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We study the default risk in incomplete information. That means we model the value of a firm by a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. This Lévy process cannot be completely observed, and another process represents the available information on the firm. We obtain a stochastic Volterra equation satisfied by the conditional density of the default time given the available information. The uniqueness of solution of this equation is proved. Numerical examples of (conditional) density are also given.

Here we consider a jump-diffusion process X which models the value of a firm. This is a Lévy process. Details on this class of processes can be found in [

We extend this approach studying the conditional law of the first passage time of Lévy process at level x given a partial information. We solve this problem using filtering theory inspired by Zakai [

This section defines the basic space in which we work and announces what we will do. Subsection 2.1 gives the model of the firm value and defines the default time. Subsection 2.2 recalls some important results in the complete information case. Subsection 2.3 defines the signal and observation process and the model for available information. Basically, it introduces the notion of filtering theory. Subsection 2.4 gives our motivation.

Let

standard Brownian motion W, a sequence of independent and identically distributed random variables

with distribution function

Poisson process with intensity ν under

probability space, we define a process X as follows:

X models a firm value and the default is modeled by the first passage time of X at a level

We suppose that X is not perfectly observable and that observation is modeled by process Q.

Let

By (5.12) page 197 of [

where

The function

to

as a default time the random variable

For a general Lévy process, Doney and Kiprianou [

Coutin and Dorobantu [

where

Our work is inspired and is in the same spirit as D. Dorobantu [

with h a Borel and bounded function and B a standard Brownian motion.

Definition 1. The process X is called the signal. The process Q is called the observation and is perfectly observed by investors.

This leads us to a filtering model and we introduce the filtering framework inspired of Zakai [

Since the function h is bounded, the Novikov condition,

define the following exponential martingale for the filtration

For a fixed maturity

Definition 2. For fixed

We also note that the law of X, so the one of

Then all the available information is represented by the filtration

where the s-algebra

D. Dorobantu [

is a

approximated by

show that the conditional law of default time

This section presents our basic model of a firm with incomplete information about its assets. More generally, we treat a continuous time setting, staying with the work of D. Dorobantu [

We recall that

Proposition 1. For all

where

And

Remark 1 Referring to [

In this subsection, we give our main results. Indeed, we first show that the conditional law of the hitting time

given the filtration

result. This type of equation is the same as the one studied in [

Theorem 1. Let

where

and G is defined in Proposition 1.

Proposition 2. If Equation (6) admits a solution, this one is unique.

Here, we give some technical and auxiliary results which are useful to prove Theorem 1 and Proposition 2.

Proposition 3. For any bounded function

By this proposition, we establish two corollaries which give a representation more accessible of the processes

_{x} = u +

Corollary 1. For all

1)

and equivalently

2)

Corollary 2. For

1)

and equivalently

2)

Proposition 4. For any

Remark 2. Equation (12) of Proposition 4 can be rewriten as:

Where

This equation is similar to the non normalized conditional distribution Equation (3.43) in A. Bain and D. Crisan [

In the same way, Equation (6) which is derived from (12) is similar to the normalized conditional distribution Equation (3.57) in A. Bain and D. Crisan [

We simulate the density of the first passage time respectively in complete information and in incomplete information. We suppose that the jump size follows a double exponential distribution, i.e, the common density of Y

is given by

Here,

information and on another hand to the values taken by the parameters m and

These four first figures (Figue 1 and

diffusion process (case of complete information). The variable

We observe that the maximum reached is greater if the drift m is positive, meaning the positive level x is more probably reached in a shorter time.

In incomplete information, the distance between the curve and axis is greater than in complete information case, this would mean that in case of incomplete information, the level x is more difficult to be reached in a short time.

The choice of the small value of

A large value of

In these last four figures (

Proposition 1

Proof. First note that, since X is a

The fact that

Secondly, for any

The

By hypothesis, we have

Then, we have for any

Now, we show the equality almost surely for all

These processes are increasing, then they are sub-martingales with respect to the filtration

We conclude that, almost surely, for all

Taking

□

Proposition 2

Proof. Let

where

We recall the expression

and remark that

Markov property implies

We use Lemma 4 with

and Lemma 7 (22) with the pair

All computations are done on the set

submartingale. Then for all

Thanks to Jensen inequality and Lemma 8 with

Concerning the numerator,

is satisfied then

So finally

Let

It follows using (13) that

Taking

Then

By Gronwall’s lemma, we deduce that

Proposition 3

Proof. Let be a process

Conditioning by

Conversely compute the expectation of the product of

Since

Finally we could replace

Proposition 4

Proof. Applying Lemma 4, it follows that

But, since the condition

that

consider for

(17). But Lemma 7 of Appendix ensures that

We apply Ito formula to the ratio of processes

satisfying the stochastic equations respectively (9) and (11):

The Itô’s formula applied to

We achieve the proof letting

Theorem 1

Proof. Let us now find a mixed filtering-integro-differential equation satisfied by the conditional probability density process defined from the representation

We fix a and t such that

By definition of G, we have

Then

By Tonelli Theorem,

Similarly

In Equation (12) of Proposition 4,

are respectively replaced by

By hypothesis, we have

For

The numerators being bounded by

position 4, which can be written again as

To express this result with

under the integral is multiplied and divided by the same term

tion, we use the filtration

Therefore, using (20) in Lemma 4, on the set

which finishes the proof. □

This paper extends the study of the first passage time for a Lévy process in [

We thank my PhD advisor Laure Coutin for her help and pointing out error. We thank too Monique Pontier for her careful reading. We thank the Editor and the referee for their comments. This work is supported by A.N.R. Masterie. This support is greatly appreciated.

WalyNgom,11, (2015) Conditional Law of the Hitting Time for a Lévy Process in Incomplete Observation. Journal of Mathematical Finance,05,505-524. doi: 10.4236/jmf.2015.55041

Lemma 1. Let be

Proof. Indeed using the law of G, we have

Since

By change of variable

□

Lemma 2. If

Proof. We have

where

□

Lemma 3. There exists some constants

Proof. The function f defined in (4) satisfies

Using the fact that if

Replacing

Let

Remark that conditionally to process N and the

Gaussian law with mean

Applying Lemma 1 we get the conditional expectation

Using the fact that

The proof is completed with Lemma 2. □

The next lemma is inspired of Jeanblanc and Rutkovski [

Lemma 4. For all

For instance with

Proof. Assume that there exists

Then,

Thus

_{0} Hence, we obtain

what is not possible. Indeed,

That means for all

Thus for any t, t,

On the set

Taking the conditional expectation with respect to

This implies that

Using Kallianpur-Striebel formula (see Pardoux [

□

The following is in [

Lemma 5. The family of

is total in the set of processes taking their values in

Let us denote by

Lemma 6. Let

Then

Proof. As in Lemma 5, the family of processes

is total in the set of processes taking their values in

Therefore, since

The equality is obtained from the fact that under

Lemma 7. Let be a process

and

For instance

Proof. Let be

The integration by parts Itô formula applied to the product

and remark that

Since X and Q are independent under

Similarly, using first

and the independence between X and Q under

Equations (23) and (24) imply that

Now let be

so

which concludes the proof. □

Lemma 8. For all

Proof. The process

From Corollary 2 (i), the process

position

Let

taking the expectation we derive

Using Gronwall’s Lemma

The proof of Lemma 8 is achieved by letting n going to infinity. □