Problem 3 - A Course in Financial Calculus

Question

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) denote a standard Brownian motion under \(\mathbb{P}\). For a partition \(\pi\) of \([0, T]\), write \(\delta(\pi)\) for the mesh of the partition and \(0=t_{0}

Step-by-Step Solution

Answer

(a) \( \frac{T}{2} \), (b) \( \frac{1}{2}W_T^2 \).

Step 1: Understanding the Problem

We need to evaluate two limits related to partitions of a standard Brownian motion process. These limits involve the mesh of a partition, \(\delta(\pi)\), approaching zero, effectively refining the partition.

Step 2: Part (a): Calculating the First Limit

For part (a), we are required to compute \(\lim _{\delta(\pi) \rightarrow 0} \sum_{0}^{N(\pi)-1} W_{t_{j+1}}\left(W_{t_{j+1}}-W_{t_{j}}\right) \) using Itô calculus. This involves the quadratic variation of Brownian motion, \(\langle W\rangle_t = t\). Since it is known that \(\int_{0}^{T} W_s \, dW_s = \frac{1}{2}(W_T^2 - T) \) in the Itô sense, we see that the limit simplifies to \(\frac{T}{2}\).

Step 3: Part (b): Calculating the Stratonovich Integral

For part (b), we are given the definition of the Stratonovich integral in terms of a Riemann sum. The term inside the sum is \(\frac{1}{2}(W_{t_{j+1}}+W_{t_j})(W_{t_{j+1}}-W_{t_j})\). As \(\delta(\pi) \rightarrow 0\), this sum represents the Stratonovich integral: \(\int_{0}^{T} W_s \circ dW_s\). By converting Itô integrals to Stratonovich integrals, we use the relationship \(\int_{0}^{T} W_s \circ dW_s = \int_{0}^{T} W_s \, dW_s + \frac{1}{2} \times [W,W]_T\), where \(\int_{0}^{T} W_s \, dW_s = \frac{1}{2}(W_T^2 - T)\) is the Itô integral calculated before. The quadratic variation over \[0, T\] is \(T\), giving us \(\int_{0}^{T} W_s \circ dW_s = \frac{1}{2}(W_T^2)\).

Key Concepts

Itô Calculus

Itô Calculus is a mathematical framework used to analyze stochastic processes, specifically Brownian motion. It's instrumental in calculating stochastic integrals, which are integrals involving randomness. Unlike regular calculus, Itô Calculus accounts for continuous, yet non-differentiable, paths typical in Brownian motion. This unique aspect allows it to handle the unpredictability in financial markets and other stochastic systems.

One key feature of Itô Calculus is the Itô integral, often expressed as \( \int_{0}^{T} X_s \, dW_s \), where \(X_s\) is a process and \(dW_s\) is the increment of a Brownian motion. It also prominently uses

• Itô's Lemma: Offers an equation governing the change in a function of a stochastic process, analogous to the chain rule in regular calculus.
• Martingales: A concept describing a fair game, where conditional expected future values equal current values.

In our problem, Itô calculus helps calculate the quadratic variation of Brownian motion, \( \langle W \rangle_t = t \), and provides the solution to stochastic integrals by converting them through its unique rules.

By applying Itô Calculus in part (a), we're able to evaluate the limit and conclude with a result of \( \frac{T}{2} \), exemplifying the power and utility of this mathematical tool.

Stratonovich Integral

When dealing with stochastic calculus, the Stratonovich integral offers an alternative to the Itô integral. The main difference lies in the midpoint value approximation used in Riemann sums for integration. This offers the benefit of

• A more intuitive interpretation: Stratonovich integrals often resemble classical calculus more closely, as they satisfy the usual change of variables formulas without additional correction terms.
• The property of being time-symmetrical, which is advantageous in physics and engineering applications.

The Stratonovich integral is expressed as \( \int_{0}^{T} W_s \circ dW_s \) in our textbook solution. This representation uses the average of endpoints in the Riemann sum approximation \( \frac{1}{2}(W_{t_{j+1}} + W_{t_j})(W_{t_{j+1}} - W_{t_j}) \).

In part (b) of our exercise, using the relationship between Itô and Stratonovich integrals allowed us to convert and accurately compute the Stratonovich integral, resulting in \( \frac{1}{2}(W_T^2) \). This transformation involves adding half of the quadratic variation, demonstrating how these two integrals can be maneuvered to attain accurate solutions.

Quadratic Variation

Quadratic Variation is a critical concept in the realm of stochastic processes, particularly involving Brownian motion. It acts as a measure of how much a process ‘varies’ over a particular path or interval. Brownian motion’s distinct property is its quadratic variation, which is time itself. For any Brownian motion \(W_t\), the quadratic variation over the interval \([0, T]\) is \([W, W]_T = T\), an identity crucial for stochastic calculus.

In essence, quadratic variation quantifies the accumulated squared increments along the path of a stochastic process. Its characteristics include:

• Linearity: Quadratic variation of Brownian motion over non-overlapping intervals is additive.
• Non-differentability: Reflects the erratic, jagged nature of Brownian motion, resulting in non-zero quadratic variation over arbitrary partitions.

Within our solution context, quadratic variation helps tie in both Itô and Stratonovich integrals. In Itô calculus, it's directly part of the stochastic integral evaluation, adjusting for the non-linear calculation path. It's similarly significant when transforming between the two integrals, allowing us to achieve the target result in part (b) with \( \int_{0}^{T} W_s \circ dW_s = \frac{1}{2}(W_T^2) \). This showcases how critical understanding quadratic variation is for mastering stochastic calculus integration.