BM {sde} | R Documentation |
Brownian motion, Brownian Bridge and Geometric Brownian motion simulators
BBridge(x=0, y=0, t0=0, T=1, N=100) BM(x=0, t0=0, T=1, N=100) GBM(x=1, r=0, sigma=1, T=1, N=100)
x |
intial value of the process at time t0 |
y |
terminal value of the process at time T |
t0 |
initial time. |
r |
the interest rate of the GBM. |
sigma |
the volatility of the GBM. |
T |
final time. |
N |
number of intervals in which to split [t0,T] . |
These functions return an invisible ts
object containing
a trajectory of the process calculated on grid of N+1
equidistant points between t0
and T
, i.e.
t[i] = t0 + (T-t0)*i/N
, i in 0:N
. t0=0
for the
Geometric Brownian Motion.
The function BBridge
returns a trajectory of the Brownian Bridge
starting in x
at time t0
and
ending at y
at time T
, i.e.
(B(t), t0 <= t <= T | B(t_0)=x, B(T)=y)
The function BM
returns
a trajectory of the translated
Brownian Motion (B(t), t>= t0 | B(t0)=x)
,
i.e. x+B(t-t0)
, for t >= t0
.
The standard Brownian motion is obtained
choosing x=0
and t0=0
(the default values).
The function GBM
returns a trajectory of the Geometric Brownian Motion
starting at x
at time t0=0
, i.e. the process
S(t) = x * exp((r-sigma^2/2)*t + sigma*B(t))
.
X |
an invisible ts object |
This package is a companion to the book Simulation and Inference for Stochastic Differential Equation, Springer, NY.
Stefano Maria Iacus
plot(BM()) plot(BBridge()) plot(GBM())