BM {Sim.DiffProc} | R Documentation |
The (S3) generic function for simulation of brownian motion, brownian bridge, geometric brownian motion, and arithmetic brownian motion.
BM(N, ...) BB(N, ...) GBM(N, ...) ABM(N, ...) ## Default S3 method: BM(N =1000,M=1,x0=0,t0=0,T=1,Dt=NULL, ...) ## Default S3 method: BB(N =1000,M=1,x0=0,y=0,t0=0,T=1,Dt=NULL, ...) ## Default S3 method: GBM(N =1000,M=1,x0=1,t0=0,T=1,Dt=NULL,theta=1,sigma=1, ...) ## Default S3 method: ABM(N =1000,M=1,x0=0,t0=0,T=1,Dt=NULL,theta=1,sigma=1, ...)
N |
number of simulation steps. |
M |
number of trajectories. |
x0 |
initial value of the process at time \code{t0}. |
y |
terminal value of the process at time \code{T} of the |
t0 |
initial time. |
T |
final time. |
Dt |
time step of the simulation (discretization). If it is |
theta |
the interest rate of the |
sigma |
the volatility of the |
... |
potentially further arguments for (non-default) methods. |
The function BM
returns a trajectory of the standard Brownian motion (Wiener process) in the time interval [t0,T]. Indeed, for W(dt) it holds true that
W(dt) = W(dt) - W(0) -> N(0,dt) -> sqrt(dt) * N(0,1), where N(0,1) is normal distribution
Normal
.
The function BB
returns a trajectory of the Brownian bridge starting at x0 at time t0 and ending
at y at time T; i.e., the diffusion process solution of stochastic differential equation:
dX(t) = ((y-X(t))/(T-t)) dt + dW(t)
The function GBM
returns a trajectory of the geometric Brownian motion starting at x0 at time t0;
i.e., the diffusion process solution of stochastic differential equation:
dX(t) = theta X(t) dt + sigma X(t) dW(t)
The function ABM
returns a trajectory of the arithmetic Brownian motion starting at x0 at time t0;
i.e.,; the diffusion process solution of stochastic differential equation:
dX(t) = theta dt + sigma dW(t)
X |
an visible |
A.C. Guidoum, K. Boukhetala.
Allen, E. (2007). Modeling with Ito stochastic differential equations. Springer-Verlag, New York.
Jedrzejewski, F. (2009). Modeles aleatoires et physique probabiliste. Springer-Verlag, New York.
Henderson, D and Plaschko, P. (2006). Stochastic differential equations in science and engineering. World Scientific.
This functions BM
, BBridge
and GBM
are available in other packages such as "sde".
op <- par(mfrow = c(2, 2)) ## Brownian motion set.seed(1234) X <- BM(M = 100) plot(X,plot.type="single") lines(as.vector(time(X)),rowMeans(X),col="red") ## Brownian bridge set.seed(1234) X <- BB(M =100) plot(X,plot.type="single") lines(as.vector(time(X)),rowMeans(X),col="red") ## Geometric Brownian motion set.seed(1234) X <- GBM(M = 100) plot(X,plot.type="single") lines(as.vector(time(X)),rowMeans(X),col="red") ## Arithmetic Brownian motion set.seed(1234) X <- ABM(M = 100) plot(X,plot.type="single") lines(as.vector(time(X)),rowMeans(X),col="red") par(op)