approximation of stochastic delay differential equations based on haar functions

The main objective of this article is to solve stochastic delay differential equations via haar wavelets‎. ‎we present fundamental concepts of stochastic process‎, ‎haar‎, ‎block pulse functions‎, ‎and their operational matrix relevant to time-delayed haar‎. ‎analytic solutions of two examples are solved for the first time to approximate two kinds of single time-delayed stochastic differential equations with additive and multiplicative noise‎. ‎this orthogonal basis function not only simplifies the problem but also speeds up the computations and lessens the computational complexity of the stochastic delay differential equations to a lower triangular system of linear algebraic equations‎. ‎the equation can be solved via forward substitution‎, ‎such as lower-upper decomposition method‎. ‎finally‎, ‎we examine the order of convergence and error analysis of two visual samples to validate the efficiency and effectiveness of the suggested procedure.