Sim.DiffProc-package: Simulation of Diffusion Processes

Description

The package Sim.DiffProc is an object created in R environment for simulation and modeling of stochastic differential equations (SDE's) the type Ito and Stratonovich. This package contains many objects, the numerical methods to find the solutions to SDE's (1, 2 and 3-dim), with a possibility for simulates a flows trajectories,with good accuracy. Many theoretical problems on the SDE's have become the object of practical research, as statistical analysis and simulation of solution of SDE's, enabled many searchers in different domains to use these equations to modeling and to analyse practical problems, in financial and actuarial modeling and other areas of application, for example modelling and simulate of dispersion in shallow water using the attractive center (Boukhetala K, 1996). We hope that the package presented here and the updated survey on the subject might be of help for practitioners, postgraduate and PhD students, and researchers in the field who might want to implement new methods.

Arguments

newcommand

\CRANpkg
\HR
\RR

href

http://CRAN.R-project.org/package=#1
http://www.usthb.dz/en/
http://www.r-project.org

pkg

Main Features

stochastic integrals: We consider a simple example to simulation Ito integral, used st.int function: $$\int_{t_{0}}^{t} W_{s}^{n} dW_{s} = \frac{1}{n+1} \left[W_{t}^{n+1} - W_{t_{0}}^{n+1}\right]- \frac{n}{2} \int_{t_{0}}^{t} W_{s}^{n-1} ds$$ And the Stratonovich integral $$\int_{t_{0}}^{t} W_{s}^{n} \circ dW_{s} = \frac{1}{n+1} \left[W_{t}^{n+1} - W_{t_{0}}^{n+1}\right]$$ R> fexpr <- expression( w^2 ) R> ito <- st.int(fexpr,type="ito",M=1,lower=0,upper=1) R> ito Ito integral: | X(t) = integral (f(s,w) * dw(s)) | f(t,w) = w^2 Summary: | Number of subintervals = 1000. | Number of simulations = 1. | Limits of integration = [0,1]. | Discretization = 0.001. R> str <- st.int(fexpr,type="str",M=1,lower=0,upper=1) R> str Stratonovich integral: | X(t) = integral (f(s,w) o dw(s)) | f(t,w) = w^2 Summary: | Number of subintervals = 1000. | Number of simulations = 1. | Limits of integration = [0,1]. | Discretization = 0.001. SDE's 1,2 and 3-dim: There are thus two widely used types of stochastic calculus, Stratonovich and Ito, 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 ${ X_{t}, t \geq 0 }$ solution to the Ito SDE: $$dX_{t} = f(t,X_{t}) dt + g(t,X_{t}) dW_{t}$$ where ${ W_{t}, t \geq 0 }$ 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: $$dX_{t} = \underline{f}(t,X_{t}) dt + g(t,X_{t}) \circ dW_{t}$$ with the modified drift coefficient $$\underline{f}(t,X_{t})= f(t,X_{t}) - \frac{1}{2} g(t,X_{t})\frac{\partial g}{\partial x}(t,X_{t})$$ 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=10,x0=1,N=1000) R> res1 Ito Sde 1D: | dX(t) = 2 * (3 - X(t)) * dt + 2 * X(t) * dW(t) Method: | Euler scheme of order 0.5 Summary: | Size of process | N = 1000. | Number of simulation | M = 10. | Initial value | x0 = 1. | Time of process | t in [0,1]. | Discretization | Dt = 0.001. R> res2 <- snssde1d(drift=f,diffusion=g,M=10,x0=1,N=1000,type="str") R> res2 Stratonovich Sde 1D: | dX(t) = 2 * (3 - X(t)) * dt + 2 * X(t) o dW(t) Method: | Euler scheme of order 0.5 Summary: | Size of process | N = 1000. | Number of simulation | M = 10. | Initial value | x0 = 1. | Time of process | t in [0,1]. | Discretization | Dt = 0.001. R> ## 2-dim sde R> fx <- expression(x-y) R> gx <- expression(2*y) R> fy <- expression(y-x) R> gy <- expression(2*x) R> res2d <- snssde2d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,M=1,N=1000,x0=1,y0=1) R> res2d Ito 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 of order 0.5 Summary: | Size of process | N = 1000. | 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) R> gx <- expression(z) R> fy <- expression(0) R> gy <- expression(1) R> fz <- expression(0) R> gz <- expression(1) R> res3d <- snssde3d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,driftz=fz, + diffz=gz,N=1000,M=100) R> res3d Ito 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 of order 0.5 Summary: | Size of process | N = 1000. | Number of simulation | M = 100. | Initial values | (x0,y0,z0) = (0,0,0). | Time of process | t in [0,1]. | Discretization | Dt = 0.001. plot3D(res3d) Estimate the parameters of 1-dim sde: Consider a process solution of the general stochastic differential equation: $$dX_{t} = f(t,X_{t},\underline{\theta}) dt + g(t,X_{t},\underline{\theta}) dW_{t}$$ The package Sim.DiffProc implements the function fitsde of estimate drift and diffusion parameters $\underline{\theta}=(\theta_{1},\theta_{2},\dots,\theta_{p})$ 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: $$dX_{t} = (\theta_{1}+\theta_{2} X_{t}) dt + \theta_{3} X^{\theta_{4}}_{t} dW_{t}$$ we estimate the vector of parameters $\underline{\theta}=(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$, 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 Random number generators (RN's) to generate 1,2 and 3-dim sde: Simulation M-sample for the random variable $X_{\tau}$ at time $t=\tau$ by a simulated 1, 2 and 3-dim sde, using the functions rsde1d, rsde2d and rsde3d. R> ## 1-dim rsde R> f <- expression( 2*(3-x) ) R> g <- expression( 1 ) R> res1d <- rsde1d(drift=f,diffusion=g,M=10,N=1000,tau=0.5412) R> res1d $SDE Ito Sde 1D: | dX(t) = 2 * (3 - X(t)) * dt + 1 * dW(t) Method: | Euler scheme of order 0.5 Summary: | Size of process | N = 1000. | Number of simulation | M = 10. | Initial value | x0 = 0. | Time of process | t in [0,1]. | Discretization | Dt = 0.001. $tau [1] 0.5412 $x [1] 1.963663 1.896083 1.548455 2.085799 1.809221 2.236625 1.130840 [8] 1.490944 2.507836 2.327291 attr(,"class") [1] "rsde1d" R> summary(res1d) Monte-Carlo Statistics for X(t) at t = 0.5412 x Mean 1.899676 Variance 0.177190 Median 1.929873 First quartile 1.613647 Third quartile 2.198918 Skewness -0.293250 Kurtosis 1.837618 Moment of order 2 0.159471 Moment of order 3 -0.021872 Moment of order 4 0.057694 Moment of order 5 -0.018334 Bound conf Inf (95%) 1.211863 Bound conf Sup (95%) 2.467214 R> ## 2 and 3-dim rsde R> example(rsde2d) R> example(rsde3d) 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: $$\tau_{(X(t),S(t))} = { t \geq 0 ; X_{t} \geq S(t) },\quad if \quad X(t_{0}) < S(t_{0})$$ $$\tau_{(X(t),S(t))} = { t \geq 0 ; X_{t} \leq S(t) },\quad if \quad X(t_{0}) > S(t_{0})$$ 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> res <- fptsde1d(drift=f,diffusion=g,boundary=St,x0=2,M=10) R> res $SDE Ito Sde 1D: | dX(t) = 0.5 * X(t) * t * dt + sqrt(1 + X(t)^2) * dW(t) Method: | Euler scheme of order 0.5 Summary: | Size of process | N = 1000. | Number of simulation | M = 10. | Initial value | x0 = 2. | Time of process | t in [0,1]. | Discretization | Dt = 0.001. $boundary -0.5 * sqrt(t) + exp(t^2) $fpt [1] 0.4159456 0.5170925 0.8002383 0.2938681 0.2186342 0.7537485 [7] 0.2830855 NA 0.8483698 0.0667398 attr(,"class") [1] "fptsde1d" R> summary(res) Monte-Carlo Statistics for the F.P.T T(S,X) = inf{t >= 0 : X(t) <= 1="" 2="" 3="" 4="" 5="" -0.5="" *="" sqrt(t)="" +="" exp(t^2)}="" na's="" mean="" 0.466414="" variance="" 0.078690="" median="" 0.415946="" first="" quartile="" 0.283085="" third="" 0.753748="" skewness="" 0.132357="" kurtosis="" 1.288982="" moment="" of="" order="" 0.069947="" 0.002922="" 0.007982="" 0.000305="" bound="" conf="" inf="" (95%)="" 0.097119="" sup="" 0.838743="" r=""> ## fpt in 2 and 3-dim sde R> example(fptsde2d) R> example(fptsde3d) For other examples see demo(Sim.DiffProc), and for an overview of this package, see vignette("SDEs"), vignette("FitSDE").

Requirements

Rversion >= 2.15.1

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.

Details

ll{ Package: Sim.DiffProc Type: Package Version: 2.8 Date: 2014-08-18 License: GPL (>= 3) Depends: R (>= 2.15.1), scatterplot3d, rgl } There are main types of functions in this package:

Computing the stochastic integrals of Ito or Stratonovich type.
Simulation of solutions to 1,2 and 3-dim stochastic differential equations of Ito or Stratonovich type, with different methods.
Estimate drift and diffusion parameters by the method of maximum pseudo-likelihood of the 1-dim stochastic differential equation.
Simulation of solutions to 1,2 and 3-dim diffusion bridge of Ito or Stratonovich type, with different methods.
Random number generators (RN's) to generate 1,2 and 3-dim sde of Ito or Stratonovich type.
First-passage-time (f.p.t) in 1,2 and 3-dim sde of Ito or Stratonovich type.
Displaying an object inheriting from class"sde"(1,2 and 3 dim).

References

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