For the \(n\) valuation times given by \(T = (t_1,\dots,t_n)'\) we simulate
from the joint distribution \((S(t_1),\ldots,S(t_n),m(t_1),\ldots,m(t_n))\)
of the finite-dimensional distribution
\((S(t_1),\ldots,S(t_n))\) and the running minimum \(m(t_i) = \min_{0 \le t \le t_i}(S_t)\)
of a restricted/truncated Geometric Brownian motion.
The Geometric Brownian Motion is conditioned at the \(n\) valuation dates \((t_1,...,t_n)\)
on \(lowerX_i \le S(t_i) \le upperX_i\) for all \(i=1,\dots,n\).
First we simulate \((S(t_1),\ldots,S(t_n))\)
from a truncated multivariate normal distribution of the returns
with mean vector
$$(\mu - \sigma^2/2) * T$$
and covariance matrix
$$\Sigma = (\min{(t_i,t_j)}\sigma^2) \\
= \left[ \begin{array}{cccc}
\min{(t_1,t_1)} \sigma^2 & \min{(t_1,t_2)} \sigma^2 & \cdots & \min{(t_1,t_n)} \sigma^2 \\
\min{(t_2,t_1)} \sigma^2 & \min{(t_2,t_2)} \sigma^2 & \cdots & \min{(t_2,t_n)} \sigma^2 \\
\vdots & & & \\
\min{(t_n,t_1)} \sigma^2 & \cdots & & \min{(t_n,t_1)} \sigma^2
\end{array} \right]
$$
and lower and upper truncation points lower=log(lowerX/S) and upper=log(upperX/S) respectively.
Given the realized prices \((S(t_1),\ldots,S(t_n))\)
we simulate the global minimum as the minimum of several Brownian Bridges as described in Beskos (2006):
We simulate the period minimum \(m_{(i-1,i)}\) between two times \(t_{i-1}\) and \(t_i\)
for all \(i=1,\dots,n\).
This minimum \(m_{(i-1,i)} | S(t_{i-1}),S(t_i)\) is the minimum of a Brownian Bridge between \(t_{i-1}\) and \(t_i\).
The global minimum is the minimum of all period minima given by
\(m_n = \min(m_{(0,1)},m_{(1,2)},\dots,m_{(n-1,n)}) = \min(m_{(i-1,i)})\)
for all \(i=1,\dots,n\).