Parallel Monte-Carlo and Moment Equations for SDEs

A.C. Guidoum1 and K. Boukhetala2

2019-05-27

The MCM.sde() function

R> MCM.sde(model, statistic, R = 1000, time, exact = NULL, names = NULL,level = 0.95, 
+         parallel = c("no", "multicore", "snow"),ncpus = getOption("ncpus", 1L), cl = NULL, ...)

The main arguments of MCM.sde() function in Sim.DiffProc package consist:

R> plot(x,index = 1,type=c("all","hist","qqplot","boxplot","CI"), ...)

This takes a MCM.sde() object and produces plots for the R replicates of the interesting quantity.

One-dimensional SDE

Itô Sde 1D:
 | dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) * dW(t)
 | t in [0,1].

MCM Based on 10 Batches of 500-Realisations at time 1

   Exact Estimate     Bias Std.Error    RMSE  CI( 2.5 % , 97.5 % )
m 1.3248   1.3228  0.00199   0.01879 0.05641 ( 1.28596 , 1.35962 )
S 1.3252   1.3609 -0.03577   0.12129 0.36562 ( 1.12321 , 1.59865 )
Stratonovich Sde 1D:
 | dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) o dW(t)
 | t in [0,1].

MCM Based on 10 Batches of 500-Realisations at time 1

    Exact Estimate     Bias Std.Error    RMSE  CI( 2.5 % , 97.5 % )
m 1.00000  1.00267 -0.00267   0.00658 0.01993 ( 0.98977 , 1.01557 )
S 0.75505  0.76701 -0.01196   0.03928 0.11844   ( 0.69002 , 0.844 )

Two-dimensional SDEs

Itô Sde 2D:
 | dX(t) = 1/mu * (theta - X(t)) * dt + sqrt(sigma) * dW1(t)
 | dY(t) = X(t) * dt + 0 * dW2(t)
 | t in [0,15].

MCM Based on 10 Batches of 500-Realisations at time 10

       Exact Estimate     Bias Std.Error    RMSE    CI( 2.5 % , 97.5 % )
m1   1.99991  2.01204 -0.01213   0.01195 0.03785   ( 1.98862 , 2.03546 )
m2  18.00009 18.05528 -0.05519   0.04671 0.15061 ( 17.96373 , 18.14683 )
S1   0.25000  0.25282 -0.00282   0.00507 0.01547   ( 0.24288 , 0.26276 )
S2   4.25005  4.43640 -0.18635   0.06818 0.27671   ( 4.30277 , 4.57003 )
C12  0.24998  0.26278 -0.01280   0.01285 0.04062   ( 0.23759 , 0.28797 )

Three-dimensional SDEs

Stratonovich Sde 3D:
 | dX(t) = mu * Y(t) * dt + sigma * Z(t) o dW1(t)
 | dY(t) = 0 * dt + 1 o dW2(t)
 | dZ(t) = 0 * dt + 1 o dW3(t)
 | t in [0,1].

MCM Based on 10 Batches of 500-Realisations at time 1

    Estimate Std.Error   CI( 2.5 % , 97.5 % )
mu1 -0.00107   0.00421 ( -0.00932 , 0.00718 )
mu2  0.00391   0.00566   ( -0.00718 , 0.015 )
mu3  0.05039   0.02058  ( 0.01005 , 0.09073 )
mu4  0.11247   0.00264   ( 0.1073 , 0.11764 )
mu5  0.24819   0.00836   ( 0.2318 , 0.26458 )
mu6 -0.00605   0.00564    ( -0.0171 , 0.005 )

The MEM.sde() function

R> MEM.sde(drift, diffusion, type = c("ito", "str"), solve = FALSE,
+         parms = NULL, init = NULL, time = NULL, ...)

The main arguments of MEM.sde() function in Sim.DiffProc package consist:

One-dimensional SDE

Itô Sde 1D:
 | dX(t) = 0.5 * 0.75^2 * X(t) * dt + 0.75 * X(t) * dW(t)
 | t in [0,1].

Moment equations: 
 | dm(t) = 0.28125 * m(t)
 | dS(t) = 0.5625 * m(t)^2 + 1.125 * S(t)

Approximation of moment at time 1
 | m(1) = 1.3248
 | S(1) = 1.3252
Stratonovich Sde 1D:
 | dX(t) = 0.5 * 0.75^2 * X(t) * dt + 0.75 * X(t) o dW(t)
 | t in [0,1].

Moment equations: 
 | dm(t) = 0
 | dS(t) = 0.5625 * S(t) + 0.5625 * m(t)^2

Approximation of moment at time 1
 | m(1) = 1
 | S(1) = 0.75505

Two-dimensional SDEs

Itô Sde 2D:
 | dX(t) = 1/1 * (2 - X(t)) * dt + sqrt(0.5) * dW1(t)
 | dY(t) = X(t) * dt + 0 * dW2(t)
 | t in [0,10].

Moment equations: 
 | dm1(t)  = 2 - m1(t)
 | dm2(t)  = m1(t)
 | dS1(t)  = 0.5 - 2 * S1(t)
 | dS2(t)  = 2 * C12(t)
 | dC12(t) = S1(t) - C12(t)

Approximation of moment at time 10                                                              
  | m1(10)  =   1.9999 | S1(10)  =  0.25 | C12(10)  =  0.24998
  | m2(10)  =  18.0001 | S2(10)  =  4.25                      

Three-dimensional SDEs

Itô Sde 3D:
 | dX(t) = 0.5 * Y(t) * dt + 0.25 * Z(t) * dW1(t)
 | dY(t) = 0 * dt + 1 * dW2(t)
 | dZ(t) = 0 * dt + 1 * dW3(t)
 | t in [0,1].

Moment equations: 
 | dm1(t)  = 0.5 * m2(t)
 | dm2(t)  = 0
 | dm3(t)  = 0
 | dS1(t)  = (0.25 * m3(t))^2 + 0.0625 * S3(t) + C12(t)
 | dS2(t)  = 1
 | dS3(t)  = 1
 | dC12(t) = 0.25 * m3(t) + 0.5 * S2(t)
 | dC13(t) = 0.25 * m3(t) + 0.5 * C23(t)
 | dC23(t) = 1

Approximation of moment at time 1                                                      
   | m1(1)  =  5 | S1(1)  =  0.11458 | C12(1)  =  0.25
   | m2(1)  =  0 | S2(1)  =  1.00000 | C13(1)  =  0.25
   | m3(1)  =  0 | S3(1)  =  1.00000 | C23(1)  =  1.00

Further reading

  1. snssdekd() & dsdekd() & rsdekd()- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations.
  2. bridgesdekd() & dsdekd() & rsdekd() - Constructs and Analysis of Bridges Stochastic Differential Equations.
  3. fptsdekd() & dfptsdekd() - Monte-Carlo Simulation and Kernel Density Estimation of First passage time.
  4. MCM.sde() & MEM.sde() - Parallel Monte-Carlo and Moment Equations for SDEs.
  5. TEX.sde() - Converting Sim.DiffProc Objects to LaTeX.
  6. fitsde() - Parametric Estimation of 1-D Stochastic Differential Equation.

References

  1. Guidoum AC, Boukhetala K (2018). Performing Parallel Monte Carlo and Moment Equations Methods for Ito and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc. Preprint submitted to Journal of Statistical Software.

  2. Guidoum AC, Boukhetala K (2019). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.4, URL https://cran.r-project.org/package=Sim.DiffProc.


  1. Department of Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ()

  2. Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ()