Modeling the stochastic dynamics of Zakaat

Abstract

In the Islamic faith, Zakaat is annually ordained on individuals whose fortunes exceed a given threshold called Nisaab, $Z_{0}$, to be distributed over the less fortunate (with fortunes, $y(t)<Z_{0}$). For $t$ in the time interval, $I=[0,T]$, with $T$ being typically one lunar year, the Zakaat value taken from an individual with a fortune $x(t)$, ($x(t)\geq Z_{0},$ for all $t$ in $I$), is proportional to the original fortune and typically prescribed at no less than, $\lambda x(T)=x(T)/40$. Usually the collection and redistribution of Zakaat monies from the Nisaab exeeding individuals, and redistribution over the less fortunate population sector is handled by a special ministry in the goverment or sanctioned agencies. To describe the dynamics of distribution of wealth among individuals of a population subject to the Zakaat system is by no means an easy task. Nevertheless, several types of models ranging from the discrete to the continuous, with theories emanating from multi-particle random walks, to systems of stochastic and periodic parabolic differential equations, can certainly be produced. In particular, one such model can be derived to yield a Black- Scholes mixed fortune-variables equation.

In this work, using tools from Stochastic Calculus and Financial Mathematics concepts, we present a derivation and analysis of the mixed dual fortune-variables stochastic model for the Zakaat problem starting with the Stochastic System,
\begin{align}
\label{1}
\nonumber dx\left( t\right)  & =a(x,t)dt+(1-\lambda )\sigma (x,t)x(t)dw_{1}\left(
t\right) , & x(t)\geq Z_{0},0\leq t\leq T, \\
dy\left( t\right)  & =b(y,t)dt+\lambda \sigma (x,t)x(t)dw_{2}\left(
t\right) , & y(t)<Z_{0},0\leq t\leq T.
\end{align}
Here,for $i=1,2$, $w_{i}\left( t\right)$ is the standard Brownian Motion (see Belgacem [1-3]), satisfying the $L^{2}$ relations, and for $\varrho ,\delta >0,$
\begin{equation}
\label{2}
dw_{1}dw_{2}=\varrho dt,\left( dw_{i}\left( t\right) \right) ^{2}=dt,\qquad
dtdw_{i}\left( t\right) =dw_{i}\left( t\right) dt=0,\qquad \left(
dw_{i}\left( t\right) \right) ^{2+\delta }=0.
\end{equation}
In this paper, which in view of the newness of the problem is meant to be of expository nature we study the effect of the value $\lambda$, taken initially to be $1/40=0.025,$ on the dynamics of system (\ref{1}) and the resulting analysis on the mixed fortune-variables (Black-Scholes Equation like) objective function,
\begin{equation}
\label{3}
F(x,y,t)=F(x,y)=x(t)-y(t).
\end{equation}