Simple Example of Stochastic Differential Equation

Stochastic Differential Equation (SDE for short) as a important aspect of solving Complex Dynamic System is related to our project, since the whole climate is one. We thought it'd be a great idea to give a simple example of it. To begin, SDE is a Differential Equation but with a randomness factor. Really similar to the technique of building a Generative Adversarial Network, when generating picture of text to speech, you certainly don't want every result to be the same, a random factor is introduced.

The following is one of the simple examples of SDE, Geometric Brownian Motion:

It consists of two main components, the former: Model Deterministic Trends and the latter: Factor for Unpredictable Events. Performing a transposing of items, making the equation:

If we neglect the randomness factor first, making the equation a simple Differential Equation and the solving it gives the following:

A exponential function (not exactly the function above, just a schematic diagram):

When adding the random factor back, creating volatility:

Pretty similar to stocks price in long term, ex:

In the long term, it would still follow the big trend; but if we take a closer look, we can see little fluctuations. Also if we plug in the second equation above to stocks, we would find:

Numerator being Gain for a short time (a day would be sufficient in this case); Denominator being Gain over all time((exactly)40 years of time span). This makes the above expression the daily return rate of a stock! Further developing this concept, the famous stock price model Black-Scholes model were created.

SDE can have a numerous use, and this is a very simple and intuitive example we thought would be relatively easy to grasp. Hope you like it!