Stochastic Processes – How to Contain a Brownian Motion Between 2 Points?

brownian motionstochastic-processes

I am interested in learning about if it is possible to mathematically define a Brownian Motion such that it can ONLY take values between two points $a$ and $b$.

Part 1: Definition

Consider a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process):

A Wiener process $W(t)$, $t \in [0, \infty)$, is a real-valued continuous-time stochastic process that satisfies the following properties:

  1. $W_0 = 0$
  2. $W$ has independent increments: for every $t > 0$, the future increments $W_{t+u} – W_t$, $u \geq 0$, are independent of the past values $W_s$, $s < t$.
  3. $W$ has Gaussian increments: $W_{t+u} – W_t$ is normally distributed with mean 0 and variance $u$, i.e., $W_{t+u} – W_t \sim \mathcal{N}(0, u)$.

Thus, in general:

$$W(t) = \sum_{i=1}^{t-1} (W_i – W_{i-1})$$
$$(W_i – W_{i-1}) \sim \mathcal{N}(0, dt)$$

Part 2: Simulating a Basic Process

Here is my attempt to simulate a Wiener Process in the R programming language:

library(ggplot2)

n <- 1000  # Number of steps
dt <- 1/n  # Time step

dW <- rnorm(n, mean = 0, sd = sqrt(dt))
W <- cumsum(dW)


df <- data.frame(time = seq(0, 1000, length.out = n), Wiener = W)

p <- ggplot(df, aes(x = time, y = Wiener)) +
    geom_line() +
    labs(title = "Wiener Process", x = "Time", y = "Value") + theme_bw()

print(p)

enter image description here

Part 3: Manually Forcing a Brownian Motion to stay between two points:

For example, imagine a coin where the probability of success itself evolves as a stochastic process. That is, the probability of getting heads on the $n_{th}$ turn depends on the history of the previous flips according to a Wiener Process. In this case, I would have to define a Wiener Process between $0$ and $1$.

From a programming perspective, I think this might be possible: Every time the Wiener Process takes a value of $0$ or $1$, you can "force" it to take a value of 0.01 or 0.99 and resume the simulation:

n <- 1000  # Number of steps
dt <- 1/n  # Time step

dW <- rnorm(n, mean = 0, sd = sqrt(dt))

W <- numeric(n)
W[1] <- 0.5  # Start in the middle of the interval [0, 1]

# Generate the Wiener process
for (i in 2:n) {
  W[i] <- W[i-1] + dW[i]
  
  # If the process hits 0 or 1, reset to 0.01 or 0.99 respectively
  if (W[i] <= 0) {
    W[i] <- 0.01
  } else if (W[i] >= 1) {
    W[i] <- 0.99
  }
}


df <- data.frame(time = seq(0, 1, length.out = n), Wiener = W)


p <- ggplot(df, aes(x = time, y = Wiener)) +
  geom_line() +
  labs(title = "Bounded Wiener Process", x = "Time", y = "Value") +
  theme_bw()


print(p)

enter image description here

Part 4: Reflected Brownian Motion

I found this link (https://en.wikipedia.org/wiki/Reflected_Brownian_motion) which looks that it might be possible to "bound" a Brownian Motion (quoting the Wikipedia Page):

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 $j = 1,2,…,d$,
  • $Z(t) \in \mathbb{R}^d_+$, $t \geq 0$.

But, 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 $(a,b)$.

My Question: How can I mathematically define the Wiener Process such that the range can only be between two numbers $a$ and $b$?

Thanks!

Note: Just to clarify – I want to ensure that the Brownian Motion NEVER exceeds a value of $a,b$ … I am not interested in the concept of the "Brownian Bridge"(https://en.wikipedia.org/wiki/Brownian_bridge) where the Brownian Motion takes a value of 0 at time=0 and time=T.

Best Answer

The most natural way to force the Wiener process $W$ to stay in the interval $[0,1]$ is via conditioning. In other words, you consider a new process $\hat{W}$ which is $W$ conditioned on $W_t \in [0,1]$ for all $t$. (This can be rigorously done using Doob's h-transform). This process is solution to the SDE $$d\hat{W}_t = dW_t + \pi \cot(\pi \hat{W}_t)dt.$$ Notice the drift is $+\infty$ when $\hat{W}_t = 0$ forcing the process to increase and similarly when $\hat{W}_t = 1$.

To simulate it, either discretize the SDE or even simpler: knowing the value $\hat{W}_{t_i}$, to get $\hat{W}_{t_{i+1}}$, keep generating random variables $Z$ with distribution $N(0,t_{i+1}-t_i)$ until $\hat{W}_{t_i} + Z \in [0,1]$; this will be $\hat{W}_{t_{i+1}}$.

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