Proving Quadratic Covariation Convergence To Zero In Probability

In the realm of stochastic processes, understanding the behavior of quadratic covariation is crucial, especially when dealing with processes like stochastic integrals. This article delves into the intricacies of proving that a quadratic covariation converges to zero in probability, focusing on the specific case of a stochastic integral driven by Brownian motion. We will explore the necessary conditions, the underlying theory, and a step-by-step approach to demonstrating this convergence. This topic is fundamental in areas such as financial modeling, where stochastic integrals are used to represent asset prices, and in the broader study of stochastic calculus.

Let's begin by defining the stochastic process under consideration. We are given a process X_t defined as a stochastic integral:

$$X_t = \int_0^t \sigma_s dW_s$$

where W represents a standard Brownian motion, and σ_s is a bounded predictable process. Here, predictability implies that the value of σ_s at time s is known just before time s. The boundedness condition ensures that |σ_s| ≤ K for some constant K and all s, which is crucial for controlling the behavior of the integral. Brownian motion, also known as a Wiener process, is a continuous-time stochastic process characterized by independent increments and a normal distribution. Its significance in stochastic calculus and financial modeling cannot be overstated, as it serves as the foundation for many models.

To provide a comprehensive understanding, we need to consider sampling times t_j,n. These sampling times partition the time interval [0, t] into smaller subintervals. The behavior of the process within these subintervals is key to understanding the convergence of the quadratic covariation. Specifically, we are interested in the normalized quadratic covariation, which involves summing the squares of the increments of the process over these subintervals.

The goal is to demonstrate that as the partition becomes finer (i.e., as n increases and the subintervals become smaller), this normalized quadratic covariation converges to zero in probability. This convergence is a powerful result, implying that the fluctuations of the process become increasingly small as we observe it at finer time scales. This is not just a theoretical curiosity; it has practical implications in areas like financial econometrics, where high-frequency data is used to estimate volatility and other parameters.

Furthermore, the predictable and bounded nature of σ_s plays a crucial role in this convergence. The predictability condition allows us to use tools from stochastic calculus, such as Itô's lemma, while the boundedness condition provides the necessary control over the integral. Without these conditions, the convergence result may not hold, highlighting the importance of these assumptions.

At the heart of our discussion lies the concept of quadratic covariation. To fully grasp its significance, we must first define it formally. For two stochastic processes, say X and Y, their quadratic covariation, denoted as [X, Y]_t, measures the accumulated co-movements of the processes up to time t. It quantifies how much the two processes tend to vary together over time. When X and Y are the same process, we simply refer to the quadratic variation of X, denoted as [X, X]_t or simply [X]_t.

In the context of our stochastic integral X_t, the quadratic variation [X]_t can be expressed as:

$$[X]_t = \int_0^t \sigma_s^2 ds$$

This integral represents the accumulated variance of the process X_t over time. The integrand, σ_s², reflects the instantaneous variance of the process at time s. Understanding this integral is crucial, as it provides a deterministic measure of the overall variability of X_t.

The quadratic covariation is a more general concept that applies to two different processes. If we consider two stochastic integrals,

$$X_t = \int_0^t \sigma_s dW_s$$

and

$$Y_t = \int_0^t \rho_s dW_s$$

their quadratic covariation is given by:

$$[X, Y]_t = \int_0^t \sigma_s \rho_s ds$$

This integral captures the co-movements of the two processes. If σ_s and ρ_s tend to have the same sign, the quadratic covariation will be positive, indicating that the processes tend to move in the same direction. Conversely, if they tend to have opposite signs, the quadratic covariation will be negative. If they are uncorrelated, the quadratic covariation will be close to zero.

Now, let's consider the normalized quadratic covariation. Given sampling times t_j,n that partition the interval [0, t], the normalized quadratic covariation is defined as a sum of the products of the increments of the processes over these subintervals. Specifically, it is given by:

$$\sum_{j=1}^n (X_{t_{j,n}} - X_{t_{j-1,n}})(Y_{t_{j,n}} - Y_{t_{j-1,n}})$$

The critical question we are addressing is: What happens to this sum as n approaches infinity, i.e., as the partition becomes finer and finer? The statement that the quadratic covariation converges to zero in probability means that this sum converges to zero in a probabilistic sense. In other words, the probability that this sum is far from zero becomes vanishingly small as n increases.

This convergence result is not just a theoretical nicety; it has profound implications in stochastic calculus and its applications. For instance, in financial modeling, it underlies the derivation of Itô's lemma and the pricing of financial derivatives. The ability to work with stochastic integrals and their quadratic covariations is fundamental to understanding and modeling complex stochastic systems.

Before diving into the proof, it is crucial to understand the concept of convergence in probability. In probability theory, convergence in probability is a mode of convergence that describes how a sequence of random variables approaches a limit. Unlike other modes of convergence, such as almost sure convergence or convergence in distribution, convergence in probability strikes a balance between being strong enough to be useful and weak enough to be widely applicable.

Formally, a sequence of random variables Z_n is said to converge in probability to a random variable Z if, for every ε > 0,

$$\lim_{n \to \infty} P(|Z_n - Z| > \epsilon) = 0$$

In simpler terms, this means that as n becomes large, the probability that Z_n differs from Z by more than any small amount ε becomes arbitrarily small. This captures the intuition that Z_n becomes increasingly close to Z in a probabilistic sense.

To appreciate the significance of this mode of convergence, let's contrast it with other types of convergence. Almost sure convergence, also known as convergence with probability 1, is a stronger mode of convergence. It requires that the sequence Z_n converges to Z for all outcomes except for a set of outcomes with probability zero. While almost sure convergence is a desirable property, it is often more challenging to prove than convergence in probability.

On the other hand, convergence in distribution is a weaker mode of convergence. It only requires that the cumulative distribution functions of Z_n converge to the cumulative distribution function of Z. Convergence in distribution does not imply that the random variables themselves become close; it only implies that their distributions become similar. This mode of convergence is useful in certain contexts, but it is not strong enough for many applications in stochastic calculus.

Convergence in probability occupies a middle ground. It is strong enough to imply convergence in distribution, but it is weaker than almost sure convergence. This makes it a versatile tool for analyzing the behavior of stochastic processes. In the context of our problem, showing that the quadratic covariation converges to zero in probability provides a meaningful statement about the behavior of the stochastic integral. It tells us that the fluctuations of the process, as measured by the normalized quadratic covariation, become small in a probabilistic sense as we observe the process at finer time scales.

Furthermore, convergence in probability is closely related to other important concepts in probability theory, such as the weak law of large numbers and the central limit theorem. These theorems provide conditions under which averages of random variables converge to their expected values or to a normal distribution. Convergence in probability plays a crucial role in the proofs of these theorems, highlighting its fundamental importance.

In the context of stochastic calculus, convergence in probability is often used to establish the convergence of stochastic integrals and other stochastic processes. It provides a way to make rigorous statements about the behavior of these processes, even when they involve random fluctuations. For example, Itô's lemma, a cornerstone of stochastic calculus, relies on convergence in probability arguments to establish the relationship between a function of a stochastic process and its stochastic differential.

To effectively demonstrate that the quadratic covariation converges to zero in probability, a strategic approach is essential. The proof often involves breaking down the problem into manageable steps and leveraging key properties of stochastic processes, particularly those related to Brownian motion and stochastic integrals. A common strategy involves the following steps:

1. Express the Normalized Quadratic Covariation: Begin by writing out the explicit expression for the normalized quadratic covariation. This involves summing the squares of the increments of the stochastic integral over the sampling intervals. Let's denote the normalized quadratic covariation as Q_n:

   $$Q_n = \sum_{j=1}^n (X_{t_{j,n}} - X_{t_{j-1,n}})^2$$

   where X_t is our stochastic integral, and t_j,n represents the sampling times.

2. Substitute the Stochastic Integral: Substitute the definition of X_t as a stochastic integral into the expression for Q_n. This will give us a sum of squares of stochastic integrals:

   $$Q_n = \sum_{j=1}^n \left( \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s dW_s \right)^2$$

   This step is crucial as it connects the quadratic covariation to the properties of the Brownian motion and the predictable process σ_s.

3. Compute the Expectation: Calculate the expected value of Q_n. This step often involves using Itô's isometry, a fundamental result in stochastic calculus that relates the expected value of the square of a stochastic integral to the integral of the square of the integrand. Applying Itô's isometry, we get:

   $$E[Q_n] = E\left[ \sum_{j=1}^n \left( \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s dW_s \right)^2 \right] = \sum_{j=1}^n E\left[ \left( \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s dW_s \right)^2 \right] = \sum_{j=1}^n \int_{t_{j-1,n}}^{t_{j,n}} E[\sigma_s^2] ds$$

   Since σ_s is a bounded process, E[σ_s²] is also bounded. This allows us to control the behavior of the sum.

4. Establish Convergence of the Expectation: Show that the expected value of Q_n converges to zero as n goes to infinity. This typically involves refining the partition of the time interval and using the boundedness of σ_s. If the partition becomes finer, the length of the subintervals [t_j-1,n, t_j,n] decreases, and the integral converges to zero:

   $$\lim_{n \to \infty} E[Q_n] = \lim_{n \to \infty} \sum_{j=1}^n \int_{t_{j-1,n}}^{t_{j,n}} E[\sigma_s^2] ds = 0$$

   This step demonstrates that, on average, the quadratic covariation becomes small as the partition becomes finer.

5. Control the Variance: To establish convergence in probability, it is not sufficient to show that the expectation converges to zero. We also need to control the variance of Q_n. This involves calculating E[Q_n²] and showing that it also converges to zero as n goes to infinity. This step often requires more advanced techniques from stochastic calculus, such as Itô's lemma and the martingale properties of stochastic integrals.

6. Apply Chebyshev's Inequality: Finally, apply Chebyshev's inequality to conclude that Q_n converges to zero in probability. Chebyshev's inequality provides an upper bound on the probability that a random variable deviates from its mean. In our case, it gives us:

   $$P(|Q_n| > \epsilon) \leq \frac{E[Q_n^2]}{\epsilon^2}$$

   Since we have shown that both E[Q_n] and E[Q_n²] converge to zero, this implies that P(|Q_n| > ε) converges to zero for any ε > 0, which is precisely the definition of convergence in probability.

By following this strategy, we can systematically demonstrate that the quadratic covariation of the stochastic integral converges to zero in probability. Each step builds upon the previous one, leveraging the properties of stochastic integrals and Brownian motion to reach the desired conclusion.

Itô's isometry is a cornerstone of stochastic calculus, providing a powerful tool for computing the expected value of stochastic integrals. Its application is pivotal in proving the convergence of quadratic covariation to zero in probability. To fully understand its role, let's delve into the theorem itself and how it simplifies calculations in our context.

Itô's isometry states that for a stochastic integral of the form:

$$I = \int_0^t f(s) dW_s$$

where W is a standard Brownian motion and f(s) is a predictable process satisfying

$$E\left[ \int_0^t f(s)^2 ds \right] < \infty$$

the following holds:

$$E[I^2] = E\left[ \left( \int_0^t f(s) dW_s \right)^2 \right] = E\left[ \int_0^t f(s)^2 ds \right]$$

In essence, Itô's isometry tells us that the expected value of the square of a stochastic integral is equal to the expected value of the integral of the square of the integrand. This seemingly simple result has profound implications, as it allows us to replace a stochastic expectation with a deterministic integral, greatly simplifying calculations.

Now, let's see how Itô's isometry is applied in the context of proving the convergence of quadratic covariation. Recall that we have the normalized quadratic covariation:

$$Q_n = \sum_{j=1}^n \left( \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s dW_s \right)^2$$

To compute the expected value of Q_n, we can apply Itô's isometry to each term in the sum:

$$E\left[ \left( \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s dW_s \right)^2 \right] = E\left[ \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s^2 ds \right]$$

This step is crucial because it replaces the expectation of the square of a stochastic integral with the expectation of a standard integral. Since σ_s is a predictable process, we can often compute or bound the expected value of the integral on the right-hand side.

Summing over all j, we get:

$$E[Q_n] = \sum_{j=1}^n E\left[ \int_{t_{j-1,n}}^{t_{j,n}} \sigma_s^2 ds \right] = \sum_{j=1}^n \int_{t_{j-1,n}}^{t_{j,n}} E[\sigma_s^2] ds$$

This expression is much easier to work with than the original expression for E[Q_n]. If σ_s is a bounded process, i.e., |σ_s| ≤ K for some constant K, then E[σ_s²] ≤ K². This allows us to bound the integral and show that it converges to zero as the partition becomes finer.

Furthermore, Itô's isometry is not only useful for computing the expectation of Q_n; it is also essential for controlling the variance of Q_n. To show convergence in probability, we need to demonstrate that both E[Q_n] and E[Q_n²] converge to zero. Computing E[Q_n²] is a more challenging task, but Itô's isometry provides a crucial tool for simplifying the calculations.

In summary, Itô's isometry is a fundamental result that simplifies the computation of expectations involving stochastic integrals. It plays a pivotal role in proving the convergence of quadratic covariation to zero in probability, allowing us to replace stochastic expectations with deterministic integrals and control the behavior of the process as the partition becomes finer.

In the context of proving the convergence of quadratic covariation, the assumptions of boundedness and predictability of the process σ_s are not merely technical conditions; they are essential for the proof to hold. These assumptions provide the necessary control over the stochastic integral and ensure that the quadratic covariation indeed converges to zero in probability. Let's examine the significance of each of these assumptions.

Boundedness

The boundedness condition states that there exists a constant K such that |σ_s| ≤ K for all s. This implies that the process σ_s does not explode or become infinitely large. The importance of this condition becomes clear when we consider the stochastic integral:

$$X_t = \int_0^t \sigma_s dW_s$$

The boundedness of σ_s ensures that the integrand in this integral remains well-behaved. Without this condition, the integral might not even be well-defined, or it might exhibit erratic behavior that prevents the quadratic covariation from converging to zero.

In particular, the boundedness of σ_s is crucial when we apply Itô's isometry. Recall that Itô's isometry states:

$$E\left[ \left( \int_0^t f(s) dW_s \right)^2 \right] = E\left[ \int_0^t f(s)^2 ds \right]$$

If σ_s is not bounded, the integral on the right-hand side might not be finite, invalidating the use of Itô's isometry. With the boundedness condition, we can ensure that:

$$E[\sigma_s^2] \leq K^2$$

This allows us to bound the expected value of the normalized quadratic covariation:

$$E[Q_n] = \sum_{j=1}^n \int_{t_{j-1,n}}^{t_{j,n}} E[\sigma_s^2] ds \leq K^2 \sum_{j=1}^n (t_{j,n} - t_{j-1,n}) = K^2 t$$

This bound is essential for showing that E[Q_n] converges to zero as the partition becomes finer.

Predictability

The predictability condition is a more subtle but equally important assumption. A process σ_s is said to be predictable if its value at time s is known just before time s. More formally, σ_s is predictable if it is measurable with respect to the σ-algebra generated by the process up to time s. This condition ensures that the stochastic integral is well-defined and that we can apply the tools of stochastic calculus.

The predictability of σ_s is crucial for Itô's lemma, which is often used to analyze the behavior of functions of stochastic integrals. Itô's lemma provides a way to compute the stochastic differential of a function of a stochastic process, and it relies heavily on the predictability of the integrand.

Furthermore, the predictability condition is essential for Itô's isometry itself. The isometry theorem only holds if the integrand is predictable. Without this condition, the equality between the expected value of the square of the stochastic integral and the expected value of the integral of the square of the integrand may not hold.

In the context of proving the convergence of quadratic covariation, the predictability of σ_s allows us to treat it as a