In part (v) of Exercise 5.12, we saw that when we change...

In part (v) of Exercise 5.12, we saw that when we change measures and change Brownian motions, correlations can change if the instantaneous correlations are random. This exercise shows that a change of measure without a change of Brownian motions can change correlations if the market prices of risk are random Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions under a probability measure $\widetilde{\mathbb{P}}$. Take $\theta_1(t)=0$ and $\theta_2(t)=W_1(t)$ in the multidimensional Girsanov Theorem, Theorem 5.4.1. Then $\widetilde{W}_1(t)=W_1(t)$ and $\widetilde{W}_2(t)=W_2(t)+\int_0^t W_1(u) d u$. (i) Because $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are Brownian motions under $\widetilde{\mathbb{P}}$, the equation $\widetilde{\mathbb{E}} \widetilde{W}_1(t)=\widetilde{\mathbb{E}} W_2(t)=0$ must hold for all $t \in[0, T]$. Use this equation to conclude that $$ \tilde{\mathbb{E}} W_1(t)=\tilde{\mathbb{E}} W_2(t)=0 \text { for all } t \in[0, T] . $$ (ii) From Itô's product rule, we have $$ d\left(W_1(t) W_2(t)\right)=W_1(t) d W_2(t)+W_2(t) d W_1(t) . $$ Use this equation to show that $$ \widetilde{\operatorname{Cov}}\left[W_1(T), W_2(T)\right]=\widetilde{\mathbb{E}}\left[W_1(T) W_2(T)\right]=-\frac{1}{2} T^2 . $$ This is different from $$ \operatorname{Cov}\left[W_1(T), W_2(T)\right]=\mathbb{E}\left[W_1(T) W_2(T)\right]=0 . $$

In part (v) of Exercise 5.12, we saw that when we change measures and change Brownian motions, correlations can change if the instantaneous correlations are random. This exercise shows that a change of measure without a change of Brownian motions can change correlations if the market prices of risk are random
Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions under a probability measure $\widetilde{\mathbb{P}}$. Take $\theta_1(t)=0$ and $\theta_2(t)=W_1(t)$ in the multidimensional Girsanov Theorem, Theorem 5.4.1. Then $\widetilde{W}_1(t)=W_1(t)$ and $\widetilde{W}_2(t)=W_2(t)+\int_0^t W_1(u) d u$.
(i) Because $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are Brownian motions under $\widetilde{\mathbb{P}}$, the equation $\widetilde{\mathbb{E}} \widetilde{W}_1(t)=\widetilde{\mathbb{E}} W_2(t)=0$ must hold for all $t \in[0, T]$. Use this equation to conclude that
$$
\tilde{\mathbb{E}} W_1(t)=\tilde{\mathbb{E}} W_2(t)=0 \text { for all } t \in[0, T] .
$$
(ii) From Itô's product rule, we have
$$
d\left(W_1(t) W_2(t)\right)=W_1(t) d W_2(t)+W_2(t) d W_1(t) .
$$

Use this equation to show that
$$
\widetilde{\operatorname{Cov}}\left[W_1(T), W_2(T)\right]=\widetilde{\mathbb{E}}\left[W_1(T) W_2(T)\right]=-\frac{1}{2} T^2 .
$$

This is different from
$$
\operatorname{Cov}\left[W_1(T), W_2(T)\right]=\mathbb{E}\left[W_1(T) W_2(T)\right]=0 .
$$

Key Concepts

Itô's Calculus and Product Rule

Itô's calculus is a cornerstone of stochastic analysis, providing a method to handle differential equations driven by Brownian motions. The product rule in Itô's calculus extends the classical product rule to stochastic processes, allowing for the computation of differentials involving products of such processes. This is particularly important in determining joint characteristics, such as covariance or correlation, under changes of measure.

Correlation Under Measure Change

Correlation under measure change refers to the possibility that the dependence structure between stochastic processes may be altered when transitioning to a different probability measure. When the market prices of risk are random, even if the underlying processes are uncorrelated under the original measure, the adjusted dynamics can exhibit non-zero correlations under the new measure.

Market Price of Risk

The market price of risk quantifies the extra return an investor requires for taking on additional risk. In the context of measure changes using Girsanov's theorem, a random market price of risk introduces randomness into the drift adjustments. This can affect various properties of the process under the new measure, including the correlation between different sources of uncertainty.

Change of Measure

Changing the probability measure is a key concept in stochastic calculus and financial mathematics. It allows practitioners to switch from the real-world probability measure to another measure under which the problem becomes more tractable, such as the risk-neutral measure used in derivative pricing. This change impacts the drift and sometimes even the correlation structure between processes.

Girsanov's Theorem

Girsanov's theorem provides a powerful framework for changing the probability measure in stochastic processes, particularly by adjusting the drift of a Brownian motion. This theorem is essential in transforming the dynamics of a process so that under the new measure, the process retains the martingale property, which is critical for applications like risk-neutral pricing.

Brownian Motion

Brownian motion is a fundamental stochastic process with continuous paths and independent normally distributed increments. It is widely used in various fields, particularly in financial modeling, to represent the unpredictable evolution of asset prices and other phenomena over time.

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