I posted a question (Forcing a Brownian Motion to Stay Between Two Values?) where I was interested in learning about if a Brownian Motion can be defined in such a way, such that it never goes below $0$ and never goes above $1$.
In the previous question (Forcing a Brownian Motion to Stay Between Two Values?), I was shown that for the range $(0,1)$, apparently this can be done with the "Triangular Wave Function".
My Problem: I am interested in knowing if this approach (i.e. Triangular Wave Function) or some other approach can be used to define a Brownian Motion, such that the Brownian Motion will always stay between some arbitrary range of $(a,b)$.
According to this Wikipedia page (https://en.wikipedia.org/wiki/Reflected_Brownian_motion), the principle of Reflection can be used to define such a Brownian Motion:
A $d$-dimensional reflected Brownian motion $Z$ is a stochastic process on $\mathbb{R}^d_+$ uniquely defined by:
- a $d$-dimensional drift vector $\mu$,
- a $d \times d$ non-singular covariance matrix $\Sigma$
- a $d \times d$ reflection matrix $R$.
Where:
- $X(t)$ is an unconstrained Brownian motion with drift $\mu$ and variance $\Sigma$
- $Z(t) = X(t) + R Y(t)$
Such that: $Y(t)$ a $d$-dimensional vector where:
- $Y$ is continuous and non-decreasing with $Y(0) = 0$,
- $Y_j$ only increases at times for which $Z_j = 0$ for $j = 1,2,...,d$,
- $Z(t) \in \mathbb{R}^d_+$, $t \geq 0$.
My Question: From the above text, it seems like $Z(t) = X(t) + R Y(t)$ is the general form for a "constrained" (i.e. reflected) Brownian Motion - however, within this formulation, I am not sure which boundaries (i.e. values of a,b) this Brownian Motion will be contained within. Furthermore, I am not sure how the matrix $R$ needs to be defined such that the Brownian Motion lies between $(a,b)$.
Can someone please show me how I can fully mathematically define a 1 dimensional Brownian Motion to only assume values between $(a,b)$?
Thanks!
