Abstract
In this paper, we study a new one-dimensional homogeneous stochastic process, termed the Square of the Brennan-Schwartz model, which is used in various contexts. We first establish the probabilistic characteristics of the model, such as the analytical expression solution to Itô’s stochastic differential equation, after which we determine the trend functions (conditional and non-conditional) and the likelihood approach in order to estimate the parameters in the drift. Then, in the diffusion coefficient, we consider the problem of parameter estimation, doing so by a numerical approximation. Finally, we present an application to population growth by the use of real data, namely the growth of the total population aged 65 and over, resident in the Arab Maghreb, to illustrate the research methodology presented.
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The authors are very grateful to the editor and referees for constructive comments and suggestions. This research has been funded by LAMSAD from ”Fonds propres de l’Université Hassan I” (Morocco) and FQM-147 from ”Plan Andaluz de Investigaciòn” (Spain).
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Appendices
Appendix A: Ergodicity and stationary distribution of the SBSDP
Here, we study the asymptotic behaviour of the process proposed in this paper, analyse the problem of ergodicity and the existence of the stationary distribution of the process and explicitly obtain its density function.
In general (see Nobile and Ricciardi (1984) and Nicolau (2005)), a diffusion process {x(t), t ≥ 0}, with state space I = (l, r) , is governed by the following SDE:
where Wt is a standard Wiener process and x is either a constant value or a random value independent of Wt. We assume that a(x) and b(x) have continuous derivatives.
Let \(\displaystyle { s(z)=\exp \left \{-{\int }_{z_{0}}^{z}\frac {2a(u)}{b^{2}(u)}du\right \}}\) be the scale density function (z0 is an arbitrary point inside I).
The speed density function is: m(u) = (b2(u)s(u))− 1. And we denote by:
where, l < x < y < r, and then if:
the process {x(t), t ≥ 0} is ergodic and has an invariant (stationary) density function which is given by:
In our diffusion, the drift and diffusion coefficient are, respectively:
and I = (0, ∞). In this case, it follows that:
and we have, for 0 < x < y < ∞
with the variable change v = u− 1/2, the latter expression can be written as
On the one hand, taking the limit as x tends to 0 in (20). We have, for β > 0, S(0, y] = ∞ (the case of β = 0 is excluded, because the process is lognormal, and this is not ergodic). On the other hand, taking the limit when y tends to ∞ in (20), we have, for \(\alpha \leq \displaystyle {\frac {\sigma ^{2}}{2}}\), S[x, ∞) = ∞. And therefore, for \(\alpha \leq \displaystyle {\frac {\sigma ^{2}}{2}}\) and β > 0,
The speed density in this case is
and we have:
and according to Gradshteyn and Ryzhik (1979) 3.18. p.317, for ν > 0 and μ > 0,
we have, for \(\alpha < \displaystyle {\frac {\sigma ^{2}}{2}}\) and β > 0,
Then, by combining the two conditions, we deduce that for \(\alpha < \displaystyle {\frac {\sigma ^{2}}{2}}\) and β > 0, the process is ergodic and its stationary density function is given by the following expression:
where \(\lambda =2-\displaystyle {\frac {4\alpha }{\sigma ^{2}}}\) and \(\mu =\displaystyle {\frac {4\beta }{\sigma ^{2}}}\), and the conditions of ergodicity in terms of λ and μ are equivalent to λ > 0 and μ > 0.
Appendix B: Approximate estimator of the diffusion coefficient of the SBSDP
To estimate the parameter σ, we used an extension of the method described by Chesney and Elliott (1995). This method has been used by Giovanis and Skiadas (1999) in a paper on a stochastic logistic innovation diffusion model studying electricity consumption in Greece and the United States, and in another on a stochastic Bass innovation diffusion model to study the growth of electricity consumption in Greece, see Skiadas and Giovanis (1997).
In our study, an approximate estimator of the σ parameter between two observations has the general form:
Then, we have
The method described by Chesney and Elliot was then applied to each time interval, and the average of these estimators for n observations of a sample path of the process is the approximate estimator. It takes the following form:
Other approximate estimators of σ can be obtained by the same procedure, for example:
Using Itô’s lemma to the transformationy = ln(x(t)) in (1) as follows:
By substituting:
Considering that
an approximate estimator of σ is:
Then, for n observations of a sample path of the process, the resulting approximate estimator has the following expression:
Alternatively, from the stochastic differential equation (1) of the variablex(t), we obtain:
and then:
Considering that
An approximate value of σ is:
Therefore, for n observations of a sample path of the process, the resulting approximate estimator has the following expression:
By applying the three expressions of \( \hat {\sigma } \) to our real data, we obtain the following results:
Similar values are obtained by all three methods.
Appendix C: Probabilistic characteristics and statistical inference of the SLDP
The SLDP from the SBSDP
In the equation (1) when β = 0, the homogeneous lognormal diffusion process is obtained as a particular case in which the infinitesimal moments are given by:
This satisfies the following Itô’s SDE:
where σ > 0 and α are real parameters, Wt is a standard Wiener process and \(x_{t_{0}}\) is fixed in \( \mathbb {R}^{*}_{+}\).
The analytical expression of the SLDP
By taking β = 0 in equation (3) of the Section 2.2, the previous SDE has a unique solution which is given analytically by the following expression:
$$ \begin{array}{@{}rcl@{}} x(t)&=&x_{t_{0}} \left( \exp\left[\left( \alpha-\frac{\sigma^{2}}{2}\right) \left( t-t_{0}\right)+\sigma\left( w(t)-w(t_{0})\right)\right]\right) \end{array} $$The trend functions of the SLDP
In the same way, by taking β = 0 in equations (4) and (5 of the Section 2.3, the conditional trend function of the process is:
$$ \begin{array}{@{}rcl@{}} \mathbb{E}(x(t)\mid x(s)=x_{s})=x_{s} e^{\alpha(t-s)}. \end{array} $$(21)and by assuming the initial condition \(\mathrm {P}\left (x(t_{0})=x_{t_{0}}\right )=1\), the trend function of the process is:
$$ \begin{array}{@{}rcl@{}} \mathbb{E}(x(t))=x_{t_{0}} e^{\alpha(t-t_{0})}. \end{array} $$(22)
Parameter estimation in the SLDP
We now determine the estimator of the parameter α of the SLDP using the method described. The estimator of the drift parameters α is obtained by the maximum likelihood method, with continuous sampling.
Likelihood estimation of the drift parameter:
The vector form of the SDE of the SLDP can be written as:
$$ \begin{array}{@{}rcl@{}} A_{t}(x(t))= x(t), \quad \theta^{*}= \alpha, \quad B_{t}(x(t))= \sigma x(t), \end{array} $$The corresponding vector HT in this case is one-dimensional and is given by:
$$ \begin{array}{@{}rcl@{}} H^{*}_{T}= \displaystyle{\frac{1}{\sigma^{2}}} {\int}_{t_{0}}^{T}\frac{dx(t)}{x(t)}, \end{array} $$and ST has the following form:
$$ \begin{array}{@{}rcl@{}} S_{T}= \frac{T-t_{0}}{\sigma^{2}} \end{array} $$Then the expression of the estimator is
$$ \begin{array}{@{}rcl@{}} \displaystyle{\hat{\alpha}=\frac{ {\int}_{t_{0}}^{T}\frac{dx(t)}{x(t)}} {T-t_{0}}}, \end{array} $$The stochastic integral in the latter expression can be transformed into Riemann integrals by using Itô’s formula and thus:
$$ \begin{array}{@{}rcl@{}} {\int}_{t_{0}}^{T}\frac{dx(t)}{x(t)}&=& \log(x_{T})-\log(x_{t_{0}}) +\frac{\sigma^{2}}{2}\left( T-t_{0}\right). \end{array} $$Therefore, the expression of the Maximum Likelihood estimator is:
$$ \displaystyle{\hat{\alpha}=\frac{\left( \log(x_{T})- \log(x_{t_{0}}\right) +\frac{\sigma^{2}}{2}\left( T-t_{0}\right)} {T-t_{0}}}. $$Approximate estimator of the diffusion coefficient
The estimator of the coefficient diffusion parameter can be approximated using a method similar to that described Section 3.2. By following the same steps and for n observations of a sample path of the process, the resulting approximate estimator is:
Computational aspects in SLDP
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Approximated likelihood estimators
As for the SBSDP, in order to use the above expression to estimate the parameter α, we must have continuous observations. Therefore, we use an alternative estimation procedure based on continuous time maximum likelihood estimators with suitable approximations of the integrals that appear in the expression. The Riemann-Stieljes integrals are approximated by means of the trapezoidal formula.
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Estimated trend functions
By applying Zehna’s theorem and by taking β = 0 in the equations (21) and (22), the estimated trend function (ETF) and estimated conditional trend function (ECTF) of the process are obtained as:
$$ \begin{array}{@{}rcl@{}} \widehat{\mathbb{E}}(x(t)/ x(s)=x_{s})=x_{s} e^{\hat{\alpha}(t-s)}, \end{array} $$$$ \begin{array}{@{}rcl@{}} \widehat{\mathbb{E}}(x(t))=x_{t_{0}} e^{\hat{\alpha}(t-t_{0})}. \end{array} $$
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Nafidi, A., Moutabir, G., Gutiérrez-Sánchez, R. et al. Stochastic Square of the Brennan-Schwartz Diffusion Process: Statistical Computation and Application. Methodol Comput Appl Probab 22, 455–476 (2020). https://doi.org/10.1007/s11009-019-09714-8
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DOI: https://doi.org/10.1007/s11009-019-09714-8
Keywords
- Brennan Schwartz diffusion Process
- Stochastic differential equation
- Statistical inference in diffusion process
- Stationary distribution
- Trend function
- Application to population growth