Stochastic Differential Equations (SDE) is amongst the most important and recent advances in applied sciences such as physics and engineering. The theory strives to model uncertainty present in several models such in economy.
In spite of the fact that several scientists in the past, maybe even today, refused to accept probability as a scientific investigation branch, this is indeed quite useful from a engineering viewpoint mainly because imprecision is almost impossible to eliminate. However, nowadays, the theory is not engineering friendly. The books are full of mathematical tricks and notation, which indeed obscure an easy understanding of the theory, which is quite important and applicable. Maybe the best way to make it simpler is applying, in special by engineers.
The roots of the theory is not completely clear, I myself used to associate to Einstein and his studies on Brownian motion (1905), but some associates it to Louis Bachelier's Theory of Speculation, publish in 1900. However, they had completely different perspective, Einstein apparently did not want a theory in mathematics, but rather a theory in physics, a new understanding of osmotic pressure in the surface of fluids. I think that the best is to give each of them the proper credits. Nonetheless, as it happens often, several new ideas before spreading out was said before. For example, the work of Einstein was not based on the work of Robert Brown (1827), but he mentioned the same: "It is possible that the movements to be discussed here are identical
with the so-called "Brownian molecular motion" ; however, the
information available to me regarding the latter is so lacking in
precision, that I can form no judgment in the matter".
Several people when hears the name for the very first time does not get the picture quite well, I was one of them. The picture that comes to us is called Random Differential Equation (RDE). RDE can be treated with our current calculus tool, it is continuous and differentiable. But stochastic process such as Brown Motion, studied by Einstein and Brown, cannot be explained just based on classical theories. The new theory needed is called stochastic calculus. The classical tools such as Riemann Integral is no longer applicable.
The new formalism was created by later scientists such as Paul Langevin and Kiyoshi Itō.
A stochastic process involves some
response variable (or a vector) that takes values varying randomly (stochastically) in some way over time or
space.
The stochastic calculus of Ito originated with
his investigation of conditions under which the local properties of a Markov
process could be used to characterize this process.
The main "trick" on stochastic differential equations is to separate time and space in the stochastic equation. Symbolically:
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| A stochastic differential equation, representative form |
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| The picture is a container with entering solution such as water with salt. .A stochastic differential equation, the upper equation is the deterministic, and the lower is the corresponding stochastic counterpart | . |
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| Studying the solution of the stochastic differential equation, expected value and variance |
Nowadays the challenge is how to apply this theory to several areas. The most prominent is medicine and life sciences, mainly due to the current trend in treating problems from them and equally their challenges placed before us.
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| A stochastic differential equation simulated using the Euler methods applied to stochastic differential equations, it has a analytical solution, see the lower equation. W is the Brownian motion |
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