Derive and understand the probability of a Brownian bridge crossing upper and lower barriers within a specified time interval | Step-by-Step Solution

Understanding Brownian Bridge Barrier Crossing Probabilities

What We're Solving:

We want to find the probability that a Brownian bridge (a special type of random walk that starts and ends at specific points) crosses either an upper barrier or a lower barrier during a given time interval. This is a classic problem in mathematical finance and physics!

The Approach:

This is like predicting whether a stock price (following a constrained random path) will hit certain "stop-loss" or "take-profit" levels before a deadline. We'll use the reflection principle and image method - clever mathematical tricks that transform this complex probability problem into something more manageable.

The key insight is that a Brownian bridge is just regular Brownian motion that's been "conditioned" to end at a specific point, which creates dependencies we need to account for.

Step-by-Step Solution:

Step 1: Set Up the Brownian Bridge

• Let B(t) be our Brownian bridge from (0,a) to (T,b)
• This means B(0) = a and B(T) = b with probability 1
• Between these points, it follows a modified random walk

Step 2: Define Your Barriers

• Upper barrier: u(t) (some function of time)
• Lower barrier: l(t) (some function of time)
• We want P(l(t) ≤ B(s) ≤ u(t) for all s ∈ [0,T])

Step 3: Transform to Standard Brownian Motion

• Convert the bridge to standard Brownian motion W(t) using:

B(t) = a + W(t) - (t/T)W(T) + (t/T)(b-a)

• This removes the "conditioning" temporarily

Step 4: Apply the Reflection Principle

• For each barrier crossing, create a "reflected" path
• The probability of crossing equals the probability of ending at the reflected endpoint
• This works because of the symmetry properties of Brownian motion

Step 5: Use the Image Method

• Place "image sources" beyond each barrier
• The probability becomes a sum over all possible reflection sequences
• Often this creates an infinite series, but many terms cancel out

Step 6: Account for the Bridge Constraint

• Re-impose the condition that B(T) = b
• This typically involves integrating over all possible intermediate paths
• Use the joint distribution of Brownian motion at multiple time points

The Framework:

The final probability typically has the form:

P = ∑(alternating series of reflected probabilities) × (bridge correction factor)

Where:

• The sum accounts for all possible barrier hitting sequences
• The correction factor ensures we end at point b
• Signs alternate based on the number of reflections (inclusion-exclusion principle)

For constant barriers u and l, this often simplifies to expressions involving:

• Gaussian error functions
• Exponential terms related to barrier distances
• Time-scaling factors

Memory Tip:

Remember "RICE" - Reflection principle creates Images that Convert complex boundary problems into Easy endpoint calculations. The Brownian bridge just adds a "leash" that pulls the path back to its endpoint!

The beauty of this approach is that it transforms a seemingly impossible continuous-time probability into a clever counting problem using reflections. Each time the process would cross a barrier, we "bounce" it back and keep track of all possible bouncing sequences!