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The purpose of this paper is to present the theorical connection between the It ô stochastic calculus and the Financial Econometrics. This paper has two contributions. First, we give the backgrounds on how the stochastic calculus is used to model the real data with the uncertainties. Finally, by using Consumer Price Index (CPI) from the Central Bank of Congo and combining the It ô stochastic calculus and the AR (1)-GARCH (1, 1) model, we estimate the stochastic volatility of inflation rate measuring efficency of monetary policy. Thus the stochastic integrals are the powerful tools of mathematical modelling and econometric analysis.

In most dynamical systems which describe processes in economics, engineering, and physics, stochastic components and random noise are included. The stochastic aspects of the models are used to capture the uncertainty about the environment in which the systems are operating. For example, there are suggestions that increased uncertainty makes fiscal policy temporarily less effective [

Therefore, the stochastic state space models and time series analysis have been both intensively and extensively developed during the past twenty years. A unified theory has been constructed during this period and the concepts and methods have been widely applied to problems in the area of engineering and communication, economics and management. Because of these developments, interest in stochastic state space model and its applications has greatly increased in econometric research.

This paper presents the stochastic integrals and numerics which permit successful mathematical modelling not only in econometrics but also in many other fields such biometrics, psychometrics, environment science, and hydrology, assuming of course that a suitable sequence of observed data is available.

For estimating the parameters of both stochastic continuous and discrete-time models, the methods of maximum likelihood are usually used by researchers because of its capacity to give the best unbaised estimators [

The purpose of this paper is to emphasize on the linkage between the theory of stochastic integrals and time series analysis used in the econometric analysis [

The structure of the paper is as follows. In Section 2 we will give the theory of stochastic integrals that is usefull to economic analysis. In Section 3 we give some stochastic differential equations used as econometric models that are used to express the economic theories. Section 4 gives some numerical methods to perform the empirical analysis. Section 5 illustrates the use of the stochastic integrals to time series econometric by estimating the stochastic volatility from the Autoregressive-Generalized Autoregressive Concoditional Heteroskedasticity model, that is, AR (1)-GARCH (1, 1) model.

Since the works of Kuyosi Itô the field of stochastic integrals attract the attention of many mathematicians and researchers [

Itô Stochastic Integrals developed here are from [

Definition 2.0.1 A process X is called adapted to the filtration ( F t ), if for all t, X ( t ) is F t -measurable.

Proposition 2.0.1. (a) X = ( X t ) , where X t is a d-dimensional measurable, F t -adapted process is a continuous semimartingale if X t is continuous and has the form

X t = X 0 + M t + B t (1)

for all t (a.s.), where E | X 0 | < ∞ , (1) M = ( M t ) is a continuous L 2 - F t -martinagle with M 0 = 0 (a.s.) and (2) ( B t ) ∈ B .

(b) If in the decompostion 1, ( M t ), is a continuous local martingale and ( B t ) belongs to B l o c , then ( X t ) will be called a continuous local semi-martingale.

Theorem 2.1. [

1) Linearity. If Ito integrals of X ( t ) and Y ( t ) are defined and α and β are some constants then

∫ 0 T ( α X ( t ) + β Y ( t ) ) d B ( t ) = α ∫ 0 T X ( t ) d B ( t ) + β ∫ 0 T Y ( t ) d B ( t ) . (2)

2) ∫ 0 T X ( t ) I ( a , b ] ( t ) d B ( t ) = ∫ a b X ( t ) d B ( t ) . The following two properties hold when the process satisfies an additional assumption

∫ 0 T E ( X 2 ( t ) ) d t < ∞ . (3)

3) Zero mean property. If condition 3 holds then

E ( ∫ 0 T X ( t ) d B ( t ) ) = 0, (4)

where E denotes expectation with respect to classical Wiener measure.

4) Isometry property. If condition 3 holds. Then

E ( ∫ 0 T X ( t ) d B ( t ) ) 2 = E ∫ 0 T X 2 ( t ) d B ( t ) (5)

Corollary 2.1.1. If X is a continuous adapted process then the Itô integral ∫ 0 T X ( t ) d B ( t ) exists. In particular, ∫ 0 T f ( B ( t ) ) d B ( t ) where f is a continuous function on R is well defined.

A consequence of the isometry property is the expectation of the product of two Itô integrals as given in the following theorem.

Theorem 2.2. [

E ( ∫ 0 T X ( t ) d B ( t ) ∫ 0 T Y ( t ) d B ( t ) ) = ∫ 0 T E ( X ( t ) Y ( t ) ) d t . (6)

where E denotes mathematical expectation.

We denote by ℝ m n all real-valued m × n matrices and by

W ( t ) = ( W 1 ( t ) , ⋯ , W n ( t ) ) ′ , t ≥ 0.

Let [ a , b ] ∈ [ 0, ∞ [ and we put

C W ( [ a , b ] ) = { f : [ a , b ] × Ω → ℝ m n | ∀ 1 ≤ i ≤ m , ∀ 1 ≤ j ≤ n : f i j ∈ C W j ( [ a , b ] ) } ,

C I W ( [ a , b ] ) = { f : [ a , b ] × Ω → ℝ m n | ∀ 1 ≤ i ≤ m , ∀ 1 ≤ j ≤ n : f i j ∈ C I W j ( [ a , b ] ) }

and C I ( [ a , b ] ) respectively.

Definition 2.2.1. [

∫ a b f ( t ) d W ( t ) = ( ∑ j = 1 n ∫ a b f i j ( t ) d W j ( t ) ) ′ 1 ≤ i ≤ m (7)

where each of the integrals on the right-hand side is defined in the sense of Itô.

Proposition 2.2.1. (Itô formula) [

F ( X t ) = F ( X 0 ) + ∑ i = 0 d ∫ 0 t ∂ F ∂ x i ( X s ) d M s i + ∑ i = 0 d ∫ 0 t ∂ F ∂ x i ( X s ) d B s i + 1 2 ∑ i = 0 d ∫ 0 t ∂ 2 F ∂ x i ∂ x j ( X s ) d 〈 M i , M j 〉 s (8)

Stratonovich Stochastic Integrals. In [

S m ( f ) = ∑ 2 k ≤ m ! 2 k k ! ( m − 2 k ) I m − 2 k ( T r k f ) . (9)

where T r denoted the iterated traces that are defined formally starting with

T r f ( s 1 , ⋯ , s m − 2 ) = ∫ f ( s 1 , ⋯ , s m − 2 , s , s ) d s .

Another approach to formula (9) using Hida’s theory of white noise. Working on ℝ m instead of ℝ + m and assuming that f is a test-function, the integral S m ( f ) may indead be rewritten as

∫ f ( s 1 , ⋯ , s m ) X ˙ s 1 ( w ) ⋯ X ˙ s m ( w ) d s 1 ⋯ d s n = 〈 f , X ˙ ⊗ n 〉

where the derivative of Brownian motion is understood in the distribution sense. In the sense of Hu and Meyer [

S ( f ) = ∑ m 1 m ! ∫ [ S ) f m ( s 1 , ⋯ , s m ) d X s 1 ( w ) ⋯ d X s m ( w ) (10)

where f is a finite sequence of coefficients f m ∈ L s 2 ( ℝ m ) and n ! = n × ( n − 1 ) × ⋯ × 1 .

Itô’s Formula for Functions of Two Variables. If two processes X and Y both possess a stochastic differential with respect to and f ( x , y ) has continuous partial derivatives up to order two, then f ( X ( t ) , Y ( t ) ) also possesses a stochastic differential.

Theorem 2.3. [

d f ( X ( t ) , Y ( t ) ) = ∂ f ∂ x ( X ( t ) , Y ( t ) ) d X ( t ) + ∂ f ∂ y ( X ( t ) , Y ( t ) ) d Y ( t ) + 1 2 ∂ 2 f ∂ x 2 ( X ( t ) , Y ( t ) ) σ X 2 ( X ( t ) ) d t + 1 2 ∂ 2 f ∂ y 2 ( X ( t ) , Y ( t ) ) σ Y 2 ( Y ( t ) ) d t + ∂ 2 f ∂ x ∂ y ( X ( t ) , Y ( t ) ) σ X ( X ( t ) ) σ Y ( Y ( t ) ) d t (11)

An important case of Itô formula is for functions of the form

Theorem 2.4. [

Stochastic Calculus. Let

Definition 2.4.1. [

The equivalence class containing X is denoted by dX and is called the stochastic differential of X. As known, by definition,

is the process

Let

(1) Addition:

(2) Product:

(3) B-multiplication: If

is defined as an element in Q. Hence

Theorem 2.5. [

(a)

(b)

(c)

(d)

for

If

where

(4) Symmetric Q-Multiplication

Theorem 2.6. [

where

Theorem 2.7. If

The stochastic integral

Theorem 2.8. [

where

Skorokhod Integral. The Skorohod integral is an extension of the Itô integral to non-adapted processes and is the adjoint of the Malliavin derivative, which is fundamentals to the stochastic calculus of variations [

Definition 2.8.1. [

Then we define the Skorohod integral of

by

where

Wick Product. The Wick product was introduced in Wick (1950) as a tool to renormalize certaint infinite quantities in quantum field theory. In stochastic analysis the Wick product was first introduced by Hida and Ikeda (1995). The Wick product is important in the study of stochastic differential equations. In general, one can say that the use of this product corresponds to and extends naturally—the use of the Itô integrals. The Wick product can be defined in the following way:

Definition 2.8.2. The Wick product

with

In the

Proposition 2.8.1. Let

Suppose

For the relation between the Wick multiplication and The Itô-Skorohod Integration we put

Here the left hand side denotes the Skorokhod integral of the Stochastic process

The objective of this section presents in short the two main types of stochastic differential equation models. The theory of stochastic differential equation is very vaste and well known by Engineers and other scientists but less known and understood among economists. For further reading the reader can see [

Example 1: Stochastic Differential Equation Model. Let

where

Theorem 3.1. [

for any given

where the constant C depends only on K and T.

The following theorem gives the solution of stochastic differential equations as Markov processes.

Theorem 3.2. [

where

Theorem 3.3 [

where

Example 2: Differential Equation with Markovian Switching Model. For economists, the economic phenomena can be governed by uncertainties and cycles. This model was developped by [

Here the state vector has two components:

Example 3: Differential with Respect to Fractional Brownian Motion Model. Let

Consider the equation on

where

Assumption 3.3.1. Let us introduce the following assumptions on the coefficients:

A1.

A2. The coefficient

where

Consider the stochastic differential equation with respect to fBm (29) on

Suppose that the coefficients

Example 4: Differential Equation with Jumps Models. In real world, some phenomena or economic policy decisions are governed under uncertainty with jumps. Therefore, stochastic differential equation with jumps modeling can be considered as a usefull econometric approach [

for

Example 5: Partial Differential Equation Models. Stochastic Partial Differential Equation Models are used as power tools of mathematical modeling in many areas [

Consider the Itô Stochastic Partial Differential Equation of the form as mentioned in [

for

with independent scalar Wiener processes

Assumption 3.3.2. [

The literature contains many existence and uniqueness theorems for mild solutions of SPDEs. Theorem below provides an existence, uniqueness, and regularity result for solutions of SPDEs with globally Lipschitz continuous coefficients in the Equation (32).

Theorem 3.4. [

In this section we give a brief review some numerical methods used in the stochastic analysis that can be usefull for economists and social scientists. These main books can help econometricians and economists to improve and understand the numerical methods for stochastic analysis [

The Euler-Maruyama Scheme. We consider a scalar Itô stochastic ordinary differential equation (SODE) [

with a standard scalar Wiener process

The simplest numerical scheme for the SODE (35) is the Euler-Maruyama Scheme given by

where one usually writes

for

The Milstein Scheme [

Numerical Methods for Stochastic Differential Equations with Jumps. The Euler scheme for SDE with jumps (30), is given by the algorithm [

for ^{th} Gaussian

In the multidimensional case with mark-indepedent jump size we obtain the k^{th} component of the Euler scheme

Methods for Stochastic Partial Differential Equations. This material is from [

Methods for SPDE with Multiplicative Noise. Two representative numerical schemes used in the literature for the Stochastic Partial Differential Equation (32) are the linear-implicit Euler and the linear-implicit Crank-Nicolson schemes [

The Euler scheme

The Crank-Nicolson scheme

for

Convergence of SPDE with Multiplicative Noise. The convergence of the exponential Euler scheme will proved under the following assumptions.

Assumption 4.0.1. (A5) (Linear operator A). there exist sequences of real eigenvalues

for all

(A6) (nonlinearity of F). The nonlinearity

for all

for all

Let Q be a nonnegative definite symmetric trace-class operator on a separable Hilbert space K,

(A7) (Cylindrical Q-Wiener process

and pairwise independent scalar

(A8) (Initial value). The random variable

The convergence theorem for SPDE model 32

Theorem 4.1. (Convergence Theorem [

holds for all

Continuous-time models are central to financial econometrics, and mathematical finance. Here we estimate the Unobserved Stochastic Volatility of Inflation Rate. The literature on discrete-time models and that on continuous-time models were developed independently, but it is possible to establish connections between the two approaches [

In time series analysis, autoregressive integrated moving average (ARIMA) models have found extensive use since the publications of Box and Jenkins (1976) [

Maximum likelihood methods are widely used for estimating stochastic volatility [

To facilitate our discussion we will specialize the general continuous time model with zero drift, i.e.

where the stochastic processes

One should note that the constant elasticity variance process (CEV) in 47 implied an autoregressive model in discrete time for

After some algebraical manipulations such as

where

where

In time series analysis, a process

(1)

(2) There exist constants

Theorem 5.1. ( [

then the infinite sum

converges almost surely (a.s.) and the process (

Another important theorem for our analysis is the secon-order stationarity of the GARCH (1, 1) process.

Theorem 5.2. Let

To estimate the parameters of these models we use the maximum likelihood method. The maximum likelihood method provides the best estimators and efficient estimators [

where the

For the student’s t-distribution, the log-likelihood contributions are of the form

where the degree of freedom _{e} where

A maximum likelihood estimator (MLE) is obtained by maximizing the likelihood on a compact subset

To select a fitted model, the Akaike (1973) information criterion (AIC), Schowrz (1978) information (SIC), the mean squared error criterion (SIC), Hannan-Quinn information criterion (HQC) are usually used, that is,

where

where

In this study we modelize the stochastic volatility of inflation rate observed by the Central Bank of Congo for the period from January 2004 to June 2018. We get the inflation rate by transforming the consumer price index (CPI) index by using log-difference transformation, that is,

In statistics, the Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. Kurtosis statistics of the inflation rate 9.23 more large than 3, and Jarque-Bera statistics indicate that inflation rate does not follow the normal distribution. With high kurtosis statistic, 9.2287, there is an indication of inflation volatility.

We use a Student statistic test of statistical significance and find that parameters estimations are all statistically significant. Results confirm that the past volatilities affect the current volatility of inflation rate. Thus, we the dynmical behavior of volatility. We restrict the constant term to a function of the GARCH parameters and the unconditional variance:

where

Mean | 0.0128 |
---|---|

Median | 0.0056 |

Maximum | 0.1139 |

Minimum | −0.0746 |

Standard Deviation. | 0.0207 |

Skewness | 1.5268 |

Kurtosis | 9.2287 |

Jarque-Bera | 346.8773 |

Sum Sq Dev | 0.4929 |

Observations | 173 |

Parameters | AR (1)-GARCH (1, 1) | Z-Statistic | Prob |
---|---|---|---|

The Mean Equations | |||

0.0011 | 7.5509 | 0.0000 | |

0.6558 | 16.8504 | 0.0000 | |

0.0032 | |||

0.4219 | |||

The Conditional Variance Equations | |||

0.00000007 | |||

0.5635 | 31.9568 | 0.0000 | |

0.4363 | 24.7967 | 0.0000 | |

0.5736 | |||

Student Distribution Parameter | 3.3461 | 10.5507 | 0.0000 |

R^{2} | 0.16 | ||

AIC | −7.1608 | ||

SIC | −7.0693 | ||

HQC | −7.1237 | ||

DW | 2.2 | ||

SQ-Stat (20) | 0.4929 | 1.0000 | |

ARCH Test | 0.0171 | 0.8963 |

volatility persistence of CPI-inflation rate is very high level,

The postestmation tests of Ljung Box (1978), Q-Stat = 3.0639, and ARCH test, 0.0171, show that there are any remaining ARCH effects in the residuals.

Since the Itô’s works, the stochastic integrals and stochastic differential equations attract the attention of many researchers in the fields of mathematical modelling. In this paper, we emphasize on the application of stochastic integrals and differential equations in the economics and finance. Comparing to discrete models, the stochastic continuous-time models have many advantages because they take into account the uncertainty. The limit of this approach is the complexity of stochastic calculus and stochastic numerical methods. As mentioned by scientists (see Wiener, Einstein, Itô) the uncertainties are anywhere and anytime; therefore the stochastic integrals must be well known and understood by all scientists.

The authors declare no conflicts of interest regarding the publication of this paper.

Mambo, L.N.K., Mabela, R.M.M., Kanyama, I.K. and Mbuyi, E.M. (2019) On the Contribution of the Stochastic Integrals to Econometrics. Applied Mathematics, 10, 1048-1070. https://doi.org/10.4236/am.2019.1012073