1. Itô Integrals with Respect to Brownian Motion

Stochastic integrals with respect to Brownian motion ³⁴ are fundamental objects in stochastic calculus, enabling the rigorous treatment of stochastic differential equations and random processes in continuous time. Unlike ordinary calculus, where integration is performed with respect to smooth functions, stochastic integration involves integrating with respect to Brownian motion, a process with infinite variation (but finite quadratic variation) and nowhere-differentiable paths.

Again as any other posts, this one is for recalling the step-by-step process to deal with Itô integrals.

1.1 The Itô Integral: Construction and Definition

1.1.1 Mathematical Framework

Let \(W(t)\) be a standard Brownian motion defined on a filtered probability space \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})\). We're looking for how to solve/analyze the integral:

where \(f(u)\) is either a deterministic function of time or an adapted stochastic process (i.e., \(f(u)\) is \(\mathcal{F}_u\)-measurable for each \(u\)).

1.1.2 Construction via Partition and Approximation

The Itô integral is defined as the limit of Riemann-like sums constructed through partitioning. Divide the interval \([s, t]\) into \(n\) subintervals:

with mesh size \(\Delta t_i = t_{i+1} - t_i\) for \(i = 0, 1, \ldots, n-1\). Then, the Itô integral is approximated by:

where:

• \(f(t_i)\) is the value of the integrand at the left endpoint of the interval \([t_i, t_{i+1}]\)
• \(W(t_{i+1}) - W(t_i)\) is the Brownian motion increment over that interval
• The limit is taken in the mean-square sense as the mesh size \(\max_i \Delta t_i \to 0\)

1.1.3 Integrability Conditions

For the integral \(\ref{eq:riemann_sum_approximation}\) to be well-defined, some conditions are required, like:

The square-integrability condition in stochastic integrability requirement, ensures that the mean-square limit in \(\ref{eq:riemann_sum_approximation}\) exists. This is weaker than requiring pathwise convergence and is the natural framework for stochastic integration.

1.1.4 Example: Exponential Integrand

For the specific case \(f(u) = e^{au}\) where \(a\) is a constant:

This integral appears frequently in the solutions to linear stochastic differential equations, including the Vasicek model.

1.2 Expected Value: The Martingale Property

1.2.1 Statement

One of the most important properties of Itô integrals is that they are martingales with zero conditional expectation:

for all \(0 \leq s < t\) and any adapted integrand \(f(u)\) satisfying the integrability conditions. This fundamental result tells us that stochastic integrals represent pure randomness with no systematic drift, as they average to zero when conditioned on past information.

1.2.2 Step-by-Step Proof for Deterministic Integrands

We will prove equation \(\ref{eq:ito_integral_zero_expectation}\) for the exponential case:

We compute the conditional expectation of the approximating sum as explained above in \(\ref{eq:riemann_sum_approximation}\).

Using the linearity of conditional expectation, we can say that

Subsequently, we can apply the tower property (law of iterated expectations):

Now, since \(e^{at_i}\) is deterministic (depends only on time, not randomness) and \(t_i\) is known at time \(t_i\), we can factor it out of the inner expectation:

The key property of Brownian motion is that increments are independent of the past. The increment \(W(t_{i+1}) - W(t_i)\) is:

• Independent of all information available at time \(t_i\) (i.e., independent of \(\mathcal{F}_{t_i}\))
• Normally distributed: \(W(t_{i+1}) - W(t_i) \sim \mathcal{N}(0, t_{i+1} - t_i)\)

Therefore:

Eventually, and substituting back:

Since every term in the sum \(\ref{eq:linearity_of_expectation}\) equals zero:

As the partition becomes finer (mesh size \(\to 0\)), the Riemann sum converges to the Itô integral:

1.2.3 Generalization to Arbitrary Integrands

The proof above generalizes to any deterministic function \(f(u)\) satisfying the integrability condition:

Moreover, for adapted stochastic processes \(H(u)\) (where \(H(u)\) is \(\mathcal{F}_u\)-measurable for each \(u\)), the same result holds under the square-integrability condition:

1.3 Variance: Itô's Isometry

While the martingale property tells us that the mean of an Itô integral is zero, we also need to characterize its variance to fully understand the distribution of stochastic integrals.

1.3.1 Itô's Isometry Theorem

For any deterministic function \(f(u)\) satisfying \(\int_s^t f(u)^2\,du < \infty\):

More generally, for the conditional variance ¹:

1.3.2 Proof Sketch for Itô's Isometry

First, let's take the approximation using riemman sum.

that, when expanding the 2nd-power we get where \(\Delta W_i = W(t_{i+1}) - W(t_i)\).

The cross terms vanish because increments at different times are independent:

Then, remaining terms are called diagonal terms, contribute with ²:

Then summing up and using the limit, we can say

as the mesh size goes to zero.

1.3.3 Example: Exponential Integrand

For \(f(u) = e^{au}\):

This variance formula in \(\ref{eq:exponential_variance}\) is crucial for computing the variance of the short rate in the Vasicek model.

1.4 Complete Distributional Characterization

Combining the zero mean property \(\ref{eq:ito_integral_zero_expectation}\) with the variance formula \(\ref{eq:ito_isometry_variance}\), when \(f(u)\) is deterministic:

This Gaussian distribution is fundamental to stochastic calculus:

• The integral is normally distributed
• Mean is zero
• Variance is the integral of the squared integrand

1.5 Connection to Other Results

• Brownian Motion: The integrator in Itô integrals; its independent increments and zero mean properties are essential

• Itô's Isometry (Full Treatment): A more detailed exposition of the variance formula and its applications

• Vasicek Model: Practical application showing how to compute both mean and variance using these properties