Credit Risk Modeling with Delayed Information ∗ Takanori Adachi †‡

Contents 1 Introduction

1

2 Market Times 2.1 The Space of Market Times . . . . . . . . . 2.2 Idempotent Market Times . . . . . . . . . 2.3 Honest Times . . . . . . . . . . . . . . . . 2.4 Examples of Market Times . . . . . . . . 2.4.1 Constantly Delayed Market Times 2.4.2 Poisson Market Times . . . . . . . 2.4.3 Starting Times for Excursions . . . 3

. . . . . . .

3 3 4 7 8 8 8 10

Market Filtrations 3.1 Definition of Market Filtrations . . . . . . . . . . . . . . . . . . . 3.2 A Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Market Filtrations and Optional Processes . . . . . . . . . . . . .

10 10 13 16

. . . . . . .

. . . . . . .

. . . . . . .

4 Conditional Expectations given Market Filtrations 4.1 Cases of Stopping Times . . . . . . . . . . . . . 4.2 Cases of Idempotent Market Times . . . . . . . 4.2.1 Hypothesis (HP) . . . . . . . . . . . . . 4.2.2 Market Times with Horizons . . . . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . .

17 18 20 20 21

1 Introduction Starting with the trailblazing work of Merton [18], a branch of credit risk modeling, called the structural approach has flourished by several authors ( See e.g. ∗ Presented at the 2nd seminar held by Credit Risk Theory Research Group, JAFEE on August 26th, 2011 † Graduate School of International Corporate Strategy, Hitotsubashi University ‡ Email: [email protected]

1

Bielecki and Rutkowski [3] ). Many of their models are defined so as to introduce components that make the model incomplete in the sense that its default time becomes a totally inaccessible stopping time. The origin of this line is the work of Duffie and Lando [8]. They link the two perspectives by introducing noise into the market’s information set. They postulate that the market can only observe the firm’s asset value plus noise at equally spaced, discrete (non-continuous) time points. Kusuoka [13] extends Duffie and Lando’s model to continuous time observations. C ¸ etin, Jarrow, Protter and Yildirim [5] simply reduce the information the market can see instead of appending noise. Giesecke’s model [10] makes the default barrier be unobservable to the market. Our approach that we present in this paper is toward the line. We focus on the market time delay as a source of the model incompleteness. Actually, there are earlier studies including the work by Lindset, Lund and Persson [16] whose model has constant lags for both managers and markets, and more recently the work by Guo, Jarrow and Zeng [11] whose model is stochastic and is based on an increasing sequence of stopping times. We enhance their approaches to the cases including not just deterministic delay but also some delay driven by (possibly non-stopping) random times. This enables us to consider a natural example of catching up to all information in a stochastically periodic manner that the continuous delayed model of GuoJarrow-Zeng does not accept. The remainder of this paper consists of three sections. In Section 2, we begin with an introduction of market times, showing the set of all market times forms a monoid as well as a complete lattice. We introduce a family of market times, called idempotent market times whose member fails to be an example of the time change utilized by Guo, Jarrow and Zeng [11]. After giving a characterization of idempotent market times, we show that the elements of idempotent market times are honest tmes. We also give few examples of market times in this section. In Section 3, we present a definition of filtrations generated by market times, showing that they are subfiltrations of the continuously delayed filtrations of Guo, Jarrow and Zeng [11]. We give a counterexample with which the subfiltration relation above becomes strict. We also investigate the relationship between market filtrations and filtrations generated by optional processes. The last material was suggested by Professor Rutkowski. In Section 4, we provide a closed form solution of the conditional expectation given a market filtration in the case that each element of the market time is a stopping time. We also try to develop tools for finding the conditional expectation when the market time is idempotent.

2

2 Market Times In this paper, all the discussion is under the probability space (Ω, G , Q ). All filtrations presented in this paper satisfy the usual condition, that is, right continuous and Q-complete.

2.1 The Space of Market Times Definition 1. [Market Times] A market time is a positive stochastic process m : R + × Ω → R + satisfying 1. m0 = 0

a.s.,

2. mt ≤ t

for all t ≥ 0

3. ms ≤ mt

for all s < t

a.s., a.s..

The market time mt represents the delayed time in the sense that if the market knows an event at time t, then the event actually happened at time mt (ahead of t) when managers learned it. So, it models the fact that the market will know the information (slightly) after the managers know it, that is, representing asymmetric information. Lindset et al. introduced the two time lags for markets and managers in [16]. Their lags are constant and not stochastically varying like ours. Definition 2. [Space of Market Times] Let M be the set of all market times. For m1 , m2 ∈ M, the composite process m1 ◦ m2 is defined by for all t ≥ 0 and ω ∈ Ω, (m1 ◦ m2 )t (ω ) = m1m2 (ω ) (ω ). t

An identity process is a process

1M

defined by for all t ≥ 0 and ω ∈ Ω,

1M t ( ω ) = t. Theorem 1. The structure hM, ◦, 1 M i forms a monoid 1 . Proof. First, we show that the process (m1 ◦ m2 ) is a market time, that is, the set M is closed under the operation ◦, by examining if it satisfies the three condition in Definition 1, which is actually straightforward. Next, we check the associativity of the operator ◦. Let m1 , m2 , m3 ∈ M. We have for all t ≥ 0 and ω ∈ Ω,

((m1 ◦ m2 ) ◦ m3 )t (ω ) = m1m2

m3t (ω )

(ω )

(ω ) = (m1 ◦ (m2 ◦ m3 ))t (ω ).

The last thing we have to check is that 1 M is an identity of the operator ◦. But, this is also straightforward. 1A

semigroup with identity.

3

In the rest of this paper, we sometimes omit the composition operator ◦ and write like m1 m2 instead of writing m1 ◦ m2 . We can see another algebraic structure in M. Theorem 2. Define a binary relation ≤ on M by for any m, n ∈ M, m≤n

iff

(∀t ∈ R + )mt ≤ nt a.s..

Then, the structure hM, ≤i forms a complete lattice 2 whose greatest element is 1 M and the least element is the process 0M which is defined by 0M t ( ω ) = 0 for all t ≥ 0 and ω ∈ Ω. Proof. It is obvious that the relation ≤ on M is a partial order, and that processes 1 M and 0M are the greatest ∨ and the least elements of M, respectively. ∧ For a given subset M ⊂ M, let M and M be processes defined by

(

∨

M )t (ω ) = sup{mt (ω ) | m ∈ M}, ∨

(

∧

M)t (ω ) = inf{mt (ω ) | m ∈ M }.

∧

Then, it is clear that both M and M are market times, and that they are the least upper bound and the greatest lower bound of M, respectively.

2.2 Idempotent Market Times Guo, Jarrow and Zeng introduced a concept of stochastic delays similar to market times when they introduced their continuously delayed filtration in [11]. Upon three conditions of Definition 1, they require an extra condition that each mt is a stopping time with respect to an adequate filtration G = {Gt }t≥0 . That is, {mt ≤ s} ∈ Gs for every pair of s, t ≥ 0. Unfortunately, we will see that this condition is too strong to accept some natural processes. The following is a definition of the family of market times that have difficulty to work as the time change processes that Guo, Jarrow and Zeng use when they define continuously delayed filtrations [11]. Definition 3. [Idempotent Market Times] A market time m = {mt }t≥0 is called idempotent if

(m ◦ m)t = mt a.s. for every t ≥ 0.

(1)

Obviously the identity market time 1 M is an idempotent market time. We will see more interesting examples of idempotent market times in Section 2.4. Proposition 1. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be an idempotent market time where mt is a G-stopping time. Then, for every pair t and s with t ≥ s, we have {mt = ms } ∈ Gs . 2 A partial ordered set having the least upper bound and the greatest lower bound for any subset. See Birkhoff [4] for further information.

4

Proof. We have the following almost surely equivalent statements, mt ≤ s ⇔ mt ≤ s ≤ t

⇔ mt = mmt ≤ ms ≤ mt ⇔ mt = ms Since G is complete and mt is a G-stopping time, we have {mt = ms } ∈ Gs . Proposition 1 says that at the current time s we can know if the information will have increased since now by any future time t, which is not realistic. This is why we exclude the condition of mt being a stopping time from Definition 1. On the other hand, one of the biggest advantages of assuming mt is a stopping time is that we can use the well-known filtration construction 3 indexed by the stopping time for creating a delayed filtration; { } Gmt := A | (∀s ≥ 0) A ∩ {mt ≤ s} ∈ Gs . (2) Next, we show a characterization of idempotent market times. Definition 4. For a random set M ⊂ R + × Ω, define a positive process m M : R + × Ω → R + by mtM (ω ) = sup{s ≤ t | (s, ω ) ∈ M}, where we use the convention sup ∅ = 0. Note that mtM is the end 4 of the random set Mt := {(s, ω ) ∈ M | s ≤ t}. Theorem 3. Let m : R + × Ω → R + be a positive process. Then, m is an idempotent market time iff there exists a random set M ⊂ R + × Ω such that m = m M . Proof. If part. It is clear that m M is a market time. So, let us show it is also idempotent. Let s := mtM . Then, s≤t

and

(∀u)u ≤ t ∧ (u, ω ) ∈ M → u ≤ s.

(3)

Now, it is enough to show that {u ≤ s | (u, ω ) ∈ M } = {u ≤ t | (u, ω ) ∈ M}. Since s ≤ t, it is obvious that LHS ⊂ RHS. Let u ∈ RHS. Then by Equation ( 3), u ≤ s. Therefore, u ∈ LHS. Only if part. For an idempotent market time m, define a random set M by M := {(mt (ω ), ω ) | t ∈ R + , ω ∈ Ω}. 3G

{

τ : = A ∈ G | (∀ t ≥ 0) A ∩ { τ ≤ t } ∈ Gt . 4 The end of a random set Γ is a random time

where we use the convention sup ∅ = 0.

(4)

}

EΓ defined by EΓ (ω ) = sup{t ∈ R + | (t, ω ) ∈ Γ},

5

Then, we have mtM (ω ) = sup{s ≤ t | (s, ω ) ∈ M}

= sup{s ≤ t | s = mu (ω ), u ∈ R + , ω ∈ Ω} = sup{mu (ω ) | u ∈ R + , ω ∈ Ω, mu (ω ) ≤ t} Since m is idempotent, mu (ω ) = mmu (ω ) (ω ) ≤ mt (ω ). Therefore, mtM (ω ) ≤ mt (ω ). On the other hand, since mt (ω ) ≤ t, mtM (ω ) ≥ mt (ω ). As the last topic of this subsection, we mention a family of market times called discretizors. Definition 5. [Discretizors] Let r be a positive number. A discretizor with the resolution r is a deterministic market time ∆r = {∆rt }t≥0 defined by ∆rt = nr,

where

n∈N

with

nr ≤ t < (n + 1)r.

This is an idempotent market time catching up with the managers’ time every r unit time. It also satisfies t − r < ∆rt ≤ t for every t ≥ 0. Discretizors are sometimes used for making a given market time that has a continuous distribution be its approximate market time with a discrete distribution. The following proposition gives a basic principle of the approximation. Proposition 2. Let m ∈ M, and t ≥ 0. 1. limr→0 ∆rt = 1 M t . 2. (∆r ◦ m)t → mt pointwise on Ω as r → 0. 3. (m ◦ ∆r )t → mt pointwise on Ω as r → 0 if every sample path of the original market time mt is left-continuous. Proof.

1. Immediate from the inequation t − r < ∆rt ≤ t.

2. Immediate from the inequation mt (ω ) − r < ∆rm (ω ) ≤ mt (ω ). t

3. Noticing that ∆rt approaching to t from left, (m ◦ ∆r )t converges to mt if mt is left-continuous. Note that the cardinality of the range of the market time ∆r ◦ m is at most countable.

6

2.3 Honest Times In Proposition 1, we were discouraged to make a market time consist of stopping times when it is idempotent. In this subsection, we revisit the issue by adopting a wider class of random times than the class of stopping times. Definition 6. [Honest Times] Let G = {Gt }t≥0 be a filtration. A random time τ is called G-honest with respect to a G-adapted process {τt }t>0 on R + if for every t > 0, τ = τt on {τ ≤ t}, i.e. τ1 {τ ≤t} = τt 1 {τ ≤t} . A random time τ is called G-honest if there exists a G-adapted process {τt }t>0 such that τ is G-honest with respect to {τt }t>0 . It is well known that every G-stopping time is G-honest (See e.g. page 373 of Protter [20] or page 384 of Nikeghbali [19] ). The following is a very nice characterization of honest times developed by Yor (Yor [22]). Theorem 4. [Yor [22]] Let G = {Gt }t≥0 be a filtration, and τ be a random time. Then, τ is a G-honest time if and only if for every u < s,

(∃ A ∈ Gs ){τ ≤ u} = A ∩ {τ ≤ s}.

(5)

Our first question in this subsection is for a given market time m = {mt }t≥0 , if there exists an honest time τ with respect to m. Here is a necessary and sufficient condition of the existence of such τ. Proposition 3. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted market time. Then, a random time τ : Ω → R + ∪ {∞} is G-honest with respect to m if and only if for every ω ∈ Ω, m∞ (ω ) := lim mt (ω ) = τ (ω ) = mτ (ω ) (ω ). t→∞

Proof. Note that the random time τ is G-honest with respect to m iff [ ] (∀t > 0)(∀ω ∈ Ω) τ (ω ) ≤ t → mt (ω ) = τ (ω ) .

(6)

Only if part: Since mt is monotonic, limt→∞ mt (ω ) = supt≥0 mt (ω ). Therefore, the result comes immediately by Equation (6). If part: Since supt≥0 mt (ω ) = τ (ω ),

(∀t > 0)mt (ω ) ≤ τ (ω ). So, it is sufficient to show mt (ω ) ≥ τ (ω ), assuming τ (ω ) ≤ t. But, by the monotonicity of mt and the assumption τ (ω ) = mτ (ω ) (ω ), we have τ ( ω ) = m τ ( ω ) ( ω ) ≤ m t ( ω ).

7

As an implication of Proposition 3, we missed the possibility of making whole market time be characterized by one honest time if the market time is unbounded. However, we have the following fairly nice theorem of asserting each mt becomes an honest time for some market times including Poisson market times. Theorem 5. Let G = {Gt }t≥0 be a filtration. If m = {mt }t≥0 is a G-adapted idempotent market time, then for every t ≥ 0, mt is a G-honest time. Proof. Define a random field {τst }t,s∈R+ by τst := mt∧s . Then, it is obvious that τst is Gs -measurable. So, all we need to show is t τs = mt on {mt ≤ s}. If s ≥ t, we have τst = mt on Ω. Hence, we concentrate on the case s < t. Now for any ω ∈ {mt ≤ s}, mt (ω ) ≤ s < t. Then, since m is idempotent, we get mt (ω ) = mmt (ω ) (ω ) ≤ ms (ω ) ≤ mt (ω ). Therefore, mt (ω ) = ms (ω ) = τst (ω ).

2.4

Examples of Market Times

We already see two sorts of concrete examples of market times, the identity market time and discretizors, which are both idempotent market times. In this subsection we show more examples including stochastic market times. Among them, Poisson market times and the starting times of Brownian excursions are idempotent market times. Therefore, they have difficulty to work as the time change processes that Guo, Jarrow and Zeng use when they define continuously delayed filtrations [11]. We will discuss the issue further in Section 2.4.2. You will find more examples of market times in Adachi [1]. 2.4.1 Constantly Delayed Market Times The following is an example of deterministic market times taken from Lindset et al. [16]. Definition 7. [Constantly Delayed Market Time] Let d be a positive constant. A market time m = {mt }t≥0 is called a constantly delayed market time with delay d if for all t ≥ 0, mt := max{t − d, 0}. 2.4.2 Poisson Market Times The second example of the stochastic market times is the market time driven by exponential interval times, which has a multiple jumps. Definition 8. [Poisson Market Time] 1. Xn ∼ i.i.d.Exp(λ) for n = 1, 2, . . . ,

8

2. Sn := ∑nk=1 Xk , 3. Nt := sup{n | Sn ≤ t}, 4. mt := S Nt . Intuitively, the exponential random variable Xn specifies an interval time between n-th and n + 1-th jumps when the market time catch up with the managers’ time. The process Nt is the Poisson process generated by those Xn ’s. The Poisson market time can be seen as a stochastic version of a discretizor where its resolution time changes statistically. You can easily verify that the Poisson market time is an idempotent market time. This means that the Poisson market time fails to be an example of the time change process that Guo, Jarrow and Zeng use when they define a continuously delayed filtration [11]. In reality, we see the situations satisfying Equation (1) occasionally including when the firm is under an audit activity by authorities, becoming all insider information available to the market. Here is a sample trajectory of a Poisson market time with λ = 10.

9

2.4.3 Starting Times for Excursions Let B = { Bt }t≥0 be a standard Brownian motion, and define a random set Z by Z = {(t, ω ) ∈ R + × Ω | Bt (ω ) = 0}. Then, the idempotent market time m Z picks the starting times for the excursions out of 0 of B.

3

Market Filtrations

In this section, we will define a filtration, called the market filtration modulated by a market time, and investigate if it can be considered as a natural extension of the continuously delayed filtration of Guo, Jarrow and Zeng [11].

3.1 Definition of Market Filtrations Let G = {Gt }t≥0 be a filtration, and assume that the market time m is Gadapted. For each time t, we will create a new σ-field Gtm for the G-adapted market time m. Since we are going to create a filtration, Gtm should contain Gsm for all s ≤ t. Now, because 0 ≤ ms ≤ s, we have a partition A=

⊕ (

A ∩ {ms = u}

)

0≤ u ≤ s

for any event A, neglecting null sets. If we know the occurrence of A by time u, then we should include the slice A ∩ {ms = u} in Gsm since in the domain {ms = u} the function ms is constant (or deterministic), having the unique value u. Suppose we pick any value v ≤ u. Since we include A ∩ {ms = v} in Gsm for A ∈ Gv ⊂ Gu , we have a motivation to include the set A ∩ {ms ≥ u} in Gsm for A ∈ Gu as well. By the above consideration, we have the following definition. Definition 9. Let G = {Gt }t≥0 be a filtration and m = {mt }t≥0 be a G-adapted market time. For t ≥ 0, the σ-fields Gtm and Gtm− are defined by { } Gtm := σ A ∩ {ms ≥ u} | 0 ≤ u ≤ s ≤ t, A ∈ Gu , (7) { } m− Gt := G0 ∨ σ A ∩ {ms > u} | 0 ≤ u < s ≤ t, A ∈ Gu . (8) Proposition 4. Let G = {Gt }t≥0 be a filtration, m = {mt }t≥0 be a G-adapted market time, and s, t ≥ 0. Then, we have 1. G0 ⊂ Gtm− ⊂ Gtm ⊂ Gt . Especially, Gtm is complete, 2. s < t

⇒

Gsm− ⊂ Gtm− ∧ Gsm ⊂ Gtm , 10

Proof.

1. If we put u := 0 in Equation 7, we get G0 ⊂ Gtm . Especially, we have G0m = G0 .

Next, for any A, s and t such that 0 ≤ u < s ≤ t and A ∈ Gu , we want to show A ∩ {ms > u} ∈ Gtm .

If u < s, there exists an integer n0 such that u + n10 < s. With this n0 , we have ∞ ( ∪ 1 ) A ∩ {ms ≥ u + } ∈ Gtm− A ∩ {ms > u} = n n=n 0

since A ∈ Gu ⊂ Gu+ 1 . n

Finally, in order to show the right most inclusion relation, we need to show A ∩ {ms ≥ u} ∈ Gt . However, it comes straightforward from the fact that A ∈ Gu ⊂ Gt and that ms is Gs -measurable. 2. Immediate. As an immediate consequence of Proposition 4, we have the following theorem. Theorem 6. [Market Filtration] Let G = {Gt }t≥0 be a filtration, m = {mt }t≥0 be a G-adapted market time, and define G m− := {Gtm− }t≥0 and G m := {Gtm }t≥0 Then they have subfiltration relations such as G m− ⊂ G m ⊂ G. We call the filtration G m the market filtration modulated by the market time m. The following proposition says that the market filtration is a natural extension of deterministic cases, which is actually one of reasons we adopt G m instead of G m− . Proposition 5. Assume that m is deterministic, i.e. there exists a deterministic function f : R + → R + such that for all t ≥ 0 and ω ∈ Ω, mt (ω ) = f (t). Then, we have for all t ≥ 0, Gtm = G f (t) . Moreover, if G is left-continuous, we have Gtm− = G f (t) . Proof. First, we show G f (t) ⊂ Gtm . Let A ∈ G f (t) . Reminding that { Ω if f (s) ≥ u, {ms ≥ u} = ∅ otherwise, we have

{ } Gtm = σ A | 0 ≤ u ≤ f (s) ≤ s ≤ t, A ∈ Gu { } ⊃ σ A | u = f (s) ≤ s ≤ t, A ∈ Gu { } = σ A | s ≤ t, A ∈ G f (s) { } = σ A | A ∈ G f (t)

= G f (t) 11

since f (s) ≤ s and f (s) ≤ f (t). Next, we show Gtm ⊂ G f (t) . All we need to show is A ∩ { f (s) ≥ u} ∈ G f (t) for 0 ≤ u ≤ s ≤ t and A ∈ Gu . If f (s) ≥ u, A ∩ { f (s) ≥ u} = A ∈ Gu ⊂ G f (s) ⊂ G f (t) . On the other hand, if f (s) < u, A ∩ { f (s) ≥ u} = ∅ ∈ G f (t) . Let’s assume that G is left-continuous. Then, we have

Gtm− = G0 ∨ σ{ A ∩ { f (s) > u} | 0 ≤ u < s ≤ t, A ∈ Gu } = G0 ∨ σ{ A | 0 ≤ u < f (s), s ≤ t, A ∈ Gu } = G0 ∨ = G0 ∨

∪

∪

{

∪

Gu | s ≤ t}

u< f (s)

{G f (s) | s ≤ t}

= G0 ∨ G f ( t ) = G f ( t ) .

Proposition 6. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted market time. Then, the process m is G m− -adapted. Especially, it is G m -adapted. Proof. In Equation 8, if we put s := t and A := Ω ∈ Gu , then we have { } { 1 } Gtm− ⊃ σ Ω ∩ {mt > u} | 0 ≤ u < t, Ω ∈ Gu = σ m− t ( u, ∞ ) | u < t . 1 On the other hand, in the case of u ≥ t, we have m− t ( u, ∞ ) = ∅ a.s. Thus, 1 m− m− since Gtm− is Q-complete. Therefore, mt is Gtm− -measurable. t ( u, ∞ ) ∈ Gt

Now, we are interested in the case when it happens to have the market time mt being a stopping time for every t. The following proposition says, in this case, our market filtration is a subfiltration of the continuously delayed filtration of Guo, Jarrow and Zeng [11]. Proposition 7. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted market time. 1. If mt is a G-optional time for all t ≥ 0, then Gtm ⊂ Gmt + , 2. If mt is a G-stopping time for all t ≥ 0, then Gtm ⊂ Gmt , where Gmt + and Gmt are the filtrations defined by { } Gτ + := A ∈ G | (∀t ≥ 0) A ∩ {τ < t} ∈ Gt , { } Gτ := A ∈ G | (∀t ≥ 0) A ∩ {τ ≤ t} ∈ Gt . Proof. Let 0 ≤ u ≤ s ≤ t and A ∈ Gu .

12

1. It is enough to show that for any r ≥ 0, A ∩ { m s ≥ u } ∩ { m t < r } ∈ Gr . In case {ms ≥ u} ∩ {mt < r } = ∅, it is trivial. So, we assume that there exists ω ∈ Ω such that ω ∈ {ms ≥ u} ∩ {mt < r }. Then, we have u ≤ ms (ω ) ≤ mt (ω ) < r. Therefore, u < r. So we have A ∈ Gu ⊂ Gr . On the other hand, since ms and mt are optional time, {ms ≥ u} = Ω − {ms < u} ∈ Gu ⊂ Gr and {mt < r } ∈ Gr . 2. It is enough to show that for any r ≥ 0, A ∩ { m s ≥ u } ∩ { m t ≤ r } ∈ Gr . And the rest of the proof is almost exactly same as that of 1 above.

Unfortunately, the opposite inclusion of the Proposition 7 (2), that is, Gmt ⊂ Gtm does not hold in general. We will see a counterexample in Section 3.2. For the detailed investigation about the situations when these two filtrations coincide, see Adachi [1].

3.2 A Counterexample In this subsection, we give a counterexample mentioned in Section 3.1, which is constructed based on Example 5.2 in Chung and Doob [7]. Let { Xt }t≥0 be a Markov process whose state space is S = {0, 1, 2} defined by the following way. Definition 10. In the following, n is any integer. 1. δn : Ω → R + (n = 0, 1, 2, . . . ) are i.i.d. positive random variables whose means are 1. 2. t0 := 0, tn+1 := tn + δn , 3. vi : Ω → {1, 2} (i = 1, 2, . . . ) are i.i.d. random variables with Q (vi = 1) = Q (vi = 2) = 12 , 4. For t ≥ 0 and i = 1, 2, . . . ,   0 Xt : = v i   3 − vi

if t ∈ [t0 , t1 ), if t ∈ [t2i−1 , t2i ), if t ∈ [t2i , t2i+1 ),

5. For k = 1, 2, τk := inf{t > 0 | Xt = k},

13

6. Assume Q (τ1 ∈ dt) = Q (τ2 ∈ dt), 7. τ := τ1 ∧ τ2 . 8. Assume Q (τ ∈ dt) = e−t dt. Then, we define a filtration G = {Gt }t≥0 and a process {mt }t≥0 . Definition 11. Let t ≥ 0. 1. Gt := σ ( Xs : 0 ≤ s ≤ t) ∨ N where N is the set of all Q-null sets. 2. mt := τ ∧ t. Proposition 8.

1. τ is a G-stopping time.

2. {mt }t≥0 is a G-adapted market time, and for each t ≥ 0, mt is a G-stopping time. Proof.

1. Note that τ = inf{t > 0 | Xt 6= 0}. Then, we have for any t > 0,

{ τ ≤ t } = { Xt 6 = 0} ∈ G t . 2. It is obvious that mt satisfies the three conditions in Definition 1. Now by noting that for any B ∈ B(R + ), { {τ ∈ B ∩ [0, t]} ∪ {τ > t} {mt ∈ B} = {τ ∧ t ∈ B} = {τ ∈ B ∩ [0, t]}

if t ∈ B, (9) if t ∈ / B.

We can easily show that both {τ ∈ B ∩ [0, t]} and {τ > t} are Gt -measurable since τ is a G-stopping time. So, m is G-adapted. If we put B := [0, s] in Equation 9, we get { Ω {mt ≤ s} = {τ ≤ s}

if t ≤ s, if t > s.

Therefore, ms is a G-stopping time. Proposition 9. For every t ≥ 0,

Gtm = G0 ∨ σ(ms : 0 ≤ s ≤ t). Proof. First, note that { } Gtm− = G0 ∨ σ A ∩ {ms > u} | 0 ≤ u ≤ s ≤ t, A ∈ Gu { } = G0 ∨ σ A ∩ {τ > u} | 0 ≤ u ≤ t, A ∈ Gu { } = G0 ∨ σ { Xr = k} ∩ {τ > u} | 0 ≤ r ≤ u ≤ t, k = 0, 1, 2 { } = G0 ∨ σ { Xr = k} ∩ { Xu = 0} | 0 ≤ r ≤ u ≤ t, k = 0, 1, 2 { } = G 0 ∨ σ { Xr = 0 } ∩ { X u = 0 } | 0 ≤ r ≤ u ≤ t { } = G0 ∨ σ { Xu = 0} | 0 ≤ u ≤ t 14

On the other hand, σ (ms : 0 ≤ s ≤ t) = σ (τ ∧ s : 0 ≤ s ≤ t) { } = σ {τ ≤ s} | 0 ≤ s ≤ t { } = σ { Xs 6 = 0} | 0 ≤ s ≤ t { } = σ { Xs = 0} | 0 ≤ s ≤ t Therefore, we have Gtm− = G0 ∨ σ (ms : 0 ≤ s ≤ t). On the other hand, since G is complete and Q has an exponential law, we get Gtm = Gtm− , which completes the proof. Definition 12. A process {Yt }t≥0 is defined by Yt := Xmt . 1. For t ≥ 0 and k = 1, 2,

Lemma 1.

{Yt = 0} = {τ > t} = { Xt = 0}, {Yt = k} = { Xτ = k} ∩ {τ ≤ t}. 2. τ ∈ Gτ . Proof.

1. For k = 0, 1, 2,

{Yt = k} = { Xτ ∧t = k} = { Xτ = k, τ ≤ t} ∪ { Xt = k, τ > t} = { Xτ = k, Xt 6= 0} ∪ { Xt = k, Xt = 0} 2. It is enough to show {τ ≤ t} ∈ Gτ for t ≥ 0. However, for any s ≤ 0,

{ τ ≤ t } ∩ { τ ≤ s } = { τ ≤ t ∧ s } ∈ Gt∧s ⊂ Gs , which completes the proof. Proposition 10. For t > 0, Yt is Gmt -measurable. Proof. We show {Yt = k } ∩ {mt ≤ s} ∈ Gs for s, t ≥ 0 and k = 0, 1, 2. Note { m t ≤ s } = { τ ∧ t ≤ s } = { τ ≤ s } ∪ { t ≤ s }. Then by Lemma 1, for k = 0,

{

{Yt = 0} ∩ {mt ≤ s} = and for k 6= 0,

{Yt = k} ∩ {mt ≤ s} =

{

{ Xt = 0} ∅

if t ≤ s, if t > s.

{ Xτ = k } ∩ { τ ≤ t } { Xτ = k } ∩ { τ ≤ s }

15

if t ≤ s, if t > s.

Proposition 11. For t > 0, Yt is not Gtm -measurable. Proof. Since Xτ 6= 0, by Lemma 1, {Yt = 1} ∪ {Yt = 2} = {τ ≤ t}. Then, for any B ∈ B[0, t],

{τ ∈ B} = {τ ≤ t} ∩ {τ ∈ B} = ({Yt = 1} ∩ {τ ∈ B}) ∪ ({Yt = 2} ∩ {τ ∈ B}) = ({ Xτ = 1} ∩ {τ ∈ B}) ∪ ({ Xτ = 2} ∩ {τ ∈ B}). Therefore for any B ∈ B[0, t] with Q (τ ∈ B) 6= 0, we have

{Yt = k} ∩ {τ ∈ B} 6= ∅ since Q (τ1 ∈ B) = Q (τ2 ∈ B). Now, suppose there exists B1 ∈ B[0, t] such that {Yt = 1} = {τ ∈ B1 }. Then, we have Q (τ ∈ B1 ) 6= 0. Therefore,

5

{Yt = 2} ∩ {τ ∈ B1 } 6= ∅, which is a contradiction since

{Yt = 1} ∩ {Yt = 2} = ∅,

Theorem 7. For every t > 0 , Gtm $ Gmt while mt is a G-stopping time.

3.3 Market Filtrations and Optional Processes In this subsection, we will investigate the relationship between market filtrations and the classical filtrations generated by optional processes. The material we treat here was given by Professor Rutkowski when he kindly suggested the author the direction of the research [21]. The following definitions are taken from Nikeghbali’s survey [19]. Definition 13. [Optional Processes] Let G = {Gt }t≥0 be a filtration. The optional σ-field with respect to G is the σ-field O G defined on R + × Ω such that

O G := σ{ X | X = { Xt }t≥0 is a G-adapted c´adl´ag process. }. An element of O G is called a G-optional set. A process X = { Xt }t≥0 is called G-optional if the map (t, ω ) 7→ Xt (ω ) is O G -measurable. The following is taken from Definition 8.4 in Nikeghbali [19]. You can see the definition in Chung [6] or Mansuy and Yor [17] as well. 5 We

may be able to put B1 := {τ (ω ) | Yt (ω ) = 1}. However, it is not trivial that B1 ∈ B[0, t].

16

Definition 14. Let τ be a random time. The σ-field G(τ ) is defined by

G(τ ) := σ{ Zτ | Z = { Zt }t≥0 is a G-optional process. }. We investigate the relationship between Gtm and G(mt ) below. Lemma 2. Let G = {Gt }t≥0 be a filtration, m = {mt }t≥0 be a G-adapted market time, and Gms ∗ be the σ-field defined by

G m s ∗ : = σ { A ∩ { m s ≥ u } | A ∈ G u }. Then, Gtm =

∨

s≤t

Gms ∗ .

Proof. Since mt ≤ t, the proof is straightforward. Proposition 12. Let Gms ∗ be the σ-field defined in Lemma 2. Then for every s ≥ 0, ∨ we have Gms ∗ ⊂ G(ms ) . Especially, Gtm ⊂ s≤t G(ms ) . Proof. Let u ≥ 0 and A ∈ Gu . All we need to show is A ∩ {ms ≥ u} ∈ G(ms ) . Define a process Zt by Zt (ω ) = 1 [u,∞[ (t)1 A (ω ). Then, obviously Zt is G-adapted and c´adl´ag. Therefore, Zt is a G-optional process. On the other hand, Zms (ω ) = 1 ⇔ ω ∈ A ∧ ms (ω ) ≥ u ⇔ ω ∈ A ∩ {ms ≥ u}. Hence,

−1 A ∩ {ms ≥ u} = Zm (1) ∈ G ( m s ) . s

It is well-known that the σ-field G(τ ) coincides with Gτ if τ is a stopping time ( See Remark 8.6 in Nikeghbali [19] ). Now let us think a market time m = {mt }t≥0 where mt is a G-stopping time for all t ≥ 0. Then we have ∨ s≤t

G(ms ) =

∨ s≤t

Gms = Gmt .

However, we know a counterexample that does not satisfy Gtm = Gmt in Theorem 7 Therefore, in general, the converse of the equation in Proposition 12, that is, G(ms ) ⊂ Gms ∗ does not hold.

4 Conditional Expectations given Market Filtrations The key technique to apply the theory of market times we have developed so far to a practical asset pricing theory is to calculate the conditional expectations given a market filtration, which we will treat in this section.

17

4.1 Cases of Stopping Times In this subsection, we present the cases when each mt of the market times are stopping times. We know that they are less interesting than non-stopping cases such as idempotent market times. However, it has some meaning to try these easier cases for the further development of the market time theory. In the following discussion, we fix a right-continuous filtration G = {Gt }t≥0 , a G-adapted market time m, and a time t ≥ 0. Our focus is to look for a characterization of EQ [ Z | Gtm ] for a given real-valued Q-integrable random variable Z. 1. For s ≥ 0, Ys := EQ [ Z | Gs ],

Definition 15.

2. A real-valued random variable Ytm is defined by for ω ∈ Ω, Ytm (ω ) = Ymt (ω ) (ω ). Lemma 3. Let Π be a unbounded infinite increasing sequence of real numbers with Π = { 0 = u 0 < u 1 < · · · < u n < u n +1 < . . . } and define For A ∈

Gtm ,

kΠk := max{un+1 − un | n = 0, 1, . . . }. define { An :=

A ∩ { m t = 0} A ∩ { u n −1 < m t ≤ u n }

and y(Π, A) :=

if if

n=0 n>0

∞

∑ 1 An Yun .

n =0

Then,

lim y(Π, A) = 1 A Ytm .

kΠk→0

Proof. Note that 6

An ∈ Gun

and

∪∞ n=0 An = A.

Now, for ω ∈ A, let Π(ω ) be the unique number satisfying ω ∈ AΠ(ω ) . Then, we have ∞ ( ) y(Π, A)(ω ) = ∑ 1 An Yun (ω ) = YuΠ(ω) (ω ). n =0

Since

ω ∈ AΠ(ω ) = A ∩ {ω 0 | uΠ(ω )−1 < mt (ω 0 ) ≤ uΠ(ω ) },

6 The first statement requires that m is a stopping time. The fact will be used in the proof of t Theorem 8.

18

we have

uΠ(ω )−1 < mt (ω ) ≤ uΠ(ω ) .

Therefore,

lim uΠ(ω ) = mt (ω ).

kΠk→0

Note that in this convergence uΠ(ω ) converges to mt (ω ) from right. On the other hand, since G is right-continuous, so is Yt (ω ) as a function of t. Hence, for all ω ∈ A, lim y(Π, A)(ω ) = lim YuΠ(ω) (ω ).

kΠk→0

kΠk→0

= Ymt (ω ) (ω ) = Ytm (ω ). And then, for all ω ∈ Ω, lim y(Π, A)(ω ) = 1 A (ω )Ytm (ω ),

kΠk→0

which proves the desired result. Theorem 8. Let G = {Gt }t≥0 be a right-continuous filtration, m be a G-adapted market time, and Z be a real-valued Q-integrable random variable. Then, EQ [ Z | Gtm ] = Ytm . Proof. Suppose we have an increasing sequence Π and { An }n∈N as defined in Lemma 3. First, we show that Ytm is Q-integrable. We have 7 ∫ Ω

|Ytm |dQ =

∞ ∫

∑

n =0 A n

|Ytm |dQ =

∞ ∫

∑

n =0 A n

|Yun |dQ

Then, by Jensen’s inequality and the fact 8 An ∈ Gun , we have ∞ ∫

∑

n =0 A n

|Yun |dQ ≤ =

∞ ∫

∑

[ ] EQ | Z | | Gun dQ

∑

| Z |dQ

n =0 A n ∞ ∫ n =0 A n

∫

=

Ω

| Z |dQ.

Therefore, Ytm is Q-integrable. 7 We

may need more consideration about the second equality. comes from the fact that mt is a stopping time. See the proof of Lemma 3.

8 This

19

Now all we need to show is that for all A ∈ Gtm , ∫

∫

A

ZdQ =

A

Ytm dQ.

Since EQ [ Z | Gun ] = Yun and An ∈ Gun , we have ∫

∞ ∫

A

ZdQ =

=

∑

ZdQ

∑

Yun dQ

n =0 A n ∞ ∫ n =0 A n ∫ ∞

∑ 1 An Yun dQ

= ∫

A n =0

= A

y(Π, A)dQ.

By Lemma 3, the last statement above converges to ∫ A

1 A Ytm dQ =

∫ A

Ytm dQ

as kΠk → 0, which completes the proof.

4.2 Cases of Idempotent Market Times We will investigate a possibility of getting a closed form of the conditional expectation given the market filtration modulated by an idempotent market time. 4.2.1 Hypothesis (HP) In this subsection, we review some hypotheses of conditional distribution and investigate the relationship between market times and one of the hypotheses. First, we present a hypothesis called the proportionality property, the hypothesis (HP) (in Li and Rutkowski [14]) or the conditional low invariance (in Jeanblanc and Song [12]). Definition 16. [Hypothesis (HP)] Let G = {Gt }t≥0 be a filtration, τ be a random time, and u, t ≥ 0. 1. Fu,t := Q (τ ≤ u | Gt ), 2. Ft := Ft,t , 3. Gt := 1 − Ft , 4. The triple (τ, Q, G ) is said to satisfy the hypothesis (HP) whenever for all 0 ≤ u < s < t, Fu,s Fs,t = Fs,s Fu,t . 20

The hypothesis (HP) is a generalization of the classic hypothesis (H) (see, for example, Elliott et al. [9] or Bielecki et al. [2]) which can be represented like Fu,s = Fu,t for all 0 ≤ u < s < t. Actually, Lemma 4.1 in [14] shows the implication (H) ⇒ (HP) is valid. The following theorem is taken from Corollary 3.1 of Li and Rutkowski [15]. Theorem 9. [Li and Rutkowski [15]] Let G = {Gt }t≥0 be a filtration, and τ be a G-honest time. Then, the triple (τ, Q, G ) satisfies the hypothesis (HP). Proof. Let 0 ≤ u < s < t. Then by Theorem 4, there exists A ∈ Gs such that {τ ≤ u} = A ∩ {τ ≤ s}. Hence, 1 {τ ≤u} = 1 A 1 {τ ≤s} . Therefore, we have [ ] [ ] Fu,s Fs,t = EQ 1 {τ ≤u} |Gs EQ 1 {τ ≤s} |Gt [ ] [ ] = EQ 1 A 1 {τ ≤s} |Gs EQ 1 {τ ≤s} |Gt [ ] [ ] = 1 A EQ 1 {τ ≤s} |Gs EQ 1 {τ ≤s} |Gt [ ] [ ] = EQ 1 {τ ≤s} |Gs EQ 1 A 1 {τ ≤s} |Gt [ ] [ ] = EQ 1 {τ ≤s} |Gs EQ 1 {τ ≤u} |Gt = Fs,s Fu,t , which completes the proof. We can conclude the following corollary. Corollary 1. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted idempotent market time. Then, for every t > 0, the triple (mt , Q, G m ) satisfies the hypothesis (HP). Proof. By Theorem 5 and Proposition 6, we can conclude that mt is a G m -honest time for every t ≥ 0. Then, by Theorem 9, we have the desired result. 4.2.2 Market Times with Horizons In this subsection, we investigate a market time with a horizon. Definition 17. [Horizon of Market Time] Let m = {mt }t≥0 be a market time. A horizon of m is a positive number T such that m T = T a.s.. The horizon is a kind of proxy of ∞ as well as a practical representation of expiring dates or the ends of accounting terms. We have the following lemma. Lemma 4. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted market time with a horizon T. Then, G Tm = G T .

21

Proof. { } G Tm = σ A ∩ {ms ≥ u} | 0 ≤ u ≤ s ≤ T, A ∈ Gu } { ⊃ σ A ∩ {m T ≥ u} | 0 ≤ u ≤ T, A ∈ Gu { } = σ A ∩ { T ≥ u} | 0 ≤ u ≤ T, A ∈ Gu { } = σ A | 0 ≤ u ≤ T, A ∈ Gu

=

∪

Gu

u≤ T

= GT . On the other hand, by Proposition 4 1, G Tm ⊂ G T . Therefore, G Tm = G T . The next theorem may become one of the key tools for calculating conditional expectations given market filtrations generated by idempotent market times. Theorem 10. Let G = {Gt }t≥0 be a filtration, and m = {mt }t≥0 be a G-adapted idempotent market time with a horizon T. Then, for every 0 ≤ u < s ≤ t < T, Q (ms ≤ u | Gtm ) = Q (ms ≤ u | G T ). Proof. By Corollary 1, (ms , Q, G m ) satisfies the hypothesis (HP). Hence, Q (ms ≤ u | Gtm )Q (ms ≤ t | G Tm ) = Q (ms ≤ t | Gtm )Q (ms ≤ u | G Tm ). Here, we have Q (ms ≤ t) = 1 since s ≤ t. Therefore, with Lemma 4, we get the desired equation.

Acknowledgements The author owes a special debt of gratitude to Professor Ryozo Miura, who took the considerable trouble to read in detail the author’s thesis on which this paper is based, providing valuable comments and corrections. Professor Hidetoshi Nakagawa found several errors in every text of early drafts by reading them quite carefully. He keeps providing the author valuable comments. Professor Marek Rutkowski kindly read one of the earliest drafts and gave the author some important hints to go further in the next step. The author would like to express his sincere appreciations to all of them.

References [1] Takanori Adachi. Credit risk modeling with market times. Master’s thesis, International Corporate Strategy, Hitotsubashi University, Tokyo, 2011.

22

[2] Tomasz R. Bielecki, Monique Jeanblanc, and Marek Rutkowski. Credit Risk Modeling, volume 2 of Osaka University CSFI Lecture Notes Series. Osaka University Press, Osaka, 2009. [3] Tomasz R. Bielecki and Marek Rutkowski. Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin Heidelberg, 2004. [4] Garrett Birkhoff. Lattice Theory. American Mathematical Society, 3rd edition, 1967. [5] Umut C ¸ etin, Robert Jarrow, Philip Protter, and Yildiray Yildirim. Modeling credit risk with partial information. Annals of Applied Probability, 14(3):1167–1178, 2004. [6] Kai Lai Chung. Some universal field equations. In S´eminaire de Probabilit´es VI, volume 258 of Lecture Notes in Mathematics, pages 90–97, Berlin Heidelberg, 1972. Springer-Verlag. [7] Kai Lai Chung and Joseph Leo Doob. Fields, optionality and measurability. American J. Mathematics, 87(2):397–424, 1965. [8] Darrell Duffie and David Lando. Term structures of credit spreads with incomplete accounting information. Econometrica, 69:633–664, 2001. [9] Robert J. Elliott, Monique Jeanblanc, and Marc Yor. On models of default risk. Mathematical Finance, 10(2):179–195, 2000. [10] Kay Giesecke. Default and information. Journal of Economic Dynamics and Control, 30:2281–2303, 2006. [11] Xin Guo, Robert A. Jarrow, and Yan Zeng. Credit risk models with incomplete information. Mathematics of Operations Research, 34(2):320–332, 2009. [12] Monique Jeanblanc and Shiqi Song. An explicit model of default time with given survival probability. Working paper, Jun 22 2010. [13] Shigeo Kusuoka. A remark on default risk models. Adv. Math. Econ., 1:69– 81, 1999. [14] Libo Li and Marek Rutkowski. Constructing random times through multiplicative systems. Working paper, Nov 15 2010. [15] Libo Li and Marek Rutkowski. Progressive enlargements of filtrations and semimartingale decompositions. Working paper, Dec 11 2010. [16] Snorre Lindset, Arne-Christian Lund, and Svein-Arne Persson. Credit spreads and incomplete information. Working paper, Mar 4 2008. [17] Roger Mansuy and Marc Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Number 1873 in Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 2006. 23

[18] Robert C. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2):449–470, 1974. [19] Ashkan Nikeghbali. An essay on the general theory of stochastic processes. Probability Surveys, 3:345–412, 2006. [20] Philip E. Protter. Stochastic Integration and Differential Equations. Number 21 in Applications of Mathematics. Springer-Verlag, Berlin Heidelberg, 2nd edition, 2004. [21] Marek Rutkowski. Private communication. Dec 27 2010. [22] Marc Yor. Grossissement d’une filtration et semi-martingales: th´eoremes g´en´eraux. In S´eminaire de Probabilit´es XII, volume 649 of Lecture Notes in Mathematics, pages 61–69, Berlin Heidelberg, 1978. Springer-Verlag.

24