Conditional Default Probability and Density

Transcription

1 Condiional Defaul Probabiliy and Densiy N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Absrac This paper proposes differen mehods o consruc condiional survival processes, i.e, families of maringales decreasing wih respec o a parameer. Condiional survival processes play a pivoal role in he densiy approach for defaul risk, inroduced by El Karoui e al.[4]. Concree examples will lead o he consrucion of dynamic copulae, in paricular dynamic Gaussian copulae. I is shown ha he change of probabiliy measure mehodology is a key ool for ha consrucion. As in Kallianpur and Sriebel [10], we apply his mehodology in filering heory o recover in a sraighforward way, he classical resuls when he signal is a random variable. 1 Inroducion The goal of his paper is o give examples of he condiional law of a random variable (or a random vecor, given a reference filraion, and mehods o consruc dynamics of condiional laws, in order o model price processes and/or defaul risk. This mehodology appears in some recen papers (El Karoui e al. [4], Filipovic e N. El Karoui Laboraoire de Probabiliés e Modèles Aléaoires, Universié Pierre e Marie Curie and Cenre de Mahémaiques Appliquées, Ecole Polyechnique nicole.elkaroui@cmap.polyechnique.fr M. Jeanblanc Laboraoire Analyse e Probabiliés, Universié d Evry-Val-D Essonne and Insiu Europlace de Finance monique.jeanblanc@univ-evry.fr Y. Jiao Laboraoire de Probabiliés e Modèles Aléaoires, Universié Paris 7. jiao@mah.jussieu.fr B. Zargari Laboraoire Analyse e Probabiliés, Universié d Evry-Val-D Essonne and Sharif Universiy of Technology. behnaz.zargari@univ-evry.fr 1

2 2 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari al. [7] and i is imporan o presen echniques o build concree examples. We have chosen o characerize he (condiional law of a random variable hrough is (condiional survival probabiliy or hrough is (condiional densiy, if i exiss. In Secion 2, we give he definiion of maringale survival processes and densiy processes. In Secion 3, we give sandard examples of condiional laws, in paricular a Gaussian model, and we give mehods o consruc oher ones. In Secion 4, we show ha, in he case of random imes (i.e., non-negaive random variables, he densiy mehodology can be seen as an exension of he Cox model, and we recall a resul which allows o consruc defaul imes having he same inensiy and differen condiional laws. We build he change of probabiliy framework in Secion 5 and show how i can be applied o filering heory for compuing he condiional law of he random variable which represens he signal. 2 Definiions Le (Ω,A,F,P be a filered probabiliy space, equipped wih a filraion F = (F 0 saisfying he usual condiions, where F A and F 0 is he rivial filraion. Le E be equal o one of he following spaces: IR, IR d, IR +, or (IR + d. A family of (P,F-maringale survival processes on E is a family of (P,F- maringales G. (θ, θ E such ha θ G (θ is decreasing, and for any θ, G. (θ is valued in [0,1]. We have used he sandard convenion for maps from IR d o IR: such a map G is decreasing if θ θ implies G(θ G( θ, where θ θ means ha i = 1,...,d, θ i θ i. A (P,F densiy process on E is a family g. (θ, θ E of non-negaive, (P,F- maringales such ha g (udu = 1,, a.s. (1 E where du denoes he Lebesgue measure on E. If here is no ambiguiy, we shall simply say a maringale survival process and a densiy process. If G is a family of maringale survival processes on E, absoluely coninuous w.r. he Lebesgue measure, i.e., G (θ = θ g (udu, he family g is a densiy process (see Jacod [9] for imporan regulariy condiions. The maringale survival process of an A -measurable random variable X valued in IR d is he family of càdlàg processes G (θ = P(X > θ F. Obviously, his is a maringale survival process (i is decreasing w.r.. θ. In paricular, assuming regulariy condiions, he non-negaive funcion g 0 such ha G 0 (θ = θ g 0 (sds is he probabiliy densiy of X. If we are given a family of densiy processes g. (θ, hen here exiss a random variable X (consruced on an exended probabiliy space such ha P(X > θ F = G (θ = g (udu, θ a.s.

3 Condiional Defaul Probabiliy and Densiy 3 where (wih an abuse of noaion P is a probabiliy measure on he exended space, which coincides wih he given probabiliy measure on F. For he consrucion, one sars wih a random variable X on Ω IR independen of F, wih probabiliy densiy g 0 and one checks ha (g (X, 0 is an F σ(x-maringale. Then, seing dq F σ(x = g (X g 0 (X dp F σ(x, one obains, from Bayes formula ha Q(X > θ F = G (θ. This consrucion was imporan in Grorud and Ponier [8] and in Amendinger [1] in an iniial enlargemen of filraion framework for applicaion o insider rading. In he specific case of random imes (non-negaive random variables, one has o consider maringale survival processes defined on IR +. They can be deduced from maringale survival processes on IR by a simple change of variable: if G is he maringale survival process on IR of he real valued random variable X and h a sricly increasing funcion from IR + o IR, hen G h (u := G (h(u defines a maringale survival process on IR + (corresponding o he change of variable Y = h 1 (X. In he case where h is differeniable, he densiy process is g h (u = g (h(uh (u. I is imporan o noe ha, due o he maringale propery, in order o characerize he family g (θ for any pair (,θ (IR + IR, i suffices o know his family for any pair (,θ such ha θ. Hence, in wha follows, we shall concenrae on consrucion for θ. In he paper, he naural filraion of a process Y is denoed by F Y. 3 Examples of Maringale Survival Processes We firs presen wo specific examples of condiional law of an F B -measurable random variable, when F B is he naural filraion of a Brownian moion B. Then we give wo large classes of examples, based on Markov processes and diffusion processes. The firs example, despie is simpliciy, will allow us o consruc a dynamic copula, in a Gaussian framework; more precisely, we consruc, for any, he (condiional copula of a family of random imes P(τ i > i,i = 1,...,n F and we can chose he parameers so ha P(τ i > i,i = 1,...,n equals a given (saic Gaussian copula. To he bes of our knowledge, here are very few explici consrucions of such a model. In Fermanian and Vigneron [5], he auhors apply a copula mehodology, using a facor Y. However, he processes hey use o fi he condiional probabiliies P(τ i > i,i = 1,...,n F σ(y are no maringales. They show ha, using some adequae paramerizaion, hey can produce a model so ha P(τ i > i,i = 1,...,n F are maringales. Our model will saisfy boh maringale condiions. In [2], Carmona is ineresed in he dynamics of prices of asses corresponding o a payoff which is a Bernoulli random variable (aking values 0 or 1, in oher words, he is looking for examples of dynamics of maringales valued in [0,1], wih a given erminal condiion. Surprisingly, he example he provides corresponds o he one

4 4 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari we give below in Secion, up o a paricular choice of he parameers o saisfy he erminal consrain. In a second example, we consruc anoher dynamic copula, again in an explici way, wih a more complicaed dependence. We hen show ha a class of examples can be obained from a Markov model, where he decreasing propery is inroduced via a change of variable. In he second class of examples, he decreasing propery is modeled via he dependence of a diffusion hrough is iniial condiion. To close he loop, we show ha we can recover he Gaussian model of he firs example wihin his framework. 3.1 A dynamic Gaussian copula model In his subsecion, ϕ is he sandard Gaussian probabiliy densiy, and Φ he Gaussian cumulaive funcion. We consider he random variable X := 0 f (sdb s where f is a deerminisic, square-inegrable funcion. For any real number θ and any posiive, one has ( P(X > θ F B = P m > θ f (sdb s F B where m = 0 f (sdb s is F B -measurable. The random variable f (sdb s follows a cenered Gaussian law wih variance σ 2 ( = f 2 (sds and is independen of F B. Assuming ha σ( does no vanish, one has ( P(X > θ F B m θ = Φ. (2 σ( In oher words, he condiional law of X given F B is a Gaussian law wih mean m and variance σ 2 (. We summarize he resul 1 in he following proposiion, and we give he dynamics of he maringale survival process, obained wih a sandard use of Iô s rule. Proposiion 1. Le B be a Brownian moion, f an L 2 deerminisic funcion, m = 0 f (sdb s and σ 2 ( = f 2 (sds. The family ( m θ G (θ = Φ σ( is a family of F B -maringales, valued in [0,1], which is decreasing w.r.. θ. Moreover ( m θ f ( dg (θ = ϕ σ( σ( db. 1 More resuls on ha model, in an enlargemen of filraion seing, can be found in Chaleya- Maurel and Jeulin in [3] and Yor [17].

5 Condiional Defaul Probabiliy and Densiy 5 The dynamics of he maringale survival process can be wrien dg (θ = ϕ ( Φ 1 (G (θ f ( σ( db. (3 We obain he associaed densiy family by differeniaing G (θ w.r.. θ, and is dynamics 1 ( g (θ = exp (m θ 2 2π σ( 2σ 2 ( dg (θ = g (θ m θ σ 2 ( f (db. (4 Le us emphasize ha, saring from (3, i is no obvious o check ha he soluion is decreasing wih respec o he parameer θ, or, as i is done in [5] and [2], o find he soluion. In he same way, he soluion of (4 wih iniial condiion a probabiliy densiy g 0, is a densiy process if and only if g (udu = 1, or equivalenly, g (θ m θ f (dθ = 0. This las equaliy reduces o σ 2 ( g (θ(m θdθ = m g (θθdθ = 0 and we do no see how o check his equaliy if one does no know he explici soluion. In order o provide condiional survival probabiliies for posiive random variables, we consider X = ψ(x where ψ is a differeniable, posiive and sricly increasing funcion and le h = ψ 1. The condiional law of X is ( m h(θ G (θ = Φ. σ( We obain and 1 ( g (θ = h (θ exp (m h(θ 2 2πσ( 2σ 2 ( ( m h(θ f ( d G (θ = ϕ σ( σ( db, d g (θ = g (θ m h(θ f ( σ( σ( db. Inroducing an n-dimensional sandard Brownian moion B = (B i,i = 1,...,n and a facor Y, independen of F B, gives a dynamic copula approach, as we presen now. For h i an increasing funcion, mapping IR + ino IR, and seing τ i =

6 6 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari (h i 1 ( 1 ρi 2 0 f i(sdb i s +ρ i Y, for ρ i ( 1,1, an immediae exension of he Gaussian model leads o P(τ i > i, i = 1,...,n F B n σ(y = Φ 1 m i h i( i ρ i Y i=1 σ i ( 1 ρi 2 where m i = 0 f i(sdb i s and σ 2 i ( = P(τ i > i, i = 1,...,n F B = i=1 (sds. I follows ha Φ 1 σ i ( fi 2 n m i h i( i ρ i y 1 ρ 2 i f Y (ydy. Noe ha, in ha seing, he random imes (τ i,i = 1,...,n are condiionally independen given F B σ(y, a useful propery which is no saisfied in Fermanian and Vigneron model. For = 0, choosing f i so ha σ i (0 = 1, and Y wih a sandard Gaussian law, we obain P(τ i > i, i = 1,...,n = n i=1 Φ h i( i ρ i y ϕ(ydy 1 ρi 2 which corresponds, by consrucion, o he sandard Gaussian copula (h i (τ i = 1 ρ 2 i X i + ρ i Y, where X i,y are independen sandard Gaussian variables. Relaxing he independence condiion on he componens of he process B leads o more sophisicaed examples. 3.2 A Gamma model Here, we presen anoher model, where he processes involved are no more Gaussian ones. Consider A (µ := 0 e2b(µ s ds where B (µ = B +µ, µ being a posiive consan. Masumoo and Yor [15] have esablished ha A ( µ = A ( µ + e 2B( µ à ( µ where à ( µ is independen of F B, wih he same law as A ( µ. The law of A ( µ is proved o be he law of 1/(2γ µ, γ µ being a Gamma random variable wih parameer µ. The survival probabiliy of A ( µ is ϒ (x = 1 1/(2x Γ (µ 0 y µ 1 e y dy, where Γ is he Gamma funcion. Then, one obains G (θ = P(A ( µ > θ F B = ϒ ( ( µ θ A e 2B( µ 1 θ>a ( µ + 1 ( µ. θ A This gives a family of maringale survival processes G, similar o (5, wih gamma srucure. I follows ha, on {θ > A ( µ }

7 Condiional Defaul Probabiliy and Densiy 7 dg (θ = 1 2 µ 1 Γ (µ e 1 2 Z (θ (Z (θ µ db where Z (θ = e2b( µ (o have ligh noaion, we do no specify ha his process Z θ A ( µ depends on µ. One can check ha G ( is differeniable w.r.. θ, so ha G (θ = θ g (udu, where g (θ = 1 θ>a ( µ 1 2 µ Γ (µ (Z (θ µ+1 e 2 1 Z (θ 2B ( µ. Again, inroducing an n-dimensional Brownian moion, a facor Y and he r.vs α i A ( µ,i + ρ i Y, where α i and ρ i are consans, will give an example of a dynamic copula. 3.3 Markov processes Le X be a real-valued Markov process wih ransiion probabiliy p T (,x,ydy = P(X T dy X = x, and Ψ a family of funcions IR IR [0,1], decreasing w.r.. he second variable, such ha Then, for any T, Ψ(x, = 1,Ψ(x, = 0. G (θ := E(Ψ(X T,θ F X = p T (,X,yΨ(y,θdy is a family of maringale survival processes on IR. While modeling (T ; x-bond prices, Filipovic e al. [6] have used his approach in an affine process framework. See also Keller-Ressel e al. [13]. Example 1. Le X be a Brownian moion, and Ψ(x,θ = e θx2 1 θ θ 0. We obain a maringale survival process on IR +, defined for θ 0 and < T as, G (θ = E [ exp( θxt 2 F X ] 1 = exp ( θx (T θ 1 + 2(T θ The consrucion given above provides a maringale survival process G(θ on he ime inerval [0,T ]. Using a (deerminisic change of ime, one can easily deduce a maringale survival process on he whole inerval [0, [: seing Ĝ (θ = G h( (θ for a differeniable increasing funcion h from [0, ] o [0,T ], and assuming ha dg (θ = G (θk (θdb, < T, one obains

8 8 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari dĝ (θ = Ĝ (θk h( (θ h (dw where W is a Brownian moion. One can also randomize he erminal dae and consider T as an exponenial random variable independen of F. Noing ha he previous G (θ s depend on T, one can wrie hem as G (θ,t and consider G (θ = G (θ,ze z dz 0 which is a maringale survival process. The same consrucion can be done wih a random ime T wih any given densiy, independen of F. 3.4 Diffusion-based model wih iniial value Proposiion 2. Le Ψ be a cumulaive disribuion funcion of class C 2, and Y he soluion of dy = a(,y d + ν(,y db, Y 0 = y 0 where a and ν are deerminisic funcions smooh enough o ensure ha he soluion of he above SDE is unique. Then, he process (Ψ(Y, 0 is a maringale, valued in [0,1], if and only if a(,yψ (y ν2 (,yψ (y = 0. (5 Proof. The resul follows by applying Iô s formula and noing ha Ψ(Y being a (bounded local maringale is a maringale. We denoe by Y (y he soluion of he above SDE wih iniial condiion Y 0 = y. Noe ha, from he uniqueness of he soluion, y Y (y is increasing (i.e., y 1 > y 2 implies Y (y 1 Y (y 2. I follows ha G (θ := 1 Ψ(Y (θ is a family of maringale survival processes. Example 2. Le us reduce our aenion o he case where Ψ is he cumulaive disribuion funcion of a sandard Gaussian variable. Using he fac ha Φ (y = yφ (y, Equaion (5 reduces o a(,y 1 2 yν2 (,y = 0 In he paricular he case where ν(, y = ν(, sraighforward compuaion leads o

9 Condiional Defaul Probabiliy and Densiy 9 Y (y = e ν 2 (sds (y + 0 e 1 s0 2 ν 2 (udu ν(sdb s. Seing f (s = ν(sexp( 2 1 s 0 ν2 (udu, one deduces ha Y (y = y m σ(, where σ 2 ( = f 2 (sds and m =: 0 f (sdb s, and we recover he Gaussian example of Subsecion Densiy Models In his secion, we are ineresed in densiies on IR + in order o give models for he condiional law of a random ime τ. We recall he classical consrucions of defaul imes as firs hiing ime of a barrier, independen of he reference filraion, and we exend hese consrucions o he case where he barrier is no more independen of he reference filraion. I is hen naural o characerize he dependence of his barrier and he filraion by means of is condiional law. In he lieraure on credi risk modeling, he aenion is mosly focused on he inensiy process, i.e., o he process Λ such ha 1 τ Λ τ is a G = F H- maringale, where H = σ( τ. We recall ha he inensiy process Λ is he only increasing predicable process such ha he survival process G := P(τ > F admis he decomposiion G = N e Λ where N is a local maringale. We recall ha he inensiy process can be recovered form he densiy process as dλ s = g s(s G s (s ds (see [4]. We end he secion giving an explici example of wo differen maringale survival processes having he same survival processes (hence he inensiies are equal. 4.1 Srucural and reduced-form models In he lieraure, models for defaul imes are ofen based on a hreshold: he defaul occurs when some driving process X reaches a given barrier. Based on his observaion, we consider he random ime on IR + in a general hreshold model. Le X be a sochasic process and Θ be a barrier which we shall precise laer. Define he random ime as he firs passage ime τ := inf{ : X Θ}. In classical srucural models, he process X is an F-adaped process associaed wih he value of a firm and he barrier Θ is a consan. So, τ is an F-sopping ime. In his case, he condiional disribuion of τ does no have a densiy process, since P(τ > θ F = 1 θ<τ for θ. To obain a densiy process, he model has o be changed, for example one can sipulae ha he driving process X is no observable and ha he observaion is a

10 10 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari filraion F, smaller han he filraion F X, or a filraion including some noise. The goal is again o compue he condiional law of he defaul P(τ > θ F, using for example filering heory. Anoher mehod is o consider a righ-coninuous F-adaped increasing process Γ and o randomize he barrier. The easies way is o ake he barrier Θ as an A - measurable random variable independen of F, and o consider τ := inf{ : Γ Θ}. (6 If Γ is coninuous, τ is he inverse of Γ aken a Θ, and Γ τ = Θ. The F-condiional law of τ is P(τ > θ F = G Θ (Γ θ, θ where G Θ is he survival probabiliy of Θ given by G Θ ( = P(Θ >. We noe ha in his paricular case, P(τ > θ F = P(τ > θ F for any θ, which means ha he H-hypohesis is saisfied 2 and ha he maringale survival processes remain consan afer θ (i.e., G (θ = G θ (θ for θ. This resul is sable by increasing ransformaion of he barrier, so ha we can assume wihou loss of generaliy ha he barrier is he sandard exponenial random variable log(g Θ (Θ. If he increasing process Γ is assumed o be absoluely coninuous w.r.. he Lebesgue measure wih Radon-Nikodým densiy γ and if G Θ is differeniable, hen he random ime τ admis a densiy process given by g (θ = (G Θ (Γ θ γ θ = g θ (θ, θ (7 = E(g θ (θ F, θ >. Example (Cox process model In he widely used Cox process model, he independen barrier Θ follows he exponenial law and Γ = 0 γ sds represens he defaul compensaor process. As a direc consequence of (7, g (θ = γ θ e Γ θ, θ. 4.2 Generalized hreshold models In his subsecion, we relax he assumpion ha he hreshold Θ is independen of F. We assume ha he barrier Θ is a sricly posiive random variable whose condiional disribuion w.r.. F admis a densiy process, i.e., here exiss a family of F B(IR + -measurable funcions p (u such ha G Θ (θ := P(Θ > θ F = p (udu. (8 θ 2 We recall ha H-hypohesis sands for any F-maringale is a G = F H maringale.

11 Condiional Defaul Probabiliy and Densiy 11 We assume in addiion ha he process Γ is absoluely coninuous w.r.. he Lebesgue measure, i.e., Γ = 0 γ sds. We sill consider τ defined as in (6 by τ = Γ 1 (Θ and we say ha a random ime consruced in such a seing is given by a generalized hreshold. Proposiion 3. Le τ be given by a generalized hreshold. Then τ admis he densiy process g(θ where g (θ = γ θ p (Γ θ, θ. (9 Proof. By definiion and by he fac ha Γ is sricly increasing and absoluely coninuous, we have for θ, G (θ := P(τ > θ F = P(Θ > Γ θ F = G Θ (Γ θ = Γ θ p (udu = θ p (Γ u γ u du, which implies g (θ = γ θ p (Γ θ for θ. Obviously, in he paricular case where he hreshold Θ is independen of F, we recover he classical resuls (7 recalled above. Conversely, if we are given a densiy process g, hen i is possible o consruc a random ime τ by a generalized hreshold, ha is, o find Θ such ha he associaed τ has g as densiy, as we show now. I suffices o define τ = inf{ : Θ} where Θ is a random variable wih condiional densiy p = g. Of course, for any increasing process Γ, τ = inf{ : Γ } where := Γ Θ is a differen way o obain a soluion! 4.3 An example wih same survival processes We recall ha, saring wih a survival maringale process G (θ, one can consruc oher survival maringale processes G (θ admiing he same survival process (i.e., G ( = G (, in paricular he same inensiy. The consrucion is based on he general resul obained in Jeanblanc and Song [11]: for any supermaringale Z valued in [0,1[, wih muliplicaive decomposiion Ne Λ, where Λ is coninuous, he family ( Z s G (θ = 1 (1 Z exp dλ s 0 < θ, θ 1 Z s is a maringale survival process (called he basic maringale survival process which saisfies G ( = Z and, if N is coninuous, dg (θ = 1 G (θ 1 Z e Λ dn. In paricular, he associaed inensiy process is Λ (we emphasize ha he inensiy process does no conain enough informaion abou he condiional law. We illusrae his consrucion in he Gaussian example presened in Secion 3.1 where we se Y = m h( σ(. The muliplicaive decomposiion of he supermaringale G = P(τ > F B is G = N exp ( 0 λ sds where

12 12 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari dn = N ϕ(y σ(φ(y dm, λ = h (ϕ(y σ(φ(y. Using he fac ha G ( = Φ(Y, one checks ha he basic maringale survival process saisfies f (ϕ(y dg (θ = (1 G (θ σ(φ( Y db, θ, G θ (θ = Φ(Y θ which provides a new example of maringale survival processes, wih densiy process g (θ = (1 G e Gs θ 1 Gs λ sds G θ λ θ, θ. 1 G θ Oher consrucions of maringale survival processes having a given survival process can be found in [12], as well as consrucions of local-maringales N such ha Ne Λ is valued in [0,1] for a given increasing coninuous process Λ. 5 Change of Probabiliy Measure and Filering In his secion, our goal is o show how, using a change of probabiliy measure, one can consruc densiy processes. The main idea is ha, saring from he (uncondiional law of τ, we consruc a condiional densiy in a dynamic way using a change of probabiliy. This mehodology is a very paricular case of he general change of measure approach developed in [4]. Then, we apply he idea of change of probabiliy framework o a filering problem (due o Kallianpur and Sriebel [10], o obain he Kallianpur-Sriebel formula for he condiional densiy (see also Meyer [16]. Our resuls are esablished in a very simple way, in a general filering model, when he signal is a random variable, and conain, in he simple case, he resuls of Filipovic e al. [7]. We end he secion wih he example of he radiional Gaussian filering problem. 5.1 Change of measure One sars wih he elemenary model where, on he filered probabiliy space (Ω,A,F,P, an A -measurable random variable X is independen from he reference filraion F = (F 0 and is law admis a densiy probabiliy g 0, so ha P(X > θ F = P(X > θ = g 0 (udu. θ

13 Condiional Defaul Probabiliy and Densiy 13 We denoe by G X = F σ(x he filraion generaed by F and X. Le (β (u, IR + be a family of posiive (P,F-maringales such ha β 0 (u = 1 for all u IR. Noe ha, due o he assumed independence of X and F, he process (β (X, 0 is a G X -maringale and one can define a probabiliy measure Q on (Ω,G X, by dq = β (XdP. Since F is a subfilraion of G X, he posiive F-maringale m β := E(β (X F = β (ug 0 (udu 0 is he Radon-Nikodým densiy of he measure Q, resriced o F wih respec o P (noe ha m β 0 = 1. Moreover, he Q-condiional densiy of X wih respec o F can be compued, from he Bayes formula Q(X B F = 1 E(β (X F E(1 B(Xβ (X F = 1 m β B β (ug 0 (udu where we have used, in he las equaliy he independence beween X and F, under P. Le us summarize his simple bu imporan resul: Proposiion 4. If X is a r.v. wih probabiliy densiy g 0, independen from F under P, and if Q is a probabiliy measure, equivalen o P on F σ(x wih Radon-Nikodým densiy β (X, 0, hen he (Q,F densiy process of X is g Q (udu := Q(X du F = 1 m β β (ug 0 (udu (10 where m β is he normalizing facor m β = β (ug 0 (udu. In paricular Q(τ du = P(τ du = g 0 (udu. The righ-hand side of (10 can be undersood as he raio of β (ug 0 (u (he change of probabiliy imes he P probabiliy densiy and a normalizing coefficien m β. One can say ha (β (ug 0 (u, 0 is he un-normalized densiy, obained by a linear ransformaion from he iniial densiy. The normalizaion facor m β = β (ug 0 (udu inroduces a nonlinear dependance of g Q (u wih respec o he iniial densiy. The example of he filering heory provides an explici form o his dependence when he maringales β (u are sochasic inegrals wih respec o a Brownian moion. Remark 1. We presen here some imporan remarks. (1 If, for any, m β = 1, hen he probabiliy measures P and Q coincide on F. In ha case, he process (β (ug 0 (u, 0 is a densiy process. (2 Le G = (G 0 be he usual righ-coninuous and complee filraion in he defaul framework (i.e. when X = τ is a non negaive r.v. generaed by F σ(τ. Similar calculaion may be made wih respec o G. The only difference is ha he condiional disribuion of τ is a Dirac mass on he se { τ}. On he se {τ > }, and under Q, he disribuion of τ admis a densiy given by:

14 14 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Q(τ du G = β (ug 0 (u 1 β (θg 0 (θdθ du. (3 This mehodology can be easily exended o a mulivariae seing: one sars wih an elemenary model, where he τ i,i = 1,...,d are independen from F, wih join densiy g(u 1,...,u d. Wih a family of non-negaive maringales β(θ 1,...,θ d, he associaed change of probabiliy provides a mulidimensional densiy process. 5.2 Filering heory The change of probabiliy approach presened in he previous subsecion 5.1 is based on he idea ha one can resric our aenion o he simple case where he random variable is independen from he filraion and use a change of probabiliy. The same idea is he building block of filering heory as we presen now. Le W be a Brownian moion on he probabiliy space (Ω,A,P, and X be a random variable independen of W, wih probabiliy densiy g 0. We denoe by dy = a(,y,xd + b(,y dw (11 he observaion process, where a and b are smooh enough o have a soluion and where b does no vanish. The goal is o compue he condiional densiy of X wih respec o he filraion F Y. The way we shall solve he problem is o consruc a probabiliy Q, equivalen o P, such ha, under Q, he signal X and he observaion F Y are independen, and o compue he densiy of X under P by means of he change of probabiliy approach of he previous secion. I is known in nonlinear filering heory as he Kallianpur-Sriebel mehodology [10], a way o linearize he problem. Noe ha, from he independence assumpion beween X and W, we see ha W is a G X = F W σ(x-maringale under P Simple case We sar wih he simple case where he dynamics of he observaion is dy = a(,xd + dw. We assume ha a is smooh enough so ha he soluion of dβ (X = β (Xa(,XdW, β 0 (X = 1 is a (P,G X -maringale, and we define a probabiliy measure Q on G X by dq = β (XdP. Then, by Girsanov s heorem, he process Y is a (Q,G X -Brownian moion, hence is independen from G0 X = σ(x, under Q. Then, we apply our change

15 Condiional Defaul Probabiliy and Densiy 15 of probabiliy mehodology, wriing wih dp = 1 β (X dq =: l (XdQ dl (X = l (Xa(,XdY, l 0 (X = 1 0 a2 (s,uds and we ge (in oher words, l (u = 1 β (u = exp( 0 a(s,udy s 1 2 from Proposiion 4 ha he densiy of X under P, wih respec o F Y, is g (u, given by P(X du F Y = g (udu = 1 m l g 0 (ul (udu where m l = E Q (l (X F Y = l (ug 0 (udu. Since and seing ( dm l = l (ua(,ug 0 (udu ( dy = m l g (ua(,udu dy â := E(a(,X F Y = g (ua(,udu, Girsanov s heorem implies ha he process B given by db = dy â d = dw + (a(,x â d is a (P,F Y Brownian moion (i is he innovaion process. From Iô s calculus, i is easy o show ha he densiy process saisfies he nonlinear filering equaion ( dg (u = g (u a(,u 1 m l dyg 0 (ya(,yl (y db = g (u(a(,u â db. (12 Remark 2. Observe ha conversely, given a soluion g (u of (12, and he process µ soluion of dµ = µ â dy, hen h (u = µ g (u is soluion of he linear equaion dh (u = h (ua(,udy General case Using he same ideas, we now solve he filering problem in he case where he observaion follows (11. Le β(x be he G X local maringale, soluion of dβ (X = β (Xσ (XdW, β 0 (X = 1 wih σ (X = a(,y,x b(,y. We assume ha a and b are smooh enough so ha β is a maringale. Le Q be defined on G X by dq = β (XdP.

16 16 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari From Girsanov s heorem, he process Ŵ defined as dŵ = dw σ (Xd = 1 b(,y dy is a (Q,G X -Brownian moion, hence Ŵ is independen from G0 X = σ(x. Being F Y -adaped, he process Ŵ is a (Q,F Y -Brownian moion, X is independen from F Y under Q, and, as menioned in Proposiion 4, admis, under Q, he probabiliy densiy g 0. We now assume ha he naural filraions of Y and Ŵ are he same. To do so, noe ha i is obvious ha FŴ F Y. If he SDE dy = b(,y dŵ has a srong soluion (e.g., if b is Lipschiz, wih linear growh hen F Y FŴ and he equaliy beween he wo filraions holds. Then, we apply our change of probabiliy mehodology, wih F Y as he reference filraion, wriing dp = l (XdQ wih dl (X = l (Xσ (XdŴ (which follows from l (X = 1 β (X and we ge ha he densiy of X under P, wih respec o FY is g (u given by wih dynamics dg (u = g (u g (u = 1 m l g 0 (ul (u ( σ (u 1 m l b(,y 1 ( a(,y,u = g (u = g (u ( a(,y,u b(,y b(,y â b(,y dyg 0 (yσ (yl (y db dyg (ya(,y,y db db. (13 Here B is a (P,F Y Brownian moion (he innovaion process given by ( a(,y,x db = dw + â d, b(,y b(,y where â = E(a(,Y,X F Y. Proposiion 5. If he signal X has probabiliy densiy g 0 (u and is independen from he Brownian moion W, and if he observaion process Y follows dy = a(,y,xd + b(,y dw, hen, he condiional densiy of X given F Y is P(X du F Y = g (udu = 1 m l g 0 (ul (udu (14

17 Condiional Defaul Probabiliy and Densiy 17 where l (u = exp( a(s,y s,u 0 b 2 (s,y s dy s 2 1 a 2 (s,y s,u 0 b 2 (s,y s ds, m l = l (ug 0 (udu, and is dynamics is given in ( Gaussian filer We apply our resuls o he well known case of Gaussian filer. Le W be a Brownian moion, X a random variable (he signal wih densiy probabiliy g 0 a Gaussian law wih mean m 0 and variance γ 0, independen of he Brownian moion W and le Y (he observaion be he soluion of dy = (a 0 (,Y + a 1 (,Y Xd + b(,y dw, Then, from he previous resuls, he densiy process g (u is of he form ( 1 a 0 (s,y s + a 1 (s,y s u exp b 2 dy 1 ( a0 (s,y s + a 1 (s,y s u 2 ds g 0 (u (s,y s 2 b(s,y s m l 0 The logarihm of g (u is a quadraic form in u wih sochasic coefficien, so ha g (u is a Gaussian densiy, wih mean m and variance γ (as proved already by Lipser and Shiryaev [14]. A edious compuaion, purely algebraic, shows ha γ 0 γ =, m 1 + γ a 2 = m (s,y s 0 0 b 2 (s,y s ds wih db = dw + a 1(,Y b(,y (X E(X F Y d. 0 0 γ s a 1 (s,y s b(s,y s db s Back o he Gaussian example Secion 3.1: In he case where he coefficiens of he process Y are deerminisic funcions of ime, i.e., dy = (a 0 ( + a 1 (Xd + b(dw he variance γ( is deerminisic and he mean is an F Y -Gaussian maringale γ( = γ γ 0 0 α2 (sds, m = m 0 + γ(sα(sdb s 0 where α = a 1 /b. Furhermore, F Y = F B. Choosing f (s = γ(sa 1(s b(s in he example of Secion 3.1 leads o he same condiional law (wih m 0 = 0; indeed, i is no difficul o check ha his choice of parameer leads o f 2 (sds = σ 2 ( = γ( so ha he wo variances are equal. The similariy beween filering and he example of Secion 3.1 can be also explained as follows. Le us sar from he seing of Secion 3.1 where X = 0 f (sdb s and inroduce G X = F B σ(x, where B is he given Brownian moion. Sandard

18 18 N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari resuls of enlargemen of filraion (see Jacod [9] show ha m s X W := B + 0 σ 2 (s f (sds is a G X -BM, hence is a G W -BM independen of X. So, he example presened in Secion 3.1 is equivalen o he following filering problem: he signal is X a Gaussian variable, cenered, wih variance γ(0 = 0 f 2 (sds and he observaion ( dy = f (Xd + f 2 (sds dw = f (Xd + σ 2 (dw. References 1. Amendinger, J. (1999: Iniial enlargemen of filraions and addiional informaion in financial markes, Phd hesis, Technischen Universiä Berlin. 2. Carmona, R. (2010: Emissions opion pricing, slides, Heidelberg. 3. Chaleya-Maurel, M. and T. Jeulin (1985: Grossissemen Gaussien de la filraion Brownienne, Lecure Noes in Mahemaics, 1118, Springer-Verlag, pp El Karoui, N., M. Jeanblanc and Y. Jiao (2010: Wha happens afer a defaul: he condiional densiy approach, Sochasic Processes and heir Applicaions, 120, pp Fermanian, J.D., and O. Vigneron (2010: On break-even correlaion: he way o price srucured credi derivaives by replicaion, Preprin. 6. Filipović, D., L. Overbeck and T. Schmid (2009: Dynamic CDO erm srucure modeling, Forhcoming Mahemaical Finance. 7. Filipovic, D., Hughson, L. and Macrina A. (2010 Condiional Densiy Models for Asse Pricing, Preprin. 8. Grorud A. and M. Ponier (2001: Asymmerical informaion and incomplee markes, Inernaional Journal of Theoreical and Applied Finance, 4, pp Jacod, J. (1985: Grossissemen iniial, hypohèse (H e héorème de Girsanov, Lecure Noes in Mahemaics, 1118, Springer-Verlag, pp Kallianpur, G. and C. Sriebel (1968. Esimaion of Sochasic Sysems: Arbirary Sysem Process wih Addiive Whie Noise Observaion Errors Ann. Mah. Sais. 39 (3, Jeanblanc, M. and S. Song (2010: Explici model of defaul ime wih given survival probabiliy, Preprin. 12. Jeanblanc, M. and S. Song (2010: Defaul imes wih given survival probabiliy and heir F-maringale decomposiion formula, Preprin. 13. Keller-Ressel, M. Papapanoleon, A. and Teichman, J. (2010: The Affine Libor Models, preprin. 14. Lipser, R.S.R. and A.N. Shiryaev (2001: Saisics of Random Processes, II Applicaions, Springer, second ediion. 15. Masumoo, H. and M. Yor (2001: A relaionship beween Brownian moions wih opposie drifs via cerain enlargemens of he Brownian filraion Osaka J. Mah. 38, pp wih 16. Meyer, P-A. (1973 Sur un problème de filraion, Séminaire de probabiliés VII, , Lecure Noes in mahemaics 321, Springer. 17. Yor, M. (1985: Grossissemen de filraions e absolue coninuié de noyaux, Lecure Noes in Mahemaics, 1118, Springer-Verlag, pp