In quantitative finance, we routinely work with stochastic differential equations, simulate price paths, and optimize portfolios. These tasks often involve mathematical abstractions, but how often do we reflect on the underlying spaces in which we define our models?
The distinction between Euclidean spaces and pre-Hilbert spaces may seem purely theoretical, yet it has practical consequences when working with continuous processes, convergence guarantees, and error measurements.
Let’s begin with Euclidean spaces. A Euclidean space, denoted ℝⁿ, consists of vectors with a finite number of components. These spaces are equipped with an inner product, defined for vectors x and y as:
\[ x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n \]
which measures angles and lengths. The associated norm is the square root of the inner product of a vector with itself, written as:
\[ \|x\| = \sqrt{x \cdot x} \]
These simple constructs are sufficient for problems involving finite-dimensional data, such as portfolio optimization, regressions, and principal component analysis.
For example, in the Markowitz model of portfolio optimization, the weights of assets are treated as vectors in ℝⁿ, where n is the number of assets. The variance of the portfolio, which serves as a risk measure, is computed using the formula:
\[ Var(w) = w^T \Sigma w \]
where \( \Sigma \) is the covariance matrix. Since the space is finite-dimensional, the optimization problem reduces to solving linear equations. Similarly, in Monte Carlo simulations, paths are discretized into steps, and calculations are performed at each time point, making Euclidean spaces sufficient for finite approximations.
Now consider what happens when we move to continuous modeling. Many financial processes, such as stock prices, are modeled as stochastic processes evolving over time. A simple example is the geometric Brownian motion described by the stochastic differential equation (SDE):
\[ dS_t = \mu S_t dt + \sigma S_t dW_t \]
The solution to this equation is a continuous function of time, given by:
\[ S_t = S_0 \exp \left( (\mu - 0.5\sigma^2)t + \sigma W_t \right) \]
While the formula looks simple, the mathematical machinery behind it cannot be captured by Euclidean spaces.
The problem lies in the fact that Brownian motion, \( W_t \), is not defined at discrete points alone but as a continuous path over time. Measuring the distance between two paths requires integrating the squared differences over an interval, leading to a norm defined by:
\[ \|f\| = \sqrt{ \int_0^T f(t)^2 dt } \]
This is the norm of an \( L^2 \) space, which belongs to the family of pre-Hilbert spaces1. Unlike Euclidean spaces, pre-Hilbert spaces can handle infinite dimensions because they treat functions as vectors.
To understand why infinite dimensions matter, consider pricing an Asian option, where the payoff depends on the average price over time. Calculating this average requires integrating the price path, which involves working in a space of functions, not just vectors. More importantly, approximating such payoffs using discrete time steps means ensuring that as the number of steps increases, the discrete approximation converges to the continuous model.
This kind of convergence is guaranteed in pre-Hilbert spaces, where completeness ensures that limits of sequences are well-defined. However, pre-Hilbert spaces are not complete2. This means that certain approximations or limits might fall outside the space, which can lead to theoretical gaps.
For this reason, when completeness becomes necessary, we switch to Hilbert spaces, which are complete pre-Hilbert spaces3.
Why Not Always Use Hilbert Spaces?
Using a Hilbert space might seem like the safest choice, but it often involves additional complexity that is not always justified for practical applications.
Finite-dimensional vs Infinite-dimensional Models
In many quantitative finance applications, we work with discrete data such as daily prices, portfolio weights, or time-stepped simulations. These naturally fit into finite-dimensional spaces, where the Euclidean framework is sufficient and computationally efficient.
Recognizing this distinction can sharpen both your implementation and theoretical insights.
Notes:
1 The prefix “pré-” in the term “préhilbertien” refers to the absence of a specific hypothesis: completeness, which is essential for many mathematical results.
2 The lack of completeness in pre-Hilbert spaces means some sequences may converge to limits outside the space, making it unsuitable for tasks requiring strict convergence guarantees.
3 Hilbert spaces are complete pre-Hilbert spaces, named after David Hilbert (1862–1943), whose work laid the foundations for functional analysis and stochastic calculus, now central to modern quantitative finance models.
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