Stochastic calculus is a branch of financial mathematics and probability that models and analyzes continuous random phenomena, such as asset prices. One of its key tools is Itô calculus, which allows differentiation of functions of stochastic processes, particularly those involving Brownian motion. This type of calculus differs significantly from classical differential calculus and requires a tailored approach to account for stochastic properties.

Brownian motion, often noted as W_t, plays a central role in finance and probability. Firstly, W_0 = 0, indicating that the motion starts at the origin. Secondly, the increments W_(t+s) - W_t are independent of the history of Brownian motion up to time t. Thirdly, for any s > 0, the variation W_(t+s) - W_t follows a normal distribution with zero mean and variance s, i.e., a N(0, s) distribution. Finally, the paths of W_t are continuous but are almost nowhere differentiable.

These properties make Brownian motion an ideal model to describe the evolution of asset prices, interest rates, or other financial or physical random phenomena.

When handling stochastic processes like W_t, it is tempting to differentiate them as if they were smooth functions in classical calculus. However, this naive approach fails because W_t is not differentiable in the conventional sense.

Consider, for example, the function t ↦ W_t^2, where W_t is Brownian motion. In the classical context, the differentiation rule would be:

d/dt [W_t^2] = 2 * W_t * (dW_t/dt).

However, since W_t is Brownian motion, this derivative does not exist in the usual sense. A different method of differentiating W_t^2 is required, which is where Itô calculus comes into play.

The Itô formula is the stochastic counterpart to the classical differentiation formula. For a smooth function g(t, W_t), meaning it is twice continuously differentiable with respect to W_t and once with respect to t, the Itô formula allows the calculation of the differential of g(t, W_t). It is expressed as follows:

dg(t, W_t) = (∂g/∂t) dt + (∂g/∂W_t) dW_t + (1/2) * (∂²g/∂W_t²) (dW_t)².

The first term is the drift (∂g/∂t) dt, which corresponds to the classical derivative with respect to time. The second term is the stochastic term (∂g/∂W_t) dW_t, representing the variation due to Brownian motion. The third term is the covariation term (1/2) * (∂²g/∂W_t²) (dW_t)², an additional term compared to classical calculus.

This last term is crucial. Unlike classical calculus, where the squares of differentials are negligible, in Itô calculus, (dW_t)² = dt. This is because the variance of Brownian motion over an infinitesimal time interval [t, t + dt] is equal to dt. This unique property means that the infinitesimal variations of W_t are "large" enough for their square not to be negligible, contributing to the dynamics of the function.

To illustrate Itô’s formula, consider g(t, W_t) = W_t^2. Let's apply Itô's formula step by step.

First, calculate the partial derivatives. The first derivative is (∂g/∂t) = 0, since g does not explicitly depend on t. The second is (∂g/∂W_t) = 2 * W_t. The third is (∂²g/∂W_t²) = 2.

Next, apply Itô's formula:

d(W_t^2) = 0 * dt + 2 * W_t * dW_t + (1/2) * 2 * (dW_t)².

Simplifying this expression yields:

d(W_t^2) = 2 * W_t * dW_t + dt [remember that (dW_t)² = dt]

The formula shows that the infinitesimal variation of W_t^2 consists of two terms: a random term 2 * W_t * dW_t and a deterministic term dt.

The reason Itô calculus differs from classical calculus is related to the nature of the stochastic integral ∫[0, t] W_s dW_s. This integral is actually a martingale, a stochastic process whose conditional expectation is equal to its current value.

A martingale has an important property: it cannot grow positively in a constant manner from zero. If this integral ∫[0, t] W_s dW_s were equal to W_t^2, then it would always be positive.

However, a martingale that starts at zero cannot remain positive at every moment unless it is identically zero. Therefore, it is impossible to have a relationship like:

W_t^2 = 2 * ∫[0, t] W_s dW_s

without the additional "t" term. It is precisely this extra term that compensates for the martingale nature of the integral.

If one integrates this differential from 0 to t, we obtain:

W_t^2 = W_0^2 + 2 * ∫[0, t] W_s dW_s + t.

In general, if W_0 = 0, we simply have:

W_t^2 = 2 * ∫[0, t] W_s dW_s + t.

This shows that W_t^2 cannot simply be written as 2 * ∫[0, t] W_s dW_s, as there is an additional "t" term arising from the covariation of Brownian motion.

The intuition behind Itô's formula is that, unlike classical calculus, the variations of Brownian motion have a "turbulent" nature. Indeed, the paths of W_t are highly irregular: they oscillate rapidly and unpredictably. Itô's formula adjusts the calculations to account for this inherent volatility.

The addition of the term (1/2) * (∂²g/∂W_t²) (dW_t)² compensates for this irregularity. In the case of W_t^2, this is manifested by the additional “t” term, which represents the accumulation of the variance of Brownian motion over time.

**About the Author**

Florian Campuzan, a graduate of Sciences Po Paris (Economic and Financial section) with a degree in Economics (money and finance), is a CFA charterholder and AMF-certified professional. He began his career in private equity and venture capital as an investment manager at Natixis before transitioning to market finance as a proprietary trader. In the early 2010s, he founded Finance Tutoring, specializing in market and corporate finance training and consulting.

For over 12 years, Florian has led finance training, advised financial institutions and industrial groups on managing risks, and prepared candidates for AMF certification and the CFA exam. Passionate about quantitative finance and the application of mathematics, he focuses on making complex concepts intuitive, believing that mastering any topic starts with understanding its core intuition.

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