Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

Abstract

Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable.

Appendices

Appendix A: Proof of Theorem 1

Proof

We only need to consider the difference between α-type and I-type processes (α=0), the difference in drift part is given by

$$ \bigl[\boldsymbol{f}\bigl(\boldsymbol{x}(t+\alpha\,\mathrm{d}{t}) \bigr) - \boldsymbol{f}\bigl(\boldsymbol{x} (t)\bigr)\bigr]\, \mathrm{d}{t} = \mathrm{O} \bigl( \boldsymbol{x}(t+\alpha\,\mathrm{d}{t}) - \boldsymbol{x}(t) \bigr )\,\mathrm{d}{t} = \mathrm{O}\bigl( \mathrm{d}{t}^{1.5}\bigr) = \mathrm{o}(\mathrm{d}{t}). $$

(21)

This means the difference in drift part is negligible. The difference in diffusion part of ith coordinate is given by

(22)

(23)

(24)

(25)

We can use interpretation of SDE over time interval t to t+α dt to get Eq. (23). Here dW(αt) is a short notation for change of Wiener Process over time α dt. Equation (24) is given by first order expansion. We get the last equality using the following facts

(26)

Here R ₁(t) and R ₂(t) are zero mean noise with standard deviation of order $\mathrm{o}(\sqrt{\mathrm{d}{t}})$. These small noise will not harm the result and can be ignored according to the following theorem.

Theorem 3

The zero mean noise in dx with standard deviation of $\mathrm{o}(\sqrt{\mathrm{d}{t}})$ can be ignored without influencing the result of stochastic integration.

Proof

Let us denote the noise term R(t). Consider the stochastic integration of the noise term over a time interval

(27)

We can find that X is a random variable with zero mean and zero variance (due to the fact that variance of R(t) is o(dt)). This means X goes to 0 by mean-square limit. □

Appendix B: Detailed Parameters of Examples

2.1 B.1 Example 1

The SDE for the I-type process is

$$ \mathrm{d}\boldsymbol{x}= -(\boldsymbol {D}+\boldsymbol{Q})\nabla\phi\,\mathrm {d}{t} + \boldsymbol{B}\,\mathrm{d}\boldsymbol{W}(t), \qquad \boldsymbol{B} \boldsymbol{B}^\tau=2\boldsymbol{D}. $$

(28)

Here

(29)

Here k is an integer vary from 1 to 10. The A-type process for Eq. (28) is the following I-type process:

(30)

where

$$ \varDelta\boldsymbol{f}= \left ( \begin{array}{c} k \\ 0 \\ \end{array} \right ). $$

(31)

2.2 B.2 Example 2

The SDE for the I-type process is

(32)

Here

(33)

It is easy to see that the above I-type process is the equivalent I-type process for the following A-type process with same B and different ϕ:

(34)

Here

$$ \phi' = \bigl(x^2+y^2\bigr)/2. $$

(35)

So its stationary distribution is

$$ \rho\propto\exp\bigl[-\bigl(x^2+y^2\bigr)/2 \bigr]. $$

(36)

This stationary distribution has only one minimum point at origin.

The A-type process for Eq. (32) is the following I-type process:

(37)

where

$$ \varDelta\boldsymbol{f}= \left ( \begin{array}{c} x \\ y \\ \end{array} \right ). $$

(38)

And its stationary distribution is

$$ \rho\propto\exp\bigl[ \log\bigl(x^2+y^2\bigr)- \bigl(x^2+y^2\bigr)/2 \bigr]. $$

(39)