Author: A.C. Guidoum and K. Boukhetala
Date: 2025-12-13

# Parallel Monte-Carlo and Moment Equations for SDEs

#### A.C. Guidoum1 and K. Boukhetala2

#### 2025-12-13

# 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:

- `model`: an object from classes `snssde1d()`, `snssde2d()` and `snssde3d()`.
- `statistic`: a function which when applied to the model (SDEs) returns a vector containing the statistic(s) of interest. Any further arguments can be passed to statistic(s) through the `...` argument.
- `R`: number of Monte Carlo replicates (`R` batches), this will be a single positive integer.
- `time`: fixed time at which the estimate of the statistic(s).
- `exact`: a named list giving the exact statistic(s), if it exists the bias calculation will be performed.
- `names`: named the statistic(s) of interest. Default `names=c("mu1","mu2",...)`.
- `level`: confidence level of the required interval(s).
- `parallel`: the type of parallel operation to be used. `"multicore"` does not work on Microsoft Windows operating systems, but on Unix is allowed and uses parallel operations. Default `parallel="no"`.
- `ncpus`: an integer value specifying the number of cores to be used in the parallelized procedure. Default is 1 core of the machine.
- `cl`: an optional parallel cluster for use if `parallel = "snow"`. Default `cl = makePSOCKcluster(rep("localhost", ncpus))`.

    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.

- `x`: an object from the class `MCM.sde()`.
- `index`: the index of the variable of interest within the output of class `MCM.sde()`.
- `type`: type of plots. Default `type="all"`.

## One-dimensional SDE

    R> set.seed(1234, kind = "L'Ecuyer-CMRG")
    R> theta = 0.75; x0 = 1
    R> fx <- expression( 0.5*theta^2*x )
    R> gx <- expression( theta*x )
    R> mod1 <- snssde1d(drift=fx,diffusion=gx,x0=x0,M=500,type="ito")
    R> mod2 <- snssde1d(drift=fx,diffusion=gx,x0=x0,M=500,type="str")
    R> ## True values of means and variance for mod1 and mod2
    R> E.mod1 <- function(t) x0 * exp(0.5 * theta^2 * t)
    R> V.mod1 <- function(t) x0^2 * exp(theta^2 * t) * (exp(theta^2 * t) - 1)
    R> E.mod2 <- function(t) x0 * exp(theta^2 * t)
    R> V.mod2 <- function(t) x0^2 * exp(2 * theta^2 * t) * (exp(theta^2 * t) - 1)
    R> ## function of the statistic(s) of interest.
    R> sde.fun1d <- function(data, i){
    +      d <- data[i, ]
    +      return(c(mean(d),var(d)))
    + }
    R> # Parallel MOnte Carlo for mod1
    R> mcm.mod1 = MCM.sde(model=mod1,statistic=sde.fun1d,R=20, exact=list(m=E.mod1(1),S=V.mod1(1)),parallel="snow",ncpus=2)
    R> mcm.mod1

    Itô Sde 1D:
     | dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) * dW(t)
     | t in [0,1] with mesh equal to 0.001

    PMCM Based on 20 Batches with 500-Realisations at time 1:

       Exact Estimate    Bias Std.Error    RMSE  CI( 2.5 % , 97.5 % )
    m 1.3248   1.3505 0.02571   0.01070 0.05327 ( 1.32952 , 1.37146 )
    S 1.3252   1.3326 0.00742   0.03637 0.15872  ( 1.2613 , 1.40386 )

    R> # Parallel MOnte Carlo for mod2
    R> mcm.mod2 = MCM.sde(model=mod2,statistic=sde.fun1d,R=20, exact=list(m=E.mod2(1),S=V.mod2(1)),parallel="snow",ncpus=2)
    R> mcm.mod2

    Stratonovich Sde 1D:
     | dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) o dW(t)
     | t in [0,1] with mesh equal to 0.001

    PMCM Based on 20 Batches with 500-Realisations at time 1:

       Exact Estimate    Bias Std.Error    RMSE  CI( 2.5 % , 97.5 % )
    m 1.7550   1.7889 0.03383   0.01418 0.07045 ( 1.76109 , 1.81667 )
    S 2.3257   2.3365 0.01081   0.06376 0.27812 ( 2.21157 , 2.46151 )

## Two-dimensional SDEs

    R> set.seed(1234, kind = "L'Ecuyer-CMRG")
    R> mu=1;sigma=0.5;theta=2
    R> x0=0;y0=0;init=c(x0,y0)
    R> f <- expression(1/mu*(theta-x), x)
    R> g <- expression(sqrt(sigma),0)
    R> OUI <- snssde2d(drift=f,diffusion=g,M=500,Dt=0.015,x0=c(x=0,y=0))
    R> ## true values of first and second moment at time 10
    R> Ex <- function(t) theta+(x0-theta)*exp(-t/mu)
    R> Vx <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
    R> Ey <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu))
    R> Vy <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
    R> covxy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu)))
    R> tvalue = list(m1=Ex(10),m2=Ey(10),S1=Vx(10),S2=Vy(10),C12=covxy(10))
    R> ## function of the statistic(s) of interest.
    R> sde.fun2d <- function(data, i){
    +   d <- data[i,]
    +   return(c(mean(d$x),mean(d$y),var(d$x),var(d$y),cov(d$x,d$y)))
    + }
    R> ## Parallel Monte-Carlo of 'OUI' at time 10
    R> mcm.mod2d = MCM.sde(OUI,statistic=sde.fun2d,time=10,R=20,exact=tvalue,parallel="snow",ncpus=2)
    R> mcm.mod2d

    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] with mesh equal to 0.015

    PMCM Based on 20 Batches with 500-Realisations at time 10:

           Exact Estimate    Bias Std.Error    RMSE
    m1   1.99991  2.00256 0.00265   0.00475 0.02087
    m2  18.00009 18.04024 0.04015   0.01598 0.08038
    S1   0.25000  0.25229 0.00229   0.00397 0.01746
    S2   4.25005  4.29577 0.04572   0.05856 0.25934
    C12  0.24998  0.25877 0.00879   0.01266 0.05588
           CI( 2.5 % , 97.5 % )
    m1    ( 1.99325 , 2.01187 )
    m2  ( 18.00892 , 18.07156 )
    S1    ( 0.24451 , 0.26007 )
    S2    ( 4.18099 , 4.41055 )
    C12   ( 0.23396 , 0.28358 )

## Three-dimensional SDEs

    R> set.seed(1234, kind = "L'Ecuyer-CMRG")
    R> mu=0.5;sigma=0.25
    R> fx <- expression(mu*y,0,0)
    R> gx <- expression(sigma*z,1,1)
    R> Sigma <-matrix(c(1,0.3,-0.5,0.3,1,0.2,-0.5,0.2,1),nrow=3,ncol=3)
    R> modtra <- snssde3d(drift=fx,diffusion=gx,M=500,type="str",corr=Sigma)
    R> ## function of the statistic(s) of interest.
    R> sde.fun3d <- function(data, i){
    +   d <- data[i,]
    +   return(c(mean(d$x),median(d$x),Mode(d$x)))
    + }
    R> ## Monte-Carlo at time = 10
    R> mcm.mod3d = MCM.sde(modtra,statistic=sde.fun3d,R=10,parallel="snow",ncpus=2)
    R> mcm.mod3d

    Stratonovich Sde 3D:
     | dX(t) = mu * Y(t) * dt + sigma * Z(t) o dB1(t)
     | dY(t) = 0 * dt + 1 o dB2(t)
     | dZ(t) = 0 * dt + 1 o dB3(t)
     | t in [0,1] with mesh equal to 0.001
     | Correlation structure:
            1.0 0.3 -0.5
            0.3 1.0  0.2
           -0.5 0.2  1.0

    PMCM Based on 10 Batches with 500-Realisations at time 1:

        Estimate Std.Error    CI( 2.5 % , 97.5 % )
    mu1 -0.06544   0.00325 ( -0.07181 , -0.05907 )
    mu2 -0.05929   0.00555 ( -0.07017 , -0.04841 )
    mu3 -0.04414   0.01664 ( -0.07675 , -0.01153 )

# The `MEM.sde()` function

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

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

- `drift`: an `R` vector of `expressions` which contains the drift specification (1D, 2D and 3D).
- `diffusion`: an `R` vector of `expressions` which contains the diffusion specification (1D, 2D and 3D).
- `corr`: the correlation coefficient ‘|corr|<=1’ of \(W_{1}(t)\) and \(W_{2}(t)\) (2D) must be an `expression` length equal 1. And for 3D \((W_{1}(t),W_{2}(t),W_{3}(t))\) an `expressions` length equal 3.
- `type`: type of SDEs to be used `"ito"` for Ito form and `"str"` for Stratonovich form. The default `type="ito"`.
- `solve`: if `solve=FALSE` only the symbolic computational of system will be made. And if `solve=TRUE` a numerical approximation of the obtained system will be performed.
- `parms`: parameters passed to `drift` and `diffusion`.
- `init`: initial (state) values of system.
- `time`: time sequence (`vector`) for which output is sought; the first value of time must be the initial time.
- `...`: arguments to be passed to `ode()` function available in deSolve package, if `solve=TRUE`.

## One-dimensional SDE

    R> fx <- expression( 0.5*theta^2*x )
    R> gx <- expression( theta*x )
    R> start = c(m=1,S=0)
    R> t = seq(0,1,by=0.001)
    R> mem.mod1 = MEM.sde(drift=fx,diffusion=gx,type="ito",solve = TRUE,parms = c(theta=0.75), init = start, time = t)
    R> mem.mod1

    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

    R> mem.mod2 = MEM.sde(drift=fx,diffusion=gx,type="str",solve = TRUE,parms = c(theta=0.75), init = start, time = t)
    R> mem.mod2

    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) = 0.5625 * m(t)
     | dS(t) = 0.5625 * m(t)^2 + 1.6875 * S(t)

    Approximation of moment at time 1
     | m(1) = 1.755
     | S(1) = 2.3257

    R> plot(mem.mod1$sol.ode, mem.mod2$sol.ode,ylab = c("m(t)"),select="m", xlab = "Time",main="",col = 2:3,lty=1)
    R> legend("topleft",c(expression(m[mod1](t),m[mod2](t))),inset = .05, col=2:3,lty=1)
    R> plot(mem.mod1$sol.ode, mem.mod2$sol.ode,ylab = c("S(t)"),select="S", xlab = "Time",main="",col = 2:3,lty=1)
    R> legend("topleft",c(expression(S[mod1](t),S[mod2](t))),inset = .05, col=2:3,lty=1)

## Two-dimensional SDEs

    R> fx <- expression(1/mu*(theta-x), x)
    R> gx <- expression(sqrt(sigma),0)
    R> start = c(m1=0,m2=0,S1=0,S2=0,C12=0)
    R> t = seq(0,10,by=0.001)
    R> mem.mod2d = MEM.sde(drift=fx,diffusion=gx,type="ito",solve = TRUE,parms = c(mu=1,sigma=0.5,theta=2), init = start, time = t)
    R> mem.mod2d

    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

    R> matplot.0D(mem.mod2d$sol.ode,main="")

## Three-dimensional SDEs

    R> fx <- expression(mu*y,0,0)
    R> gx <- expression(sigma*z,1,1)
    R> RHO <- expression(0.75,0.5,-0.25)
    R> start = c(m1=5,m2=0,m3=0,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0)
    R> t = seq(0,1,by=0.001)
    R> mem.mod3d = MEM.sde(drift=fx,diffusion=gx,corr=RHO,type="ito",solve = TRUE,parms = c(mu=0.5,sigma=0.25), init = start, time = t)
    R> mem.mod3d

    Itô Sde 3D:
     | dX(t) = 0.5 * Y(t) * dt + 0.25 * Z(t) * dB1(t)
     | dY(t) = 0 * dt + 1 * dB2(t)
     | dZ(t) = 0 * dt + 1 * dB3(t)
     | t in [0,1].
     | Correlation structure: E(dB1dB2) = 0.75 * dt
                            : E(dB1dB3) = 0.5 * dt
                            : E(dB2dB3) = -0.25 * dt

    Moment equations:
     | dm1(t)  = 0.5 * m2(t)
     | dm2(t)  = 0
     | dm3(t)  = 0
     | dS1(t)  = 0.0625 * S3(t) + 0.0625 * m3(t)^2 + C12(t)
     | dS2(t)  = 1
     | dS3(t)  = 1
     | dC12(t) = 0.1875 * m3(t) + 0.5 * S2(t)
     | dC13(t) = 0.125 * m3(t) + 0.5 * C23(t)
     | dC23(t) = -0.25

    Approximation of moment at time 1
       | m1(1)  =  5 | S1(1)  =  0.11458 | C12(1)  =   0.2500
       | m2(1)  =  0 | S2(1)  =  1.00000 | C13(1)  =  -0.0625
       | m3(1)  =  0 | S3(1)  =  1.00000 | C23(1)  =  -0.2500

    R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("m1","m2","m3"))
    R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("S1","S2","S3"))
    R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("C12","C13","C23"))

# 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 (2020). “Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc”. Journal of Statistical Software, 96(2), 1–82. https://doi.org/10.18637/jss.v096.i02

1. Department of Mathematics and Computer Science, Faculty of Sciences and Technology, University of Tamanghasset, Algeria, E-mail (acguidoum@univ-tam.dz)↩︎

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