Question

A function \(b\left(x_{1}, \ldots, x\right)\) is said to be harmonic if it satisfies the condition $$ \sum_{i=1}^{n} \frac{\partial^{2} h}{\partial x_{i}^{2}}=0 $$ It is subharmonic if it satisfies the condition $$ \sum_{i=1}^{n} \frac{\partial^{2} h}{\partial x_{i}^{2}}>0 $$ Let \(W_{1}, \ldots, W\), be independent standard Wiener processes, and define the process \(X\) by \(X(t)=b\left(W_{1}(t), \ldots, W_{n}(t)\right) .\) Show that \(X\) is a martingale (submartingale) if \(b\) is harmonic (subharmonic).

Short answer

Question: Show that the process \(X(t)\) is a martingale or submartingale if the function \(b\) is harmonic or subharmonic respectively. Answer: We applied Ito's lemma to \(X(t)\) and then calculated the conditional expectation. If \(b\) is harmonic, the conditional expectation of \(dX(t)\) given the information up to time \(t\) is zero, making \(X(t)\) a martingale. If \(b\) is subharmonic, the conditional expectation of \(dX(t)\) given the information up to time \(t\) is greater than or equal to zero, making \(X(t)\) a submartingale.

Step-by-step solution

Apply Ito's lemma to X(t)

First, let's apply Ito's lemma to \(X(t) = b(W_1(t), \ldots, W_n(t))\). Ito's lemma states that if \(X(t) = f(W_1(t), \ldots, W_n(t))\) where \(W_i(t)\) are independent standard Wiener processes, then: $$ dX(t) = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} dW_i(t) + \frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 f}{\partial x_i^2} dt. $$ Plugging in \(X(t) = b(W_1(t), \ldots, W_n(t))\): $$ dX(t) = \sum_{i=1}^{n} \frac{\partial b}{\partial x_i} dW_i(t) + \frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} dt. $$

Calculate the conditional expectation

Now, we'll calculate the conditional expectation \(\mathbb{E}[dX(t) \mid \mathcal{F}_{t}]\). Remember that the Wiener process increments \(dW_i(t)\) have mean zero and are independent of the information up to time \(t\), \(\mathcal{F}_t\). So, their conditional expectation is zero: $$ \mathbb{E}\left[\sum_{i=1}^{n} \frac{\partial b}{\partial x_i} dW_i(t) \mid \mathcal{F}_{t}\right] = 0. $$ The remaining term in the total derivative of \(X(t)\) is: $$ \frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} dt. $$ Now, let's consider the two cases: (a) when \(b\) is harmonic and (b) when \(b\) is subharmonic.

Step 3a: Case when b is harmonic

If \(b\) is harmonic, by definition: $$ \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} = 0. $$ Taking the conditional expectation of this equality given \(\mathcal{F}_{t}\), we get: $$ \mathbb{E}[dX(t) \mid \mathcal{F}_{t}] = \mathbb{E}\left[\sum_{i=1}^{n} \frac{\partial b}{\partial x_i} dW_i(t) \mid \mathcal{F}_{t}\right] + \mathbb{E}\left[\frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} dt \mid \mathcal{F}_{t}\right] = 0, $$ which implies that \(X(t)\) is a martingale.

Step 3b: Case when b is subharmonic

If \(b\) is subharmonic, by definition: $$ \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} > 0. $$ Taking the conditional expectation of this inequality given \(\mathcal{F}_{t}\), we get: $$ \mathbb{E}[dX(t) \mid \mathcal{F}_{t}] = \mathbb{E}\left[\sum_{i=1}^{n} \frac{\partial b}{\partial x_i} dW_i(t) \mid \mathcal{F}_{t}\right] + \mathbb{E}\left[\frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 b}{\partial x_i^2} dt \mid \mathcal{F}_{t}\right] \geq 0, $$ which implies that \(X(t)\) is a submartingale. In conclusion, we've shown that if \(b\) is harmonic, the process \(X(t)\) is a martingale, and if \(b\) is subharmonic, the process \(X(t)\) is a submartingale.

Key concepts

Harmonic Function

The term 'harmonic function' may sound like it relates to music, but in mathematics, it has a quite different meaning. A harmonic function refers to a solution to Laplace's equation, which in multiple dimensions is represented as the sum of second partial derivatives equaling zero, \( \sum_{i=1}^{n} \frac{\partial^{2} h}{\partial x_{i}^{2}}=0 \). This condition suggests a sort of balance, as if the function is 'evenly spread out' over the space it occupies, with no 'net curvature' at any point.

Harmonic functions have some fascinating properties. They are infinitely differentiable, and they also exhibit the mean value property, which means that the value of a harmonic function at any point is the average of its values over any spherical surface centered at that point. Imagine a smooth landscape where every hill and valley averages out in the immediate vicinity - that's what a harmonic function sorts of like, but in possibly higher dimensions.

In the context of the given exercise, a harmonic function is associated with the potential for a process to be a martingale, which we'll examine in more detail when discussing stochastic calculus.

Ito's Lemma

Ito's lemma is fundamental to stochastic calculus — it's sort of like the chain rule from calculus, but for stochastic processes. The lemma provides a way to compute the differential of a function of a Wiener process, which is a continuous-time stochastic process also known as Brownian motion.

According to Ito's lemma, if you have a function \( f \) applied to a Wiener process, then the differential of this function is a combination of partial derivatives and increments of the Wiener process along with a correction term due to the stochastic nature of the increments. In the exercise, this lemma is used to transform the function \( b \) of Wiener processes into a form where we can evaluate its expected change over time and thus determine whether the process \( X(t) \) is a martingale or submartingale.

Wiener Process

The Wiener process is a mathematical model for random movement, named after the American mathematician Norbert Wiener. In finance and physics, it's known for modeling stock price movements or particle diffusion. If you imagine an extremely unpredictable, continuously moving particle that doesn't seem to have any particular direction or speed, that's what the Wiener process looks like statistically.

A key characteristic of the Wiener process is its increments, which are independent and normally distributed with a mean of zero. This means that the future path of the process doesn't depend on its history — essentially, it has no memory. In the step-by-step solution, noting that a Wiener process has mean zero increments is crucial for proving that the conditional expectation is zero, which links the nature of the Wiener process directly to the concept of a martingale.

Subharmonic Function

While 'subharmonic' might seem to suggest a weaker version of a harmonic function, in mathematics, it actually has a distinct definition. Subharmonic functions satisfy a condition where the sum of their second partial derivatives is strictly positive, \( \sum_{i=1}^{n} \frac{\partial^{2} h}{\partial x_{i}^{2}}>0 \). These are functions that have no local maxima -- intuitively, there are no 'peaks' in this landscape, and it can only plateau or dip down into valleys.

Subharmonic functions come into play when looking at processes that might, on average, drift upwards over time instead of remaining flat. This characteristic aligns with the concept of a submartingale in stochastic processes, which we encountered in the exercise as having a non-negative expected increase, epitomizing a tendency to rise over time.

Stochastic Calculus

Stochastic calculus is the branch of mathematics that deals with calculus methods applied to stochastic processes — in other words, it's how we do calculus when randomness is involved. This field is incredibly important in finance for option pricing, risk management, and modeling stock prices, as well as in many other domains like physics and engineering.

At its core, stochastic calculus allows to model the evolution of systems over time under uncertainty. The exercise we're exploring applies concepts from stochastic calculus to demonstrate that certain functions of Wiener processes exhibit martingale or submartingale properties under specific conditions. Stochastic calculus arms you with the tools to describe these sophisticated models and help grasp how they behave.

Conditional Expectation

Conditional expectation is a predictive concept; it's like looking into a crystal ball but with a solid mathematical foundation. It's the expected value of a random variable given that certain conditions are true — or, in more casual terms, it’s what you’d predict on average knowing some part of the story ahead of time.

In the realm of stochastic calculus, conditional expectation is crucial because it helps us measure how a stochastic process is expected to behave in the future, based on the information available now. The exercise illustrates this notion by calculating conditional expectations under the framework that the increments of the Wiener process are independent and have zero mean, thus relying on these expectations to verify if a process is a martingale or a submartingale.