BrownianBridgeMinimum: Distribution of the Minimum of a Brownian Bridge

Description Usage Arguments Details Value Author(s) References Examples

Description

Density function and random generation of the minimum m_T=\min_{t_0 ≤ t ≤ T} of a Brownian Bridge B_t between time t_0 and T.

Usage

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 rBrownianBridgeMinimum(n = 100, t0 = 0, T = 1, a = 0, b = 0, sigma = 1)
 dBrownianBridgeMinimum(x, t0 = 0, T = 1, a = 0, b = 0, sigma = 1)

Arguments

n

the number of samples to draw

x

a vector of minimum values to calculate the density for

t0

start time

T

end time

a

start value of the Brownian Bridge (B(t0)=a)

b

end value of the Brownian Bridge (B(T)=b)

sigma

volatility p.a., e.g. 0.2 for 20%

Details

rBrownianBridgeMinimum() simulates the minimum m(T) for a Brownian Bridge B(t) between t0 <= t <= T, i.e. a Brownian Motion W(t) constraint to W(t_0)=a and W(T)=b.
The simulation algorithm uses the conditional density f(m(T) = x | B(t_0)=a, B(T)=b) and is based on the exponential distribution given by Beskos et al. (2006), pp.1082–1083, which we generalized to the sigma^2<>1 case.

The joint density function m(T) and W(T) is (see Beskos2006, pp.1082–1083 and Karatzas2008, p.95):

f_{m(T),W(T)}(b,a) = \frac{2 \cdot (a-2b)}{√{2π} σ^3 √{T^3}} \cdot \exp{≤ft\{ -\frac{(a-2b)^2}{2σ^2 T} \right\}}

With the density of W(T)

f_{W(T)}(a) = \frac{1}{√{2π} σ √{T}} \cdot \exp{≤ft\{ -\frac{a^2}{2σ^2 T} \right\}}

it follows for the conditional density of the minimum m(T) | W(T)=a

f_{m(T)|W(T)=a}(b) = \frac{2 \cdot (a-2b)}{σ^2 T} \cdot \exp{≤ft\{ -\frac{(a-2b)^2}{2σ^2 T} + \frac{a^2}{2σ^2 T} \right\}}

Value

simBrownianBridgeMinimum() returns a vector of simulated minimum values of length n.

densityBrownianBridgeMinimum returns a vector of length length(x) with density values

Author(s)

Stefan Wilhelm wilhelm@financial.com

References

Beskos, A.; Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective Exact Simulation of Diffusion Sample Paths with Applications Bernoulli, 12, 1077–1098

Karatzas/Shreve (2008). Brownian Motion and Stochastic Calculus, Springer, p.95

Examples

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  # simulate 1000 samples from minimum distribution
  m <- rBrownianBridgeMinimum(n = 1000, t0 = 0, T = 1, a = 0.2, b = 0, sigma = 2)
    
  # and compare against the density
  x  <- seq(-6, 0, by=0.01)
  dm <- dBrownianBridgeMinimum(x, t0 = 0, T = 1, a = 0.2, b = 0, sigma = 2)
  
  plot(density(m))
  lines(x, dm, lty=2, col="red")

fExpressCertificates documentation built on May 2, 2019, 11:02 a.m.