High-order strong methods for stochastic differential equations with colored noises
Shuanglin Sun Yun-An Yan




 
School of Business, Ludong University, Shandong 264025, China
yunan@ldu.edu.cn
School of Physics and Optoelectronic Engineering,
  Ludong University, Shandong 264025, China


The key difficulty to develop efficient high-order methods for integrating
stochastic differential equations lies in the calculations of the 
multiple stochastic integrals. This letter suggests a scheme to compute 
the stochastic integrals for the colored noises based on the white noise 
representation. The multiple stochastic integrals involving one 
and two stationary noises can be conveniently generated 
together with noises using the discrete Fourier transformation.
Based on the calculated stochastic integrals, we obtain simple fourth-order 
and third-order strong methods for equations with a single and multiple noises,
respectively. Numerical tests verify the accuracy of the suggested methods.

Keywords: Stochastic differential equations, 
  Strong stochastic methods, 
  Quantum dissipative dynamics, 
  Magnus expansion, 
  Colored noises 


High-order strong methods for stochastic differential equations with colored noises
    Yun-An Yan
    December 30, 2023
===================================================================================





§ INTRODUCTION


We consider the initial value problems with the linear 
stochastic differential equations (SDEs)

    d X(t)/dt = A X(t) + ∑_j=1^K B_j X(t) ξ_j(t)

where X(t) is an M-dimensional state vector, 
A and B_j are M× M matrices, 
K is the number of noises, 
and ξ_j(t) are stationary colored Gaussian noises with zero means
ξ_j(t)=0 and two-time correlation functions 
ξ_j(t)ξ_k(s) = α_jk(t-s). 
Here we focus on the noises with continuous correlation functions α_jk(t).

Linear SDEs arise in simulating the dynamics of social or physical 
systems perturbed by noises. For open physical systems, the 
dissipative dynamics is often described by system-plus-environment model and the 
effect of the environment on the evolution of the system can be characterized by
noises. With such a paradigm, the stochastic approach becomes a powerful
tool to study quantum dissipative dynamics as well as 
quantum measurements <cit.>.
Various stochastic methods are available in the literature, such as
the stochastic Liouville equation <cit.> and
the stochastic Schrödinger's 
equation <cit.>.
As most stochastic simulations focused on dynamics of bosonic systems,
the stochastic description was recently extended to fermionic quantum 
dissipation <cit.>. For social systems, linear SDEs are used 
to simulate the stock price in the Black-Scholes model and 
become the basis for mathematical finance <cit.>.

Due to the wide applications and the importance of SDEs, it has attracted extensive 
attentions to  develop efficient numerical integrators for stochastic equations.
However, high-order integrators have to deal with the multiple stochastic integrals 
and become more sophisticated than its deterministic counterpart.
To reduce the complexity in stochastic integration, 
the average-converged weak methods were suggested as useful alternatives to
the pathwise strong methods <cit.>. 
With the Brownian path, one can also develop variable step-size 
approaches <cit.>.
In order to obtain simpler integrators,
the idea of Runge-Kutta methods originally designed 
for ordinary differential equations were also applied to the stochastic 
cases <cit.>.

For linear SDEs, the linearity allows us to develop more 
efficient and yet simpler integrators. For example, while strong order 1 Milstein 
method <cit.> is widely used and high strong order 
integrators become sophisticated for general SDEs <cit.>, 
high strong order schemes for linear equations are much simpler based on the Magnus 
formula <cit.>.
When the noise is of the Ornstein-Uhlenbeck type, we can even derive efficient, deterministic  
hierarchical approaches to solve the stochastic average 
X(t) <cit.>.

In simulating physical systems, one often needs to use colored noise to represent
the effect of the environment <cit.>. 
In this case, one can of course use the integrators designed for the white noises 
to simulate the dynamics. But more efficient schemes could be developed by 
utilizing the noise correlation. For example, Milshtein and Tret'yakov suggested 
a 5/2 strong order method for colored noises <cit.>. 
Honeycutt even put forward a fourth-order strong method
for Ornstein-Uhlenbeck noises <cit.>.

In this letter, we take advantage of the correlation 
for the colored noises and suggest an efficient way to 
calculate the multiple stochastic integrals with the 
discrete Fourier transformation. The current approach 
results in a simple third-order strong integrator for SDEs 
with multiple colored noises and a fourth-order method 
for SDEs with a single noise. 



§ CALCULATIONS OF MULTIPLE STOCHASTIC INTEGRALS FOR COLORED NOISES


For simplicity, we extend lower bound of the summation in Eq. (<ref>) to zero and 
rewrite the equation as 
Ẋ = ∑_j=0^K B_j X(t) ξ_j(t) 
so that B_0≡ A and ξ_0(t)=1.
In numerical simulations, we normally discretize the time interval [0,T_f]
with a uniform grid {t_0,t_1,⋯,t_N} where N is the 
number of time steps, Δ t is the step size, and t_j=jΔ t.
For linear SDEs, one can employ the Magnus expansion
to find an approximation for X(t+Δ t) 
based on X(t) <cit.>,

    X(t+Δ t) = e^Ω(t, Δ t)X(t).

Up to the third order, Ω(t, Δ t) reads

    Ω(t, Δ t) = ∑_j B_j J_j(t) 
      +∑_j<k [B_j, B_k] K_kj(t)
      + 1/6∑_j,k<l[B_j, [B_k, B_l]] K_lkj(t),

where Js represents the multiple stochastic integral,

    J_j(t)    = ∫^t+Δ t_t dt_1 ξ_j(t_1),
           J_jk(t) = ∫^t+Δ t_t dt_1 ξ_k(t_1)∫^t_1_t dt_2 ξ_j(t_2),
      
          
      J_jkl(t) = ∫^t+Δ t_t dt_1 ξ_l(t_1)∫^t_1_t dt_2 ξ_k(t_2)
        ∫^t_2_t dt_3 ξ_j(t_3),

and

    K_kj(t)    =    J_kj(t)-J_jk(t),
    
      K_lkj(t)    =    J_lkj(t)-J_jlk(t)-J_klj(t)+J_jkl(t).

Note that for white noises, one should be cautious about the calculations of 
multiple stochastic integrals because the Itô integration and the Stratonovich integration will
yield different results. But here these two methods will 
produce the same results since ξ_j(t) are continuous functions of time.

The stationary colored noises can be represented with white noises,
that is,

    ξ_j(t) = ∫ dW(ω) e^-iω tΓ_j(ω),

where W(ω) is a Wiener process in the frequency domain. 
In this letter, the integration range is always [-∞,∞] 
with respect to the Wiener process W(ω) and omitted in the expression. 
One may verify that ξ_j(t) satisfies the desired correlations
if Γ_j(ω) Γ^∗_k(ω) is 
the inverse Fourier transformation
of α_jk(t). In practice, the circulant embedding 
method of the correlations may be used to solve the discretized approximation 
for Γ_j(ω) <cit.>.

With the noise representation, one can directly calculate the 
multiple integrals involved in Eq. (<ref>). 
In a third-order simulation, J_j and J_jk are 
all the needed multiple stochastic integrals, and can be 
calculated with the white noise representation of ξ_j(t) for j≠ 0,

    J_j(t) = ∫ dW(ω) e^-iω tΓ_j(ω)/ω[1-e^-i ωΔ t],
          K_j0(t) =∫ dW(ω) e^-iω tΓ_j(ω)/ω^2[2-i ωΔ t
      -e^-i ωΔ t(2+i ωΔ t)],
          K_jk(t) =∫ dW(ω_1)∫ dW(ω_2) Δ t^2/2Γ_j(ω_1)
        
            ×
      e^-iω_1 t[ℋ(ω_1 Δ t, ω_2 Δ t) Γ_k(ω_2) e^-iω_2 t
      +ℋ(ω_1 Δ t, -ω_2 Δ t) Γ_k(-ω_2) e^iω_2 t ],

where ℋ(ω_1,ω_2) is the kernel 

    ℋ(x, y) = e^-i (x+y)/x y (x+y)[ x (1+e^i x)(1-e^i y)
         -y (1-e^i x)(1+e^i y)
      ].

In numerical simulations, we usually employ the discrete Fourier 
transformation to calculate the integrals. Then the computational
effort will be O(N^2log(N)) for a straightforward implementation of Eq. (<ref>).
To reduce the computational effort in noise generation, 
we approximate ℋ(x,y) with a degenerate kernel,

    ℋ(x,y) ≈∑_j=0^N_f-1ϵ_j ψ_j(x) φ_j(y),

where N_f is the number of terms for the approximation, 
ϵ_j, ψ_j(x) and ψ_j(x) are the eigenvalues, 
the left and right eigenfunctions of the kernel, respectively.
Note that here the vector elements are counted from 0.
The degenerate kernel can be obtained with a singular value decomposition 
after discretizing ℋ(x,y).
As Fig. (<ref>) shows, a six-term kernel can well approximate ℋ(x,y) 
and the maximum absolute error will be as low as the order of 10^-12. 
With such a degenerate kernel approximation, the computational effort
for producing J_jk-J_kj is only six times of that for the noise.

Note that K_lkj(t) is of the fourth order with respect to Δ t^4.
We thus can construct a third-order strong method for SDEs with colored noises 
with a second-order Magnus expansion. 
For SDEs with a single noise, the O(Δ t^4) multiple stochastic integrals 
are K_100(t) and K_101(t),
which can also be calculated with Eq. (<ref>). The result 
for K_100(t) reads

    K_100(t) =∫ dW(ω) e^-iω t/2 ω^3[
        6 ωΔ t (1+e^-i ωΔ t) 
        + i (12-ω^2 Δ t^2) (1-e^-i ωΔ t)
      ]Γ_j(ω).

The expression for K_101(t) assume the same form as Eq. (<ref>) 
for K_jk(t), but with the kernel 
ℋ(x,y) replaced by ℋ̃(x,y) Δ t. 
The kernel ℋ̃(x,y) reads

    ℋ̃(x,y)    =   e^-i (x+y)/x^2 y^2 (x+y)^2{[e^iy (i + y) - i 
          + e^ix (y-i + i e^iy)] x^3
          - 3 i (1 - e^ix) (1 + e^iy) x y^2
            
          
          + [1 + 2 e^iy + e^ix (2 + e^iy)] x^2 y^2
          + [(1 + e^iy) (-2 i + x + e^ix (2 i + x))] y^3
        }.

Again, the kernel ℋ̃(x,y) can be approximated 
with a degenerate one to reduce the computational efforts.

Upon the calculated stochastic integrals, Eq. (<ref>) is 
transformed into an ordinary differential equation 

    dX(s)/d s = ℒ(t)X(s)

for t≤ s ≤ t+Δ t. 
Note that the matrix ℒ(t)=Ω(t,Δ t)/Δ t
is constant in the time interval [t, t+Δ t]
so that Eq. (<ref>) can be integrated with the low-storage Runge-Kutta methods 
specifically designed for autonomous linear equations <cit.>.
We thus obtain a third-order strong method for multiple-noise SDEs 
and a fourth-order strong method for single-noise SDEs.





§ NUMERICAL TESTS AND DISCUSSION








We will discuss the performance of the stochastic integrators 
with two uncorrelated, colored noises ξ_1(t) and ξ_2(t) satisfying the correlation
α_11(t) = 2 e^-π t/2, α_22(t) = 5 e^-π t/√(2)
and α_12(t)=0.

The first numerical example is the stochastically-driven quantum harmonic oscillator  
with A = -i â^†â and B_1 = -iâ^† in Eq. (<ref>). 
Here â^†(â) is the creation (annihilation) operator
of the oscillator. The dynamics is propagated with the third-order method 
for the lowest 20 levels with the initial condition X_0(0)=1 and X_j(0)=0 (j≠0).
This model is analytically solvable and the solutions are
X_n(t) = e^-i n (t + π/2) L^n_1(t,0)/√(n!),
where L_1(t,s) = ∫^t_s dτ exp(i τ)ξ_1(τ). 
The function L_1(t+Δ t, t) may be generated together with the 
noise ξ_1(t) and the multiple stochastic integrals, that is,

    L_1(t+Δ t, t)    =   ∫ dW(ω) e^-iω tΓ_1(ω)/ω^2-1{-i ω 
     + [i ωcos(Δ t) -sin(Δ t)] 
     e^-i wΔ t}
            
     -i ∫ dW(ω) e^-iω tΓ_1(ω)/ω^2-1{[cos(Δ t) 
     + i ωsin(Δ t) - 1] e^-i ωΔ t}.

We would like to stress that the same white noise W(ω) 
should be used to calculate the stochastic integrals 
in Eqs. (<ref>)-(<ref>) and (<ref>). Note that 
with the spirit of Eq. (<ref>) we can apply the 
current method to propagate time-dependent SDEs. 

The stochastic equation is integrated to the final time T_f = 2 π 
with 2^n (1≤ n≤ 18) steps using the same stochastic trajectory. 
This task can be done in a recursive manner. First, a 2^18-step propagation 
is performed with the stochastic integrals J_1, K_10 
and L_1 generated from the discrete Fourier transformation.  
Then the step size is doubled and stochastic integrals with 2^n-1 steps
are calculated based on those with 2^n steps.
The matrix exponential function in Eq. (<ref>) is calculated with 
the fourth-order low-storage Runge-Kutta method <cit.>.

We employ two methods to analyze the errors for X_1(T_f) and X_2(T_f). 
First, the results with the smallest step size are used as the reference. 
Second, the identity X_2(t) = X^2_1(t)/√(2) for the solutions is utilized 
to verify the results. The errors are depicted in Fig. <ref>, which clearly 
shows the errors are of O(Δ t^3.5). 

The second example is the simplest non-commutative two-state system with
A=-iσ_z/2, B_1=(σ_x+iσ_y)/2 and the initial condition
X(0)= (0, 1)^T.
Here σ_x, σ_y and σ_z are Pauli matrices.
This model is analytically solvable and the solutions are
X_0(t) = e^-i t/2 L_1(t,0) and X_1(t) = e^i t/2. 
Fig. (<ref>) illustrates the errors for the two-state model with 
different step sizes. Because the evolution of X_1(t) in this model is 
deterministic, its accuracy is controlled solely by the method to calculate 
the matrix exponential. The plots show that the errors are O(Δ t^4) as 
it should for a fourth-order method in calculating the matrix exponential.
Different from X_1, the propagation of X_0(t) depends on the stochastic integrals,
whose errors change with the order of stochastic integrator.
Here we display the results with and without incorporating the 
triple stochastic integrals and the orders are four and three, respectively, 
which are the expected results.

The last example is a two-state system driven by two noises, whose
dynamics is described by A=iσ_z, B_1=iσ_x, and B_2=iσ_y.
For such a model, a fourth-order integrator is complicated and the
third-order strong method is used to propagate the system with 
the initial state X(0)= (1, 0)^T. This model is not analytically solvable 
and the results with 2^18 steps will serve as the reference.
The errors with different step sizes are presented in Fig. <ref>,
which shows that the errors approximately scale as O(Δ t^3.2).



§ CONCLUSIONS

Magnus expansion is a useful tool to develop simple and efficient
strong methods for linear stochastic equations and then the 
key step is the calculations of multiple stochastic integrals.
Here we have suggested a method to generate the 
multiple stochastic integrals for colored noises based on
the white noise representation. The integrals J_k, 
2 J_010-J_001 - J_100, J_jk-J_kj, 
and 2 J_101-J_110 - J_011
can be conveniently calculated with discrete Fourier transformation.
While the calculations of the first two integrals are straightforward, 
the last two integrals should be handled with degenerate kernel approximation 
to reduce computational efforts.
Once the multiple integrals are calculated, the matrix exponential function 
resulted from the Magnus expansion can be evaluated with Runge-Kutta methods. 
The above procedure gives a fourth-order strong method
for single-noise SDEs and 
a third-order one for multiple-noise SDEs. 

The current approach is simple and easy to implement. Furthermore, 
although the method is derived for autonomous linear stochastic differential 
equations, it can be readily extended to general equations as well. 



§ ACKNOWLEDGMENTS


The authors acknowledge the support from the National
Natural Science Foundation of China under grant No. 21373064.









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