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1. Variation and Quadratic Variation of Functions

In classical calculus, smooth functions have bounded variation, which allows us to define integration using Riemann-Stieltjes theory. However, stochastic processes like Brownian motion behave very differently — they have unbounded variation but finite quadratic variation. This fundamental property is what makes stochastic calculus distinct from ordinary calculus and is the foundation of Itô calculus 123.

This document provides a rigorous treatment of variation concepts, building from basic definitions to the remarkable properties of Brownian motion's quadratic variation.

1.1 What is Variation of a Function?

The variation of a function measures how much the function "moves up and down" over an interval. Intuitively, it quantifies the total distance traveled by the function, accounting for all direction changes.

1.1.1 Informal Motivation

Consider a function \(f: [a, b] \to \mathbb{R}\) and a partition of the interval:

\[ \begin{equation} a = t_0 < t_1 < t_2 < \cdots < t_n = b \label{eq:basic_partition} \end{equation} \]

As we move from \(t_i\) to \(t_{i+1}\), the function changes by \(f(t_{i+1}) - f(t_i)\). The total variation accumulates all these changes (in absolute value), while the quadratic variation accumulates the squares of these changes.

Why Study Variation?

Variation characterizes the "roughness" or "irregularity" of a function:

  • Smooth functions: Bounded variation, zero quadratic variation
  • Functions with jumps: Finite variation, finite quadratic variation at jumps
  • Brownian motion: Infinite variation, finite quadratic variation

Understanding variation is essential for determining what type of calculus applies to a given process.

1.2 First-Order (Total) Variation

1.2.1 Definition

The (total) variation of a function \(f: [a, b] \to \mathbb{R}\) is defined as:

\[ \begin{equation} V_a^b(f) = \sup_{\Pi} \sum_{i=0}^{n-1} |f(t_{i+1}) - f(t_i)| \label{eq:total_variation_definition} \end{equation} \]

where the supremum is taken over all partitions \(\Pi: a = t_0 < t_1 < \cdots < t_n = b\) of the interval \([a, b]\).

Interpretation

The total variation \(V_a^b(f)\) represents the total distance traveled by the function along the vertical axis as the input moves from \(a\) to \(b\). If you imagine walking along the graph of \(f(t)\), the total variation measures the cumulative vertical displacement.

1.2.2 Properties

A function \(f\) is said to have bounded variation if \(V_a^b(f) < \infty\).

Key properties of functions with bounded variation:

  1. Decomposition: Any function of bounded variation can be written as the difference of two monotone increasing functions:
\[ \begin{equation} f(t) = f_1(t) - f_2(t) \label{eq:jordan_decomposition} \end{equation} \]

  1. Differentiability: Functions of bounded variation are differentiable almost everywhere (Lebesgue's theorem)

  2. Integration: If \(f\) has bounded variation, the Riemann-Stieltjes integral \(\int g(t)\,df(t)\) is well-defined for continuous \(g\)

1.2.3 Examples

1.2.3.1 Monotone Increasing Function

Let \(f(t) = t\) on \([0, 1]\).

Consider an arbitrary partition

\[ \begin{equation} 0 = t_0 < t_1 < t_2 < \cdots < t_n = 1 \label{eq:monotone_partition} \end{equation} \]

Then, compute the variation sum. Since \(f\) is increasing, \(f(t_{i+1}) - f(t_i) > 0\), so:

\[ \begin{equation} \sum_{i=0}^{n-1} |f(t_{i+1}) - f(t_i)| = \sum_{i=0}^{n-1} (f(t_{i+1}) - f(t_i)) = f(1) - f(0) = 1 \label{eq:monotone_variation_sum} \end{equation} \]

This is a telescoping sum that collapses to the total change. Then taking the supremum, and since this equals 1 for every partition:

\[ \begin{equation} V_0^1(f) = 1 < \infty \label{eq:monotone_total_variation} \end{equation} \]
Result: Bounded Variation

Monotone functions have bounded variation equal to the total change \(|f(b) - f(a)|\).

1.2.3.2 Differentiable Function

Let \(f(t) = \sin(t)\) on \([0, 2\pi]\). For a continuously differentiable function:

\[ \begin{equation} V_a^b(f) = \int_a^b |f'(t)|\,dt \label{eq:smooth_variation_formula} \end{equation} \]

After this, apply to sine

\[ \begin{equation} V_0^{2\pi}(\sin) = \int_0^{2\pi} |\cos(t)|\,dt \label{eq:sine_variation} \end{equation} \]

Eventually, evaluate the integral breaking into regions where \(\cos(t)\) has constant sign:

\[ \begin{equation} \int_0^{2\pi} |\cos(t)|\,dt = \int_0^{\pi/2} \cos(t)\,dt - \int_{\pi/2}^{3\pi/2} \cos(t)\,dt + \int_{3\pi/2}^{2\pi} \cos(t)\,dt = 1 + 2 + 1 = 4 \label{eq:sine_variation_result} \end{equation} \]
Result: Finite Variation

Continuously differentiable functions have bounded variation given by the integral of \(|f'(t)|\).

1.2.3.3 Brownian Motion - Infinite Variation

Let \(W(t)\) be a standard Brownian motion on \([0, T]\).

We will show that \(W(t)\) has unbounded variation almost surely, i.e., \(V_0^T(W) = \infty\) a.s.

Consider a uniform partition, by dividing \([0, T]\) into \(n\) equal subintervals:

\[ \begin{equation} t_i = \frac{iT}{n}, \quad i = 0, 1, \ldots, n \label{eq:uniform_partition_brownian} \end{equation} \]

Then, compute the expected variation sum

\[ \begin{equation} \mathbb{E}\left[\sum_{i=0}^{n-1} |W(t_{i+1}) - W(t_i)|\right] \label{eq:expected_variation_sum} \end{equation} \]

Last but not least, use properties of Brownian increments in which each increment \(W(t_{i+1}) - W(t_i) \sim \mathcal{N}(0, T/n)\), whereas a standard normal variable \(Z \sim \mathcal{N}(0, \sigma^2)\):

\[ \begin{equation} \mathbb{E}[|Z|] = \sigma\sqrt{\frac{2}{\pi}} \label{eq:absolute_normal_expectation} \end{equation} \]

Therefore:

\[ \begin{equation} \mathbb{E}[|W(t_{i+1}) - W(t_i)|] = \sqrt{\frac{T}{n}} \cdot \sqrt{\frac{2}{\pi}} = \sqrt{\frac{2T}{\pi n}} \label{eq:brownian_increment_expectation} \end{equation} \]

that aggregated, and taking into account the sum us a linear operator on expectations, the sum over all increments is depicted by:

\[ \begin{equation} \mathbb{E}\left[\sum_{i=0}^{n-1} |W(t_{i+1}) - W(t_i)|\right] = n \cdot \sqrt{\frac{2T}{\pi n}} = \sqrt{\frac{2Tn}{\pi}} \label{eq:total_expected_variation} \end{equation} \]

Last, taking the limit as \(n \to \infty\) we see it is infinite.

\[ \begin{equation} \lim_{n \to \infty} \sqrt{\frac{2Tn}{\pi}} = \infty \label{eq:infinite_variation_limit} \end{equation} \]
Critical Result: Unbounded Variation

Brownian motion has unbounded (infinite) variation on any non-trivial interval. This is why:

  • Classical Riemann-Stieltjes integration fails for Brownian motion
  • We cannot use ordinary calculus: \(dW(t)/dt\) does not exist
  • A new theory (stochastic calculus) is required

This infinite variation property distinguishes stochastic processes from smooth functions.

1.3 Quadratic Variation

Since Brownian motion has infinite variation, we need a different measure of its "roughness." The quadratic variation turns out to be the right concept.

1.3.1 Definition

The quadratic variation of a function \(f: [a, b] \to \mathbb{R}\) over the interval \([a, b]\) is defined as:

\[ \begin{equation} [f, f]_a^b = \lim_{|\Pi| \to 0} \sum_{i=0}^{n-1} (f(t_{i+1}) - f(t_i))^2 \label{eq:quadratic_variation_definition} \end{equation} \]

where \(|\Pi| = \max_i(t_{i+1} - t_i)\) is the mesh size of the partition, and the limit is taken as the mesh size goes to zero.

For stochastic processes, this limit is typically understood in the probability or mean-square sense.

Why Quadratic?

Instead of summing \(|f(t_{i+1}) - f(t_i)|\) (first-order variation), we sum squares \((f(t_{i+1}) - f(t_i))^2\). This seemingly minor change has profound consequences for stochastic processes.

1.3.2 Notation

Common notations for quadratic variation include:

  • \([f, f]_a^b\) or \([f]_a^b\): Interval notation
  • \(\langle f \rangle_t\) or \([f]_t\): Process notation (quadratic variation up to time \(t\))

1.3.3 Properties

  • For smooth functions: If \(f\) is continuously differentiable, then:
\[ \begin{equation} [f, f]_a^b = 0 \label{eq:smooth_quadratic_variation_zero} \end{equation} \]
  • For jump processes: Quadratic variation accumulates only at jump points
  • For continuous martingales: Quadratic variation is a key characteristic measuring the "randomness"

1.3.4 Example

1.3.4.1 Smooth Function - Zero Quadratic Variation

Let \(f(t) = t^2\) on \([0, 1]\). Consider a partition with mesh size \(|\Pi|\).

\[ \begin{equation} 0 = t_0 < t_1 < \cdots < t_n = 1, \quad |\Pi| = \max_i(t_{i+1} - t_i) \label{eq:smooth_partition} \end{equation} \]

Then, compute the quadratic variation sum

\[ \begin{equation} \sum_{i=0}^{n-1} (f(t_{i+1}) - f(t_i))^2 = \sum_{i=0}^{n-1} (t_{i+1}^2 - t_i^2)^2 \label{eq:quadratic_sum_smooth} \end{equation} \]

By the mean value theorem, there exists \(\xi_i \in (t_i, t_{i+1})\), then square and bound, such that:

\[ \begin{equation} f(t_{i+1}) - f(t_i) = f'(\xi_i)(t_{i+1} - t_i) = 2\xi_i(t_{i+1} - t_i) \label{eq:mean_value_application} \end{equation} \]
\[ \begin{equation} (f(t_{i+1}) - f(t_i))^2 = 4\xi_i^2(t_{i+1} - t_i)^2 \leq 4(t_{i+1} - t_i)^2 \leq 4|\Pi|(t_{i+1} - t_i) \label{eq:squared_increment_bound} \end{equation} \]

since \(\xi_i \in [0, 1]\) and \((t_{i+1} - t_i) \leq |\Pi|\). Eventually, sum and take limit.

\[ \begin{equation} \sum_{i=0}^{n-1} (f(t_{i+1}) - f(t_i))^2 \leq 4|\Pi| \sum_{i=0}^{n-1} (t_{i+1} - t_i) = 4|\Pi| \cdot 1 \to 0 \label{eq:quadratic_variation_limit_smooth} \end{equation} \]

as \(|\Pi| \to 0\).

Result: Zero Quadratic Variation

For smooth functions (continuously differentiable), the quadratic variation is zero. The key is that squared increments decay like \(O(|\Pi|^2)\), which vanishes as the mesh size decreases.

1.3.5 Formal Notation

In differential notation:

\[ \begin{equation} d[W]_t = dt \quad \Rightarrow \quad (dW(t))^2 = dt \label{eq:differential_quadratic_variation} \end{equation} \]

This is often written symbolically as:

\[ \begin{equation} dW \cdot dW = dt \label{eq:symbolic_quadratic_variation} \end{equation} \]

This is the foundation of Itô's calculus multiplication rules.

1.4 Summary and Key Insights

1.4.1 Total Variation vs. Quadratic Variation

Property Smooth Functions Brownian Motion
First-order variation Finite Infinite
Quadratic variation Zero Finite (\(= t\))
Calculus type Classical Stochastic (Itô)
Integration Riemann-Stieltjes Itô integral
Chain rule correction No second-order term \(\frac{1}{2}f''\sigma^2\,dt\) term

1.4.2 Fundamental Formula

The single most important formula in stochastic calculus:

\[ \begin{equation} (dW(t))^2 = dt \label{eq:fundamental_quadratic_variation} \end{equation} \]

This is not an approximation — it's an exact statement about the quadratic variation of Brownian motion, and it's the foundation of all stochastic calculus.

Bottom Line

Quadratic variation is the key that unlocks stochastic calculus.

Without it: - We couldn't distinguish between smooth and rough processes - Itô's formula would reduce to the classical chain rule - Stochastic integrals would behave like ordinary integrals - We couldn't model genuine randomness in continuous time

1.4.3 Generalization: Covariation

For two processes \(X(t)\) and \(Y(t)\), the quadratic covariation is:

\[ \begin{equation} [X, Y]_t = \lim_{|\Pi| \to 0} \sum_{i=0}^{n-1} (X(t_{i+1}) - X(t_i))(Y(t_{i+1}) - Y(t_i)) \label{eq:quadratic_covariation} \end{equation} \]

This extends the concept to correlation between stochastic processes.

Building confidence through rigorous validation

  1. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 6th edition, 2013. URL: https://link.springer.com/book/10.1007/978-3-642-14394-6. ↩

  2. Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus. Springer, 2nd edition, 1991. ↩

  3. Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 3rd edition, 1999. URL: https://link.springer.com/book/10.1007/978-3-662-06400-9. ↩