Riemann–Stieltjes Integration

• Riemann–Stieltjes integration is a generalization of the Riemann integral that integrates a function against the increments of another, accommodating discontinuities and irregular behaviors.
• It extends classical properties like linearity and integration by parts, and offers a framework for handling functions of bounded variation and regulated integrators.
• The technique enables precise numerical quadrature with explicit error bounds and applies to stochastic calculus, differential equations, and multivariable integration.

The Riemann–Stieltjes integral generalizes the classical Riemann integral by allowing integration of a function $f$ (the "integrand") against the increments of another function $g$ (the "integrator" or "measure-generating function") rather than against Lebesgue or Riemann measure. It plays a fundamental role in analysis, probability, numerical quadrature, and the study of differential equations with irregular or generalized driving signals. Its definition, existence theory, and properties have led to extensive applications, ranging from stochastic calculus and probability theory to the numerical approximation of integrals and the study of rough or stochastic dynamics.

1. Definition and Fundamental Properties

Let $f,\,g:[a,b]\to\mathbb{R}$ be bounded real-valued functions. The Riemann–Stieltjes integral of $f$ with respect to $g$ over $[a,b]$ is defined, if it exists, as the common limit of Riemann–Stieltjes sums

$$S(P,f,g) = \sum_{i=1}^n f(\xi_i)\;[g(x_{i})-g(x_{i-1})]$$

as the mesh of the partition $P = \{a=x_0< x_1<\cdots< x_n=b\}$ tends to zero, for all choices of sample points $\xi_i\in[x_{i-1},x_i]$. Existence is guaranteed under several sufficient conditions, notably if one function is continuous and the other is of bounded variation on $[a,b]$ (Alomari, 2014, Niang et al., 2020).

The classical integration by parts formula holds: $$\int_a^b f\,dg + \int_a^b g\,df = f(b)g(b) - f(a)g(a)$$ when $f$ and $g$ are both of bounded variation.

Linearity, additivity on intervals, and monotonicity properties all extend from the Riemann setting. Existence criteria permit $g$ to have jump discontinuities, fractal structure, or to be only regulated, making the Riemann–Stieltjes formalism flexible for modeling integrators with a range of regularity (Niang et al., 2020, Pouso, 2011).

2. Existence Criteria and Main Theorems

Classical existence

• If $f$ is continuous and $g$ is of bounded variation, or vice versa, the Riemann–Stieltjes integral exists (Alomari, 2014, Pouso, 2011).
• If $g$ is continuously differentiable and $f$ is Riemann integrable, then the reduction

$\int_a^b f(x)\,dg(x) = \int_a^b f(x)g'(x)\,dx$

holds (Pouso, 2011).

• If $g$ is a step function, $\int_a^b f\,dg$ exists if and only if $f$ has left and right limits at the jump points of $g$ (Niang et al., 2020).

Bounded variation, truncated and $p$-variation

The existence theory reaches further with the use of total variation or truncated variation. If $f$ and $g$ are regulated with no common discontinuities, and appropriate truncated variations at suitably chosen scales decay rapidly enough, then the Riemann–Stieltjes integral exists and is controlled quantitatively (Łochowski, 2014):

Given sequences $(n_k), (\epsilon_k)\searrow 0$, if the sum

$$S = \sum_{k=0}^\infty 2^k n_{k-1}TV_{(\epsilon_k)}(g;[a,b]) + \sum_{k=0}^\infty 2^k \epsilon_k TV_{(n_k)}(f;[a,b])$$

is finite, where $TV_{(\delta)}$ denotes $\delta$-truncated variation, then the integral exists and $|\int_a^b f\,dg - f(a)[g(b)-g(a)]|\leq S$.

When $f\in V^p$, $g\in V^q$ for $p,q>1$, $1/p+1/q>1$, the Young–Loeve theorem ensures existence, and the improved Loéve–Young inequality provides explicit estimates (Łochowski, 2014).

Stochastic and pathwise generalizations

For random processes (e.g., fractional Brownian motion), versions of the Riemann–Stieltjes integral can be defined pathwise when the processes possess joint increment controls or satisfy joint Hölder conditions, even when classical $p$-variation is infinite (Yaskov, 2015, Chen et al., 2016). For instance, if $X$ and $Y$ are Hölder processes of orders $\alpha$ and $\beta$ with $\alpha+\beta>1$, the pathwise Riemann–Stieltjes integral $\int f(X_t)\,dY_t$ exists for locally finite variation $f$ (Chen et al., 2016).

3. Explicit Formulas and Computation

Reduction to Riemann integrals

If $g$ is an indefinite Riemann integral $g(x) = c + \int_a^x h(t)\,dt$ for some Riemann–integrable $h$, and $f$ is bounded with $fh$ Riemann integrable, then

$\int_a^b f(x)\,dg(x) = \int_a^b f(x)h(x)\,dx$

(Pouso, 2011).

Discrete and step integrators

If $g$ is a pure step function with jumps at $(x_h)$ of size $\Delta g(x_h)$, and $f$ is continuous at these points,

$\int_a^b f(x)\,dg(x) = \sum_h \Delta g(x_h)f(x_h)$

(Niang et al., 2020).

Change of variable

For an invertible, continuous, strictly monotone $\phi:[a,b]\to J$, the substitution formula applies: $$\int_{\phi(a)}^{\phi(b)} f(v)dY(v) = \int_{a}^{b} f(\phi(x))\,d(Y\circ\phi)(x)$$ Generalizations to the non-invertible case allow composition with non-one-to-one substitutions, provided the induced integrator is a well-defined indefinite integral (Torchinsky, 2019).

4. Numerical Quadrature and Error Bounds

Riemann–Stieltjes integrals admit quadrature approximations via two- and three-point rules, generalizations of classical Newton–Cotes and Gaussian quadrature. For functions of bounded variation and Lipschitz or Hölder integrators, explicit $L^p$ error estimates and sharp inequalities are established (Alomari et al., 2018, Alomari, 2014, Alomari, 2014):

• General quadrature functional $D_\alpha(f,u;x)$ interpolates midpoint, trapezoid, and Simpson's rules.
• Error terms are explicitly bounded using triangle-type and Beesack–Wirtinger inequalities, involving $L^p$ norms and Hölder exponents (Alomari et al., 2018).
• For the two-point Gauss–Legendre rule, error estimates depend sharply on smoothness and variation of $f$ and $g$ (Alomari, 2014).
• Weighted Ostrowski and generalized trapezoid inequalities provide optimal bounds in terms of total variation and (weighted) increments (Alomari, 2014).

Modified Riemann sums—with subintervals shrunk or resampled—also converge to linear transformations of the Riemann–Stieltjes integral under explicit hypotheses, allowing for partial or data-limited integration schemes, change-of-integrator constructs, or quadrature under physical constraints (Torchinsky, 2019).

5. Extensions: Multivariable, Time Scales, and Special Integrators

The Riemann–Stieltjes formalism extends naturally to double integrals, with the bi-variation of the integrand and integrator governing existence. Specialized inequalities (Hermite–Hadamard, Trapezoid, Grüss, Ostrowski) and tailored cubature rules for double Riemann–Stieltjes integrals exploit properties such as bounded bi-variation, coordinatewise monotonicity, or Lipschitz continuity (Alomari, 2016).

The theory also generalizes to the calculus on time scales, merging discrete and continuous analysis. All foundational properties (linearity, additivity, integration by parts, substitution) extend, with the forward and backward graininess dictating the elementary increments and integration by parts adjustments (0903.1224).

Comparisons between Riemann–Stieltjes and Lebesgue–Stieltjes integrals, especially with step or jump-discontinuous integrators, clarify the respective requirements and offer insight into the expressive power and limitations of each approach in probability and measure theory (Niang et al., 2020).

6. Applications, Limitations, and Open Problems

The Riemann–Stieltjes integral is critical for:

• Expressing and computing expectations with respect to cumulative distribution functions with discrete or continuous components (Niang et al., 2020).
• Defining and numerically approximating integrals driven by rough, non-classically smooth, or jump processes (Łochowski, 2014, Yaskov, 2015).
• Modeling feedback and memory effects in differential equations and stochastic processes beyond the semimartingale/Itô framework (Yaskov, 2015, Chen et al., 2016).
• Analyzing convergence and positivity properties (including necessary and sufficient conditions, such as bounded variation) with respect to varying regularity of integrand and integrator (Lukkarinen et al., 2012).

However, the classical theory requires suitable regularity—bounded variation, regulated functions, or suitable integrability conditions. Insufficient regularity can lead to the non-existence of the integral or loss of key properties such as positivity (Lukkarinen et al., 2012, Łochowski, 2014).

Open questions include further generalizations to singular continuous integrators, vector-valued settings, extensions of numerical quadrature to higher-order schemes for Riemann–Stieltjes and related rough integrals, as well as sharper characterizations of existence and stability in stochastic and rough regimes (Pouso, 2011, Yaskov, 2015, Łochowski, 2014).