2) . The following two properties hold when the process satisfies an additional assumption
3) Zero mean property. If condition 3 holds then
where E denotes expectation with respect to classical Wiener measure.
4) Isometry property. If condition 3 holds. Then
Corollary 2.1.1. If X is a continuous adapted process then the Itô integral exists. In particular, where f is a continuous function on 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.  Let and be regular adapted processes, such that and . Then
where E denotes mathematical expectation.
We denote by all real-valued matrices and by
Let and we put
Definition 2.2.1.  If belongs to , then the stochastic integral with respect to W is the m-dimensional vector defined by
where each of the integrals on the right-hand side is defined in the sense of Itô.
Proposition 2.2.1. (Itô formula)   Let be a d-dimensional continuous semimartingale. Let , that is, let be bounded and continuous and have bounded, continuous derivatives of orders 1 and 2. Then,
Stratonovich Stochastic Integrals. In , the multidimensional Stratonovich integrals can be expressed by the following formula using Itô integrals
where denoted the iterated traces that are defined formally starting with
Another approach to formula (9) using Hida’s theory of white noise. Working on instead of and assuming that f is a test-function, the integral may indead be rewritten as
where the derivative of Brownian motion is understood in the distribution sense. In the sense of Hu and Meyer , a Stratonovich integral is given in rigorous form as
where f is a finite sequence of coefficients and .
Itô’s Formula for Functions of Two Variables. If two processes X and Y both possess a stochastic differential with respect to and has continuous partial derivatives up to order two, then also possesses a stochastic differential.
Theorem 2.3.  Let have continuous partial derivatives up to order two (a function) and X, Y be Itô processes, then
An important case of Itô formula is for functions of the form.
Theorem 2.4.    Let be twice continuously differentiable in x, and continuously differentiable in t (a function) and x be an Itô process, then
Stochastic Calculus. Let and and. We denote by Q the totality of quasimartingales.
Definition 2.4.1.  For, we say that X and Y are equivalent and write if, with probability one,
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, and. We introduce the following operations in dQ .
(1) Addition: for.
(2) Product: for where and are the martingale parts of X and i respectively.
(3) B-multiplication: If and, then
is defined as an element in Q. Hence is defined from and dX. We define an element of dQ by.
Theorem 2.5.  The space dQ with the operations (1), (2) and (3) is a commutative algebra over B, i.e., a commutative ring with the operations (1) and (2) satisfying the relations
for and. We also have that, and.
If and, then and
where and are elements in B defined by and, respectively. If and, then is a d-dimensional Wiener process. Such a system of martingales is called a d-dimensional Wiener martingale.
(4) Symmetric Q-Multiplication
Theorem 2.6.   The space dQ with the operations (1), (2), (3) and (4) is a commutative algebra over Q; we have for,
where denotes Stratonovich product.
Theorem 2.7. If and, then for we have
The stochastic integral is called the Stratonovich integral or the Fisk integral or sometimes the Fisk-Stratonovich symmetric integral. Indeed, we have the following theorem:
Theorem 2.8.  For every X and Y in Q,
where denotes a partition and
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.  Assume that
Then we define the Skorohod integral of denoted by
where represents the Kronecker product.
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 of to elements
with is defined by
In the cas the basis independence of the Wick product can be seen from the following formulation of Wick multiplication in terms of multiple Itô integrals.
Proposition 2.8.1. Let. Assume that have the following representation in terms of multiple Itô integrals:
For the relation between the Wick multiplication and The Itô-Skorohod Integration we put for simplicity. One of the most stricking features of the Wick product is its relation to Itô-Skorokhod Integration. In short, this relation can be expressed as
Here the left hand side denotes the Skorokhod integral of the Stochastic process (which coincides with the Itô integral if is adapted), while the right hand side is to be interpreted as an -valued (Pettis) integral. The relation 22 explains why the Wick product is so natural and importnat in stochastic calculus.
3. Stochastic Differential Equations Models
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 be a diffusion in n dimensions described by the multidimensional stochastic differential equation
where is matrix valued function, is d-dimensional Brownian motion and and X and are n-dimensional vector valued functions. The vector and the matrix are the coefficients of the stochastic differential equation.
Theorem 3.1.  (Uniqueness and Existence of Solution) If the coefficients are locally Lipschitz in X with a constant independent of t; that is, for every N, there is a constant K depending only on T and N such that for all and all,
for any given the strong solution to stochastic differentional Equation (27) is unique. If in addition to condition 24 the linear growth condition holds
is independent of B, and, then the strong solution exists and is unique on, moreover,
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.  (The solution of SDEs as Markov processes) If equation 27 satisfies the conditions of the existence and uniqueness theorem 3.1, the solution of the equation for arbitrary initial values is a Markov process on the interval whose initial probability distribution at the instant to is the distribution of C and whose transition probabilities are given by
where is the solution of equation.
Theorem 3.3  (The solution of SDEs as Diffusion processes) The condition of the existence and uniqueness theorem 3.1 are satisfied for the SDE
where, , and is a matrix. If in addition, the functions and are continuous with respect to t, the solution is a d-dimensional diffusion process on with drift vector and diffusion matrix. In particular, the solution of an autonomous SDE is always a homogeneous diffusion process on.
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  as hybrid models. Consider the Stochastic Differential Equation with Markovian Switching of the form
Here the state vector has two components: and. The first one is normally referred to as the state while the second one is regarded as the mode. In its operation, the system will switch from one mode to another in random way, and the switching among the modes governed by the Markov chain.
Example 3: Differential with Respect to Fractional Brownian Motion Model. Let be a m-dimensional fractional Brownian motion of Hurst parameter. This means that the components of B are independent fractional Brownian motions with the same Hurst parameter H. For further reading see   .
Consider the equation on
where is an m-dimensional random variable.
Assumption 3.3.1. Let us introduce the following assumptions on the coefficients:
A1. is differentiable in x, and there exists some constants and for every there exist such that the following properties hold:
A2. The coefficient satisfies for every
where), with and for some constant.
Consider the stochastic differential equation with respect to fBm (29) on where the process B is a d-dimensional fBm with Hurst parameter and is an m-dimensional random variable.
Suppose that the coefficients are measurable functions satisfying conditions A1 and A2, where the constants and may depend on, and,. Fix such that a uniue continuous solution such that for all. Moreover the solution is Holder continuous of order.
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 . Consider a one-dimensional SDE, d = 1, in the form
for, with, and an -adapted one-dimensional Wiener process. We assume an an -adapted Poisson measure with mark space and with intensity measure, where is a given probability distribution function for the realizations of the marks. Consider a one-dimensional SDE with Jumps (30) in integral form, is of the form
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, where, is an infinite dimensional Wiener process of the form
with independent scalar Wiener processes, and. Note that (Laplacian with Dirichlet boundary conditions) and in one spatial dimension has sample paths which are only -Hölder continuous. Here the family, , is an orthonormal basis in,.
Assumption 3.3.2.  (1) Linear operator A. Let be a finite or countable set. In addition, let be a family of real numbers with and let be an orthonormal basis of H. The linear operator is given by for all with. (2) Drift term F. Let be real numbers with and let be a globally Lipschitz continuous mapping. (3) Diffusion term B. Let be real numbers with and let be a globally Lipschitz continuous mapping. (4) Initial value: Let and be real numbers and let be an -measurable mapping with.
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.  Let Assumptions 3.3.2 (1)-(4) be fulfilled. Then there exists a unique of the Equation (32) that is predictable stochastic process
for all. In addition,.
4. Numerical Methods for Stochastic Differential Equations
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 numerical methods for stochastic ordinary differential equations can be summarized as follows.
The Euler-Maruyama Scheme. We consider a scalar Itô stochastic ordinary differential equation (SODE) 
with a standard scalar Wiener process. The SODE (35) is in fact a symbolic representation for the stochastic integral equation
The simplest numerical scheme for the SODE (35) is the Euler-Maruyama Scheme given by
where one usually writes
for and where with is an arbitrary partition of.
The Milstein Scheme . The another useful numerical scheme for the SODE (35) is the Milstein Scheme given by
Numerical Methods for Stochastic Differential Equations with Jumps. The Euler scheme for SDE with jumps (30), is given by the algorithm   ,
for with initial value. Here is the length of the time interval and is the nth Gaussian distributed increment of the Wiener process W, , represents the total number of jumps of Poisson random measure up to time t, which is Poisson distributed with mean.
In the multidimensional case with mark-indepedent jump size we obtain the kth 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 and. Here it is necessary to assume that for all in Assumptions 2 in order to ensure that is inversible for every.
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 and orthonormal eigenfunctions of such that the linear operator is given by
for all with.
(A6) (nonlinearity of F). The nonlinearity is two times continuously Fréchet differentiable and its derivatives satisfy the following conditions
for all, , and, and
for all, where is a positive constant.
Let Q be a nonnegative definite symmetric trace-class operator on a separable Hilbert space K, be an ONB in K diagonalizing Q, and let the correspoing eigenvalues be. Let, , be a sequence of independent Brownian motion defined on filtered probability space. The process is called a Q-Wiener process in K.
(A7) (Cylindrical Q-Wiener process) There exist a sequence of positive real numbers and a real number such that
and pairwise independent scalar -adapted Wiener process for. The cylindrical Q-Wiener process is given formally by
(A8) (Initial value). The random variable satisfies, where is given in A7.
The convergence theorem for SPDE model 32
Theorem 4.1. (Convergence Theorem  ) Suppose that Assumptions 3 (A5)-(A8) are satisfied. Then there is a constant such that
holds for all, where is the solution of SPDE 32, is the numerical solution given by 42, for, and is the constant given in Assumption A8.
5. Application to Stochastic Volatility Estimation
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, and are adapted. Here is a stationary process with nonnegative values and is called the stochastic volatility. The is the speed of adjustment of y to its long-run mean, , and is a positive scalar. And also is a standard Wiener process.
One should note that the constant elasticity variance process (CEV) in 47 implied an autoregressive model in discrete time for, namely:
After some algebraical manipulations such as, and and, , we have this hybrid model that has the autoregssive model and the generalized autoregressive condintionally heteroscedastic models, i.e. the AR (1)-GARCH (1, 1) Model with the mean equation      ,
where, with following a t-Student distribution and the variance equation that can be presented as follows
where is a vector of two standard dimensional Brownian motions that are independent with zero mean and unit variance, and are defined on probability space.
In time series analysis, a process is called a GARCH(p,q) process if its first two conditional moments exist and satisfy 
(2) There exist constants, and such that
Theorem 5.1. (  Strict stationarity of the strong GARCH (1, 1) process) if
then the infinite sum
converges almost surely (a.s.) and the process () defined by is the unique strictly stationary solution of the model. This solution is nonanticipative and ergodic. If and, there exists no strictly stationary solution.
Another important theorem for our analysis is the secon-order stationarity of the GARCH (1, 1) process.
Theorem 5.2. Let. If, a nonanticipative and second-order stationary solution to the GARCH(1,1) model does not exist. If, the process () defined by (2.13) is second-order stationary. More precisely () is a weak, white noise. Moreover, there exists no other second-order stationary and nonanticipative solution.
To estimate the parameters of these models we use the maximum likelihood method. The maximum likelihood method provides the best estimators and efficient estimators   - . The density f of the strong write noise is assumed known. This assumption is obviously very strong. Conditionally on the -field generated by, the variable has the density. It follows that given the observations, and the initial values, the conditional likelihood is defined by
where the are recursively, defined for, by
For the student’s t-distribution, the log-likelihood contributions are of the form
where the degree of freedom controls the tail behavior and log denotes the natural logarithm, that is, loge where. The t-distribution approaches the normal as and denotes the Gamma function.
A maximum likelihood estimator (MLE) is obtained by maximizing the likelihood on a compact subset of the parameter space     that is,
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 refers to the number of estimated model parameters.
where is the log-likelihood, k is the number of parameters, and n is the number of observations. Among a finite set of models; the model with the lowest criteria is preferred.
6. Empirical Results
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,. The operations of taking logarithms and differencing are standard time series tools for coering a data set into looking stationary (Resnick, 2007); therefore our variable is stationary. The inflation rate measures how fast prices are rising  . For the period under analysis Table 1 shows that the mean, the maximum, and minimum inflation rates are 1.3, 11.4, −7.5 percentages respectively. (ii) With the Jarque-Bera statistic, 346.8773, it indicates that the inflation rate does not follow the normal distribution. It is well known that the fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis. The Skewness statistics is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. The Skewness of 1.52 indicates the moderate level.
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 is the unconditional variance of the residuals, that is,.
Table 2 raises tree isues. First, in the mean equation, the coefficient measuring the persistence of inflation rate is high. This means that the monthly last inflation contributes to current rate by 66 percents. Secondly, the stochastic
Table 1. Summary statistics.
Table 2. Results of estimation.
volatility persistence of CPI-inflation rate is very high level, , this means that the past volatility information contributes to current volatility of inflation rate at 100 percents. Therefore the purchasing power of congolese householders is also volatile.
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.
7. Concluding Remarks
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.
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