Applied Probability:
Law of Iterated Logarithm for Extensions of Brownian motion

This is a joint work with Prof. Mayru Chen in the Department of Applied Mathematics, NSYSU.

This research has been published in The 29th South Taiwan Statistics Conference: Applied Statistics Section.

Math Work

The main topic about this research is to find the asymptotic behavior of Brownian motion and its extensions such as fractional Brownian motion. The Law of Iterated Logarithm (LIL) is to describe the magnitude of a stochastic process while time goes to infinty. Below is the mathematical fomular for LIL for standard Brownian motion.

$$\limsup\limits_{t\rightarrow \infty} \frac{B(t)}{\sqrt{2t \ln \ln t}} = 1\quad \mathrm{a.s.}$$

The stochastic processes I was investigating are: - Maximum Process of Absolute Value of Standard Brownian Motion $M{|B|}(t) = \max\limits{s\leqslant t}{|B(s)|}$ - Reflected Fraction Brownian Motion $$Q{B{H}}(t)=BH(t)-ct+\max(Q{B{H}}(0),-\inf\limits{s\in [0,t]}(BH(s)-cs)),$$ where $BH$ is the standard fractional Brownian motion - Maximum Process of Reflected Fraction Brownian Motion $M{Q{B{H}}}(t) = \max\limits{s\leqslant t}{Q{B{H}}(s)}$

In this paper, we provide the exact form of the limiting functions for these three processes and rigorous mathematical proof.

Simulation

Before we dive right into the mathematical part, we need to know which limiting functions for each stochastic process. We can use mathematical deduction, or we can use simulation to find out the answewr.

In the simulation folder, I provide the r code to simulate the extensions of Brownian motion. The source code of generation of standard Brownian motion comes from the r package sde. Below is the simulations of the stochastic processes I mentioned above.

There are two ways to simulate the $\limsup$ function: - Strictly follow the mathematical definition (Primary) - Clustering

For the first method, since we can't simulate while t goes to infinty, we can run the simulation while t goes to zero based on the following expression. $$\limsup\limits{t\rightarrow \infty} \frac{B(t)}{\sqrt{2t \ln \ln t}} = 1\quad \mathrm{a.s.}\iff\limsup\limits{t\rightarrow 0} \frac{B(t)}{\sqrt{2t \ln \ln \frac{1}{t}}} = 1\quad \mathrm{a.s.}$$ After collecting finite data points of the ratio of the stochastic process and target function. Plot those data points with histogram. Below is the an example of the histogram with standard Brownian motion.

We can see that the mean of the data points is very close to 1.

For the second method, we encounter some trouble in the simulation. By the definiton of $\limsup$, the value should be the greatest cluster points. The definition of cluster points $x$ is: $$\forall \epsilon >0, N_{\epsilon}(x) \text{ contains infinitely many points}.$$ With this definition, we can use DBSCAN as our clustering method. However, since we do not know how many cluster points within our data, which is a pre-defined parameter for this particular algorithm, the greatest cluster point obtained from this method is fairly unstable.