Problem 4 - A Course in Financial Calculus

Question

Let \(\left\\{W_{t}^{i}\right\\}_{t \geq 0}, i=1, \ldots, n\), be independent Brownian motions. Show that \(\left\\{R_{t}\right\\}_{t \geq 0}\) defined by $$ R_{t}=\sqrt{\sum_{i=1}^{n}\left(W_{t}^{i}\right)^{2}} $$ satisfies a stochastic differential equation. The process \(\left\\{R_{t}\right\\}_{t \geq 0}\) is the radial part of Brownian motion in \(\mathbb{R}^{n}\) and is known as the \(n\)-dimensional Bessel process.

Step-by-Step Solution

Answer

\( R_t \) satisfies the SDE: \[ dR_t = \frac{1}{R_t} \sum_{i=1}^{n} W_t^i dW_t^i + \frac{n - 1}{2R_t} dt \].

Step 1: Define the Process

Consider each \( W_t^i \) as an independent Brownian motion for \( i = 1, \ldots,\ n \). The process \( R_t \) is defined as \( R_{t}=\sqrt{\sum_{i=1}^{n}\left(W_{t}^{i}\right)^{2}} \), which represents the distance from the origin for the Brownian path in \( \mathbb{R}^{n} \).

Step 2: Apply Itô's Formula

We apply Itô's formula to \( R_t^2 = \sum_{i=1}^{n} (W_t^i)^2 \). For each \( W_t^i \), the Itô formula gives \( d(W_t^i)^2 = 2W_t^i dW_t^i + (dW_t^i)^2 = 2W_t^i dW_t^i + dt \). Therefore, \[ d(R_t^2) = \sum_{i=1}^{n} 2W_t^i dW_t^i + n \, dt \].

Step 3: Differentiate \( R_t = \sqrt{R_t^2} \)

To find \( dR_t \), use the chain rule taking the differential of \( R_t = \sqrt{R_t^2} \), yielding \[ dR_t = \frac{1}{2\sqrt{R_t^2}} d(R_t^2) = \frac{1}{R_t} \sum_{i=1}^{n} W_t^i dW_t^i + \frac{n}{2 R_t} dt \].

Step 4: Simplify the Stochastic Differential Equation (SDE)

The term \( \sum_{i=1}^{n} W_t^i dW_t^i \) can be expressed directly as part of the stochastic term. Since \( dW_t^i \) are orthogonal, this simplifies the stochastic integral to a single term. Thus, the SDE for \( R_t \) becomes \[ dR_t = \frac{1}{R_t} \sum_{i=1}^{n} W_t^i dW_t^i + \frac{n - 1}{2R_t} dt \]. Converting this to an SDE for \( R_t \), we recognize it as the form of the \( n \)-dimensional Bessel process.

Key Concepts

Brownian Motion

The concept of Brownian Motion is foundational in understanding stochastic differential equations. Imagine a particle moving randomly in space, where at any point in time, its next move is unpredictable. This random motion, discovered by botanist Robert Brown, is termed as Brownian Motion.
In mathematical terms, Brownian Motion is a continuous-time stochastic process, indexed by time, typically represented as \( \{W_t\} \), where \( t \ge 0 \). It has several key properties:

• Independence: The increments are independent from each other.
• Stationary Increments: The process has increments \( W_{t+s} - W_s \) that are normally distributed with mean zero and variance \( t \).
• Continuous Paths: The paths of \( \{W_t\} \) are continuous; there are no sudden jumps.

Because of these properties, Brownian Motion serves as a fundamental building block for modeling randomness in finance, physics, and other fields.

Bessel Process

The Bessel process is a specific type of stochastic process that can be characterized by its association with the radial part of Brownian Motion. When dealing with Brownian Motion in an \( n \)-dimensional space, we often focus on the distance of a particle from the origin.
This distance is formally the \( n \)-dimensional Bessel process, noted as \( \{R_t\}_{t \ge 0} \). It's defined by \[ R_{t} = \sqrt{\sum_{i=1}^{n} (W_t^i)^2} \] where each \( W_t^i \) is an independent Brownian motion. The Bessel process describes how this distance evolves over time. It is particularly important in areas where the magnitude of a vector, rather than its direction, is of interest.

• Bessel processes are used in various fields like finance to model stock prices and in physics to represent diffusion processes.
• For \( n = 1 \), it is similar to the absolute value of a one-dimensional Brownian motion.
• As \( n \) increases, the behavior of the process changes, with different implications for stability and behavior at large \( t \).

Itô's Formula

Itô's Formula is a powerful tool in stochastic calculus, similar to the chain rule in standard calculus but applicable to stochastic processes. It is used to find the differential of a function of a stochastic process, which is crucial when dealing with equations involving Brownian Motion.
Consider a process \( X_t \) driven by Brownian Motion \( W_t \). If \( f \) is a twice continuously differentiable function, Itô's formula helps us represent \( f(X_t) \) by
\[ df(X_t) = \frac{\partial f}{\partial x}(X_t) \, dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}(X_t) \, d\langle X \rangle_t \] This formula is crucial because it accounts for the randomness and variance inherent in each infinitesimal step \( dX_t \) caused by Brownian Motion. In our context, Itô’s formula allowed us to express the dynamics of \( R_t^2 =\sum_{i=1}^{n} (W_t^i)^2 \), which then helps derive the stochastic differential equation (SDE) for the Bessel process.

Radial Part of Brownian Motion

The radial part of Brownian Motion refers to the distance of the Brownian path from the origin in a multi-dimensional space. In essence, it focuses on how far a particle, which moves randomly according to Brownian Motion, is from a fixed point over time.
Mathematically, for Brownian motions \( \{W_t^i\} \) in \( \mathbb{R}^n \), the radial part is represented by \( R_t = \sqrt{\sum_{i=1}^{n} (W_t^i)^2} \). This provides an essential way to study the spatial dynamics of stochastic processes.

• The radial part is crucial in translating complex multi-dimensional motion into a more easily manageable one-dimensional process.
• It’s widely applicable in fields concerning diffusion processes which involve particle movement over time from a source point.
• Understanding the radial part assists in solving equations related to heat distribution, finance models, and other scientific phenomena where spherical symmetry is involved.

By recognizing the radial part as a Bessel process, we uncover a robust framework for analyzing how such systems evolve under random influences.