Special cases of the stochastic differential equation reproducing 1/f noise

Considerable part of stochastic models available on Physics of Risk website (ex., Agent based herding model of financial markets or Long-range memory stochastic model of return) are related to the general class of stochastic differential equations derived by our group [1, 2]. The general form of this class is the following stochastic differential equation:

\begin{equation} \mathrm{d} x = \left(\eta - \frac{\lambda}{2} \right)x^{2 \eta -1} \mathrm{d} t + x^\eta \mathrm{d} W . \label{sde} \end{equation}

In our talks at various scientific events and on Physics of Risk itself we frequently say that this equation also encompasses other widely known stochastic processes. Thus further in this text we will show some of the relations between this class and some widely known stochastic processes.

Variable transformation

One of the most simple ways to related two stochastic processes is to transform the modeled variable. The formula for the transformation, via Ito's lemma, is as follows [3]:

\begin{equation} \mathrm{d} y(x) = \left[ A_x(x) \partial_x y(x) +\frac{1}{2} B_x^2(x) \partial^2_x y(x) \right] \mathrm{d} t +B_x(x) \partial_x y(x) \mathrm{d} W , \end{equation}

here x is a stochastic process from which we start, y is a stochastic process which we are willing to obtain, A is a drift function of the "primary" stochastic differential equation (describing stochastic process x), while B is a diffusion function of the same stochastic differential equation.

The only problem is to select the correct relations between the two stochastic processes From the above variable transformation formula we can require that the new drift or diffusion functions would take certain shapes. In most cases it will be rather hard to introduce the two requirements. We have used this approach in our recent paper [4] to relate \eqref{sde} and Bessel process. We have set up the requirement for the diffusion function of the resulting process:

\begin{equation} x^\eta \partial_x y(x) = \pm 1 . \end{equation}

This differential equation provides two possible transformation! Yet the Bessel process is defined only for positive real numbers, thus we have only one meaningful Lamperti transformation:

\begin{equation} \ell:x \mapsto y(x) = \frac{1}{(\eta-1) x^{\eta-1}} , \end{equation}

which transforms \eqref{sde} to a Bessel process:

\begin{equation} \mathrm{d} y = \left( \nu + \frac{1}{2} \right)\frac{\mathrm{d} t}{y} + \mathrm{d} W , \end{equation}

where \( \nu = \frac{\lambda - 2 \eta +1}{2 (\eta -1)} \) is an index of the Bessel process.

Variable transformation method might not be useful in the most cases, yet if one is lucky enough. He will be able to obtain very strong evidence for the full equivalence of the processes.

Introducing exponential restrictions and selecting parameter values

This method allows to obtain correspondence between two stochastic processes in another way - by introducing specific conditions and selecting specific parameter sets. If one shows that stochastic processes agree using this method then the less general process might be said to be a separate case of more general process.

The aforementioned Bessel process can be obtained from \eqref{sde} by setting \( \eta =0 \):

\begin{equation} \mathrm{d} x = - \frac{\lambda}{2} \frac{\mathrm{d} t}{x} +\mathrm{d} W . \end{equation}

The index of such Bessel process is \( \nu =-\frac{\lambda+1}{2} \).

It is also very easy to obtain the squared Bessel process, one just has to set that \( \eta = 0.5 \):

\begin{equation} \mathrm{d} x = \frac{1-\lambda}{2} \mathrm{d} t + \sqrt{x}\mathrm{d} W . \end{equation}

The index of such squared Bessel process is \( \nu = - \lambda \). Note that for the full analogy between the processes one should also transform time.

More complex CEV and CIR processes might be obtained by introducing exponential diffusion restrictions into \eqref{sde}:

\begin{equation} \mathrm{d} x = \left[\eta - \frac{\lambda}{2} +\frac{m}{2} \left( \frac{x_{min}}{x} \right)^m - \frac{m}{2}\left( \frac{x}{x_{max}} \right)^m \right] x^{2 \eta -1}\mathrm{d} t + x^\eta \mathrm{d} W . \end{equation}

Thus, if \( \eta=1/2 \), \( m=1 \), \( x_{min}=0 \) and \( x_{max} = 1 \), one can obtain CIR process:

\begin{equation} \mathrm{d} x = k (\theta -x) \mathrm{d} t + \sqrt{x}\mathrm{d} W , \end{equation}

where \( k = 1/2 \) and \( \theta = 1 - \lambda \).

While if \( \eta = \lambda /2 \), \( m = 2 \eta -2 \), \( x_{min} = 1 \) and \( x_{max} \rightarrow \infty \), one can obtain CEV process:

\begin{equation} \mathrm{d} x = \mu x \mathrm{d} t + x^\eta \mathrm{d} W , \end{equation}

where \( \mu = \eta -1 \). The relation of our stochastic models to the CEV process is more broadly discussed in [5, 6].