In this paper, we intend to solve special kind of ordinary DIFFERENTIAL EQUATIONs which is called Heun EQUATIONs, by converting to a corresponding STOCHASTIC DIFFERENTIAL EQUATION (S.D.E.). So, we construct a STOCHASTIC linear EQUATION system from this EQUATION which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding DIFFERENTIAL EQUATIONs.

The purpose of this article is modeling the behavior of stock price using STOCHASTIC DIFFERENTIAL EQUATIONs. The data for this study include daily observations of the total stock market index, the index of the top 50 companies and the index of the 30 largest companies in the Tehran Stock Exchange. The data are daily from March 25, 2006 to April 15, 2015.The geometric Brownian motion and geometric Brownian motion with nonlinear GARCH are used to modeling the behavior of price index. The results of this study includes the following: (1) According to the log likelihood function, geometric Brownian motion with nonlinear GARCH in the three groups studied data has better performance than the geometric Brownian motion. (2) Based on the model of STOCHASTIC DIFFERENTIAL EQUATIONs with STOCHASTIC volatility, the total market index is more influenced by the good news. (3) The impact of the bad news on the index of the 30 largest companies is more than the impact of the good news. (4) The unconditional variance of the total stock market index has two structural breaks; the unconditional variance of index of the top 50 companies has one structural break and no structural breaks in the unconditional variance of index of the 30 largest companies.

In this paper, efforts have been made so the suitable STOCHASTIC DIFFERENTIAL EQUATION for Tehran Stock Exchange total index modeling to be chosen meticulously. To do so, first the necessity of STOCHASTIC models and derived STOCHASTIC calculus methods are explained, then the most important STOCHASTIC EQUATIONs in finance (including geometric Brownian motion, model with jump term, nonlinear GARCH, variance-gamma model, Vasicek and Heston) are illustrated. Then the suitable model is selected using a practical approach and considering capability of each model to estimate value at risk and prediction of total index by Monte Carlo simulation method.Back test’s and fitness function’s results show that the model with a jump term is the most suitable one for value at risk calculation and nonlinear GARCH model is best for prediction of Tehran Stock Exchange total index.

This paper proves the existence and uniqueness of quadratic mean almost periodic mild solutions for a class of STOCHASTIC DIFFERENTIAL EQUATIONs in a real separable Hilbert space. The main technique is based upon an appropriate composition theorem combined with the Banach contraction mapping principle and an analytic semigroup of linear operators.

In this study, overall index of Tehran Stock Exchange is modeled by Heston STOCHASTIC DIFFERENTIAL EQUATIONs and its performance is measured. To do this, after a brief introduction of STOCHASTIC DIFFERENTIAL EQUATIONs, Heston model is explained in more detail and parameters of this model based on the data of Tehran Stock Exchange overall index is estimated. In this way, Fokker-Plank theorem is used to find probability distribution function of Heston model and Gauss-Hermit method is used to estimate an indefinite integral. Finally we calculate value at risk of Tehran Stock Exchange overall index by Monte Carlo methods based on Heston model and we compare this with Geometric Brownian model as a widely used model by means of back test approach. These tests show superior performance of Heston model.

The occurrence of financial crises in recent decades has caused a lot of damage to the economy as well as economic enterprises in many countries. The Extreme Value Approach is a new approach to the phenomenon of financial crisis, which has been able to analyze the events that are less likely to occur but the damage caused by them is significant. In this study, we use the Extreme Value theory and STOCHASTIC DIFFERENTIAL EQUATIONs to find a new method for estimating the more precisely the value at risk. For this purpose, after estimating the parameters of the STOCHASTIC DIFFERENTIAL EQUATIONs, which includes the geometric Brownian motion, the geometric Brownian motion with the jump, the nonlinear GARCH model, and the Heston model, simulate the Monte Carlo simulations of future paths and then use peak over threshold approach, to estimate the value We at risk. The results of the simultaneous use of STOCHASTIC DIFFERENTIAL EQUATIONs and Extreme value theory are compared with historical simulations and variance-covariance approaches for value at risk. The results of Back-test techniques on value at risk indicate the superiority of the Heston model in estimation of value at risk.

In this paper, an extension of DIFFERENTIAL Transformation Method (DTM) which is an analytical-numerical method for solving the fuzzy partial DIFFERENTIAL EQUATION (FPDE) by using the strongly generalized differentiability concept is investigated. The proposed algorithm is illustrated by numerical example.