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 It<U+00F4> 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.
Stochastic integrals:
We consider a simple example to simulation It<U+00F4> 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> f <- expression( w )
R> It<U+00F4> <- st.int(f,type="Ito",M=500,lower=0,upper=1)
R> It<U+00F4>
It<U+00F4> 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(It<U+00F4>)
Monte-Carlo Statistics for integral(f(s,w) * dw(s)) at time t = 1
| f(t,w) = wMean 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 It<U+00F4>, 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 It<U+00F4> 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=1000,x0=1)
R> res1
It<U+00F4> 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
It<U+00F4> 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
It<U+00F4> 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
It<U+00F4> 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=0.5Monte-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:
$$dX_{t} = f(t,X_{t},\underline{\theta}) dt + g(t,X_{t},\underline{\theta}) dW_{t}$$
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
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:
$$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})$$
$$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})$$
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
It<U+00F4> 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.
R version >= 3.0.0
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.
| Package: | Sim.DiffProc |
| Type: | Package |
| Version: | 4.3 |
| Date: | 2018-11-28 |
| 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:
Simulation of solution to 1,2 and 3-dim stochastic differential equations of It<U+00F4> and Stratonovich types, with different methods.
Simulation of solution to 1,2 and 3-dim diffusion bridge of It<U+00F4> and Stratonovich types, with different methods.
Simulation the first-passage-time (f.p.t) in 1,2 and 3-dim sde of It<U+00F4> and Stratonovich types.
Calculate symbolic ODE's of moment equations (means and variances-covariance) for 1,2 and 3-dim SDE's.
Monte-Carlo replicates of a statistic applied to 1,2 and 3-dim SDE's at any time t.
Computing the basic statistics (mean, var, median, ...) of the processes at any time t using the Monte Carlo method.
Random number generators (RN's) to generate 1,2 and 3-dim sde of It<U+00F4> and Stratonovich types.
Approximate the transition density 1,2 and 3-dim of the processes at any time t.
Approximate the density of first-passage-time in 1,2 and 3-dim SDE's.
Computing the stochastic integrals of It<U+00F4> and Stratonovich types.
Estimate drift and diffusion parameters by the method of maximum pseudo-likelihood of the 1-dim stochastic differential equation.
Converting Sim.DiffProc objects to LaTeX.
Displaying an object inheriting from class "sde" (1,2 and 3 dim).
Argyrakisa, P. and G.H. Weiss (2006). A first-passage time problem for many random walkers. Physica A. 363, 343--347.
Aytug H., G. J. Koehler (2000). New stopping criterion for genetic algorithms. European Journal of Operational Research, 126, 662--674.
Boukhetala, K. (1994). Simulation study of a dispersion about an attractive centre. In proceedings of 11th Symposium Computational Statistics, edited by R.Dutter and W.Grossman, Wien , Austria, 128--130.
Boukhetala, K. (1996). Modelling and simulation of a dispersion pollutant with attractive centre. ed by Computational Mechanics Publications, Southampton ,U.K and Computational Mechanics Inc, Boston, USA, 245--252.
Boukhetala, K. (1998a). Estimation of the first passage time distribution for a simulated diffusion process. Maghreb Math.Rev, 7(1), 1--25.
Boukhetala, K. (1998b). Kernel density of the exit time in a simulated diffusion. les Annales Maghrebines De L ingenieur, 12, 587--589.
Ding, M. and G. Rangarajan. (2004). First Passage Time Problem: A Fokker-Planck Approach. New Directions in Statistical Physics. ed by L. T. Wille. Springer. 31--46.
Ait-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. The Journal of Finance, 54, 1361--1395.
Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica. 70, 223--262.
Roman, R.P., Serrano, J. J., Torres, F. (2008). First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Computational Statistics and Data Analysis. 52, 4132--4146.
Roman, R.P., Serrano, J. J., Torres, F. (2012). An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function. Applied Mathematics and Computation, 218, 8408--8428.
Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24, 211-229.
Gardiner, C. W. (1997). Handbook of Stochastic Methods. Springer-Verlag, New York.
Friedman, A. (1975). Stochastic differential equations and applications. Volume 1, ACADEMIC PRESS.
Henderson, D. and Plaschko,P. (2006). Stochastic differential equations in science and engineering. World Scientific.
Croissant, Y. (2014). Ecdat: Data sets for econometrics. R package version 0.2-5.
Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5, 177--188.
Allen, E. (2007). Modeling with It<U+00F4> stochastic differential equations. Springer-Verlag.
Jedrzejewski, F. (2009). Modeles aleatoires et physique probabiliste. Springer-Verlag.
Iacus, S.M. (2008). Simulation and inference for stochastic differential equations: with R examples. Springer-Verlag, New York.
Iacus, S.M. (2014). sde: Simulation and Inference for Stochastic Differential Equations. R package version 2.0.13.
Brouste, A. et al. (2014). The yuima Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations. Journal of Statistical Software, 57(4).
Kloeden, P.E, and Platen, E. (1989). A survey of numerical methods for stochastic differential equations. Stochastic Hydrology and Hydraulics, 3, 155--178.
Kloeden, P.E, and Platen, E. (1991a). Relations between multiple It<U+00F4> and stratonovich integrals. Stochastic Analysis and Applications, 9(3), 311--321.
Kloeden, P.E, and Platen, E. (1991b). Stratonovich and It<U+00F4> stochastic taylor expansions. Mathematische Nachrichten, 151, 33--50.
Kloeden, P.E, and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York.
Oksendal, B. (2000). Stochastic Differential Equations: An Introduction with Applications. 5th edn. Springer-Verlag, Berlin.
B.L.S. Prakasa Rao. (1999). Statistical Inference for Diffusion Type Processes. Arnold, London and Oxford University press, New York.
Kutoyants, Y.A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.
Sorensen, H. (2000). Inference for Diffusion Processes and Stochastic Volatility Models. Ph.D. thesis, Department of Mathematical Sciences, University of Copenhagen.
Sorensen, H. (2002). Estimation of diffusion parameters for discretely observed diffusion processes. Bernoulli, 8, 491--508.
Sorensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: a survey. International Statistical Review, 72, 337--354.
Platen, E. (1980). Weak convergence of approximations of It<U+00F4> integral equations. Z Angew Math Mech. 60, 609--614.
Platen, E. and Bruti-Liberati, N. (2010). Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer-Verlag, New York.
SaIt<U+00F4>, Y, and Mitsui, T. (1993). Simulation of Stochastic Differential Equations. The Annals of the Institute of Statistical Mathematics, 3, 419--432.
Risken, H. (2001). The Fokker Planck Equation : Methods of Solutions and Applications. 2nd edition, Springer Series in Synergetics.
Dacunha, D.C. and Florens, D.Z. (1986). Estimation of the Coefficients of a Diffusion from Discrete Observations. Stochastics. 19, 263--284.
Dohnal, G. (1987). On estimating the diffusion coefficient. J. Appl.Prob., 24, 105--114.
Genon, V.C. (1990). Maximum constrast estimation for diffusion processes from discrete observation. Statistics, 21, 99--116.
Protter, P. (2005). Stochastic Integration and Differential Equations. 2nd edn. Springer-Verlag, New York.
Bladt, M. and Sorensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working Paper, University of Copenhagen.
Ozaki, T. (1992). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach. Statistica Sinica, 2, 25-83.
Shoji, L., Ozaki, T. (1998). Estimation for nonlinear stochastic differential equations by a local linearization method. Stochastic Analysis and Applications, 16, 733-752.
Nicolau, J. (2004). Introduction to the estimation of stochastic differential equations based on discrete observations. Autumn School and International Conference, Stochastic Finance.
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.
sde, yumia, QPot, DiffusionRgqd, fptdApprox.