Some Notes about the Continuous-in-Time Financial Model

HAL Id: hal-01584982

https://hal.archives-ouvertes.fr/hal-01584982

Submitted on 11 Sep 2017

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Some Notes about the Continuous-in-Time Financial Model

Tarik Chakkour

To cite this version:

Tarik Chakkour. Some Notes about the Continuous-in-Time Financial Model. Abstract and Applied Analysis, Hindawi Publishing Corporation, 2017, 2017, pp.6985820. �10.1155/2017/6985820�. �hal- 01584982�

Some notes about the continuous-in-time financial model

Tarik Chakkour

Univ. Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France February 20, 2017

Abstract. In this paper, we investigate the properties of operators in the continuous-in-time model which is designed to be used for the finances of public institutions. These operators are involved in the inverse problem of this model. We discuss this inverse problem in Schwartz space that we prove the uniqueness theorem.

Keywords: Inverse problem; ill-posed problems; adjoint operators; Schwartz space; financial model.

1 Introduction

Linear inverse problems arise whenever throughout engineering and the mathematical sciences. In most applications, these problems are ill-conditioned or underdetermined. Consequently, over the last two decades, the theory and practice of inverse problems is rapidly growing, if not exploding in many scientific domains. The fundamental reason is that solutions to inverse problems describe important properties of solution in this theory and the development of sophisticated numerical techniques for its treating on a level of high complexity. We mention the paper [9] introduced by Hadamard in the field of ill-posed problems.

We built in previous work [5, 3] the continuous-in-time model which is designed to be used for the finances of public institutions. This model uses measures over time interval to describe loan scheme, reimbursement scheme and interest payment scheme. Algebraic Spending Measure ˜σand Loan measure

˜

κ_E are financial variables involved in the model. Measure ˜σis defined such that the difference between spendings and incomes required to satisfy the current needs. Assuming that measures ˜σ and ˜κE are absolutely continuous with respect to the Lebesgue measure dt. This means that they readσ(t)dt and κE(t)dt, wheret is the variable inR. We call thatσandκE are time densities.

LetF andGbe normed spaces. Throughout this paper,L:F→Gis a continuous linear application (in short, an operator). We say that the following problem:

Givenσ∈G, findκE∈F such thatσ=L[κE],

is well-posed ifLis invertible and its inverseL⁻¹ : G→F is continuous. In other words, the problem is said to be well-posed if

∀σ∈G,∃!κE ∈F :σ=L[κE]; (1)

the solutionκEdepends continuously onσ. (2)

Existence and uniqueness of a solution for all g ∈ G (condition (1)) is equivalent to surjectivity and injectivity of L, respectively. Stability of the solution (condition (2)) amounts to continuity of L⁻¹. Conditions (1) and (2) are referred to as the Hadamard conditions. A problem which is not well-posed is said to be ill-posed. OperatorL links between Algebraic Spending Density σand Loan Desnity κE. If this operator is not invertible, solutions of the posed inversion problem can be brought.

In the recent papers [4, 5], we study the inverse problem stability of the continuous-in-time model.

We discuss this study with determining Loan Measure ˜κEfrom Algebraic Spending Measure ˜σin Radon measure space, i.e. F = M([t_I,Θmax −Θγ]), G = M([t_I,Θmax]), and in Hilbert space, i.e. F = L²([t_I,Θ_max−Θ_γ]), G =L²([t_I,Θ_max]) when they are density measures. For this inverse problem we prove the uniqueness theorem in [4], we obtain a procedure for constructing the solution and provide necessary and sufficient conditions for the solvability of the inverse problem.

We are motivated by a recently developed nonlinear inverse scale in Schwartz space. We refer the reader to [1, 10], for applications of fast inversion formulas to inverse problems. Frank Bauer & Mark A.

Lukas inverstigate in [1] some different frameworks for regularization of linear inverse problems when error is expected to be decreased at infinity. In the paper [10], P. C. Hansen investigates the approximation properties of regularized solutions to discrete ill-posed associated with the average decay to zero faster than the generalized singular values.

We show in this paper some results of this inverse problem in Schwartz space. We sketch the theo- retical results that justify the mathematical well-posedness under some assumptions. The main result of this paper is to study the existence and uniqueness of solutions. We give an overview of properties for operatorL, describing the computation of its image.

The rest of this paper is arranged as follows. In section 2 we introduce the definition of operatorLand others, and the mathematical properties of these operators is shown. We treat in section 3 the spectrum of some operators involved in the model by determining the inverse of operator under some hypothesis.

It is followed by enrichment of the model of variable rate in section 4. In section 5, we examine the concept of ill-posedness in Schwartz space in order to obtain interesting and useful solutions.

2 Properties of operators

This section is devoted to explore mathematical properties of some operators involved in the model.

Those properties will be useful for some aspects of the model implementation to come in the following.

These operators are acting on measures over R. For that, we will also consider that M([t_I,Θ_max]) is the set of Radon Measures overR, supported in [t_I,Θ_max]. In the sequel, we consider the case when all measures are density measures. The purpose is to compute the adjoint of these operators. We will be able to use some specific mathematical tools as inner product.

We proceed by denoting L²([t_I,Θmax]) the space of square-integrable functions overRhaving their support in [t_I,Θmax]. We state the Repayment Pattern Densityγas follows:

γ∈L²([0,Θγ]), (3)

where Θγ is a positive number such that:

Θ_γ <Θ_max−t_I. (4)

We recall that we have shown the balanced equation given by equality (14) in [5]. This equality consists in writing Loan Density κ_E as a sum of Algebraic Spending Density σ and densities associated with quantities that have to be repaid or paid. This equality yields with convolution equality defined by (9) in [5] to express densityσ:

σ(t) =κ_E(t)−(κ_E? γ)(t)−α Z t

tI

(κ_E−κ_E? γ)(s)ds −α Z Θmax

t

ρ^I_K(s)ds −ρ^I_K(t). (5) From this, linear term of density σ is defined by linear operator L acting on Loan Density κE ∈ L²([t_I,Θmax−Θγ]) given by:

L[κE](t) =κE(t)−(κE? γ)(t)−α Z t

tI

(κE−κE? γ)(s)ds. (6)

The aim here is to compute operatorL^∗. For that, we will compute inner product D

L[κE], κF

E

for any densitiesκ_E and κ_F in respectively spacesL²([t_I,Θ_max−Θ_γ]) andL²([t_I,Θ_max]) defined by:

DL[κE], κ_FE

= Z Θmax

tI

L[κE](x)κ_F(x)dx,

= Z Θmax

tI

κ_E(x)κ_F(x)dx− Z Θmax

tI

(κ_E? γ)(x)κ_F(x)dx

−α Z Θ_max

tI

Z x

tI

κE(y)dy

!

κF(x)dx+α Z Θ_max

tI

Z x

tI

κE? γ(y)dy

!

κF(x)dx.

(7)

We will simplify inner product D

L[κE], κF

E

given by relation (7) in function of four terms. Since first term

Z Θ_max

tI

κE(x)κF(x)dx is already simplified, we simplify the second one as follows:

Z Θ_max

tI

(κE? γ)(x)κF(x)dx= Z Θ_max

tI

Z x

tI

γ(x−y)κE(y)dy

!

κF(x)dx,

= Z Θmax

tI

Z Θmax

tI

γ(x−y)κ_E(y)1{y≤x}dy

!

κ_F(x)dx,

= Z Θmax

tI

κE(y)

Z Θmax

y

γ(x−y)κF(x)dx

! dy.

(8)

Next, we simplify the third one as follows:

Z Θmax

tI

Z x

tI

κE(y)dy

!

κF(x)dx= Z Θmax

tI

Z Θmax

tI

κE(y)1{y≤x}dy

!

κF(x)dx,

= Z Θ_max

tI

κE(y)

Z Θ_max

y

κF(x)dx

! dy.

(9)

The last one is simplified as follows:

Z Θ_max

tI

Z x

tI

κE? γ(y)dy

!

κF(x)dx= Z Θ_max

tI

Z x

tI

Z y

tI

γ(y−t)κE(t)dt

! dy

!

κF(x)dx,

= Z Θ_max

t_I

Z Θ_max

t_I 1{y≤x}

Z Θ_max

t_I

γ(y−t)κE(t)1{t≤y}dt

! dy

!

κF(x)dx,

= Z Θ_max

tI

Z Θ_max

tI

κ_E(t)1{t≤x}

Z Θ_max

tI

γ(y−t)1{t≤y≤x}dy

! dt

!

κ_F(x)dx,

= Z Θmax

tI

Z Θmax

tI

κ_F(x)1{t≤x}

Z Θmax

tI

γ(y−t)1{t≤y≤x}dy

! dx

!

κ_E(t)dt.

(10)

According to relations (8), (9) and (10), expression (7) of inner productD

L[κE], κF

E reads DL[κE], κF

E

= Z Θmax

tI

κE(y) κF(y)− Z Θmax

y

γ(x−y)κF(x)dx−α Z Θmax

y

κF(x)dx

+α Z Θ_max

y

κF(x) Z x

y

γ(z−y)dz

! dx

! dy.

(11)

Since operatorL^∗is one such that

∀κE∈L²([t_I,Θmax−Θγ]),∀κF ∈L²([t_I,Θmax]),D

L[κE], κF

E

=D

κE,L^∗[κF]E

, (12)

and according to relation (11), operatorL^∗ is defined by:

L^∗[κF](y) =κF(y)− Z Θ_max

y

γ(x−y)κF(x)dx−α Z Θ_max

y

κF(x)dx+α Z Θ_max

y

κF(x) Z x

y

γ(z−y)dz

! dx,

=κF(y)− Z Θ_max

y

κF(x) γ(x−y) +α−α Z x

y

γ(z−y)dz

! dx.

(13) Defining linear operatorDacting on Initial Debt Repayment Densityρ^I_K∈L²([t_I,Θmax]) as:

D[ρ^I_K](t) =−α Z Θ_max

t

ρ^I_K(s)ds−ρ^I_K(t). (14) The integration by parts states that inner product D

D[ρ^I_K], κF

E

is computed for any densities ρ^I_K and κF inL²([t_I,Θmax]) as follows:

DD[ρ^I_K], κF

E

= Z Θ_max

tI

D[ρ^I_K](x)κF(x)dx,

=−α Z Θ_max

tI

Z Θ_max

x

ρ^I_K(y)dy

!

κF(x)dx− Z Θ_max

tI

ρ^I_K(x)κF(x)dx,

=α Z Θ_max

t_I

ρ^I_K(x)

Z Θ_max

x

κF(y)dy

! dx+α

Z Θ_max

t_I

ρ^I_K(y)dy

! Z Θ_max

t_I

κF(y)dy

!

− Z Θmax

tI

ρ^I_K(x)κ_F(x)dx,

= Z Θ_max

tI

ρ^I_K(x) α Z Θ_max

x

κ_F(y)dy+α Z Θ_max

tI

κ_F(y)dy−κ_F(x)

! dx.

(15) From this, operatorD^∗ is defined by:

D^∗[κF](x) =α Z Θ_max

x

κF(y)dy+α Z Θ_max

tI

κF(y)dy−κF(x). (16) Operator L we set out in relation (6) considered a constant rate. Nevertheless, if we consider in it a function α that depends on t, the model becomes a financial model with variable rate. The only modification to make is to enrich (6) and (14) by writing:

L[κE](t) =κE(t)−(κE? γ)(t)−α(t) Z t

t_I

(κE−κE? γ)(s)ds. (17)

D[ρ^I_K](t) =−α(t) Z Θmax

t

ρ^I_K(s)ds−ρ^I_K(t). (18)

Once this enrichment is done, by using a variable rate makes operator D^∗ be expressed in terms of densitiesαandγ:

L^∗[κF](y) =κF(y)− Z Θmax

y

κF(x) γ(x−y) +α(x)−α(x) Z x

y

γ(z−y)dz

!

dx. (19)

By definitionα(x) is the rate at timex. OperatorD^∗given by relation (16) can be rewritten adding this enrichment:

D^∗[κF](x) =α(x) Z Θ_max

x

κF(y)dy+α(x) Z Θ_max

tI

κF(y)dy−κF(x). (20) Lemma 2.1. The image of operator L is such that:

Im(L)^⊥={0}. (21)

Proof. In order to show equality (21), we will show that the kernel of operatorL^∗is reduced to null set because of following property:

Im(L)^⊥= Ker(L^∗). (22)

According to (13), if densityκF is in Ker(L^∗), then, we get the following equation:

κ_F(y)− Z Θmax

y

κ_F(x) γ(x−y) +α−α Z x

y

γ(z−y)dz

!

dx= 0. (23)

Derivating, we get the ODE that the solutionκF is expressed as follows:

κ⁰_F(y)−(α+γ(0))κ_F(y) = 0. (24) The general solution to (24) is given by:

κF(y) =κF(t_I)e−(α+γ(0))(y−tI), (25) where initial conditionκF(t_I) stands for the value of densityκF at initial time t_I. On the other hand, equation (23) is equivalent to:

κ_F(y)−(α+γ(0)) Z Θmax

y

κ_F(x)dx

| {z }

First term I₁

+ Z Θmax

y

−γ(x−y) +γ(0) +α Z x

y

γ(z−y)dz

! dx

| {z }

Second term I₂

= 0. (26)

Conversely, we will show that densityκF is zero. In the first place, replacing density κF given by (25) in first termI1of equation (26), we get:

κF(y)−(α+γ(0)) Z Θmax

y

κF(x)dx= −ακF(t_I)

α+γ(0) e−(α+γ(0))(Θ_max−tI). (27) Secondly, the second termI₂ is a constant function due to its derivative which equals zero:

Z Θmax

y

−γ(x−y) +γ(0) +α Z x

y

γ(z−y)dz

! dx

!⁰

= 0− −γ(0) +γ(0) +α Z y

y

γ(z−y)dz

! ,

= 0.

(28) Consequently, the second termI2 is equal to a real constantC to be determined:

Z Θ_max

y

−γ(x−y) +γ(0) +α Z x

y

γ(z−y)dz

!

dx=C. (29)

The initial condition is obtained from integral equation defined by (29) with replacingy by Θmax, which implies that constantC is zero. It is concluded that:

∀y∈[t_I,Θ_max], Z Θmax

y

−γ(x−y) +γ(0) +α Z x

y

γ(z−y)dz

!

dx= 0. (30)

Then, relations (26), (27) and (30) yield the following equality:

−ακ_F(t_I)

α+γ(0)e−(α+γ(0))(Θmax−tI)= 0. (31) From this, initial densityκF(t_I) or loan rateαis zero since exponential function is positive. It follows that ifκF(t_I) is zero, then, densityκF is zero, which is obtained from relation (25). In this case:

Ker(L^∗) ={0}. (32)

If loan rateαis zero, then, according to (23), we obtain following integral equation:

κF(y)− Z Θ_max

y

κF(x)γ(x−y)dx= 0, (33)

where expression of densityκF is determined from equality (25) as:

κF(y) =κF(t_I)e^{−γ(0)(y−tI)}. (34)

If densityκF given by (34) is coupled with (33), then,κF(t_I) is zero allowing that densityκF is zero in this case. We showed that densityκF is zero in both cases. From this, we can deduce that (32) is true, proving the lemma.

3 Spectrum of operators

It is well known that the integral operators [11, 2] possess a very rich structure theory, such that these operators played an important role in the study of operators on Hilbert Spaces. The paper [6] and book [7] by M.I. Gil’ deal with the spectra of a class of linear non-selfadjoint operators containing the Volterra operators. Since this operator is involved in the model, we use it in order to study the spectrum of some operators. It is shown in [8] that the spectrum of Volterra composition operator is consisting of zero only.

This section is devoted to explore the spectrum of some operators involving the spectrum of Volterra.

Defining linear operator ˜V :L²([t_I,Θmax−Θγ])→L²([t_I,Θmax]) by operator that acting on Loan Density κ_E

V[κ˜ E](t) =κE? γ(t) +α Z t

tI

(κE−κE? γ)(s)ds. (35)

The canonical injectionId_L2([tI,Θ_max−Θγ])→L²([tI,Θ_max])is defined fromL²([t_I,Θmax−Θγ]) toL²([t_I,Θmax]) as:

Id_L2([tI,Θmax−Θγ])→L²([tI,Θmax])[κE](t) = [κE](t). (36) which is decomposed as a sum of operatorsLand ˜V given by relations (6) and (35), respectively:

L[κE] + ˜V[κE] =Id_L2([t_I,Θ_max−Θγ]→L²([t_I,Θ_max]). (37) Theorem 3.1. If densityγ has upper boundM−2|α|over its support:

sup

z∈[0,Θγ]

(

|γ(z)|

)

=M−2|α|, (38)

whereM is a positive real satisfying:

2|α|< M < 1

Θmax−t_I, (39)

then, operator Lis invertible, where its inverse L⁻¹ is given by:

L⁻¹[κ_E](x) =J◦ X

k≥0

( ˜V ◦J)^(k)

!

[κ_E](x), (40)

whereJ is an operator defined by:

J : L²([t_I,Θmax]) → L²([t_I,Θmax−Θγ])

κE 7→ κE

(41) Proof. Since operator ˜V is defined from L²([t_I,Θ_max−Θ_γ]) to L²([t_I,Θ_max]), and by using definition (41) of operatorJ, we get:

V ◦˜ J(L²([t_I,Θ_max]))⊂L²([t_I,Θ_max]). (42) From this, we get:

∀k∈N,( ˜V ◦J)^(k)(L²([t_I,Θ_max]))⊂L²([t_I,Θ_max]), (43) which implies that:

J◦ X

k≥0

( ˜V ◦J)^(k)

!

(L²([t_I,Θ_max]))⊂L²([t_I,Θ_max−Θ_γ]). (44)

Relation (44) shows that equality (40) is consistent due to the inverse of operatorLis inL²([t_I,Θmax− Θγ]). Next, we can show that operator ˜V ◦J can be written in the form:

V ◦˜ J[κE](x) = Z x

tI

k(x, y)κE(y)dy, (45)

where kernelkis defined as:

k(x, y) =γ(x−y) +α−α Z x

y

γ(t−y)dt. (46)

Following equality

{(y, t)∈R²/t_I≤y≤t, t_I≤t≤x}={(y, t)∈R²/t_I≤y≤x, y≤t≤x}, (47) yields with Fubini-Tonelli theorem to obtain:

( ˜V ◦J)⁽²⁾[κ_E](x) = Z x

tI

k(x, t)( ˜V ◦J)[κ_E](t)dt,

= Z x

tI

k(x, t) Z t

tI

k(t, y)κ_E(y)dy

! dt,

= Z x

tI

Z x

y

k(x, t)k(t, y)dt

!

κ_E(y)dy.

(48)

Consequently, operator ( ˜V ◦J)⁽²⁾ is written in following form:

( ˜V ◦J)⁽²⁾[κE](x) = Z x

tI

k₂(x, y)κE(y)dy, (49)

where

k2(x, y) = Z x

y

k(x, t)k(t, y)dt. (50)

We can verify by induction forn≥2 that the recurrence expresses each operator ( ˜V ◦J)⁽ⁿ⁾as an integral operator which is written in following form:

( ˜V ◦J)⁽ⁿ⁾[κ_E](x) = Z x

tI

k_n(x, y)κ_E(y)dy, (51)

where kernelk_n is given as:

kn(x, y) = Z x

y

k(x, t)k_n−1(t, y)dt. (52)

Now, we will show thatM is a maximum of kernelkover [t_I,Θ_max]×[t_I,Θ_max−Θ_γ] with using equality (38):

|k(x, y)| ≤ |γ(x−y)|+|α|

Z Θ_γ

0

γ(z)dt

+|α|,

≤M−2|α|+|α|+|α|,

≤M.

(53)

By a recurrence starting with initial staten= 1 is true, it is easy to prove that:

|kn(x, y)| ≤Mⁿ(x−y)ⁿ⁻¹

(n−1)! . (54)

Ifn= 1, inequality (54) is true becauseM is the maximum of the kernelkover [t_I,Θmax]×[t_I,Θmax−Θγ].

Assuming that inequality (54) is true for a casen, and showing a casen+ 1. According to (52), we get:

|kn+1(x, y)| ≤ Z x

y

|k(x, t)||kn(t, y)|dt,

≤M× Mⁿ (n−1)!

Z x

y

(t−y)ⁿ⁻¹dt,

≤M× Mⁿ

(n−1)! ×(x−y)ⁿ

n ,

≤Mⁿ⁺¹(x−y)ⁿ n! .

(55)

Sincex−y∈[0,Θ_γ], we use inequality (54) to get:

|kn(x, y)| ≤Mⁿ Θⁿ⁻¹_γ

(n−1)!. (56)

Applying Cauchy-Schwarz inequality at ( ˜V ◦J)⁽ⁿ⁾defined by (51), we get following equality:

|( ˜V ◦J)⁽ⁿ⁾[κ_E](x)|²≤ Z x

tI

|k_n(x, y)|²dy

! kκ_Ek²

L²([t_I,Θ_max−Θγ]). (57) From this and using inequality (56), we get:

|( ˜V ◦J)⁽ⁿ⁾[κ_E](x)|²≤ Mⁿ (n−1)!

!2

Z x

tI

(x−y)²⁽ⁿ⁻¹⁾dy

!

kκ_Ek²_L2([tI,Θ_max−Θγ]),

≤ Mⁿ (n−1)!

!2

×(x−t_I)²ⁿ⁻¹

2n−1 kκEk²_L2([tI,Θ_max−Θγ]).

(58)

By integration each terms of (59) over interval [t_I,Θmax], we get:

k( ˜V ◦J)⁽ⁿ⁾[κE]k²_L2([tI,Θ_max])≤ Mⁿ (n−1)!

!2

×(Θmax−t_I)²ⁿ

2n(2n−1) kκEk²_L2([tI,Θ_max−Θγ]). (59) Consequently,

k( ˜V ◦J)⁽ⁿ⁾k_L2([tI,Θmax])≤ Mⁿ (n−1)!

!

×(Θmax−t_I)ⁿ

√2n . (60)

Inequality (60) gives:

k( ˜V ◦J)⁽ⁿ⁾kⁿ¹

L²([tI,Θ_max])≤M(Θmax−t_I)× √

2n(n−1)!

!−_n¹

. (61)