Optimization of Estimates and Comparison of Their Efficiency under Stochastic Methods and Its Application in Financial Models

In this paper, we first introduce stochastic differential equations and provide the definition and basic theories of Monte Carlo and quasi-Monte Carlo methods. We specifically focus on Sobel and Halton sequences. We utilize simulations under these methods to compare their efficiency in obtaining solutions. The results indicate that the approximation achieved by the Sobel sequence is significantly superior to other stochastic methods. Next, we examine the efficiency of random and quasi-random methods, considering both geometric Brownian movement and the price index of the Tehran stock (equal weight and weight-value). Our findings demonstrate that the quasi-Monte Carlo method outperforms other approaches in terms of efficiency.