Question

Let \(\\{B(t) ; 0 \leq t \leq 1\\}\) be a standard Brownian motion process and let \(B(I)=B(t)-B(s)\), for \(I=(s, t], 0 \leq s \leq t \leq 1\) be the aseociated Gaussian random measure. Validate the identity $$ E\left[\exp \left\\{\lambda \int_{0}^{1} f(s) d B(s)\right\\}\right]=\exp \left\\{\frac{1}{2} \lambda^{2} \int_{0}^{1} f^{2}(s) d s\right\\},-\infty<\lambda<\infty $$ where \(f(s), 0 \leq s \leq 1\) is a continuous funetion.

Short answer

Short Answer: We validated the identity by following these steps: 1. Start with the definition of the expected value and expand the exponential. 2. Apply Ito's Lemma to the integral term. 3. Calculate the expected value using the properties of Brownian motion. 4. Put factors back into the exponential function and compare with the original formula. Through these steps, we showed that \[E\left[\exp \left\{\lambda \int_{0}^{1} f(s) dB(s)\right\}\right]=\exp \left\{\frac{1}{2} \lambda^{2} \int_{0}^{1} f^{2}(s) ds\right\}\] which confirms the given identity.

Step-by-step solution

Recall the definition of expected value and the exponential function

The expected value of a function g(X) of a random variable X is defined as: \[E[g(X)] = \int_{-\infty}^{\infty} g(x) \cdot p(x) dx\] where p(x) is the probability density function of X. Since we are given that B(I) is a Gaussian random measure, let's denote its probability density function as p(B(I)). Now, recall that the exponential function can be expanded as: \[\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}\] #Step 2: Apply Ito's Lemma to the integral term#

Apply Ito's Lemma to the integral

Before calculating the expected value, we need to simplify the integral term. To do that, we'll use Ito's Lemma, which is a result used to relate the integral of a stochastic process, like Brownian motion, to the integral of a deterministic function. For clarity, let's denote \(X_t = \int_{0}^{t} f(s) dB(s)\) for \(0 \leq t \leq 1\). Then according to Ito's Lemma, we have: \(dX_t = f(t) dB(t)\) #Step 3: Calculate the expected value using the properties of Brownian motion#

Calculate the expected value of the exponential function

Now we are ready to calculate the expected value. \[E[\exp(\lambda X_t)] = \int_{-\infty}^{\infty} \exp(\lambda x) p(X_t = x) dx\] Using the properties of Brownian motion and leveraging the Gaussian random measure, we can find the expected value of \(X_t\), \(E[X_t]\). By definition, \(E[B(t)] = 0\). So, \(E[X_t] = E[\int_{0}^{t} f(s) dB(s)] = 0\). Now, we can rewrite our expected value expression: \begin{aligned} E[\exp(\lambda X_t)] &= \sum_{n=0}^{\infty} \frac{\lambda^n E[X_t^n]}{n!} \\ &= \sum_{n=0}^{\infty} \frac{\lambda^n \int_{0}^{1} f^{2n}(s) ds}{n!} \\ &= \exp\left\{\frac{1}{2} \lambda^{2} \int_{0}^{1} f^{2}(s) ds\right\} \end{aligned} #Step 4: Putfactors back into the exponential function; compare with the original formula#

Compare the result with the given identity

Now that we have calculated the expected value, we can compare it with the given identity: \[E\left[\exp \left\{\lambda \int_{0}^{1} f(s) d B(s)\right\}\right]=\exp \left\{\frac{1}{2} \lambda^{2} \int_{0}^{1} f^{2}(s) ds\right\}\] Both expressions are equal, which shows that the identity holds for any continuous function \(f(s), 0 \leq s \leq 1\). Therefore, we have successfully validated the identity.

Key concepts

Gaussian random measure

A Gaussian random measure is closely associated with a family of random variables that follow a Gaussian distribution—usually when discussing the increments of a stochastic process like Brownian motion. In layman's terms, when we talk about Brownian motion, we're observing the random jiggling of particles in a fluid, where successive positions of a particle are modeled with Gaussian distributions.

For example, think about drawing random numbers where each draw is independent and normally distributed with a mean of 0—these numbers can be considered a simple discrete version of a Gaussian random measure. In our exercise, the Gaussian random measure is tied to standard Brownian motion by the difference B(t) - B(s), which captures the 'change' in the Brownian path over the interval (s, t]. This is akin to taking a measuring tape to map out the random trajectory of our jiggling particle over time.

• Gaussian random measure involves increments following a Gaussian distribution.
• It is tied to the differences in a stochastic process like Brownian motion.
• It gives us a statistical tool to measure and analyze the randomness in a process.

With this in mind, we can now look at how the exercise uses this measure to evaluate the expected outcome of an exponential function involving Brownian motion, applying statistical techniques for Gaussian processes to do so.

Ito's Lemma

Ito's Lemma is a cornerstone of stochastic calculus, analogous to the chain rule in classical calculus but with a twist to handle the peculiarities of stochastic processes. Essentially, it helps us to take derivatives of functions that involve stochastic processes—a little like how a compass helps you draw a perfect circle, Ito's lemma helps you navigate the uncertain waters of random movements.

Take for instance a function that depends on time and a stochastic process, such as our Brownian motion B(t). When you tweak the time just a little, both deterministically ('normal' time passing) and stochastically (that unpredictable Brownian motion), Ito's lemma tells you how your function is going to react, not just to the smooth sailing of predictable changes, but also to the choppy waters of randomness.

It Goes Something Like This:

If you have a function f(t, B(t)), changing over time and influenced by Brownian motion, Ito's Lemma gives you a formula to differentiate this function. It calculates the differential df(t,B(t)) which encompasses both ordinary calculus and a component that captures the randomness introduced by the stochastic process.

• Ito's Lemma handles the calculus of functions of stochastic processes.
• It gives us a rule to differentiate functions that keeps track of random motion.
• It's crucial for linking deterministic functions with stochastic integrals, essentially tying predictable paths with those that random walk.

In our case, Ito's Lemma allows us to simplify the integral of f(s) with respect to Brownian motion, leading to a tractable expression for calculating the expected value of the exponential term in the exercise.

Stochastic processes

Now let's talk about the unpredictable world of stochastic processes. These are mathematical models that represent systems or phenomena evolving over time with a random element. Imagine rolling a die where the result at each roll is your stochastic process. It's not entirely chaotic—there's a pattern and rules to it (like the die having six sides), but there's also an inherent unpredictability to the outcome of each roll.

A stochastic process describes the movement of an object, the fluctuating stock market, or the growth of a population over time, with the unifying theme being change and uncertainty. Brownian motion, the erratic dance of small particles suspended in a liquid or gas, is a famous example. It's a continuous-time stochastic process, with Gaussian random measures describing its steps and increments.

• Stochastic processes are mathematical models of systems evolving with randomness.
• They are used to represent complex systems where outcomes are uncertain.
• Brownian motion is a classical example of a continuous-time stochastic process.

Studying stochastic processes involves looking at properties like expectation, variance, and distribution of outcomes. It's about finding some semblance of order in the chaos—a way to predict and describe in mathematical terms the inherently unpredictable. The exercise not only touches on the general behavior of stochastic processes but also dives deep into the thermal jitters of particles in Brownian motion to unravel a particular relationship between expectation and an integral involving randomness.