Solutions of nonlinear stochastic differential
equations with long-range power-law distributions
Julius Ruseckas, Vygintas Gontis, and Bronsilovas Kaulakys
Institute of Theoretical Physics and Astronomy, Vilnius University, Lithuania
March 17, 2011
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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1/f noise
Our research is related to the 1/f noise problem and long-range
processes.
1/f noise
a type of noise whose power spectral density S(f ) behaves like
S(f ) ∼ 1/f β ,
β is close to 1
Fluctuations of signals exhibiting 1/f behavior of the power spectral
density at low frequencies have been observed in a wide variety of
physical, geophysical, biological, financial, traffic, Internet,
astrophysical and other systems.
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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Example of 1/f noise
107
S(f)
106
105
104
103
102 -6
10
10-5
10-4
f [Hz]
10-3
10-2
Power spectral density of trading activity (number of trades per 1 min).
for ABT stock on NYSE
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Nonlinear stochastic differential equations
March 17, 2011
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Notes about 1/f noise
A pure 1/f power spectrum is physically impossible because the
total power would be infinity.
We search for a model where the spectrum of signal has 1/f β
behavior only in some intermediate region of frequencies,
fmin f fmax , whereas for small frequencies f fmin the
spectrum is bounded.
The behavior of spectrum at frequencies fmin f fmax is
connected with the behavior of the autocorrelation function at
times 1/fmax t 1/fmin .
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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Notes about 1/f noise
Often 1/f noise is defined by a long-memory process,
characterized by S(f ) ∼ 1/f β as f → 0.
This long-range dependence property is equivalent to similar
behavior of autocorrelation function C(t) as t → ∞
This behavior of the autocorrelation function is not necessary for
obtaining required form of the power spectrum in a finite interval of
the frequencies which does not include zero
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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1/f noise
1/f noise is intermediate between white noise, S(f ) ∼ 1/f 0 and
Brownian motion S(f ) ∼ 1/f 2
In contrast to the Brownian motion generated by the linear
stochastic equations, the signals and processes with 1/f
spectrum cannot be understood and modeled in such a way.
Goal
to find a simple nonlinear stochastic differential equation (SDE)
generating signals exibiting 1/f noise
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Application to the description of trading
Time series of financial data exhibit highly nontrivial statistical
properties. Many of these properties appear to be universal.
Trading activity, trading volume, and volatility are stochastic
variables with the long-range correlation. The autocorrelation of
the volatility decays only slowly as a power law.
Probability distribution functions (PDFs) of return and trading
activity have fat tails exhibiting power-law decay.
Proposed equations can exhibit both power-law PDF and
power-law spectrum.
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Justification of SDE
If S(f ) ∼ f −β then power spectral density has a scaling property
S(af ) = a−β S(f )
Wiener-Khintchine theorem
Z +∞
C(t) =
S(f ) cos(2πft) df
−∞
Autocorrelation function C(t) has scaling property
C(at) ∼ aβ−1 C(t)
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Justification of SDE
Autocorrelation function can be written as
Z
Z
C(t) = dx dx 0 xx 0 P0 (x)Px (x 0 , t|x, 0)
P0 (x) is the steady state PDF
Px (x 0 , t|x, 0) is the transition probability
The transition probability can be obtained from the solution of the
Fokker-Planck equation with the initial condition
Px (x 0 , 0|x, 0) = δ(x 0 − x).
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Justification of SDE
Let us assume that
Steady state PDF has power-law form
P0 (x) ∼ x −ν
Trasnsition probability has a scaling property
P(ax 0 , t|ax, 0) = a−1 P(x 0 , a2(η−1) t|x, 0)
Then the autocorrelation function will have the required scaling
with
ν−3
β =1+
2(η − 1)
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Justification of SDE
To get the required scaling of transition probability:
SDE will contain only powers of x
The diffusion coefficient will be of the form x 2η
The drift term is fixed by the requirement that the steady-state
PDF should be x −ν
Proposed SDE
dx = σ 2 (η − ν/2)x 2η−1 dt + σx η dWt
B. Kaulakys and J. Ruseckas, Phys. Rev. E 70, 020101(R) (2004).
B. Kaulakys and J. Ruseckas, V. Gontis, and M. Alaburda, Physica A 365, 217 (2006).
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Restriction of diffusion
Because of the divergence of the power-law distribution and the
requirement of the stationarity of the process, the SDE should be
analyzed together with the appropriate restrictions of the diffusion
in some finite interval.
When diffusion is restricted, scaling properties are only
approximate, but 1/f spectrum remains in a wide interval of
frequencies.
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Restriction of diffusion
Possible forms of restriction:
Reflective boundary conditions at x = xmin and x = xmax
Exponential restriction of the diffusion
m m xmin m m
x
ν
2
x 2η−1 dt + σx η dWt
−
dx = σ η − +
2
2
x
2 xmax
Steady state PDF:
P0 (x) ∼ x
J. Ruseckas et al. (Lithuania)
−ν
m xmin m
x
exp −
−
x
xmax
Nonlinear stochastic differential equations
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Restriction of diffusion
q-exponential steady-state PDF
dx = σ 2 (η − ν/2)(x + x0 )2η−1 dt + σ(x + x0 )η dWt
P0 (x) ∼ exp1+1/ν (−νx/x0 )
Reflective boundary condition at x = 0
B. Kaulakys and M. Alaburda, J. Stat. Mech. 2009, P02051 (2009).
q-Gaussian steady-state PDF
dx = σ 2 (η − ν/2)(x 2 + x02 )η−1 xdt + σ(x 2 + x02 )η/2 dWt
P0 (x) ∼ exp1+2/ν (−νx 2 /2x02 )
B. Kaulakys, M. Alaburda, and V. Gontis, AIP Conf. Proc. 1129, 13 (2009).
V. Gontis, B. Kaulakys, and J. Ruseckas, AIP Conf. Proc. 1129, 563 (2009).
V. Gontis, J. Ruseckas, and A. Kononovičius, Physica A, 389, 100 (2010).
q-exponential function: expq (x) ≡ (1 + (1 − q)x)1/(1−q)
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Nonlinear stochastic differential equations
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Trading return
The distribution of normalized return r per 1 min is close to q-Gaussian.
0
10
-1
P(r)
10
10-2
10-3
10-4
10-5
-20 -15 -10 -5
0
r
5
10
15
20
Normalized trading return per 1 min for ABT stock on NYSE
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Nonlinear stochastic differential equations
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Connection with other equations
For some choces of parameters our SDE takes the form of well-known
SDE’s considered in econopysics and finance.
η = 0 and σ = 1 corresponds to the Bessel process
dx =
δ−11
dt + dWt
2 x
of dimension δ = 1 − ν
η = 1/2, σ = 2 corresponds to the squared Bessel process
√
dx = δdt + 2 x dWt
of dimension δ = 2(1 − ν)
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Nonlinear stochastic differential equations
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Connection with other equations
SDE with exponential restriction with η = 1/2, xmin = 0 and m = 1
gives Cox-Ingersoll-Ross (CIR) process
√
dx = k (θ − x)dt + σ x dWt
where k = σ 2 /2xmax , θ = xmax (1 − ν)
When ν = 2η, xmax = ∞ and m = 2η − 2 then we get the Constant
Elasticity of Variance (CEV) process
dx = µxdt + σx η dWt
2(η−1)
where µ = σ 2 (η − 1)xmin
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Numerical simulation
60
50
P(x)
I(t)
40
30
20
10
0
0
0.2 0.4 0.6 0.8
t
1
1.2 1.4 1.6
102
0
10
10-2
-4
10
10-6
10-8
10-10
10-12
10-14
10-16
10-18
0.1
1
10
100
1000 10000
x
Typical signal
Distribution of x
1
10
0
10
Used parameters: ν = 3, η = 5/2,
xmin = 1.0, xmax = 103 .
1/f spectrum.
-1
S(f)
10
-2
10
-3
10
10-4
10-5
0.1
1
10
100
1000
10000
f
Power spectral density
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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Analytically solvable case
A CEV process:
3
dx = µxdt + σx 2 dWt
where µ = σ 2 xmin /2, η = 3/2, ν = 3 and xmax = ∞
Transition probability is
r
x
1
xmin
1
1 −µt
xmin
0
exp
µt −
+ e
Px (x , t|x, 0) =
(1 − e−µt ) x 05
2
(1 − e−µt ) x 0 x
!
xmin
1
√
× I1
sinh 12 µt
xx 0
The steady-state probability distribution has the form
2
x −3 exp(−xmin /x)
P0 (x) = xmin
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Nonlinear stochastic differential equations
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Analytically solvable case
The autocorrelation function
2
C(t) = −xmin
eµt ln 1 − e−µt
2 ln(µt)
When µt 1 we get C(t) ≈ −xmin
The power spectral density
γ + ψ(iω/µ)
2
S(f ) = −4xmin Re
µ − iω
where γ is the Euler’s constant and ψ(·) is the digamma function.
2 /f
When ω µ then the power spectral density is S(f ) ≈ xmin
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Analytically solvable case
101
0
10
10
10-4
S(f)
P(x)
0
10-1
-8
10
10-12
10-1
10-2
100
101
102
103
104
10-3 -2
10
10-1
100
101
102
f
x
Probability distribution function P0 (x) and power spectral density S(f )
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
March 17, 2011
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1/f noise and eigenvalues of the F-P equation
Solutions of the Fokker-Planck equation having the form
P(x, t) = Pλ (x)e−λt determine eigenfunctions Pλ (x) and
eigenvalues λ
The power spectral density
S(f ) = 4
X
λ
λ
X2 ,
2
λ + ω2 λ
Z
xmax
Xλ =
xPλ (x) dx
xmin
The shape of the power spectral density depends on the behavior
of the eigenfunctions and the eigenvalues
Expression for the power spectral density resembles the models of
1/f noise using the sum of the Lorentzian spectra. The
Lorentzians can arise from the single nonlinear stochastic
differential equation
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Nonlinear stochastic differential equations
March 17, 2011
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1/f noise and eigenvalues of the F-P equation
Z
S(f ) ≈ 4
λ
X 2 D(λ) dλ ∼
λ2 + ω 2 λ
Z
λmax
λmin
1
1
dλ
λβ−1 λ2 + ω 2
The largest contribution make the terms corresponding to the
eigenvalues λ obeying the condition λmin λ λmax , where
2(η−1)
,
λmax = σ 2 xmax
2(η−1)
,
λmax = σ 2 xmin
λmin = σ 2 xmin
λmin = σ 2 xmax
2(η−1)
,
η>1
2(η−1)
,
η<1
When λmin ω λmax then the leading term in the expansion in the
power series of ω is
S(f ) ∼ ω −β ,
J. Ruseckas et al. (Lithuania)
β<2
Nonlinear stochastic differential equations
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1/f noise and eigenvalues of the F-P equation
The shape of the power spectral density depends on the behavior
of the eigenfunctions and the eigenvalues in terms of the function
Xλ2 D(λ).
One obtains 1/f β behavior of the power spectral density when
function Xλ2 D(λ) is proportional to λ−β for a wide range of
eigenvalues λ
J. Ruseckas and B. Kaulakys, Phys. Rev. E 81, 031105 (2010).
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Summary
We obtain a class of nonlinear SDEs, giving the power-law
behavior of the power spectral density in any desirably wide range
of frequencies
and power-law steady state distribution of the signal intensity.
The equations, as special cases, contain the well-known SDEs in
economics and finance.
One of the reasons for the appearance of the 1/f spectrum is the
scaling property of the SDE.
The power spectral density may be represented as a sum of the
Lorentzian spectra.
J. Ruseckas et al. (Lithuania)
Nonlinear stochastic differential equations
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Thank you for your attention!
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