ M µ + θe θ M M, M E θ M µ + θe θ M M, M = θ = dµ d M, M. Wih θ = dµ dm,m and 1 + θ M 1 + θ + M >, one can easily see ha he process ZY is a local opional maringale. Remark 5. Implici in he definiion HY is he fac ha X and x are he defaul-free evoluion of he componens of he asse Y over is life and H is he defaul process. his allows us o suppose ha H and X, x are orhogonal. In realiy, X and x can be observed in isolaion of H. In defaulable markes, we observe he process X + HX x. he above calculaion of local maringale deflaor assumes knowledge of x, X and H, and ha hey are relaed in a paricular way. his is done o prove ha local opional maringale deflaors can be consruced for defaulable asse Y. Acknowledgmens: We wish o hank reviewers and ediors for heir commens which have improved his work. his research is suppored by Naural Sciences and Engineering Research Council of Canada NSERC discovery gran 591. Auhor Conribuions: he wo auhors conribue equally o his aricle. Conflics of Ineres: he auhors declare no conflic of ineres. Appendix A. Opional Calculus his secion is based on he following papers Galchuk 198, Le Ω, F, F = F, P,,, where F F, F s F, s, be a complee probabiliy space F conains all P null ses. However, he family F is no assumed o be complee, righ or lef coninuous. We inroduce OF and PF as he opional and predicable σ-algebras on Ω,,, respecively. OF is generaed by all F adaped processes whose rajecories are righ coninuous and have a limi from he lef. PF is generaed by all F adaped processes whose rajecories are lef coninuous and have limis on he righ. A random process X = X,, is said o be opional if i is OF-measurable. In general, an opional process has righ and lef limis bu is no necessarily righ or lef coninuous in F. A random

19 Risks 217, 5, of 21 process, X,,, is predicable if X PF. Furhermore, in general, a predicable process has righ and lef limis bu may no necessarily be righ or lef coninuous in F. For eiher opional or predicable processes, we can define he following processes: X = X and X + = X +, X = X such ha X = X X and + X = + X such ha + X = X + X. In addiion, a process is srongly predicable, which is X in P s F, if X PF and X + OF. On an unusual sochasic basis, hree canonical ypes of sopping imes exis: predicable sopping imes, S F, are such ha S is F measurable for all. oally inaccessible sopping imes, F, are such ha is F measurable for all ; however, we noe ha < is no necessarily F measurable since F is no righ coninuous. Finally, oally inaccessible sopping imes in he broad sense, U F +, are such ha U is F + measurable and, since F + is righ coninuous, U < is also F + measurable for or all. A process X = X belongs o he space J loc if here is a localizing sequence of sopping imes in he broad sense, R n, n N, R n F +, R n a.s. such ha X1,Rn J for all n, where J is a space of processes and J loc is an exension of J by localizaion. A process A = A is increasing if i is non-negaive, is rajecories do no decrease and for any, and he random variable A is F -measurable. Le V + F, P V + for shor be he collecion of increasing processes. An increasing process A is inegrable if EA < and locally inegrable if here is a sequence R n F +, n N, R n a.s. such ha EA Rn+ < for all n N. he collecion of such processes is denoed by A + A + loc, respecively. A process A = A, R +, is a finie variaion process if i has finie variaion on every segmen,, R + a.s., ha is VarA <, for all R + a.s., where VarA = s< + A s + + dar s and A r is a righ coninuous par of A. We shall denoe by VF, P V for shor he se of F adaped finie variaion processes. A process A = A of finie variaion belongs o he space A of inegrable finie variaion processes if E VarA <. A process A = A belongs o A loc if here is a sequence R n F +, n N, R n, such ha A1,Rn A for any n N, i.e., for any n, E VarA Rn <. A finie variaion or an increasing process A can be decomposed o A = A r + A g = A c + A d + A g, where A c is coninuous, A r is a righ coninuous, A d is discree righ coninuous, and A g is discree lef coninuous. A process M = M is an opional maringale supermaringale, submaringale if M OF, he random variable M 1 < is inegrable for any sopping ime F and here exiss an inegrable random variable µ F such ha M = Eµ F respecively, M Eµ F, M Eµ F a.s. for any sopping ime F wih <. Le MF, P M for shor denoe he se of opional maringales and M loc he se of opional local maringales. If M is local opional maringale, hen i can be decomposed o M = M r + M g, where M r = M c + M d, M c is coninuous, M d is righ coninuous and M g is lef coninuous local opional maringales. M d and M g are orhogonal o each oher and o any coninuous local maringale. An opional semimaringale X = X can be decomposed o an opional local maringale and an opional finie variaion processes; X = X + M + A, where M M loc and A V. A semimaringale X is called special if he above decomposiion exiss wih a srongly predicable process A A loc. Le SF, P denoe he se of opional semimaringales and S pf, P he se of special opional semimaringales. If X S pf, P, hen he semimaringale decomposiion is unique. By opional maringale decomposiion and decomposiion of predicable processes, we can decompose a semimaringale furher o X = X + X r + X g wih X r = A r + M r, X g = A g + M g and M r = M c + M d, where A r and A g are finie variaion processes ha are righ and lef coninuous, respecively M r M r loc righ coninuous local maringales, Md M d loc discree righ coninuous local maringales and M g M g loc a lef coninuous local maringales. his decomposiion is useful for defining inegraion wih respec o opional semimaringales. A sochasic inegral wih respec o opional semimaringale was defined as ϕ X = ϕ s dx s = + ϕ s dx r s + ϕ s dx g s+. A1

20 Risks 217, 5, 56 2 of 21 he sochasic inegral wih respec o he finie variaion processes or srongly predicable processes A r over, and A g over, are inerpreed as usual, in he Lebesgue sense. he inegral + ϕ s dms r over, is our usual sochasic inegral wih respec o RCLL local maringale, whereas φ s dm g s+ over, is Galchouk sochasic inegral wih respec o lef coninuous local maringale. In general, he sochasic inegral wih respec o opional semimaringale X can be defined as a bilinear form Y = f, g X = f X r + g Xg such ha f X r = + f s dx r s, g X g = g s dx g s+, A2 where Y is again an opional semimaringale f PF, and g OF. Noe ha he sochasic inegral over opional semimaringales is defined on a much larger space of inegrands, he produc space of predicable and opional processes, PF OF. he operaor o denoe he sochasic opional inegral while he operaor o denoe he regular sochasic inegral wih respec o RCLL semimaringales, and he operaor for he Galchouk sochasic inegral g X g wih respec o lef coninuous semimaringales. For any semimaringales X and Y, heir quadraic variaion is defined as X, Y = X r, Y r + X g, Y g where References X r, X r = X c, X c + X s 2, X g, X g = + 2 X s. <s s< Abdelghani, Mohamed Nabeel On he General heory of Opional Sochasic Processes and Financial Markes Modeling. Ph.D. hesis, Universiy of Albera, Edmonon, AB, Canada. Abdelghani, Mohamed Nabeel, and Alexander V. Melnikov Financial Markes in he Conex of he General heory of Opional Processes. Cham: Springer Inernaional Publishing, pp Abdelghani, Mohamed Nabeel, and Alexander V. Melnikov. 217a. A comparison heorem for sochasic equaions of opional semimaringales. Sochasics and Dynamics doi:1.1142/s Abdelghani, Mohamed Nabeel, and Alexander V. Melnikov. 217b. On linear sochasic equaions of opional semimaringales and heir applicaions. Saisics & Probabiliy Leers 125: Arzner, Philippe, and Freddy Delbaen Defaul risk insurance and incomplee markes. Mahemaical Finance 5: Belanger, Alain, Seven E. Shreve, and Dennis Wong. 21. A unified model for credi derivaives. Mahemaical Finance. Available online: hps://pdfs.semanicscholar.org/646f/77a7d32dd49a6a552dcdd aaf387ec.pdf accessed on 21 Ocober 217 Bielecki, omasz R., and Marek Rukowski. 22. Credi Risk: Modeling, Valuaion and Hedging. Berlin: Springer Science & Business Media. Bingham, Nicholas H., and Rüdiger Kiesel Risk-Neural Valuaion: Pricing and Hedging of Financial Derivaives. Berlin: Springer Science & Business Media. Black, Fischer, and Myron Scholes he pricing of opions and corporae liabiliies. he Journal of Poliical Economy 81: Cox, John C., and Sephen A. Ross he valuaion of opions for alernaive sochasic processes. Journal of Financial Economics 3: Delbaen, Freddy, and Freddy Schachermayer. 26. he Mahemaics of Arbirage. Berlin: Springer Science & Business Media. Dellacherie, Claude Seminaire de Probabilies XI Lecure Noes in Mahemaics. Chaper Deux Remarques sur la Separabilie Opionelle; Berlin: Springer, vol. 581, pp Duffie, Darrel, and Kenneh J. Singleon An economeric model of he erm srucure of ineres-rae swap yields. he Journal of Finance 52: Duffie, Darrel, and Kenneh J. Singleon Modeling erm srucures of defaulable bonds. he Review of Financial Sudies 12:

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