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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 15 (2010), Paper no. , pages 1743–1771. Journal URL

http://www.math.washington.edu/~ejpecp/

A class of

F

_{-doubly stochastic Markov chains}

∗

Jacek Jakubowski

Institute of Mathematics, University of Warsaw Banacha 2, 02-097 Warszawa, Poland

and

Faculty of Mathematics and Information Science Warsaw University of Technology Plac Politechniki 1, 00-661 Warszawa, Poland

E-mail:jakub@mimuw.edu.pl

Mariusz Niew˛

egłowski

Faculty of Mathematics and Information Science, Warsaw University of Technology Plac Politechniki 1, 00-661 Warszawa, Poland

E-mail:M.Nieweglowski@mini.pw.edu.pl

Abstract

We define a new class of processes, very useful in applications, F_{-doubly stochastic Markov}

chains which contains among others Markov chains. This class is fully characterized by some martingale properties, and one of them is new even in the case of Markov chains. Moreover a predictable representation theorem holds and doubly stochastic property is preserved under natural change of measure.

Key words: F_{-doubly stochastic Markov chain, intensity, Kolmogorov equations, martingale}

characterization, sojourn time, predictable representation theorem.

AMS 2000 Subject Classification:Primary 60G99; Secondary: 60G55, 60G44, 60G17, 60K99. Submitted to EJP on March 29, 2010, final version accepted September 28, 2010.

∗_{Research supported in part by Polish KBN Grant P03A 034 29 “Stochastic evolution equations driven by Lévy noise”}

1

Introduction

Our goal is to find a class of processes with good properties which can be used for modeling

differ-ent kind of phenomena. So, in Section 2 we introduce a class of processes, which we callF_-doubly

stochastic Markov chains. The reason for the name is that there are two sources of uncertainty in their definition, so in analogy to Cox processes, called doubly stochastic Poisson processes, we chose

the name “F_{-doubly stochastic Markov chains”. This class contains Markov chains, compound}

Pois-son processes with jumps inZ_{, Cox processes, the process of rating migration constructed by Lando}

[16]and also the process of rating migration obtained by the canonical construction in Bielecki and

Rutkowski [3]. We stress that the class of F_{-doubly stochastic Markov chains contains processes}

that are not Markov. In the following we use the shorthand “F_{–DS Markov chain” for the “}F_-doubly

stochastic Markov chain”. Note that anF-doubly stochastic Markov chain is a different object than a

doubly stochastic Markov chain which is a Markov chain with a doubly stochastic transition matrix.

On the end of this section we give examples of F-doubly stochastic Markov chains. Section 3 is

devoted to investigation of basic properties ofF_{-doubly stochastic Markov chains. In the first part}

we prove that anF_{–DS Markov chainC is a conditional Markov chain and that any}F_{-martingale is}

aF_∨FC_{-martingale. This means that the immersion property for(}F_,F_∨FC_{), so called hypothesis}

H, holds. Moreover, the family of transition matrices satisfies the Chapman-Kolmogorov equations.

In the second part and until the end of the paper we restricted ourselves to a class ofF_{–DS Markov}

chains with values in a finite setK ={1, . . . ,K}. We introduce the notion of intensity of anF_–DS

Markov chain and formulate conditions which ensure its existence. In section 4 we prove that an

F_{–DS Markov chain C with intensity is completely characterized by the martingale property of the}

compensated process describing the position ofC as well as by the martingale property of the

com-pensated processes counting the number of jumps of C from one state to another (Theorem 4.1).

The equivalence between points iii) and iv) in Theorem 4.1 in a context of Markov chains has not yet been known according to the best of our knowledge. In a view of the above characterizations,

theF–DS Markov chains can be described as the class of processes that behave like time

inhomoge-neous Markov chains conditioned onF_∞. Moreover these equivalences and the fact that the class of

F-DS Markov chains contains the most ofFconditionalF∨FC Markov chains used for modeling in

finance, indicate that the class ofF_{-DS Markov chains is a natural, and very useful in applications,}

subspace ofF_conditionalF_∨FC _{Markov chains. Next, an}F_{–DS Markov chain with a given}

inten-sity is constructed. Section 5 is devoted to investigation of properties of distribution ofC and the

distribution of sojourn time in fixed state junder assumption thatC does not stay in jforever. We

find some conditional distributions which among others allows to find a conditional probability of transition from one state to another given that transition occurs at known time. In Section 6 a kind of predictable representation theorem is given. Such theorems are very important for applications,

for example in finance and backward stochastic differential equations (see Pardoux and Peng[18]

and El Karoui, Peng and Quenez[7]). By the way we prove thatF_{–DS Markov chain with intensity}

and arbitraryF_{adapted process do not have jumps at the same time. Our results allows to describe}

and investigate a credit risk for a single firm. In such case the predictable representation theorem

(Theorem 6.5) generalize the Kusuoka theorem[14]. In the last section we study how replacing the

probability measure by an equivalent one affects the properties of anF–DS Markov chain.

Summing up, the class ofF_{–DS Markov chains is a class with very good and desirable properties in}

used for modeling a credit rating migration process and which contains processes usually taken for this purpose. This allows us to include rating migration in the process of valuation of defaultable claims and generalize the case where only two states are considered: default and non-default (for

such generalizations see Jakubowski and Niew˛egłowski[12]). These processes can also be applied

in other fields where system evolves in a way depending on random environment, e.g., in insurance.

2

Definition and examples

In this section we introduce and investigate a new class of processes, which will be calledF_-doubly

stochastic Markov chains. This class contains among others Markov chains and Cox processes. We

assume that all processes are defined on a complete probability space (Ω,F,P). We also fix a

filtrationF_{satisfying usual conditions, which plays the role of a reference filtration.}

Definition 2.1. A càdlàg process C is called anF_{–doubly stochastic Markov chain with state space}

K ⊂Z={. . . ,−1, 0, 1, 2, . . .}if there exists a family of stochastic matricesP(s,t) = (p_i,j(s,t))i,j∈K for 0≤s≤t such that

1) the matrixP(s,t)isF_t–measurable, andP(s,·)isF_{progressively measurable,}

2) for anyt ≥s≥0 and everyi,j∈ K we have

P(C_t= j| F_∞∨ F_sC)✶_{C

s=i}=✶{Cs=i}pi,j(s,t). (2.1)

The processPwill be called the conditional transition probability process ofC.

The equality (2.1) implies that P(t,t) = I _{a.s. for all t ≥ 0. Definition 2.1 extends the notion}

of Markov chain with continuous time (when F∞ is trivial). A process satisfying 1) and 2) is

called a doubly stochastic Markov chain by analogy with Cox processes (doubly stochastic Poisson processes). In both cases there are two sources of uncertainty. As mentioned in the Introduction, we

use the shorthand “F_{–DS Markov chain” for the “}F_{–doubly stochastic Markov chain”. Now, we give}

a few examples of processes which areF_{–DS Markov chains.}

Example 1. (Compound Poisson process) LetX be a compound Poisson process with jumps inZ_,

i.e., X_t = PNt

i=1Yi, where N is a Poisson process with intensity λ, Yi is a sequence of independent

identically distributed random variables with values inZ_{and distribution ν, moreover}(Yi)i≥1 and

N are independent. By straightforward calculations we see that:

a)X is anF_{–DS Markov chain with}F=FN _and

p_i,j(s,t) =ν⊗(Nt−Ns)_(j−i).

b)X is anF_{–DS Markov chain with respect to} F_{being the trivial filtration, and with deterministic}

transition matrix given by the formula

pi,j(s,t) =

∞

X

k=0

ν⊗k(j−i)[λ(t−s)]

k

k! e

−λ(t−s)_.

Example 2. (Cox process) The processC with càdlàg trajectories such that

P(Ct−Cs=k| F∞∨ FsC) =e

−Rt

sλudu Rt

s λudu

k

k! (2.2)

for someF–adapted processλsuch thatλ≥0,Rt

0λsds<∞for allt ≥0 we call a Cox process. This

definition implies that

P(Ct−Cs=k| F∞∨ FsC) =P(Ct−Cs=k| F∞),

so the increments and the past (i.e. FC

s ) are conditionally independent givenF∞. Therefore for

j≥i,

P(Ct = j| F∞∨ FsC)✶{Cs=i}=✶{Cs=i}P(Ct−Cs= j−i| F∞∨ F

C s )

=✶_{C

s=i}e

−Rt

s λudu Rt

s λudu

j−i

(j−i)! .

Thus

p_i,j(s,t) =

  

Rt sλudu

j−i

(j−i)! e

−R_stλudu _{for j≥i,}

0 for j<i,

satisfy conditions 1) and 2) of Definition 2.1. A Cox processC is therefore an F–DS Markov chain

with K = N_{. Usually in a definition of Cox process there is one more assumption on intensity,}

namelyR∞

0 λsds=∞a.s. Under this assumptionC has properties similar to Poisson process and is

called conditional Poisson process (or doubly stochastic Poisson process). So our definition of Cox process is a slight generalization of a classical one.

Example 3. (Time changed discrete Markov chain) Assume that ¯C is a discrete time Markov chain with values inK ={1, . . . ,K}, N is a Cox process and the processes(C¯_k)_k≥0 and(N_t)_t≥0 are

inde-pendent and conditionally indeinde-pendent givenF_∞. Then the processC_t := C¯_N

t is anF–DS Markov

chain (see Jakubowski and Niew˛egłowski[11, Theorem 7 and 9]).

Example 4. (Truncated Cox process) Simple calculations give us that the processC_t:=minN_t,K ,

whereN is a Cox process andK∈N_{, is an}F_{–DS Markov chain with state spaceK ={0, . . . ,K}.}

3

Basic properties

3.1

General case

In this subsection we consider the case of an arbitrary countable state space K. We study basic

properties of transition matrices and martingale invariance property of F with respect to F∨FC.

Moreover we prove that the class of F_{-DS Markov chains is a subclass of} F _conditional F_∨FC

Markov chains.

For the rest of the paper we assume thatC₀=i₀for somei₀∈ K. We start the investigation ofF_–DS

Markov chains from the very useful lemma describing conditional finite–dimensional distributions

Lemma 3.1. If C is anF_{–DS Markov chain, then}

Proof. The proof is by induction onn. For n=1 the above formula obviously holds. Assume that it holds forn, arbitrary 0≤u₀ ≤. . . ≤u_n and(i₀, . . . ,i_n)∈ Kn+1. We will prove it for n+1 and

left hand side of (3.1) is equal to

E ✶¦_C

F_{–DS Markov chain, we obtain}

I=E

and this completes the proof.

Remark 3.2. Of course, if (3.1) holds, then condition 2) of Definition 2.1 ofF_{–DS Markov chain is}

satisfied. Therefore(3.1)can be viewed as an alternative to equality(2.1).

As a consequence of our assumption thatC₀=i₀ and Lemma 3.1 we obtain

The following hypothesis is standard for many applications e.g., in credit risk theory.

HYPOTHESIS H: For every bounded F_∞–measurable random variable Y and for each t ≥ 0 we have

E(Y | F_t∨ F_tC) =E(Y | F_t).

It is well known that hypothesis H for the filtrations F _and FC _{is equivalent to the martingale}

invariance property of the filtrationF_{with respect to}F∨FC _{(see Brémaud and Yor}[4]or Bielecki

and Rutkowski[3, Lemma 6.1.1, page 167]) i.e. anyF_{martingale is an}F∨FC _{martingale. From}

results in[4]one can deduce that hypothesis H implies (3.3). So, by Proposition 3.3, we can expect

that for a class ofF_{-DS Markov chains hypothesis H is satisfied. Indeed,}

Proposition 3.4. If C is anF_{–DS Markov chain then hypothesis H holds.}

Proof. According to[11, Lemma 2] we know that hypothesis H is equivalent to (3.3). This and Proposition 3.3 complete the proof.

Now, we will show that eachF_{–DS Markov chain is a conditional Markov chain (see[3, page 340]}

for a precise definition). For an example of a process which is anF_conditionalF_∨FC _{Markov chain}

and is not anF_{–DS Markov chain we refer to Section 3 of Becherer and Schweizer}[2].

Proposition 3.5. Assume that C is anF_{–DS Markov chain. Then C is an}F_conditionalF_∨FC_Markov

chain.

Proof. We have to check that fors≤t,

P(Ct=i| Fs∨ FsC) =P(Ct=i| Fs∨σ(Cs)).

By the definition of anF_{–DS Markov chain,}

P(Ct=i| Fs∨ FsC) =E(E(✶{Ct=i} | F∞∨ F

C

s )| Fs∨ FsC)

=E X

j∈K

✶_{Cs=j}pj,i(s,t)|Fs∨ F

C s

= X

j∈K

✶_{Cs=j}E €

p_j,i(s,t)|F_s∨ F_sCŠ=I.

But

I= X

j∈K

✶_{Cs=j}E €

p_j,i(s,t)| F_sŠ

since hypothesis H holds (Proposition 3.4), and this ends the proof.

Now we define processesHi, which play a crucial role in our characterization of the class ofF_–DS

Markov chains:

H_ti:=✶_{C

t=i} (3.4)

fori∈ K. The process H_ti tells us whether at time t the processC is in statei or not. Let H_t := (Hα_t)⊤_α∈K, where⊤denotes transposition.

We can express condition (2.1) in the definition of anF_{–DS Markov chain, fort}≤u, in the form

or equivalently

E(H_uj | F_∞∨ F_tC) = X

i∈K

H_tip_i,j(t,u)

and so (2.1) is equivalent to

E€H_u| F∞∨ FtC

Š

=P(t,u)⊤H_t. (3.5)

The next theorem states that the family of matricesP(s,t) = [p_i,j(s,t)]_i,j∈K satisfies the Chapman-Kolmogorov equations.

Theorem 3.6. Let C be anF_{–DS Markov chain with transition matrices P}(s,t). Then for any u≥t≥s we have

P(s,u) =P(s,t)P(t,u) a.s., (3.6)

so on the setC_s=i we have

p_i,j(s,u) = X

k∈K

p_i,k(s,t)p_k,j(t,u).

Proof. It is enough to prove that (3.6) holds on each setCs=i ,i∈ K. So we have to prove that

H_s⊤P(s,u) =H_s⊤P(s,t)P(t,u).

By the chain rule for conditional expectation, equality (3.5) and the fact that P(t,u) is F_∞–

measurable it follows that fors≤t≤u,

P(s,u)⊤Hs =E

€

Hu| F∞∨ FsC

Š

=E€E€Hu| F∞∨ FtC

Š

| F_∞∨ FC s

Š

=E€P(t,u)⊤H_t| F∞∨ F_sC

Š

=P(t,u)⊤E€H_t | F∞∨ F_sC

Š

=P(t,u)⊤P(s,t)⊤Hs= (P(s,t)P(t,u))⊤Hs,

and this completes the proof.

3.2

The case of a finite state space

From this subsection, until the end of the paper we restrict ourselves to a finite setK, i.e. K =

{1, . . . ,K}, withK<∞. It is enough for the most of applications, e.g., in finance to model markets with rating migrations we use processes with values in a finite set. In this caseH_t:= (H1_t . . . ,H_tK)⊤. We recall the standing assumption thatC₀=i₀for somei₀∈ K.

The crucial concept in this subsection and in study of properties of F_{–DS Markov chains is the}

concept of intensity, analogous to that for continuous time Markov chains.

Definition 3.7. We say that an F_{–DS Markov chainC has an intensity if there exists an}F_–adapted

matrix-valued processΛ = (Λ(s))_s≥0= (λi,j(s))s≥0 such that:

1) Λis locally integrable, i.e. for anyT >0

Z

]]0,T]]

X

i∈K

2) Λsatisfies the conditions:

λ_i,j(s)≥0 ∀i,j∈ K,i6= j, λ_i,i(s) =−X

j6=i

λ_i,j(s) ∀i∈ K, (3.8)

the Kolmogorov backward equation: for allv≤t,

P(v,t)−I=

Z t

v

Λ(u)P(u,t)du, (3.9)

the Kolmogorov forward equation: for allv≤t,

P(v,t)−I₌ Z t

v

P(v,u)Λ(u)du. (3.10)

A processΛsatisfying the above conditions is called an intensity of theF_{–DS Markov chainC.}

It is not obvious that if we have a solution to the Kolmogorov backward equation then it also solves the Kolmogorov forward equation. This fact follows from the theory of differential equa-tions, namely we have

Theorem 3.8. Assume thatΛis locally integrable. Then the random ODE’s

d X(t) =−Λ(t)X(t)d t, X(0) =I_{, (3.11)}

d Y(t) =Y(t)Λ(t)d t, Y(0) =I_{, (3.12)}

have unique solutions, and in addition X(t) = Y−1(t). Moreover, Z(s,t) := X(s)Y(t) is a unique solution to the Kolmogorov forward equation(3.10)and to the Kolmogorov backward equation(3.9).

Proof. The existence and uniqueness of solutions of the ODE’s (3.11) and (3.12) follows by standard arguments. To deduce thatX(t) =Y−1(t)we apply integration by parts to the productX(t)Y(t)of finite variation continuous processes and get

d(Y(t)X(t)) = Y(t)d X(t) + (d Y(t))X(t)

= Y(t) (−Λ(t)X(t)d t) +Y(t)Λ(t)X(t)d t=0.

From Y(0) =X(0) =I _{we haveY(t)X(t) =}I_{, which means thatX(t) is a right inverse matrix of}

Y(t). It is also the left inverse, since we are dealing with square matrices.

Now we check that Z(s,t) are solutions to the Kolmogorov backward equation and also the

Kol-mogorov forward equation. Indeed,

dsZ(s,t) = (d X(s))Y(t) =−Λ(s)X(s)Y(t)ds=−Λ(s)Z(s,t)ds,

and

d_tZ(s,t) =X(s)d Y(t) =X(s)Y(t)Λ(t)d t=Z(s,t)Λ(t)d t.

Corollary 3.9. If anF_{–DS–Markov chain C has intensity, then the conditional transition probability}

process P(s,t)is jointly continuous at(s,t)for s≤t.

Proof. This follows immediately from Theorem 3.8, since

P(s,t) =X(s)Y(t)

and both factors are continuous insandt, respectively.

Theorem 3.8 gives us existence and uniqueness of solutions to (3.9) and (3.10). Next proposition provides the form of these solutions.

Proposition 3.10. Let a matrix process(Λ(s))_s≥0satisfies conditions(3.7)and(3.8). Then the solution to(3.9)is given by the formula

P(v,t) =I₊

∞

X

n=1

Z t

v

Z t

v1

. . .

Z t

vn−1

Λ(v₁). . .Λ(v_n)d v_n. . .d v₁,

and the solution to(3.10)is given by

P(v,t) =I+

∞

X

n=1

Z t

v

Z v1

v

. . .

Z vn−1

v

Λ(v_n). . .Λ(v₁)d v_n. . .d v₁.

Proof. It is a special case of Theorem 5 in Gill and Johansen [8], see also Rolski et al. [20, § 8.4.1].

Proposition 3.11. Let P = (P(s,t)), 0 ≤ s ≤ t, be a family of stochastic matrices such that the matrix P(s,t)is F_t–measurable, and P(s,·) isF_{–progressively measurable. Let Λ = (Λ(s))s≥0 be an} F_{–adapted matrix-valued locally integrable process such that the Kolmogorov backward equation(3.9)}

and Kolmogorov forward equation(3.10)hold. Then

i) For each s∈[0,t]there exists an inverse matrix of P(s,t)denoted by Q(s,t).

ii) There exists a version of Q(·,t) such that the process Q(·,t) is a unique solution to the integral (backward) equation

dQ(s,t) =Q(s,t)Λ(s)ds, Q(t,t) =I_{. (3.13)}

This unique solution is given by the following series:

Q(s,t) =I₊

∞

X

k=1 (−1)k

Z t

s

Z t

u1

. . .

Z t

uk−1

Λ(u_k). . .Λ(u₁)du_k. . .du₁. (3.14)

iii) There exists a version of Q(s,·) such that the process Q(s,·) is a unique solution to the integral (forward) equation

dQ(s,t) =−Λ(t)Q(s,t)d t, Q(s,s) =I_{. (3.15)}

This unique solution is given by the following series:

Q(s,t) =I₊

∞

X

k=1 (−1)k

Z t

s

Z u1

s

. . .

Z uk−1

s

Proof. i) From Theorem 3.8 it follows that P(s,t) = X(s)Y(t), where X, Y are solutions to the

random ODE’s (3.11), (3.12) and moreoverY =X−1. Therefore the matrix P(s,t)is invertible and

its inverseQ(s,t)is given byQ(s,t) =X(t)Y(s).

ii) We differentiateQ(s,t)with respect to the first argument and obtain

d_sQ(s,t) =X(t)d Y(s) =X(t)Y(s)Λ(s)ds=Q(s,t)Λ(s)ds.

MoreoverQ(t,t) =X(t)Y(t) =I_{. SoQ}(·,t)solves (3.13). Uniqueness of solutions to (3.13) follows by standard arguments based on Gronwall’s lemma. Formula (3.14) is derived analogously to a similar formula forP(s,t)in § 8.4.1, page 348 of Rolski et al.[20].

iii) The proof of iii) is analogous to that of ii).

In the next theorem we prove that under some conditions imposed on the conditional transition

probability process P, an F–DS Markov chain C has intensity. Using this intensities we can

con-struct martingale intensities for different counting processes building in natural way from F–DS

Markov chains (see Theorem 4.1). Therefore, Theorem 3.12 is in a spirit of approaches of Del-lacherie (Meyer’s Laplacian see DeDel-lacherie[6], Guo and Zeng[9]) and of Aven[1]. Theorem 3.12 generalizes forF_{–DS Markov chains results from[1].}

Theorem 3.12(Existence of Intensity). Let C be anF_{–DS–Markov chain with conditional transition}

probability process P. Assume that

1) P as a matrix–valued mapping is measurable, i.e.

P:(R₊×R₊×Ω,B(R₊×R₊)⊗ F)→(RK×K_,B(RK×K)).

2) There exists a version of P which is continuous in s and in t. 3) For every t≥0the following limit exists almost surely

Λ(t):=lim

h↓0

P(t,t+h)−I

h , (3.17)

and is locally integrable. ThenΛis the intensity of C.

Proof. By assumption 3 the processΛis well defined and by 1) it is(R₊×Ω,B(R₊)⊗F)measurable.

By assumption 3, Λ(t) is F_t+–measurable, but F _{satisfies the usual conditions, so Λ(t) is Ft–}

measurable. It is easy to see that (3.8) holds.

It remains to prove that equations (3.9) and (3.10) are satisfied. Fixt. From the assumptions and

the Chapman–Kolmogorov equations it follows that forv≤v+h≤t,

P(v+h,t)−P(v,t) = P(v+h,t)−P(v,v+h)P(v+h,t)

= −(P(v,v+h)−I_)P(v+h,t),

so

P(v+h,t)−P(v,t)

h =−

(P(v,v+h)−I₎

Therefore ∂_∂+_vP(v,t) exists for a.e. v and is(R_+×R_+×Ω,B(R_+×R_{+)⊗ F) measurable. Using}

assumptions 2 and 3 we finally have

∂+

∂vP(v,t) =−Λ(v)P(v,t), P(t,t) =I. (3.18)

Since elements of P(u,t) are bounded by 1, andΛis integrable over [v,t](by assumption 3), we see that ∂_∂+

uP(u,t)is Lebesgue integrable on[v,t], so (see Walker[21])

I−P(v,t) =

Z t

v

∂+

∂uP(u,t)du.

Hence, by (3.18), we have

P(v,t)−I₌ Z t

v

Λ(u)P(u,t)du,

and this is exactly the Kolmogorov backward equation (3.9).

Similar arguments apply to the case of right derivatives ofP(v,t)with respect to the second variable. Since forv≤t≤t+h,

P(v,t+h)−P(v,t) =P(v,t)(P(t,t+h)−I_),

we obtain

∂+

∂tP(v,t) =P(v,t)Λ(t), P(v,v) =I,

which gives (3.10),

P(v,t)−I₌ Z t

v

P(v,u)Λ(u)du.

Now, we find the intensity for the processes described in Examples 3 and 4.

Example 5. IfCt =min

Nt,K , whereN is a Cox process with càdlàg intensity process ˜λ, thenC

has the intensity process of the form

λ_i,j(t) =

  

−λ˜(t) fori= j∈ {0, . . .K−1}; ˜

λ(t) for j=i+1 withi∈ {0, . . .K−1};

0 otherwise.

Example 6. If anF_{–DS Markov chain C is defined as in Example 3 with a discrete time Markov}

chain ¯C with a transition matrix Pand a Cox processN with càdlàg intensity process ˜λ, then

P(s,t) =e(P−I)

Rt sλ˜(u)du

(see Theorem 9 in[11]), so the intensity of C is given by

4

Martingale properties

We prove that under natural assumptions, belonging of a process X to the class of F-DS Markov

chains is fully characterized by the martingale property of some processes strictly connected withX.

These nice martingale characterizations allow to check by using martingales whenever process is an

F_{-DS Markov chain. To do this, we introduce a filtration}Gb_{= (G}b_{t)t≥0, where}

b

G_t:=F_∞∨ F_tC. (4.1)

Theorem 4.1. Let(C_t)_t≥0 be aK–valued stochastic process and(Λ(t))_t≥0be a matrix valued process satisfying(3.7)and(3.8). The following conditions are equivalent:

i) The process C is anF_{–DS Markov chain with intensity processΛ.}

ii) The processes

M_ti:=H_ti−

Z

]]0,t]]

λ_Cu,i(u)du, i∈ K, (4.2)

areGb _{local martingales.}

iii) The processes Mi,j defined by

M_ti,j:=Hi_t,j−

Z

]]0,t]]

H_siλ_i,j(s)ds, i,j∈ K, i6= j, (4.3)

where

H_ti,j:=

Z

]]0,t]]

H_ui₋d H_uj, (4.4)

areGb _{local martingales.}

iv) The process L defined by

Lt:=Q(0,t)⊤Ht, (4.5)

where Q(0,t)is a unique solution to the random integral equation

dQ(0,t) =−Λ(t)Q(0,t)d t, Q(0, 0) =I_{, (4.6)}

is aGb _{local martingale.}

Proof. Denoting byM the vector valued process with coordinatesMi, we can writeM as follows

Mt:=Ht−

Z

]]0,t]]

Λ⊤(u)Hudu. (4.7)

"i)⇒ii)" Assume that C is anF_{–DS Markov chain with intensity process (Λ(t))t≥0. Fix t ≥0 and}

set

N_s:=P(s,t)⊤H_s for 0≤s≤t. (4.8)

The processC satisfies (3.5), which is equivalent to N being a Gb _{martingale for 0}≤s≤ t. Using

integration by parts and the Kolmogorov backward equation (3.9) we find that

d N_s= (d P(s,t))⊤H_s+P⊤(s,t)d H_s (4.9)

Hence, using Q(s,t) (the inverse of P(s,t); we know that it exists, see Proposition 3.11iii)), we conclude that

M_s−M0=

Z

]]0,s]]

Q⊤(u,t)P⊤(u,t)d M_u=

Z

]]0,s]]

Q⊤(u,t)d N_u.

Therefore, by theGb martingale property ofN, we see thatM is aGb local martingale.

"ii)⇒i)" Assume that the processM associated withC andΛis aGb _{martingale. Fixt≥0. To prove}

thatCis anF_{–DS Markov chain it is enough to show that for some process(P(s,t))0≤s≤tthe process}

N defined by (4.8) is aGb _{martingale on} [0,t]. Let P(s,t):= X(s)Y(t) withX, Y being solutions

to the random ODE’s (3.11) and (3.12). We know that P(·,t) satisfies the integral equation (3.9)

(see Theorem 3.8). We also know thatP(s,t) isF_t–measurable (Remark 3.10) and continuous in

t, henceF_{progressively measurable. Using the same arguments as before, we find that (4.9) holds.}

So, using the martingale property of M we see that N is a local martingale. The definition of N

implies thatN is bounded (sinceH andPare bounded, see Last and Brandt[17, §7.4]). Therefore

N has an integrable supremum, so it is a Gb martingale, which implies that C is anF–DS Markov

chain with transition matrixP. From Theorem 3.8 it follows thatΛis the intensity matrix process

ofC.

"ii)⇔ iii)" and "iii)⇔ i v)" These equivalences follows from Lemmas 4.3 and 4.4 below, respec-tively, withA=Gb _{given by (4.1).}

The proof is complete.

Remark 4.2. The equivalence between i) and ii) in the above proposition corresponds to a well known martingale characterization of a Markov chain with a finite state space. In a view of this character-ization of F_{–DS Markov chains with intensities, they can be described as the class of processes that}

conditioned onF∞behave as time inhomogeneous Markov chains with intensities. Hence, we can also

see that the nameF_{–doubly stochastic Markov chain well describe this class of processes. The}

equiva-lence between iii) and iv) in a context of Markov chains has not yet been known according to the best of our knowledge.

The equivalence between points ii), iii) and iv) in Theorem 4.1 is a simple consequence of slightly more general results, which we formulate in two separate lemmas. It is worth to note that equiva-lences in lemmas below follow from general stochastic integration theory and in the proofs we do not use the doubly stochastic property.

Lemma 4.3. LetA_{be a filtration such thatΛand H are adapted to}A_{. Assume thatΛsatisfies(3.7)}

and(3.8). The processes Mi, defined by(4.2)for i ∈ K, areA_{local martingales if and only if for all}

i,j∈ K, i6= j, the processes Mi,jdefined by(4.3)areA_{local martingales.}

Proof. ⇒Fixi6= j, i,j∈ K. Using the definition ofM_ti,j andM_ti we have

M_ti,j=

Z

]]0,t]]

H_ui₋d H_uj−

Z

]]0,t]]

H_uiλ_i,j(u)du=

Z

]]0,t]]

H_ui₋d H_uj−

Z

]]0,t]]

H_uiλ_C

u,j(u)du

=

Z

]]0,t]]

H_ui₋d H_uj−

Z

]]0,t]]

H_ui₋λ_Cu,j(u)du=

Z

]]0,t]]

HenceMi,jis anA_{local martingale, sinceM}j _{is one andH}i

u−is a bounded process.

⇐Assume that theM_ti,j areA_{local martingales for alli}6= j, i,j∈ K. First notice that Hi can be obtained fromHj,i by the formula

H_ti=H₀i +X

and thereforeMi is anA_{local martingale for eachi∈ K as a finite sum of}A_{local martingales.}

The processHi,jdefined by (4.4) counts the number of jumps from stateito jover the time interval

]]0,t]]. Indeed, it is easy to see that

Hi_t,j= X 0<u≤t

H_ui₋H_uj.

Lemma 4.4. LetA_{be a filtration such thatΛand H are adapted to}A_{. Assume thatΛsatisfies(3.7)}

and (3.8). The processes Mi, i ∈ K, are A_{local martingales if and only if the process L defined by}

(4.5)is anA_{local martingale.}

Proof. Integration by parts formula gives

indeed

d Lt=Q(0,t)⊤d Ht+d(Q(0,t)⊤)Ht =Q(0,t)⊤d Ht−Q(0,t)⊤Λ(t)⊤Htd t=Q(0,t)⊤d Mt.

We know thatQ isA_{predictable and locally bounded. So, if M is an}A_{local martingale, then L is}

also anA_{local martingale.}

On the other hands, takingP(0,t)as an unique solution to the integral equation

d P(0,t) =P(0,t)Λ(t)d t, P(0, 0) =I

and noting thatP(0,t)is the inverse ofQ(0,t)(see Proposition 3.11) we have

P(0,t)⊤d L_t =P(0,t)⊤Q(0,t)⊤d M_t=d M_t.

Therefore, if Lis anA_{local martingale, thenM is also an}A_{local martingale.}

Corollary 4.5. If C is anF_{–DS Markov chain, then M}i _areG_{local martingales with}G_t=F_t∨ FC t .

Proof. This follows from the fact that theMi are adapted toG, andGis a subfiltration ofGb.

Remark 4.6. The process C obtained by the canonical construction in[3]is an F_{–DS Markov chain.}

This is a consequence of Theorem 4.1, becauseΛin the canonical construction is bounded and calcula-tions analogous to those in[3, Lemma 11.3.2 page 347]show that Mi areGb _{martingales. In a similar}

way one can check that if C is a process of rating migration given by Lando[16], then Mi areGb _local

martingales, so C is anF_{–DS Markov chain.}

The martingaleM is orthogonal to all square integrableF_martingales.

Proposition 4.7. Let C be an F_{–DS Markov chain. The martingale M given by (4.7) is strongly}

orthogonal to all square integrableF_martingales.

Proof. Denote by N an arbitrary Rd_{–valued, square integrable}F _{martingale. Since C is an} F_–DS

Markov chain we have, by Proposition 3.5, thatH hypothesis holds and thereforeN is alsoF∨FC

martingale. SinceM is anF∨FC _{martingale, we need to show that the processN}j_Mi _{is an}F∨FC

martingale for everyi∈ K and j∈ {1, . . . ,d}. Fix arbitraryt≥0 and anys≤t, then

E N_tjM_tiF_s∨ F_sC=EN_tjE€M_tiF_∞∨ F_sCŠ F_s∨ F_sC

=E

N_tjM_siF_s∨ F_sC=M_siE

N_tjF_s∨ F_sC=N_sjM_si

and hence the result follows.

Now we construct an F_{–DS Markov chain with intensity given by an arbitrary}F _adapted

matrix-valued locally bounded stochastic process which satisfies condition (3.8).

Theorem 4.8. Let(Λ(t))_t≥0be an arbitraryF_{adapted matrix-valued stochastic process which satisfies}

Proof. We assume that on a probability space(Ω,F,P)with a filtrationF_{we have a family of Cox}

processesNi,j fori,j∈ K with intensities(λ_i,j(t))such that theNi,j are conditionally independent

givenF∞(otherwise we enlarge the probability space). We construct on(Ω,F,P)anF–DS Markov

chainC with intensity(Λ(t))_t≥0 and given initial state i0. It is a pathwise construction inspired by

Lando[16]. First, we define a sequence(τ_n)_n≥1of jump times of C and a sequence(C¯n)n≥0 which

describes the states of rating after change. We define these sequences by induction. We put

¯

C₀=i₀, τ₁:= min

j∈K \C¯0

infnt>0 :∆NC¯0,j

t >0

o

Ifτ₁=∞, thenC never jumps so ¯C₁ :=i₀. Ifτ₁ <∞, then we put ¯C₁:= j, where jis the element

ofK \_C¯_{0 for which the above minimum is attained. By conditional independence of N}i,j _given

F_∞, the processesNi,jhave no common jumps, so ¯C1 is uniquely determined. We now assume that

τ1, . . . ,τk, ¯C1, . . . , ¯Ck are defined,τk<∞and we constructτk+1 as the first jump time of the Cox

processes afterτk, i.e.

τk+1:= min

j∈K \C¯k

inf

n

t> τk:∆N

¯

Ck,j

t >0

o

.

Ifτ_k+1 =∞then ¯Ck+1 =C¯k, and ifτk+1 <∞we put ¯Ck+1:= j, where j is the element ofK \C¯k

for which the above minimum is attained. Arguing as before, we see thatτ_k+1 and ¯C_k+1 are well

defined.

Having the sequences(τ_n)_n≥0 and(C¯n)n≥0 we define a processC by the formula

Ct:=

∞

X

k=0

✶[τk,τk+1)(t)C¯k. (4.11)

This processCis càdlàg and adapted to the filtrationA_{= (At)t≥0, whereAt:=Ft∨}W

i6=jF Ni,j

t

,

and hence it is also adapted to the larger filtrationAe = (Af_t)_t≥0,Af_t:=F_∞∨W_i6=jF_tNi,j. Notice

thatHi,j defined by (4.4) is equal to

Hi_t,j=

Z

]]0,t]]

✶{i}(Cs−)d Nsi,j a.s.

The processesN_ti,j−R

]]0,t]]λi,j(s)dsareAelocal martingales (since they are compensated Cox’s

pro-cesses, see e.g.[3]). Likewise, eachMi,j defined by (4.3) is anAe local martingale, since

M_ti,j=Hi_t,j−

Z

]]0,t]]

H_siλ_i,j(s)ds=H_ti,j−

Z

]]0,t]]

H_si₋λ_i,j(s)ds

=

Z

]]0,t]]

✶{i}(Cs−)d(Nsi,j−λi,j(s)ds).

Recall thatGb_t=F_∞∨ FC

t , soGb ⊆Ae. Therefore eachMi,jis also aGb local martingale, sinceMi,j is

b

Remark 4.9. Suppose that in the above constructionτ_k<∞a.s. and C_τ

k =i. Then

a)τ_k+1<∞a.s. provided

∃j∈ K \ {i}

Z ∞

0

λ_i,j(s)ds=∞ a.s.,

b)τ_k+1=∞a.s. provided

∀j∈ K \ {i}

Z ∞

τk

λ_i,j(s)ds=0 a.s.

5

Distribution of

C

and sojourn times

In this section we investigate properties of distribution ofC and the distribution of sojourn time in

a fixed state j under assumption thatC does not stay in j forever. These properties give, among

others, some interpretation to

λ_j(t):= X

l∈K \{j}

λ_j,l(t) =−λ_j,j(t) (5.1)

and to the ratios λ_λi,j

i . The first result says that the sojourn time of a given state has an exponential

distribution in an appropriate time scale.

Proposition 5.1. Let C be anF_{–DS Markov chain with intensityΛ,τ0=0, and}

τ_k=inf¦t> τ_k−1:C_t 6=C_τ

k−1,Ct−=Cτk−1

©

. (5.2)

Ifτk<∞a.s., then the random variable

Ek:=

Z τk

τk−1

λCτ_k−1(u)du (5.3)

is independent of Gb_τk−1 and E_k is exponentially distributed with parameter equal to 1. Moreover, (E_i)_1≤i≤kis a sequence of independent random variables.

Proof. For arbitrary j∈ K define the process that counts number of jumps from j

N_tj:= X

l∈K \{j}

H_tj,l.

Let

e

Y_tj:=N_tj−

Z t

0

H_uj₋λ_j(u)du.

e

Yj is aGb _{local martingale by Theorem 4.1. Moreover, a sequence of stopping times defined by}

σ_n:=inf

¨

t≥0 :N_tj≥n∨

Z t

0

H_uj₋λ_j(u)du≥n

reducesYej. In what follows we denote

Y_tn:=Ye_tj_∧σ

n.

The definition ofσ_nimplies thatYnis a boundedGb martingale for everyn. Using the Itô lemma on

the interval]τ_k−1,τ_k]we get

is a uniformly integrableGb martingale. Therefore, by the Doob optional sampling theorem we have

E

Applying Lebesgue’s dominated convergence theorem and the fact thatσ_n↑+∞yields

E

As a consequence of the Proposition 5.1 we have the following result which states that the exit times are in some sense exponentially distributed. This is a generalization of the well known property of Markov chains.

Corollary 5.2. Let C be an F_{–DS Markov chain with intensity} Λ and let τ_k < ∞ a.s., where τ_k is defined by(5.2). Then

P€τk−τk−1>t|Gbτk−1

Š

=e−

Rτ_k−1+t

τ_k−1 λCτ_k−1(u)du_{. (5.5)}

Moreover, on the set¦C_τk−1=i©we have

Z ∞

τk−1

λ_i(u)du=∞. (5.6)

Proof. Because

τ_k−τ_k−1=inf

(

t≥0 :

Z τk−1+t

τk−1

λ_Cτ

k−1(u)du≥Ek

)

,

we obtain (5.5). Indeed

P€τ_k−τ_k−1≥t|Gb_τ

k−1

Š

=P

Z τk−1+t

τk−1

λ_C

τ_k−1(u)du≤Ek

bGτk−1

!

=e−

Rτ_k−1+t

τ_k−1 λCτ_k−1(u)du_.

(5.5) yields (5.6) by letting t→ ∞.

The next proposition describes the conditional distribution of the vector(C_τ

k,τk−τk−1)givenGbτk−1.

Proposition 5.3. Let C be an F_{-DS Markov chain with an intensity} Λ and τk be given by (5.2). If

τk<∞a.s., then we have

P€C_τk = j,τ_k−τ_k−1≤t

bG_τk−1Š=

Z t

0

e−

Ru

0λi(v+τk−1)d v_λ

i,j(τk−1+u)du (5.7)

on the set¦C_τ

k−1=i

©

.

Proof. First we note that RHS of (5.7) is well defined. Fix arbitrary j ∈ K. Let Nj be the process that counts number of jumps ofC to the state j, i.e.

N_tj:= X

i∈K \{j}

H_ti,j,

and

Y_tj:=N_tj−

Z t

0

X

i∈K \{j}

Then Yj is a Gb _{local martingale by Theorem 4.1. Therefore there exist a sequence of}Gb _stopping

times σ_n ↑ +∞ such that (Y_tj_∧σ

n)t≥0 is an uniformly integrable martingale. By Doob’s optional

sampling theorem we have on the set¦C_τ

k−1=i

we can pass to the limit on the LHS of (5.8) and obtain

lim

By a monotone convergence theorem and fact thatσ_n↑+∞we obtain

lim

On the other hand

lim

n→∞I2(n) =0, a.s.

Summing up, we have proved the following equality holds on the set¦C_τ

k−1=i

=

In the last equality we use (5.6). The proof is now complete.

Remark 5.4. Using Corollary 5.2 in the same way as in the last part of the proof of Proposition 5.3 we obtain on the set¦Cτk−1=i

i(τk) can be treated as a conditional probability of transition from i to j given

that transition occurs at the timeτk. The next corollary confirms this suggestion.

Proof. In a view of Proposition 5.3 and Corollary 5.2 we have on the set¦C_τ

k−1=i

©

:

P€Cτk ∈d x,τk−τk−1∈d t bG_τ

k−1

Š

= X

j∈K \{i}

δ_{j}(d x)e−

Rt

0λi(v+τk−1)d vλ_i,j(t+τ_k−1)d t

= X

j∈K \{i}

δ_{j}(d x)λi,j(t+τk−1)

λ_i(t+τ_k−1) e

−Rt

0λi(v+τk−1)d vλ_i(t+τ_k−1)d t

= X

j∈K \{i}

δ_{j}(d x)λi,j(t+τk−1)

λ_i(t+τ_k−1) P

€

τ_k−τ_k−1∈d t

bG_τ

k−1

Š

,

which implies (5.10).

6

Predictable representation theorem

In this section we prove a predictable representation theorem in the form useful for applications. At

first we study a hypothesis which often appears in literature as hypothesis that a givenρavoidsF

stopping times. This hypothesis says that for everyF_{stopping timeσit holdsP(ρ=σ) =0.}

Proposition 6.1. Everyτ_kdefined by(5.2)such thatτ_k<∞a.s. avoidsF_{stopping times.}

Proof. Fixk and letσbe an arbitraryF_{stopping time. Using the random variable Ek+1 defined by}

(5.3) we have

P(τ_k+1=σ) =P

Z τk+1

τk

λ_C

τ_k(u)du=

Z σ

τk

λ_C

τ_k(u)du

!

=E ‚

E ‚

✶§

Ek+1=

Rσ

τ_kλCτ_k(u)du

ª_F

∞∨ FτCk ŒŒ

=E ‚

E

✶_{Ek+1=x} _x=Rσ

τ_kλCτ_k(u)du

Œ

=0.

The third equality follows from measurability of R_τσ

kλCτk(u)du with respect to F∞∨ F

C

τk and

in-dependence of E_k+1 and F_∞∨ F_τC

k. Moreover P(Ek+1 = x) = 0 since random variable Ek+1 has

an exponential distribution (see Proposition 5.1), so the fourth equality holds, and the assertion follows.

This proposition immediately implies

Corollary 6.2. AnF_{–DS Markov chain with intensity and an arbitrary}F_{adapted process do not have}

jumps at the same time.

Remark 6.3. The assertion of Proposition 4.7 follows immediately from Corollary 6.2, since[M,N] =0

Now we precise the notion of predictable representation property for square integrable martingales

(although this definition has sense for local martingales, see Klebaner[13, Chapter 8§12], Cont and

Tankov[5, Chapter 9§2]).

Definition 6.4. Let N be a square integrable martingale. We say thatthe predictable representation property(PRP) holds forN if for everyT >0 and everyF_TN measurable square integrable random

variableX there exists anFN _{predictable stochastic processZ such that}

X =EX+

Z T

0

Z_t⊤d Nt. (6.1)

Assume that PRP holds forN and letC be anFN_{-DS Markov chain. It turns out that the PRP holds}

for an(N,M)where M is given by (4.7), so M is a martingale which characterizes C. Indeed, the

following form of a predictable representation theorem holds:

Theorem 6.5. Assume that the predictable representation property holds for N and let C be anFN_–DS

Markov chain. Then the predictable representation property holds for(N,M), where M is a martingale given by

M_t:=H_t−

Z

]]0,t]]

Λ⊤(u)H_udu.

Proof. Proposition 3.4 implies thatNis anFN_∨FC _{martingale and, by Theorem 4.1, the processM}

isFN _∨FC _{martingale. For every square integrable F}N

T ∨ F C

T measurable random variable X and

T >0 we have to find anFN_∨FC _{predictable stochastic process such that (6.1) holds with(N,M)}

instead ofN. Fix arbitraryT >0. In the proof we will use a monotone class theorem. Let

N =

X ∈L2(F_TN∨ F_TC):X =EX+

Z

]]0,T]]

R⊤_s d Ns+

Z

]]0,T]]

S⊤_s d Ms,

whereR,S areFN_∨FC _{predictable stochastic processes}

.

We claim thatN is a monotone vector space. Obviously, it is a vector space over Rand 1 ∈ N.

From closedness of the space of integrals follows that the bounded limit of monotone sequence of

elements fromN belongs toN. Let

M :=

X : X =Y

n

Y

k=1

✶n

C_tk=ik

o_{whereY ∈L}∞_(FN

T), (6.2)

(t₁, . . . ,t_n)∈[0,T]n,(i₁, . . . ,i_n)∈ Kn:n∈N

.

It is easy to see thatM is a multiplicative class. To finish the proof it is enough to prove thatM ⊂ N,

because by a monotone class theorem,N contains all bounded functions that are measurable with

respect to theσ-algebra generated byM i.e. L∞(F_TN∨ F_TC)⊂ N. Hence, by standard arguments

Therefore, we need to derive predictable representation for an arbitrary random variable inM. Fix

together with invertibility ofP(0,t). Define a bounded martingaleNmby

N_tm=E P(0,tm)eimY

By assumption,Nmadmits the following predictable decomposition:

N_Tm=N₀m+

Z

]]0,T]]

(Rm_t)⊤d N_t

for someF-predictable stochastic processRm. Hence, using (4.10) and Proposition 6.1, we have

Therefore,

By (6.4), (6.3), (6.5) form−1 and properties of stochastic integral we have

Ztm=Ztm−1+

Thus defining processesR,S by

Rt:=

7

Change of probability and doubly stochastic property

Now, we investigate how changing the probability measure to an equivalent one affects the

proper-ties of anF_{–DS Markov chain. We start from a lemma}

Lemma 7.1. LetQ,Pbe equivalent probability measures with density factorizing as

dQ

dP

F∞∨F_TC∗

:=η₁η₂, (7.1)

whereη₁ is anF_∞–measurable strictly positive random variable andη₂ is anF_∞∨ F_TC∗–measurable

strictly positive random variable integrable underP. Let(η₂(t))_t∈[0,T∗_]be defined by the formula

η2(t):=EP(η2| F∞∨ FtC), η2(0) =1. (7.2)

Then (N(t))_t∈[0,T∗_] is a Gb martingale (resp. local martingale) under Q if and only if

(N(t)η2(t))t∈[0,T∗_]is aGb martingale (resp. local martingale) underP.

Proof. ⇒ By the abstract Bayes rule and the fact that η1 is F∞ measurable and hence also Gu

measurable for allu≥0, we obtain, fors<t,

N(s) = E_Q€N(t)|Gb_sŠ=EP

€

N(t)η₁η₂|Gb_sŠ

EP

€

η1η2|Gbs

Š =E_P N(t)EP

€

η₁η₂|Gb_tŠ

EP

€

η1η2|Gbs

Š bG_s

!

= E_P N(t)EP

€

η₂|Gb_tŠ

E_P€η₂|Gb_sŠ

bG_s

!

=E_P

N(t)η2(t)

η₂(s)

bG_s

=EP

€

N(t)η₂(t)|Gb_sŠ

η₂(s) .

⇐The proof is similar.

The next lemma is a standard one, describing the change of compensator of a pure jump martingale under a change of probability measure. Although it can be proved using Girsanov-Meyer theorem (see e.g. [10],[15],[19]), we give a short self-contained proof.

Lemma 7.2. Let C be anF_{–DS Markov chain underPwith intensity(λi,j)and suppose thatη2defined}

by(7.2)satisfies

dη₂(t) =η₂(t−)



 X

k,l∈K:k6=l

κ_k,l(u)d M_uk,l



 (7.3)

with someG_{predictable stochastic processesκi,j, i,j∈ K, such thatκi,j>−1. Then}

e

M_ti,j=H_ti,j−

Z

]]0,t]]

H_ui(1+κ_i,j(u))λ_i,j(u)du, (7.4)