A Stochastic Differential Equation (SDE) is a mathematical model that describes the evolution of a system over time, taking into account both deterministic and random effects. It is a differential equation with a stochastic term, meaning that the solution of the equation is not fully predictable due to the influence of randomness.
Unlike a regular Differential Equation, a Stochastic Differential Equation incorporates a stochastic term, which accounts for the random fluctuations in the system being modeled. This makes the solution of the equation more uncertain and requires the use of probability theory to analyze and solve it.
Stochastic Differential Equations have a wide range of applications in various fields such as physics, finance, biology, and engineering. They are commonly used to model complex systems that involve both deterministic and random factors, such as stock prices, population dynamics, and chemical reactions.
There are several methods for solving Stochastic Differential Equations, including the Euler-Maruyama method, the Milstein method, and the Runge-Kutta method. These methods involve discretizing the equation and approximating the solution at different time points using random numbers.
One of the main challenges in solving Stochastic Differential Equations is the inherent uncertainty and randomness in the system being modeled. This makes it difficult to obtain an exact solution and requires the use of numerical methods. Additionally, the presence of a stochastic term can make the equations more complex and computationally expensive to solve.