Problem 13 - A Course in Financial Calculus

Question

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) denote standard Brownian motion under \(\mathbb{P}\) and define \(\left\\{M_{t}\right\\}_{r \geq 0}\) by $$ M_{t}=\max _{0 \leq s \leq t} W_{s} $$ Suppose that \(x \geq a .\) Calculate (a) \(\mathbb{P}\left[M_{t} \geq a, W_{t} \geq x\right]\), (b) \(\mathbb{P}\left[M_{t} \geq a, W_{t} \leq x\right]\).

Step-by-Step Solution

Answer

(a) \(2(1 - \Phi(\frac{a}{\sqrt{t}}))(1 - \Phi(\frac{x-a}{\sqrt{t}}))\), (b) \(2(1 - \Phi(\frac{a}{\sqrt{t}}))\Phi(\frac{x-a}{\sqrt{t}})\)."

Step 1: Understanding the Problem

We need to calculate two probabilities involving maximum \(M_t\) and the position \(W_t\) of a standard Brownian motion up to time \(t\). The problem specifies conditions based on \(x\) and \(a\), where \(x \geq a\).

Step 2: Probability (a) Setup

For part (a), we need \(\mathbb{P}[M_t \geq a, W_t \geq x]\). This means \(W_s\) has hit \(a\) at least once before or by time \(t\), and at time \(t\) the Brownian motion is at least \(x\).

Step 3: Use Reflection Principle

The reflection principle helps here: if \(M_t \geq a\), then there exists a time \(s \leq t\) such that \(W_s = a\). Thereafter, \(W_t\) behaves like standard Brownian motion starting from \(a\).

Step 4: Calculate Probability Using Normality

Conditionally on \(M_t \geq a\), \(W_t\) is distributed as a normal random variable with mean \(0\) and variance \(t\), starting from \(a\), so find the probability\[\mathbb{P}(W_t \geq x | W_t = a) = \mathbb{P}(N(0, t) \geq x-a) = 1 - \Phi\left(\frac{x-a}{\sqrt{t}}\right)\]

Step 5: Combine Events for Probability (a)

The probability \(\mathbb{P}(M_t \geq a) = 2(1 - \Phi(a/\sqrt{t}))\).Using independent increments,\[\mathbb{P}(M_t \geq a, W_t \geq x) = \mathbb{P}(M_t \geq a) \times \mathbb{P}(W_t \geq x) = 2\left(1 - \Phi\left(\frac{a}{\sqrt{t}}\right)\right) \left(1 - \Phi\left(\frac{x-a}{\sqrt{t}}\right)\right)\]

Step 6: Probability (b) Setup

For part (b), we need \(\mathbb{P}[M_t \geq a, W_t \leq x]\). This is similar to (a) but requires \(W_t \leq x\) at time \(t\).

Step 7: Calculate Probability for (b)

Using the same reflection principle and conditional distribution, we find:\[\mathbb{P}(W_t \leq x | W_t = a) = \Phi\left(\frac{x-a}{\sqrt{t}}\right)\]

Step 8: Combine Events for Probability (b)

Using the results from previous steps, we calculate:\[\mathbb{P}(M_t \geq a, W_t \leq x) = \mathbb{P}(M_t \geq a) \times \mathbb{P}(W_t \leq x) = 2\left(1 - \Phi\left(\frac{a}{\sqrt{t}}\right)\right) \Phi\left(\frac{x-a}{\sqrt{t}}\right)\]

Key Concepts

Reflection Principle

The reflection principle is a fascinating concept in the study of Brownian motion. Imagine a path of Brownian motion, i.e., the random walk path of a particle under stochastic processes, with a certain maximum threshold it can hit. When we apply the reflection principle, we consider this maximum point and "reflect" paths that cross it as if they were "mirrored" across this threshold.
This principle aids in understanding how long or how high a path can go in relation to a given point, such as when calculating probabilities like the one for the maximum position reached during Brownian motion.

• Think of it as a flip or reflection over a line where the maximum is the line itself.
• The principle helps determine the distribution of future path positions based on those that have crossed this line at least once.

This helps connect past maximum thresholds with future probabilities, forming a bridge between random paths that have and have not crossed certain bounds.

Probability Calculation

Calculating probabilities in the context of Brownian motion involves understanding how likely it is for a path to reach certain points. In the scenario given, we are interested in how often our Brownian path exceeds a specific maximum and ends up in a certain position at a fixed time.
There are several steps involved in this probability calculation:

• First, establish the conditions under which these probabilities are measured, such as time and initial position.
• Next, make use of statistical distributions like the normal distribution to find probabilities of surpassing specific values.

The goal is to translate the random nature of these paths into quantifiable probabilities, making it possible to predict behaviors over time.

Normal Distribution

Brownian motion is tightly linked to the normal distribution, one of the most crucial tools in probability and statistics. In this context, when a Brownian particle strikes a point or exceeds a maximum, its distribution from that point onward is consistent with a normal distribution.
The normal distribution, characterized by its bell-shaped curve, assumes a mean (average) value, and the spread is described by its variance. Here is why it's important:

• The mean of a Brownian motion's position over time remains at 0, with potential upward or downward shifts based on movement.
• Variance helps understand deviations from the mean; it increases over time, showing wider spread due to the unpredictable nature of each step.

To calculate the probability of the position being above or below certain values, we use cumulative normal distribution functions, denoted as \( \,\Phi\,.\)

Maximum of Brownian Motion

Understanding the maximum of Brownian motion is vital for analyzing paths in stochastic processes. The maximum value reached by a Brownian path up to a specific time adds another layer to predicting paths and calculating related probabilities.
When considering maximum values:

• The maximum path value resembles an anchor point for calculations; paths are either reflected at or studied from it.
• The maximum gives context for how paths evolve, particularly when we examine those that surpass certain thresholds.

This focus on maximum values is not just about reaching a peak but understanding the distribution of paths after hitting this peak and how likely they are to reach other significant points.