Riemann-Stieltjes Integral

Motivation: why this matters in quant finance

Many finance quantities are accumulated against something other than calendar time. Portfolio value accumulates against price changes. Bond cash flows accumulate against a distribution of payment dates. Stochastic calculus eventually defines integrals such as $\int_0^T H_t\,dS_t$ , where the integrator is a price process, not $dt$ .

The Riemann-Stieltjes integral is the deterministic predecessor of this idea. It replaces the ordinary integral $\int f(x)\,dx$ with $\int f(x)\,dg(x)$ , where increments of $g$ decide how much weight each region receives. If $g$ jumps, the integral picks up point masses. If $g$ is smooth, it reduces to an ordinary weighted integral.

This note is not the full Itô integral. It is the bridge: it teaches the idea of integrating against an accumulated quantity before randomness and quadratic variation enter.

The informal idea

An ordinary Riemann integral sums height times width:

\sum_i f(x_i^*)\Delta x_i.

A Riemann-Stieltjes integral sums height times change in another function:

\sum_i f(x_i^*)\Delta g_i = \sum_i f(x_i^*)\left(g(x_i)-g(x_{i-1})\right).

If $g$ grows quickly on one interval, that interval receives more weight. If $g$ is flat, that interval contributes nothing. If $g$ jumps, the jump contributes a discrete amount.

Formal definition

Let $f,g:[a,b]\to\mathbb{R}$ and let $P$ be a partition $a=x_0<\cdots<x_n=b$ . Choose sample points $x_i^*\in[x_{i-1},x_i]$ . The Riemann-Stieltjes sum is

\sum_{i=1}^n f(x_i^*)\left(g(x_i)-g(x_{i-1})\right).

If these sums converge to the same limit as the mesh of the partition goes to zero, regardless of sample points, the limit is the Riemann-Stieltjes integral:

\int_a^b f(x)\,dg(x).

A common sufficient condition is: $f$ is continuous and $g$ has bounded variation on $[a,b]$ .

Key properties

Reduction to ordinary integration

If $g$ is continuously differentiable, then

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

So the Stieltjes integral generalises ordinary integration rather than replacing it.

Jumps create point masses

If $g$ jumps by $\Delta g(c)$ at $c$ , then the integral includes $f(c)\Delta g(c)$ , depending on convention and continuity side. This is why Stieltjes notation is natural for distributions with atoms.

Integration by parts

When the relevant integrals exist,

\int_a^b f\,dg + \int_a^b g\,df = f(b)g(b)-f(a)g(a).

This is the deterministic ancestor of product rules in stochastic calculus. Itô's formula changes this identity by adding a quadratic-variation correction.

Distribution functions

If $F$ is a cumulative distribution function, then $\int h(x)\,dF(x)$ represents expectation with respect to that distribution. Discrete masses and continuous densities are handled in one notation.

Worked examples

Example 1: smooth integrator

Let $g(x)=x^2$ on $[0,1]$ and $f(x)=x$ . Since $g'(x)=2x$ ,

\int_0^1 x\,d(x^2)=\int_0^1 2x^2\,dx=\frac{2}{3}.

Example 2: jump integrator

Let $g(x)=\mathbf{1}_{x\ge c}$ on $[0,1]$ . Then $g$ is flat except for a unit jump at $c$ , and

\int_0^1 f(x)\,dg(x)=f(c).

The integral has selected the value at the jump point. This is the deterministic analogue of integrating against a point mass.

Example 3: expectation notation

For a random variable $X$ with CDF $F$ , expectation can be written as

\mathbb{E}[h(X)]=\int h(x)\,dF(x).

If $F$ has both a density and jumps, this single formula covers both the continuous and discrete pieces.

Common confusions and pitfalls

" $dg$ always means $g'(x)dx$ ." Only when

g

is smooth enough. The point of Stieltjes integration is that

g

may have jumps or singular growth.

"This is already the Itô integral." No. The Itô integral handles random integrators of unbounded variation such as Brownian motion. The Riemann-Stieltjes integral usually requires bounded variation of the integrator.

"The sample-point convention never matters." With jumps, left- and right-continuity conventions can matter. Financial cash-flow timing is exactly where that detail becomes economic.

"Bounded variation is a technical afterthought." It is the condition that makes pathwise deterministic integration work. Brownian paths violate it, which is why stochastic integration needs new machinery.

Where this goes next

Introduction to Integration: The ordinary Riemann integral that Stieltjes integration generalises.
Integration by Parts: The deterministic product rule mirrored by Stieltjes integration.
Probability Space: The measure-theoretic setting for expectations written as integrals.
Itô's Lemma: The stochastic-calculus analogue, where quadratic variation changes the product rule.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 5 definite-integral construction supplies the Riemann-sum background; the Riemann-Stieltjes extension is beyond this single-variable source and is included for quant-finance integration against cumulative processes.