Problem 17 - A Course in Financial Calculus

Question

Let \(\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\) be the natural filtration associated with a \(\mathbb{P}\)-Brownian motion, \(\left\\{W_{t}\right\\}_{t \geq 0}\). Show that if \(X\) is an \(\mathcal{F}_{T}\)-measurable random variable with \(\mathbb{E}[|X|]<\infty\) and \(\mathbb{P}^{*}\) is a probability measure equivalent to \(\mathbb{P}\), then the process $$ M_{t} \triangleq \mathbb{E}^{\mathbb{P}^{*}}\left[X \mid \mathcal{F}_{t}\right] $$ is a \(\left(\mathbb{P}^{*},\left\\{\mathcal{F}_{t}\right\\}_{0 \leq t \leq T}\right)\)-martingale.

Step-by-Step Solution

Answer

\(M_t\) is a martingale under \(\mathbb{P}^{*}\).

Step 1: Understanding the Given Components

We have a Brownian motion \(\left\{W_{t}\right\}_{t \geq 0}\) under measure \(\mathbb{P}\), with its natural filtration \(\left\{\mathcal{F}_{t}\right\}_{t \geq 0}\). The random variable \(X\) is \(\mathcal{F}_{T}\)-measurable with \(\mathbb{E}[|X|]<\infty\), meaning \(X\) depends on the information up to time \(T\). \(\mathbb{P}^{*}\) is another probability measure equivalent to \(\mathbb{P}\). We need to show that the process \(M_t = \mathbb{E}^{\mathbb{P}^{*}}[X | \mathcal{F}_t]\) forms a martingale under \(\mathbb{P}^{*}\).

Step 2: Defining the Martingale Property

A process \(M_t\) is a \(\left(\mathbb{P}^{*},\left\{\mathcal{F}_{t}\right\}_{0 \leq t \leq T}\right)\)-martingale if for all \(0 \leq s \leq t \leq T\), \(\mathbb{E}^{\mathbb{P}^{*}}[M_t | \mathcal{F}_s] = M_s\). This property essentially means that the conditional expectation of the future value, given the past, equals the present value.

Step 3: Applying Measurability and Expectations

Since \(X\) is \(\mathcal{F}_{T}\)-measurable, \(X\) is independent of any information happening after \(T\). Therefore, for any time \(t\), \(M_t = \mathbb{E}^{\mathbb{P}^{*}}[X | \mathcal{F}_t]\) involves conditioning on all \(\mathcal{F}_t\) information, which captures the dependency on the path of \(W\) up to time \(t\).

Step 4: Showing Martingale Equality

To show the martingale property, consider \(\mathbb{E}^{\mathbb{P}^{*}}[M_t | \mathcal{F}_s] = \mathbb{E}^{\mathbb{P}^{*}}[\mathbb{E}^{\mathbb{P}^{*}}[X | \mathcal{F}_t] | \mathcal{F}_s]\). By the tower property of conditional expectation, this simplifies to \(\mathbb{E}^{\mathbb{P}^{*}}[X | \mathcal{F}_s]\). Thus, \(\mathbb{E}^{\mathbb{P}^{*}}[M_t | \mathcal{F}_s] = M_s\), showing \(M_t\) is a \(\left(\mathbb{P}^{*},\left\{\mathcal{F}_{t}\right\}_{0 \leq t \leq T}\right)\)-martingale.

Key Concepts

Brownian Motion

Brownian Motion is an intriguing concept in mathematics, famous for its unpredictable path. Imagine an extremely drunk person trying to walk; their path would zigzag unpredictably. Similarly, Brownian motion, denoted here by \( \{W_t\}_{t \geq 0} \), represents random motion with continuous paths.

• This motion is critical in probability theory and finance.
• It serves as the mathematical modeling of stock prices or physical phenomena like particle diffusion.
• Brownian motion starts at zero, meaning \( W_0 = 0 \).

Each step in a Brownian path is independent of others, making it a Markov process, thus memoryless. The increments \( W_{t+s} - W_s \) are independent and normally distributed with mean zero and variance \( t \). This property allows scientists and financiers to predict future states to some extent.

Probability Measure

Understanding Probability Measure involves appreciating how we assign 'likelihood' to events in probability space. It's like using a ruler to measure weight in terms of probability mass.

• A probability measure \( \mathbb{P} \) assigns values from 0 to 1 to events in a sample space.
• Here, it's used to define how likely outcomes of a stochastic process, like a Brownian motion, may be.

In our context, there are two probability measures, \( \mathbb{P} \) and \( \mathbb{P}^{*} \). Both are equivalent, meaning they assign zero probability to the same sets of outcomes, but they might distribute probability mass differently over some events. This equivalency allows for changing perspectives on how the outcome was achieved, offering insightful interpretations.

Filtration

Filtration \( \{\mathcal{F}_{t}\}_{t \geq 0} \) is a sequence of \( \sigma \)-algebras which expand as time progresses, similar to accumulating more pieces of a puzzle as time continues.

• Each \( \mathcal{F}_{t} \) represents the information available up to time \( t \).
• In the context of Brownian Motion, it contains all events influenced by the motion up to \( t \).

Think of it as a growing database continually updated with more information over time. Filtrations are vital in defining which pieces of information are known at each point, and they play a key role in martingale theory and financial models. In simpler terms, they 'filter' past knowledge, essential for predicting future changes.

Conditional Expectation

Conditional Expectation intuitively extends expectation by computing the expected value of a random variable given known events or information. Consider it as calculating an average based on certain pre-known conditions.

• It's denoted by \( \mathbb{E}[X | \mathcal{F}_t] \), representing the expected result of \( X \) having knowledge up to time \( t \).
• In martingale settings, it helps evaluate what we anticipate an outcome to be considering past events.

Conditional expectation uses the concept of 'information up to now' to refine expectations. This ensures refined and targeted predictions, essential in areas such as risk management and decision-making, where knowing specifics ahead impacts present strategies.