Question

The object of this exercise is to give an argument for the formal identity $$ d W_{1} \cdot d W_{2}-0 $$ when \(W_{1}\) and \(W_{2}\) are independent Wiener processes. Let us therefore fix a time \(t\), and divide the interval \([0, t]\) into equidistant points \(0=t_{0}

Short answer

Answer: The significance of showing that the formal identity \(dW_1 \cdot dW_2 = 0\) holds for two independent Wiener processes is that it confirms the orthogonality between the two processes. It means that the changes or increments of one Wiener process have no correlation or influence on the changes or increments of the other process. This is important in modeling and simulating systems with multiple independent noise sources, such as in finance, physics, and engineering applications.

Step-by-step solution

Calculate the expectation of \(Q_n\)

First, we need to show that the expectation of \(Q_n\) is zero. The expectation of \(Q_n\) can be written as: $$ E[Q_n] = E\left[\sum_{k=1}^{n}\Delta W_1(t_k) \cdot \Delta W_2(t_k)\right] $$ Since \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) are independent, we have: $$ E[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] = E[\Delta W_1(t_k)] \cdot E[\Delta W_2(t_k)] $$ The expectations of \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) are both \(0\) since the increments of Wiener processes have zero mean. Therefore, the expectation of their product is also zero: $$ E[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] = 0 \cdot 0 = 0 $$ Finally, we have: $$ E[Q_n] = \sum_{k=1}^{n} E[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] = \sum_{k=1}^{n} 0 = 0 $$ So, the expectation of \(Q_n\) is indeed \(0\).

Calculate the variance of \(Q_n\)

Now, we need to show that the variance of \(Q_n\) goes to \(0\). Recall that the variance of a sum of independent random variables is equal to the sum of the variances of the individual variables. Thus, the variance of \(Q_n\) can be written as: $$ Var[Q_n] = Var\left[\sum_{k=1}^{n}\Delta W_1(t_k) \cdot \Delta W_2(t_k)\right] = \sum_{k=1}^{n} Var[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] $$ Since \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) are independent, the variance of their product can be written as: $$ Var[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] = E[\Delta W_1^2(t_k)] \cdot E[\Delta W_2^2(t_k)] - E^2[\Delta W_1(t_k)] \cdot E^2[\Delta W_2(t_k)] $$ As mentioned earlier, the expectations of \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) are both \(0\). Therefore, the second term in the expression above is zero. In addition, since \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) are increments of Wiener processes, we know that their variances are equal to the length of the interval they cover, which in this case is \(\frac{t}{n}\): $$ Var[\Delta W_1(t_k)] = Var[\Delta W_2(t_k)] = \frac{t}{n} $$ The first term in the expression for the product's variance can now be calculated as: $$ E[\Delta W_1^2(t_k)] \cdot E[\Delta W_2^2(t_k)] = \frac{t}{n} \cdot \frac{t}{n} = \frac{t^2}{n^2} $$ Therefore, the variance of the product \(\Delta W_1(t_k) \cdot \Delta W_2(t_k)\) is equal to \(\frac{t^2}{n^2}\). Finally, we have: $$ Var[Q_n] = \sum_{k=1}^{n} Var[\Delta W_1(t_k) \cdot \Delta W_2(t_k)] = \sum_{k=1}^{n} \frac{t^2}{n^2} = n \cdot \frac{t^2}{n^2} = \frac{t^2}{n} $$ As \(n \rightarrow \infty\), we have \(\frac{t^2}{n} \rightarrow 0\), so the variance of \(Q_n\) indeed goes to \(0\).

Conclusion

In conclusion, we have shown that \(Q_n\), the sum of products of increments of two independent Wiener processes, has an expectation of \(0\) and a variance that goes to \(0\) as \(n \rightarrow \infty\). This implies that \(Q_n\) converges to \(0\) in the \(L^2\) sense, which is the desired result to verify the formal identity \(dW_1 \cdot dW_2 = 0\).

Key concepts

Independence of Wiener processes

When discussing the properties of Wiener processes, also known as Brownian motion, independence takes a central stage. Wiener processes are a mathematical model used to describe the seemingly random movement of particles suspended in a fluid, but they also serve as a cornerstone in financial mathematics and other applied fields.

Independence between two Wiener processes means that the movements of one process do not affect or provide any information about the movements of the other. This concept is akin to flipping two coins; the result of one flip has no bearing on the result of the other. In the context of the given exercise, we consider two independent Wiener processes, represented by \(W_{1}\) and \(W_{2}\).

The product of differentials of two independent Wiener processes is expected to be zero. This is because the increments of each Wiener process have a mean of zero and are independent over non-overlapping intervals. The exercise demonstrates this by analyzing the sum of products of increments of these two processes. As we have shown in the solution, the expectation of this sum, denoted by \(Q_n\), is zero, reinforcing the independence of the Wiener processes at hand.

The implication here is profound. When modeling phenomena with multiple sources of uncertainty, their independence allows us to consider each source on its own merits without worrying about cross-correlations. This assumption of independence is a pillar of many models in stochastic calculus and is fundamental when dealing with complex systems.

Convergence in L2

Convergence in \(L^2\) is a type of convergence in probability theory that is particularly relevant when working with Wiener processes. It is based on the idea of mean square convergence, a stronger form of convergence compared to convergence in probability or almost surely.

In simple terms, a sequence of random variables converges in the \(L^2\) sense to a random variable if the expected value of their squared differences tends to zero. In our exercise, we are interested in showing that the sequence \(Q_n\) converges in \(L^2\) to zero.

This kind of convergence is significant because it implies not only convergence in probability but also that the second moments (the variances) of the random variables are becoming stable. It's analogous to a group of archers becoming better over time: not only are their arrows hitting closer to the target on average (convergence in mean), but the spread of their arrows is also getting tighter (stabilizing variance).

The solution steps clarify how both the expectation and the variance of \(Q_n\) approach zero as \(n\) grows. This demonstrates \(L^2\) convergence and cements the argument that the product of the differentials of two independent Wiener processes converges to zero in the mean square sense as more subdivisions are considered over a given interval.

Increments of Wiener processes

Increments of Wiener processes are foundational to understanding the behavior of these stochastic processes. An increment of a Wiener process over an interval is the difference in the process's value at the end and start of the interval.

One key property is that increments over non-overlapping intervals are independent and normally distributed with a mean of zero and variance equal to the interval's length. This property aligns with the process's continuous paths and stationary increments, which is central to defining the process. In the step-by-step solution of our exercise, the increments are denoted by \(\Delta W_1(t_k)\) and \(\Delta W_2(t_k)\) and are crucial to deriving the expectation and variance of the sequence \(Q_n\).

The increments' independence and distribution are used to calculate \(E[Q_n]\) and \(Var[Q_n]\), supporting the result that \(Q_n\) converges to zero in \(L^2\). This characteristic is not just a mathematical curiosity; it has practical implications in fields such as finance, where Wiener process increments underpin the modeling of asset prices in the Black-Scholes model for options pricing. Understanding the nature of these increments allows us to better comprehend and predict the behavior of various phenomena modeled by Wiener processes.