Description Usage Arguments Details Value Author(s) References See Also Examples
Calculate and numerical approximation of moment equations (Symbolic ODE's of means and variances-covariance) at any time for SDE's (1,2 and 3 dim) for the two cases Ito and Stratonovich interpretations.
1 2 3 4 5 6 7 8 |
drift |
drift coefficient: an |
diffusion |
diffusion coefficient: an |
corr |
the correlation coefficient '|corr|<=1' of W1(t) and W2(t) (2d) must be an expression length equal 1. And for 3d (W1(t),W2(t),W3(t)) an expressions length equal 3. See examples. |
type |
type of process |
solve |
if |
parms |
parameters passed to |
init |
the initial (state) values for the ODE system. for 1-dim (m=x0,S=0), 2-dim (m1=x0,m2=y0,S1=0,S2=0,C12=0) and for 3-dim (m1=x0,m2=y0,m3=z0,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0), see examples. |
time |
time sequence (vector) for which output is wanted; the first value of time must be the initial time. |
object, at |
an object inheriting from class |
... |
potentially arguments to be passed to methods, such as |
The stochastic transition is approximated by the moment equations, and the numerical treatment is required to solve these equations from above with given initial conditions.
An overview of this package, see browseVignettes('Sim.DiffProc')
for more informations.
Symbolic ODE's of means and variances-covariance. If solve=TRUE
approximate the moment of SDE's at any time.
A.C. Guidoum, K. Boukhetala.
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. doi:10.18637/jss.v096.i02
Rodriguez R, Tuckwell H (2000). A dynamical system for the approximate moments of nonlinear stochastic models of spiking neurons and networks. Mathematical and Computer Modelling, 31(4), 175–180.
Alibrandi U, Ricciardi G (2012). Stochastic Methods in Nonlinear Structural Dynamics, 3–60. Springer Vienna, Vienna. ISBN 978-3-7091-1306-6.
MCM.sde
Monte-Carlo methods for SDE's.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | library(deSolve)
## Example 1: 1-dim
## dX(t) = mu * X(t) * dt + sigma * X(t) * dW(t)
## Symbolic ODE's of mean and variance
f <- expression(mu*x)
g <- expression(sigma*x)
res1 <- MEM.sde(drift=f,diffusion=g,type="ito")
res2 <- MEM.sde(drift=f,diffusion=g,type="str")
res1
res2
## numerical approximation of mean and variance
para <- c(mu=2,sigma=0.5)
t <- seq(0,1,by=0.001)
init <- c(m=1,S=0)
res1 <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t)
res1
matplot.0D(res1$sol.ode,main="Mean and Variance of X(t), type Ito")
plot(res1$sol.ode,select=c("m","S"))
## approximation at time = 0.75
summary(res1,at=0.75)
##
res2 <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t,type="str")
res2
matplot.0D(res2$sol.ode,main="Mean and Variance of X(t), type Stratonovich")
plot(res2$sol.ode,select=c("m","S"))
## approximation at time = 0.75
summary(res2,at=0.75)
## Comparison:
plot(res1$sol.ode, res2$sol.ode,ylab = c("m(t)"),select="m",xlab = "Time",
col = c("red", "blue"))
plot(res1$sol.ode, res2$sol.ode,ylab = c("S(t)"),select="S",xlab = "Time",
col = c("red", "blue"))
## Example2: 2-dim
## dX(t) = 1/mu*(theta-X(t)) dt + sqrt(sigma) * dW1(t),
## dY(t) = X(t) dt + 0 * dW2(t)
## Not run:
para=c(mu=0.75,sigma=0.1,theta=2)
init=c(m1=0,m2=0,S1=0,S2=0,C12=0)
t <- seq(0,10,by=0.001)
f <- expression(1/mu*(theta-x), x)
g <- expression(sqrt(sigma),0)
res2d <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t)
res2d
## Exact moment
mu=0.75;sigma=0.1;theta=2;x0=0;y0=0
E_x <- function(t) theta+(x0-theta)*exp(-t/mu)
V_x <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
E_y <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu))
V_y <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
cov_xy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu)))
##
summary(res2d,at=5)
E_x(5);E_y(5);V_x(5);V_y(5);cov_xy(5)
matplot.0D(res2d$sol.ode,select=c("m1"))
curve(E_x,add=TRUE,col="red")
## plot
plot(res2d$sol.ode)
matplot.0D(res2d$sol.ode,select=c("S1","S2","C12"))
plot(res2d$sol.ode[,"m1"], res2d$sol.ode[,"m2"], xlab = "m1(t)",
ylab = "m2(t)", type = "l",lwd = 2)
hist(res2d$sol.ode,select=c("m1","m2"), col = c("darkblue", "red", "orange", "black"))
## Example3: 2-dim with correlation
## Heston model
## dX(t) = mu*X(t) dt + sqrt(Y(t))*X(t) * dW1(t),
## dY(t) = lambda*(theta-Y(t)) dt + sigma*sqrt(Y(t)) * dW2(t)
## with E(dw1dw2)=rho
f <- expression( mu*x, lambda*(theta-y) )
g <- expression( sqrt(y)*x, sigma*sqrt(y) )
RHO <- expression(rho)
res2d <- MEM.sde(drift=f,diffusion=g,corr=RHO)
res2d
## Numerical approximation
RHO <- expression(0.5)
para=c(mu=1,lambda=3,theta=0.5,sigma=0.1)
ini=c(m1=10,m2=2,S1=0,S2=0,C12=0)
res2d = MEM.sde(drift=f,diffusion=g,solve=TRUE,parms=para,init=ini,time=seq(0,1,by=0.01))
res2d
matplot.0D(res2d$sol.ode,select=c("m1","m2"))
matplot.0D(res2d$sol.ode,select=c("S1","S2","C12"))
## Example4: 3-dim
## dX(t) = sigma*(Y(t)-X(t)) dt + 0.1 * dW1(t)
## dY(t) = (rho*X(t)-Y(t)-X(t)*Z(t)) dt + 0.1 * dW2(t)
## dZ(t) = (X(t)*Y(t)-bet*Z(t)) dt + 0.1 * dW3(t)
## with E(dw1dw2)=rho1, E(dw1dw3)=rho2 and E(dw2dw3)=rho3
f <- expression(sigma*(y-x),rho*x-y-x*z,x*y-bet*z)
g <- expression(0.1,0.1,0.1)
RHO <- expression(rho1,rho2,rho3)
## Symbolic moments equations
res3d = MEM.sde(drift=f,diffusion=g,corr=RHO)
res3d
## Numerical approximation
RHO <- expression(0.5,0.2,-0.7)
para=c(sigma=10,rho=28,bet=8/3)
ini=c(m1=1,m2=1,m3=1,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0)
res3d = MEM.sde(drift=f,diffusion=g,solve=T,parms=para,init=ini,time=seq(0,1,by=0.01))
res3d
summary(res3d,at=0.25)
summary(res3d,at=0.50)
summary(res3d,at=0.75)
plot(res3d$sol.ode)
matplot.0D(res3d$sol.ode,select=c("m1","m2","m3"))
matplot.0D(res3d$sol.ode,select=c("S1","S2","S3"))
matplot.0D(res3d$sol.ode,select=c("C12","C13","C23"))
plot3D(res3d$sol.ode[,"m1"], res3d$sol.ode[,"m2"],res3d$sol.ode[,"m3"], xlab = "m1(t)",
ylab = "m2(t)",zlab="m3(t)", type = "l",lwd = 2,box=F)
## End(Not run)
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