Some Notes about the Continuous-in-Time Financial Model

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.

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 [1] introduced by Hadamard in the field of ill-posed problems.

We built in previous work [2, 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 and Loan Measure are financial variables involved in the model. Measure is defined such that the difference between spending and incomes required to satisfy the current needs. Assume that measures and are absolutely continuous with respect to the Lebesgue measure . This means that they read and , where is the variable in . We call and time densities.

Let and be normed spaces. Throughout this paper, is a continuous linear application (in short, an operator). We say that the following problem: is well-posed if is invertible and its inverse : is continuous. In other words, the problem is said to be well-posed ifExistence and uniqueness of a solution for all (condition (2)) are equivalent to surjectivity and injectivity of , respectively. Stability of the solution (condition (3)) amounts to continuity of . Conditions (2) and (3) are referred to as the Hadamard conditions. A problem which is not well-posed is said to be ill-posed. Operator links between Algebraic Spending Density and Loan Density . If this operator is not invertible, solutions of the posed inversion problem can be brought.

In the recent papers [2, 4], we study the inverse problem stability of the continuous-in-time model. We discuss this study with determining Loan Measure from Algebraic Spending Measure in Radon measure space, that is, and , and in Hilbert space, that is, and , 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 [5, 6], for applications of fast inversion formulas to inverse problems. Bauer and Lukas investigate in [5] some different frameworks for regularization of linear inverse problems when error is expected to be decreased at infinity. In the paper [6], Hansen investigates the approximation properties of regularized solutions to discrete ill-posed. They 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 theoretical 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 operator , describing the computation of its image.

The rest of this paper is arranged as follows. In Section 2 we introduce the definition of operator and others, and the mathematical properties of these operators are 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 . For that, we will also consider that is the set of Radon Measures over , supported in . 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 the space of square-integrable functions over having their support in . We state the Repayment Pattern Density as follows:where is a positive number such thatWe recall that we have shown the balanced equation given by equality () in [2]. This equality consists in writing Loan Density as a sum of Algebraic Spending Density and densities associated with quantities that have to be repaid or paid. This equality yields with convolution equality defined by () in [2] to express density :From this, linear term of density is defined by linear operator acting on Loan Density given byThe aim here is to compute operator . For that, we will compute inner product for any densities and in, respectively, spaces and defined byWe will simplify inner product given by relation (8) in function of four terms. Since first term is already simplified, we simplify the second one as follows:Next, we simplify the third one as follows:The last one is simplified as follows:According to relations (9), (10), and (11), expression (8) of inner product readsSince operator is one such thatand, according to relation (12), operator is defined byDefine linear operator acting on Initial Debt Repayment Density asThe integration by parts states that inner product is computed for any densities and in as follows:From this, operator is defined byOperator we set out in relation (7) considered a constant rate. Nevertheless, if we consider in it a function that depends on , the model becomes a financial model with variable rate. The only modification to make is to enrich (7) and (15) by writingOnce this enrichment is done, using a variable rate makes operator expressed in terms of densities and :By definition is the rate at time . Operator given by relation (17) can be rewritten adding this enrichment

Lemma 1. The image of operator is such that

Proof. In order to show equality (22), we will show that the kernel of operator is reduced to null set because of the following property:According to (14), if density is in , then, we get the following equation:Deriving, we get the ODE that the solution is expressed as follows:The general solution to (25) is given bywhere initial condition stands for the value of density at initial time . On the other hand, (24) is equivalent toConversely, we will show that density is zero. In the first place, replacing density given by (26) in first term of (27), we getSecondly, the second term is a constant function due to its derivative which equals zero:Consequently, the second term is equal to a real constant to be determined:The initial condition is obtained from integral equation defined by (30) with replacing by , which implies that constant is zero. It is concluded thatThen, relations (27), (28), and (31) yield the following equality:From this, initial density or loan rate is zero since exponential function is positive. It follows that if is zero, then, density is zero, which is obtained from relation (26). In this case:If loan rate is zero, then, according to (24), we obtain following integral equation:where expression of density is determined from equality (26) asIf density given by (35) is coupled with (34), then, is zero allowing zero density . We showed that density is zero in both cases. From this, we can deduce that (33) is true, proving the lemma.

3. Spectrum of Operators

It is well known that the integral operators [7, 8] possess a very rich structure theory, such that these operators played an important role in the study of operators on Hilbert Spaces. The paper [9] and book [10] by Gil’ deal with the spectra of a class of linear non-self-adjoint 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 [11] 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 by operator that is acting on Loan Density ,The canonical injection is defined from to aswhich is decomposed as a sum of operators and given by relations (7) and (36), respectively:

Theorem 2. If density has upper bound over its support:where is a positive real satisfyingthen, operator is invertible, where its inverse is given bywhere is an operator defined by

Proof. Since operator is defined from to and by using definition (42) of operator , we getFrom this, we getwhich implies thatRelation (45) shows that equality (41) is consistent due to the inverse of operator being in . Next, we can show that operator can be written in the formwhere kernel is defined asFollowing equalityyields with Fubini-Tonelli theorem to obtainConsequently, operator is written in following form:whereWe can verify by induction for that the recurrence expresses each operator as an integral operator which is written in following form:where kernel is given asNow, we will show that is a maximum of kernel over with using equality (39): By a recurrence starting with initial state being true, it is easy to prove thatIf , inequality (55) is true because is the maximum of the kernel over . Assume that inequality (55) is true for a case and show a case . According to (53), we get Since , we use inequality (55) to getApplying Cauchy-Schwarz inequality at defined by (52), we get following equality:From this and using inequality (57), we getBy integrating each term of (60) over interval , we get Consequently, Inequality (61) gives Since the series of defined termsis convergent, then quantity converges absolutely in . Consequently, exists and is finite in . We recall that we have Since we have for all integer , inequality (61) implies that Coupling inequality (64) with the fact that converges to 0 due to (40), we getComposing operators defined by (38) with operator , we get According to (66) and (67), we getThat is,achieving equality (41) of the lemma.

Let us characterize under assumption of Theorem 2 a spectrum set of operator . Defining spectrum as a set of reals such that is not invertible. Formally:Defining spectral radius of operator asNote that spectrum is included in a disk of center and radius . That is to sayNow we will compute spectral radius of operator using following equality:Since we have shown that converges to 0, equality (73) shows that spectral radius is zero:Consequently, spectrum is reduced to null set:

4. Extensions in the Model of Variable Rate

We built in [2] the financial models that are used on simplified problems in order to show how they can be used in reality. This section is devoted to enrich the model in order to account for this reality. In particular, we will express the Algebraic Spending Density in the model with variable rate. The mathematical consistency of this density is analyzed.

The Current Debt Field is related to Loan Measure and Repayment Measure by the following Ordinary Differential Equation:The solution of this ODE is expressed as follows: Since the Interest Payment Density is related to the Current Debt Field by a proportionality relation:the Interest Payment Density can be expressed in terms of Loan Density :Since Density readswhere operators and are defined in relations (18) and (19), respectively, we get equality (6). From this, we get the following Ordinary Differential Equation:Defining the Current Debt at time which is associated with the amount borrowed at time . is related to Loan Density using Repayment Pattern Density by the following Ordinary Differential Equation:with initial condition that expresses that The Current Debt at time which is related to the borrowed amount at time is the borrowed amount at time . The solution of this differential equation is expressed asWe will use the expression (81) of density in order to express Algebraic Spending Density in the model with variable rate. Quantity is defined such that the difference between spending and income at time is associated with the amount borrowed at time . It is a time density with respect to both variables and . Relation (81) yieldswhere is the Interest Payment Density at time which is associated with the borrowed amount at time , and where is a repayment scheme at time which is associated with the borrowed amount at time .

In what follows, we will show that the definition of Algebraic Spending Density is consistent with the definition of Algebraic Spending which is given in relation (6). Indeed, Algebraic Spending Density can be expressed in terms of . By integration over variable (which describes the borrowed time), from , Algebraic Spending Density can be defined as follows:where density is to be determined, which is Algebraic Spending Density associated with the known Current Debt at initial time .Replacing first term in relation (86) by and using the fact that the Interest Payment Density is expressed in [2] as follows:we get: The integration of Density over variable givesExpression (88) and consequently definition of Borrowed Time Related Algebraic Spending Density are consistent with the expression of Algebraic Spending Density given in relation (6). Indeed, if the rate is fixed with worth , (88) writes where density can be expressed as follows:We will justify the expression (91) of density as follows. Indeed, replacing time by initial time in expressions (6) and (91) of Density , we obtain the same expression defined by

5. Inverse Problem of the Model in

Denoting the Schwartz space consists of smooth functions whose derivatives (including the function itself) decay at positive infinity faster than any power. We say, for short, that Schwartz functions are rapidly decreasing. We state the Repayment Pattern Density as follows:We use the Fourier Transform which are operators acting on densities over . Operators stand for the Fourier Transform, and stands for the Inverse Fourier Transform.

Lemma 3. If function is in and if satisfies relation (93), then we have the following equality:

Proof. Integrating by parts states thatFrom this, we get the following equality:Using definition (18) of operator , equality in (96) is multiplied by density to giveReplacing in relation (97) by which is possible because of (18), we obtain the following equality:Applying Fourier Transform to each term of equality (98), we obtain equality (94), proving the lemma.

Lemma 4. Assuming (93), that implies thatis achieved and if function given by relation (7) satisfiesand for a negative rate then, and is such thatIf does not satisfy the equality in relation (100), then, has an infinite limit in .

Proof. As and the fact that , we getIndeed, under assumption (101) the product of a bounded function due to the negative rate by a function in is a function in . Then, using an order 1 Taylor expansion of , we obtain the following expansion of function :Since operator satisfies equality in relation (100), relation (104) givesAccording to (93), function is Taylor expanded in until the order to obtainAccording to relations (94), (105), and (106), we get equality (102). Moreover, according to relation (94), is a finite quantity outside of . Hence, it is concluded that is in . On the other hand, if equality in relation (100) is not satisfied, then, according to relations (94), (104), and (106), has an infinite limit at . From this, the proof of the lemma is achieved.

Theorem 5. If Repayment Pattern Density satisfies relation (93) such thatand if Initial Debt Repayment Density is in , then for any Algebraic Spending Density in satisfying the following equality:there exists an unique Loan Density in which is given in terms of by

Proof. According to relation (107), we getIn the first place, we will show that the left term given by (110) is in . Indeed, since is stable by Fourier Transform, assumption (93) states that is in . From this, quantity is in due to its stability under power and infinite sum. Secondly, we will show that density is in , where density is defined byBecause densities and are in , density is rapidly decreasing. Thus, for an integer there exists a positive constant such thatSince the derivative of density is equal to , inequality (112) givesIntegrate inequality (113) over to getConsequently, for an integer there exists a positive constant such thatInequality (115) states that is rapidly decreasing. Furthermore, since the derivative of density is equal to the derivative of density ,and the fact that space is stable under the operation of derivation, for any integer function is rapidly decreasing. Consequently, is in , achieving the second point of proof. We conclude thatbecause it is the product of two functions in .

Conflicts of Interest

The author declares that there are no conflicts of interests.