Show that when w=h / g and h and g are each dependent on n...

Show that when $w=h / g$ and $h$ and $g$ are each dependent on $n$ Wiener processes, the $i$ th component of the volatility of $w$ is the $i$ th component of the volatility of $h$ minus the $i$ th component of the volatility of $g$. Use this to prove the result that if $\sigma_{U}$ is the volatility of $U$ and $\sigma_{V}$ is the volatility of $V$ then the volatility of $U / V$ is $\sqrt{\sigma_{U}^{2}+\sigma_{V}^{2}-2 \rho \sigma_{U} \sigma_{V}}$. (Hint: Use the result in Footnote 7.)

Key Concepts

Covariance and Correlation in SDEs

When combining or comparing stochastic processes, covariance measures the degree to which two processes move together, while correlation normalizes this measure to reflect the strength and direction of their linear relationship. In assessing the volatility of a ratio of two processes, the interaction between their volatilities is quantified by a term involving the product of their individual volatility components and their correlation, which adjusts the overall uncertainty in the resulting process.

Itô's Lemma

Itô's lemma is a cornerstone of stochastic calculus that generalizes the chain rule to functions of stochastic processes. By accommodating the non-negligible quadratic variation present in Wiener processes, it allows for the precise calculation of the differential of a function—such as a ratio of two stochastic processes—by incorporating both the drift and diffusion components arising from the individual processes.

Quotient Rule in Stochastic Calculus

When dealing with the ratio of two stochastic processes, the differential must account for the contributions from both the numerator and the denominator via a modified quotient rule derived from Itô's lemma. This rule illustrates that the volatility of the ratio can be expressed as the difference between the volatilities of the numerator and denominator in each component, with the overall variance emerging from the squared differences and their correlation.

Wiener Process

A Wiener process, or Brownian motion, is a continuous-time stochastic process characterized by having independent, normally distributed increments and continuous paths. It serves as a fundamental building block in stochastic modeling, particularly in the design of stochastic differential equations (SDEs) that describe random shocks or fluctuations in various fields such as finance and physics.

Volatility in Stochastic Processes

Volatility in the context of stochastic processes represents the instantaneous standard deviation or measure of dispersion of the process’s increments. It is typically defined as the coefficient of the stochastic term in an SDE, indicating how much uncertainty or variation is introduced by the underlying random shocks driven by Wiener processes.

Transcript

00:01 Now here on this problem, we are told that we have a random variable y with a moment generating function m of t.

00:09 We're also told that we have u that's equal to a, y, plus b.

00:15 And we'd like to find its moment generating function and then it's expected value.

00:21 Now, the moment generating function for you is equal to the moment generating function of ay plus b, which means that this is the expected value of e to the ay plus b.

00:45 T.

00:47 E to the a .y plus b .t.

00:49 I use our properties of exponents here.

00:51 This is equal to the expected value of e to the ayt times e to the bt.

01:04 Now since e to the bt is just a constant, this means this is e to the bt times the expected value of.

01:12 Of e to the ayt.

01:18 Now e to the ayt is just the moment generating function of y evaluated at a .t.

01:26 And so the moment generating function of u is equal to e to the bt as a moment generating function of y at a .t.

01:34 And that's what we wanted to show here.

01:37 Now we want to use this to find the variance and the expected value for you.

01:45 Now the expected value of you is equal to the moment generating function.

01:49 Of u's derivative at zero.

01:54 First, let's find our derivative here.

01:58 Using the product rule, this is equal to b, e to the b, t, times m of a, t, plus a, m prime of a t, e to the b t.

02:17 And so m, u prime of zero is b, e to the zero, m of zero, plus a, m prime of zero, plus a, m prime of zero, e to the zero.

02:37 E to the 0 and m of 0 are both 1, and so that term there just becomes b.

02:42 M prime of 0 is the expected value of y.

02:47 And so this becomes a times mu, because mu is the expected value of y.

02:55 And so the expected value of u is b plus a mu.

03:01 That's the expected value of mu.

03:04 Now for the variance, we need the second derivative of you.

03:09 And so we take the derivative of the first derivative...

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