Sim.DiffProc-package {Sim.DiffProc}R Documentation

Simulation of Diffusion Processes

Description

A package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations. It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms Itô and Stratonovich. Statistical analysis with parallel Monte-Carlo and moment equations methods of SDE's. Enabled many searchers in different domains to use these equations to modeling practical problems in financial and actuarial modeling and other areas of application, e.g., modeling and simulate of first passage time problem in shallow water using the attractive center (Boukhetala K, 1996) ISBN:1-56252-342-2.

Details

Package: Sim.DiffProc
Type: Package
Version: 4.4
Date: 2019-05-27
License: GPL (>= 2)
Depends: R (>= 2.15.1)
Imports: Deriv (>= 3.8.0), MASS (>= 7.3-30), parallel
Suggests: deSolve (>= 1.11), knitr (>= 1.10), rgl (>= 0.93.991), rmarkdown (>= 0.8), scatterplot3d (>= 0.3-36), sm (>= 2.2-5.3)
Classification/MSC: 37H10, 37M10, 60H05, 60H10, 60H35, 60J60, 65C05, 68N15, 68Q10

There are main types of functions in this package:

  1. Simulation of solution to 1,2 and 3-dim stochastic differential equations of Itô and Stratonovich types, with different methods.

  2. Simulation of solution to 1,2 and 3-dim diffusion bridge of Itô and Stratonovich types, with different methods.

  3. Simulation the first-passage-time (f.p.t) in 1,2 and 3-dim sde of Itô and Stratonovich types.

  4. Calculate symbolic ODE's of moment equations (means and variances-covariance) for 1,2 and 3-dim SDE's.

  5. Monte-Carlo replicates of a statistic applied to 1,2 and 3-dim SDE's at any time t.

  6. Computing the basic statistics (mean, var, median, ...) of the processes at any time t using the Monte Carlo method.

  7. Random number generators (RN's) to generate 1,2 and 3-dim sde of Itô and Stratonovich types.

  8. Approximate the transition density 1,2 and 3-dim of the processes at any time t.

  9. Approximate the density of first-passage-time in 1,2 and 3-dim SDE's.

  10. Computing the stochastic integrals of Itô and Stratonovich types.

  11. Estimate drift and diffusion parameters by the method of maximum pseudo-likelihood of the 1-dim stochastic differential equation.

  12. Converting Sim.DiffProc objects to LaTeX.

  13. Displaying an object inheriting from class "sde" (1,2 and 3 dim).

Main Features

Stochastic integrals:

We consider a simple example to simulation Itô integral, used st.int function:

int(w^n * dw(s),0,T) = w^(n+1) /n+1 - n/2 int(w^(n-1) ds,0,T)

And the Stratonovich integral

int(w^n o dw(s),0,T) = w^(n+1) /n+1

  R> f <- expression( w )
  R> Itô <- st.int(f,type="Ito",M=500,lower=0,upper=1)
  R> Itô
   Itô integral:
        | X(t)   = integral (f(s,w) * dw(s))
        | f(t,w) = w
   Summary:
        | Number of subinterval | N = 1001.
        | Number of simulation  | M = 500.
        | Limits of integration | t in [0,1].
  R> summary(Itô)
   Monte-Carlo Statistics for integral(f(s,w) * dw(s)) at time t = 1
   | f(t,w) = w
                         
  Mean              0.01330
  Variance          0.51102
  Median           -0.28645
  Mode             -0.42772
  First quartile   -0.44666
  Third quartile    0.22534
  Minimum          -0.55198
  Maximum           4.38802
  Skewness          2.27133
  Kurtosis          9.27393
  Coef-variation   53.75783
  3th-order moment  0.82972
  4th-order moment  2.42178
  5th-order moment  7.60355
  6th-order moment 26.72897 
  
  R> str <- st.int(f,type="str",M=500,lower=0,upper=1)
  R> str
  Stratonovich integral:
        | X(t)   = integral (f(s,w) o dw(s))
        | f(t,w) = w
  Summary:
        | Number of subinterval | N = 1001.
        | Number of simulation  | M = 500.
        | Limits of integration | t in [0,1].
  R> summary(str)
   Monte-Carlo Statistics for integral (f(s,w) o dw(s)) at time t = 1
   | f(t,w) = w
                          
   Mean               0.55655
   Variance           0.66663
   Median             0.21223
   Mode               0.08249
   First quartile     0.04269
   Third quartile     0.79322
   Minimum            0.00000
   Maximum            6.70508
   Skewness           2.72896
   Kurtosis          13.34205
   Coef-variation     1.46702
   3th-order moment   1.48532
   4th-order moment   5.92908
   5th-order moment  26.87087
   6th-order moment 138.27901  
  

SDE's 1,2 and 3-dim:

There are thus two widely used types of stochastic calculus, Stratonovich and Itô, differing in respect of the stochastic integral used. Modelling issues typically dictate which version in appropriate, but once one has been chosen a corresponding equation of the other type with the same solutions can be determined. Thus it is possible to switch between the two stochastic calculi. Specifically, the processes X(t) solution to the Itô SDE:

dX(t) = f(t,X(t)) dt + g(t,X(t)) dW(t)

where W(t) is the standard Wiener process or standard Brownian motion, the drift f(t,X(t)) and diffusion g(t,X(t)) are known functions that are assumed to be sufficiently regular (Lipschitz, bounded growth) for existence and uniqueness of solution; has the same solutions as the Stratonovich SDE:

dX(t) = bar_f(t,X(t)) dt + g(t,X(t)) o dW(t)

with the modified drift coefficient

bar_f(t,X(t)) = f(t,X(t)) - 1/2 g(t,X(t)) gx(t,X(t))

The following examples for different methods of simulation of SDEs (1,2 and 3-dim) use the snssde1d, snssde2d and snssde3d functions.

  R> ## 1-dim sde
  R> f <- expression(2*(3-x) )
  R> g <- expression(2*x)
  R> res1 <- snssde1d(drift=f,diffusion=g,M=1000,x0=1)
  R> res1
  Itô Sde 1D:
        | dX(t) = 2 * (3 - X(t)) * dt + 2 * X(t) * dW(t)
  Method:
        | Euler scheme with order 0.5
  Summary:
        | Size of process       | N  = 1001.
        | Number of simulation  | M  = 1000.
        | Initial value         | x0 = 1.
        | Time of process       | t in [0,1].
        | Discretization        | Dt = 0.001.	
  R> res2 <- snssde1d(drift=f,diffusion=g,M=1000,x0=1,type="str")
  R> res2
  Stratonovich Sde 1D:
        | dX(t) = 2 * (3 - X(t)) * dt + 2 * X(t) o dW(t)
  Method:
        | Euler scheme with order 0.5
  Summary:
        | Size of process       | N  = 1001.
        | Number of simulation  | M  = 1000.
        | Initial value         | x0 = 1.
        | Time of process       | t in [0,1].
        | Discretization        | Dt = 0.001.
		
  R> ## 2-dim sde
  R> fx <- expression(x-y, y-x)
  R> gx <- expression(2*y, 2*x)
  R> res2d <- snssde2d(drift=fx,diffusion=gx,x0=c(1,1))
  R> res2d
  Itô Sde 2D:
        | dX(t) = X(t) - Y(t) * dt + 2 * Y(t) * dW1(t)
        | dY(t) = Y(t) - X(t) * dt + 2 * X(t) * dW2(t)
  Method:
        | Euler scheme with order 0.5
  Summary:
        | Size of process       | N  = 1001.
        | Number of simulation  | M  = 1.
        | Initial values        | (x0,y0) = (1,1).
        | Time of process       | t in [0,1].
        | Discretization        | Dt = 0.001.
  R> plot2d(res2d)
		
  R> ## 3-dim sde
  R> fx <- expression(y, 0, 0)
  R> gx <- expression(z, 1, 1)
  R> res3d <- snssde3d(drift=fx,diffusion=gx,M=1000)
  R> res3d
  Itô Sde 3D:
        | dX(t) = Y(t) * dt + Z(t) * dW1(t)
        | dY(t) = 0 * dt + 1 * dW2(t)
        | dZ(t) = 0 * dt + 1 * dW3(t)
  Method:
        | Euler scheme with order 0.5
  Summary:
        | Size of process       | N  = 1001.
        | Number of simulation  | M  = 1000.
        | Initial values        | (x0,y0,z0) = (0,0,0).
        | Time of process       | t in [0,1].
        | Discretization        | Dt = 0.001.
  plot3D(res3d) 
  

Bridge SDE's 1,2 and 3-dim:

Simulation of bridge SDEs (1,2 and 3-dim) with bridgesde1d, bridgesde2d and bridgesde3d functions.

  R> ## 1-dim bridge sde
  R> f <- expression( 2*(1-x) )
  R> g <- expression( 1 )
  R> mod1 <- bridgesde1d(drift=f,diffusion=g,x0=2,y=1,M=1000)
  R> mod1
  Itô Bridges Sde 1D:
        | dX(t) = 2 * (1 - X(t)) * dt + 1 * dW(t)
  Method:
        | Euler scheme with order 0.5
  Summary:
        | Size of process       | N = 1001.
        | Crossing realized     | C = 843 among 1000.
        | Initial value         | x0 = 2.
        | Ending value          | y = 1.
        | Time of process       | t in [0,1].
        | Discretization        | Dt = 0.001.	
  R> summary(mod1) ## Monte-Carlo statistics at T/2=0.5
  
        Monte-Carlo Statistics for X(t) at time t = 0.5
                Crossing realized 843 among 1000                         
   Mean              1.31263
   Variance          0.18352
   Median            1.30504
   Mode              1.46713
   First quartile    1.02722
   Third quartile    1.60984
   Minimum          -0.22080
   Maximum           2.83339
   Skewness          0.01722
   Kurtosis          3.19888
   Coef-variation    0.32636
   3th-order moment  0.00135
   4th-order moment  0.10773
   5th-order moment  0.00645
   6th-order moment  0.11233
  R> plot(mod1)
  R> den <- dsde1d(mod1)
   Density of X(t-t0)|X(t0)=x0 at time t = 1

  Data: x (843 obs.);     Bandwidth 'bw' = 0.2339

             x                f(x)        
  Min.   :0.29822   Min.   :0.01913  
  1st Qu.:0.64911   1st Qu.:0.13600  
  Median :1.00000   Median :0.55258  
  Mean   :1.00000   Mean   :0.70988  
  3rd Qu.:1.35089   3rd Qu.:1.28369  
  Max.   :1.70178   Max.   :1.70511
  R> plot(den)
  
  R> ## 2 and 3-dim Bridge sde
  R> example(bridgesde2d)
  R> example(bridgesde3d) 
  

Estimate the parameters of 1-dim sde:

Consider a process solution of the general stochastic differential equation:

dX(t) = f(t,X(t),theta) dt + g(t,X(t),theta) dW(t)

The package Sim.DiffProc implements the function fitsde of estimate drift and diffusion parameters theta=(theta1,theta2,...,thetap) with different methods of maximum pseudo-likelihood of the 1-dim stochastic differential equation.

An example we use a real data, fit with the CKLS model:

dX(t) = (theta1+theta2*X(t)) dt + theta3 * X(t)^theta4 dW(t)

we estimate the vector of parameters theta=(theta1,theta2,theta3,theta4), using Euler pseudo-likelihood.

  R> ## 1-dim fitsde  
  R> data(Irates)
  R> rates <- Irates[,"r1"]
  R> rates <- window(rates, start=1964.471, end=1989.333)
  R> fx <- expression(theta[1]+theta[2]*x)
  R> gx <- expression(theta[3]*x^theta[4]) 
  R> ## theta = (theta1,theta2,theta3,theta4), p=4
  R> fitmod <- fitsde(rates,drift=fx,diffusion=gx,pmle="euler",start = list(theta1=1,
                      theta2=1,theta3=1,theta4=1),optim.method = "L-BFGS-B")
  R> fitmod
  Call:
  fitsde(data = rates, drift = fx, diffusion = gx, pmle = "euler", 
     start = list(theta1 = 1, theta2 = 1, theta3 = 1, theta4 = 1), 
     optim.method = "L-BFGS-B")
  Coefficients:
      theta1     theta2     theta3     theta4 
   2.0769516 -0.2631871  0.1302158  1.4513173 
  R> summary(fitmod)
  Pseudo maximum likelihood estimation
  Method:  Euler
  Call:
  fitsde(data = rates, drift = fx, diffusion = gx, pmle = "euler", 
     start = list(theta1 = 1, theta2 = 1, theta3 = 1, theta4 = 1), 
     optim.method = "L-BFGS-B")
  Coefficients:
           Estimate Std. Error
  theta1  2.0769516 0.98838467
  theta2 -0.2631871 0.19544290
  theta3  0.1302158 0.02523105
  theta4  1.4513173 0.10323740

  -2 log L: 475.7572 
  R> coef(fitmod)
     theta1     theta2     theta3     theta4 
  2.0769516 -0.2631871  0.1302158  1.4513173 
  R> logLik(fitmod)
  [1] -237.8786
  R> AIC(fitmod)
  [1] 483.7572
  R> BIC(fitmod)
  [1] 487.1514
  R> vcov(fitmod)
                theta1        theta2        theta3        theta4
  theta1  0.9769042534 -1.843596e-01 -2.714334e-04  0.0011374342
  theta2 -0.1843595796  3.819793e-02  5.169849e-05 -0.0002165286
  theta3 -0.0002714334  5.169849e-05  6.366061e-04 -0.0025457493
  theta4  0.0011374342 -2.165286e-04 -2.545749e-03  0.0106579616
  R> confint(fitmod,level=0.95)
              2.5 %    97.5 %
  theta1  0.13975321 4.0141499
  theta2 -0.64624812 0.1198740
  theta3  0.08076388 0.1796678
  theta4  1.24897569 1.6536589
  

Transition density and Random number generators (RN's) for 1,2 and 3-dim sde:

Simulation M-sample for the random variable X(at) at time t=at by a simulated 1, 2 and 3-dim sde, using the functions rsde1d, rsde2d and rsde3d. And dsde1d, dsde2d and dsde3d returns a kernel approximate of transitional densities.

  R> f <- expression(-2*(x<=0)+2*(x>=0))
  R> g <- expression(0.5)
  R> res1 <- snssde1d(drift=f,diffusion=g,M=50,type="str",T=10)
  R> x <- rsde1d(res1,at=10)
  R> x
    [1] -17.64115  21.67111 -19.00162 -20.21546  20.65829  19.59535
    [7] -20.00676 -18.75649 -19.04453 -15.55535 -18.75077  18.89528
   [13] -22.99474 -19.66526 -19.75898  22.02310 -19.68301 -19.08581
   [19] -19.15081 -19.24476 -22.24332  17.74989  19.88449 -18.17091
   [25] -18.65697  19.08473 -17.81218  19.58453  19.27531 -21.88292
   [31]  19.03283 -19.29196  21.99163  20.12123  21.09657 -20.20252
   [37]  20.85097 -19.41987 -18.67530 -19.36289  19.50057  16.30538
   [43]  19.34247 -17.97358  22.81003 -18.40051 -18.47490 -21.86839
   [49] -21.32638 -18.96264
  R> summary(x)
      Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   -22.990 -19.350 -18.440  -4.436  19.330  22.810
  R> den <- dsde1d(res1,at=10)
  R> ## 2 and 3-dim rsde
  R> example(dsde2d)
  R> example(dsde3d)
  

First-passage-time (f.p.t) in 1,2 and 3-dim sde

The functions fptsde1d (fptsde2d and fptsde3d for 2 and 3-dim) returns a random variable tau(X(t),S(t)) "first passage time", is defined as:

tau(X(t),S(t))={t>=0; X(t) >= S(t)}, if X(t0) < S(t0)

tau(X(t),S(t))={t>=0; X(t) <= S(t)}, if X(t0) > S(t0)

And dfptsde1d, dfptsde2d and dfptsde3d returns a kernel density approximation for first passage time. with S(t) is through a continuous boundary (barrier).

  R> f <- expression( 0.5*x*t )
  R> g <- expression( sqrt(1+x^2) )
  R> St <- expression(-0.5*sqrt(t)+exp(t^2))
  R> mod <- snssde1d(drift=f,diffusion=g,x0=2,M=1000)
  R> fptmod <- fptsde1d(mod,boundary=St)
  R> fptmod
   Itô Sde 1D:
        | dX(t) = 0.5 * X(t) * t * dt + sqrt(1 + X(t)^2) * dW(t)
        | t in [0,1].
   Boundary:
        | S(t) = -0.5 * sqrt(t) + exp(t^2)
   F.P.T:
        | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -0.5 * sqrt(t) + exp(t^2) }
        | Crossing realized 738 among 1000.
  R> summary(fptmod)

   Monte-Carlo Statistics of F.P.T:
   |T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -0.5 * sqrt(t) + exp(t^2) }
                        
   Mean             0.47742
   Variance         0.07348
   Median           0.44831
   Mode             0.18582
   First quartile   0.23746
   Third quartile   0.71321
   Minimum          0.03002
   Maximum          0.98877
   Skewness         0.22793
   Kurtosis         1.79959
   Coef-variation   0.56778
   3th-order moment 0.00454
   4th-order moment 0.00972
   5th-order moment 0.00134
   6th-order moment 0.00158
  R> den <- dfptsde1d(mod,boundary=St)
  R> den
    Kernel density for the F.P.T of X(t)
    T(S,X) = inf{t >= 0 : X(t) <= -0.5 * sqrt(t) + exp(t^2)}

     Data: fpt (738 obs.);    Bandwidth 'bw' = 0.0828

          x                f(x)       
   Min.   :-0.2095   Min.   :0.0019  
   1st Qu.: 0.1458   1st Qu.:0.2163  
   Median : 0.5010   Median :0.5307  
   Mean   : 0.5010   Mean   :0.7029  
   3rd Qu.: 0.8563   3rd Qu.:1.1943  
   Max.   : 1.2116   Max.   :1.8548
  R> ## fpt in 2 and 3-dim sde
  R> example(dfptsde2d)
  R> example(dfptsde3d)  
  

For other examples see demo(Sim.DiffProc), and for an overview of this package, see browseVignettes('Sim.DiffProc') for more informations.

Requirements

R version >= 3.0.0

Licence

This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package.

Author(s)

A.C. Guidoum acguidoum@usthb.dz and K. Boukhetala kboukhetala@usthb.dz (Dept. Probability and Statistics, USTHB, Algeria).
Please send comments, error reports, etc. to the author via the addresses email.

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F C Klebaner, F.C. (2005). Introduction to stochastic calculus with application. 2nd edn. Imperial College Press (ICP).

Henderson, D. and Plaschko, P. (2006). Stochastic differential equations in science and engineering. World Scientific.

See Also

sde, yumia, QPot, DiffusionRgqd, fptdApprox.


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