There are \(n\) assets satisfying the following stochastic differential equations: $$ d S_{i}=\sigma_{i} S_{i} d X_{i}+\mu_{i} S_{i} d t \text { for } i=1, \ldots, n $$ The Wiener processes \(d X_{i}\) satisfy $$ \mathcal{E}\left[d X_{i}\right]=0, \quad \mathcal{E}\left[d X_{i}^{2}\right]=d t $$ as usual, but the asset price changes are correlated with $$ \mathcal{E}\left[d X_{i} d X_{j}\right]=\rho_{i j} d t $$ where \(-1 \leq \rho_{i j}=\rho_{j i} \leq 1\). Derive Itô's lemma for a function \(f\left(S_{1}, \ldots, S_{n}\right)\) of the \(n\) assets \(S_{1}, \ldots, S_{n}\).

Stochastic Differential Equations (SDEs) are a type of mathematical equation used to model the evolution of systems that are subject to random fluctuations, often called 'noise'. They are a cornerstone of stochastic calculus and have a wide range of applications, particularly in the fields of finance, physics, and biology.

In finance, SDEs are used to model the random behavior of asset prices, interest rates, and other market variables. An SDE represents how a quantity changes over time, considering both deterministic trends, like a stock's expected return, and stochastic trends, like the volatility of the stock market. The standard form of an SDE might look something like \(dS = \mu S dt + \sigma S dX\), where \(S\) is the stock price, \(\mu\) is the drift coefficient, \(\sigma\) is the volatility coefficient, and \(dX\) represents the increment of a Wiener process or standard Brownian motion.

In the given exercise, a series of assets are modeled using SDEs that account for both a deterministic part associated with the expected return \(\mu_i S_i dt\) and a stochastic part related to the asset's volatility \(\sigma_i S_i dX_i\). Each asset's price is influenced by a separate Wiener process \(X_i\), reflecting the notion that markets are influenced by factors that can't be predicted.

Multivariable calculus extends the principles of calculus to functions of several variables. It involves concepts like partial derivatives and multiple integrals, which are essential for analyzing how multivariate functions change over time or across different conditions.

When dealing with financial models involving SDEs, such as the pricing of financial derivatives, multivariable calculus becomes indispensable. We often want to know how a function that depends on multiple underlying assets, say \(f(S_1, S_2, ..., S_n)\), will evolve as each of these assets undergoes random changes over time. To understand this, we employ partial derivatives to calculate the rate of change of the function with respect to each asset.

For instance, the partial derivative \(\frac{\partial f}{\partial S_i}\) measures how the function \(f\) changes with a small change in the asset price \(S_i\), holding all other variables constant. Similarly, the second-order mixed partial derivative \(\frac{\partial^2 f}{\partial S_i \partial S_j}\) indicates how sensitive the function is to simultaneous changes in two different variables \(S_i\) and \(S_j\). These computations form the backbone of Itô's Lemma, allowing us to express the differential \(df\) in terms of these partial derivatives and the differentials of the individual stochastic processes.

A Wiener process, also known as a Brownian motion, is a mathematical model used to describe the random movement of particles in fluid, but it also underpins the random walk theory used in financial mathematics. It's a continuous-time stochastic process characterized by its mean, variance, and independence of increments.

In the realm of financial derivatives and SDEs, the Wiener process is used to model the 'noise' or randomness that affects asset prices. Its main properties are that the increment \(dX\) over any time interval has a mean of zero (\(\mathcal{E}[dX]=0\)) and a variance equal to the length of the time interval (\(\mathcal{E}[dX^2]=dt\)). This makes it a fundamental building block in the model of asset price dynamics.

Moreover, in a multi-asset framework, as presented in the exercise, the correlation between two Wiener processes \(dX_i\) and \(dX_j\) becomes crucial. The correlation coefficient \(\rho_{ij}\), which falls between -1 and 1, tells us how much two asset prices movements are related to each other, allowing for the possibility of asset prices moving in tandem or opposite directions. This correlation impacts the joint behavior of asset prices and is taken into account when applying Itô's Lemma in multivariable setups.

Financial derivatives are financial instruments whose value is derived from the performance of underlying assets such as stocks, bonds, commodities, or market indices. Common types of derivatives include options, futures, and swaps. They play a vital role in financial markets, as they allow investors to hedge risks, speculate on price movements, or gain exposure to specific markets without directly owning the underlying assets.

The pricing of financial derivatives is often based on mathematical models that use stochastic differential equations to capture the dynamics of the underlying assets. Itô's Lemma, an essential tool in this context, enables us to link the changes in the value of the derivative to the changes in the underlying asset or assets.

In the exercise, we are concerned with a derivative that depends on multiple assets, each described by an SDE. Itô's Lemma provides us with a framework for understanding how small changes in the underlying asset prices translate to changes in the derivative's value. Incorporating the critical variables of drift, volatility, and correlation between the assets, we can derive a robust formula that helps in the decision-making process for both pricing and risk management of financial derivatives.