Analysis by using ODE - 趣味の研究

We analyze the stochastic model including stochastic model of Collatz process.

The step is

１.Derive the formula of the characteristic function of the first passage time, and probability function of the passage frequency.

2)Example of geomeric Brownian motion

3)Apply to general linear stochastic differntial equentions(including Collatz stochastic model)

First, we think about

$dX_t=\mu (X_t, t)dt+\sigma (X_t,t)dW_t$

In Collatz stochastic model,

$\mu=-\log(\frac{4}{3})$

$\sigma=\frac{\log(3)}{2}$

We suppose $\mu, \sigma$ are independent of "t".

The Fokker-Planck equation to this stochastic process is

$\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}(\mu (x)p(x,t))+\frac{{\partial}^2}{\partial x^2}(\frac{\sigma(x)^2}{2}p(x,t))$ eq(1)

When we think about the first passage time in the case there is absorption barrier at x = b, we solve the Fokker-Planck equation under

$p(b,t)=0$ .

The probability of first passage time is

$f(t)=-\frac{\partial}{\partial t}\int_{0}^{\infty}p(x,t)dx$

Here,

$p(x,0)=\delta(x-\xi)$ .

Next, we derive the characteristic function of $f(t)$ , and define the function g(x)

$g(x)=\int_{0}^{\infty}\exp(ikt)p(x,t)dt$

We derive the differential equation for g(x).

We multiply $\exp(ikt)$ for eq(1), and intetegrating from b to $\infty$ , get

$-\delta(x-\xi)=ikg(x)-\frac{d}{dx}(\mu(x)g(x) )+\frac{d^2}{dx^2}(\frac{\sigma(x)^2}{2}g(x) )$

This is the problem to solve the Green function of the second ODE.

On the other side, integrating eq(1) $x=b\sim\infty$ , using the boundary condition $p(b,t)=p(\infty,t)=0$ ,

$\int_{0}^{\infty}\frac{\partial p(x,t)}{\partial t}dx=-\frac{\sigma(b)^2}{2}\frac{\partial}{\partial x}p(x,t)|_{x=b}$

We multiply $-\exp(ikt)$ for this equation, and integrating $t=0\sim\infty$ , we derive

$\int_{0}^{\infty}\exp(ikt)f(t)dt=\frac{\sigma(b)^2}{2}\frac{dg(x)}{dx}|_{x=b}$

That is, we solve the ODE

$\frac{d^2}{dx^2}(\frac{\sigma(x)^2}{2}g(x) )-\frac{d}{dx}(\mu(x)g(x) )+ikg(x)=-\delta(x-\xi)$ ,

and we evaluate the characteristic function of the first passage time using the formula shown below.

$\int_{0}^{\infty}\exp(ikt)f(t)dt=\frac{\sigma(b)^2}{2}\frac{dg(x)}{dx}|_{x=b}$