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Bounded-variation Riemann-Stieltjes theory does not construct the Brownian Ito integral

Statement refuted

The inference "the pathwise Riemann--Stieltjes construction for a bounded-variation integrator defines the Brownian Ito integral $\int_{0}^{T} H d B$ " is false. Almost surely, Brownian paths have infinite total variation on every nondegenerate compact interval, so the hypothesis of the Riemann--Stieltjes existence theorem for every continuous integrand fails; and for the explicit integrand $f (x) = x$ , $g = B$ , the Riemann--Stieltjes sums along dyadic partitions do not converge to a common limit, because the left-endpoint rule gives $\frac{1}{2} (B_{t}^{2} - t)$ while the right-endpoint rule gives $\frac{1}{2} (B_{t}^{2} + t)$ . The counterexample refutes only the bounded-variation construction; it does not assert that no pathwise integral of any other kind exists, and it does not claim that the Ito integral fails to exist.

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ Brownian motion, $t > 0$ , the dyadic partitions $t_{k} = k t / 2^{n}$ of $[0, t]$ , and the integrand $f (x) = x$ .

[F1]

Almost surely, on every nondegenerate compact interval $[a, b]$ the variation sums of the Brownian path are unbounded, so the path is not of bounded variation there. Brownian paths have infinite total variation Bounded variation and total variation on an interval

[F2]

If the integrator $g$ has bounded variation on $[a, b]$ and the integrand $f$ is continuous there, then the Riemann--Stieltjes sums converge along partitions of mesh tending to $0$ , independently of the evaluation points, to the Riemann--Stieltjes integral $\int_{a}^{b} f d g$ . A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions

[F3]

Along the dyadic partitions of $[0, t]$ , $\sum_{k} (B_{t_{k + 1}} - B_{t_{k}})^{2} \to t$ uniformly almost surely, and the two telescoping identities $2 \sum_{k} B_{t_{k}} Δ_{k} = B_{t}^{2} - \sum_{k} Δ_{k}^{2}$ and $2 \sum_{k} B_{t_{k + 1}} Δ_{k} = B_{t}^{2} + \sum_{k} Δ_{k}^{2}$ hold with $Δ_{k} = B_{t_{k + 1}} - B_{t_{k}}$ . Uniform dyadic Brownian quadratic variation process Quadratic variation along a partition sequence

[F4]

AC is declared for the ambient interfaces. The Axiom of Choice

Counterexample

1.1

By [F1] there is an event of probability one on which the Brownian path is of unbounded variation on every nondegenerate compact interval; on that event the hypothesis " $g$ has bounded variation" of [F2] fails for $g = B$ and every interval $[0, t]$ with $t > 0$ , so the Riemann--Stieltjes existence theorem for a general continuous integrand is not available pathwise.

F1F2given

1.2

For the specific integrand $f (x) = x$ and the dyadic partitions, the two evaluation rules give the sums $L_{n} : = \sum_{k} B_{t_{k}} Δ_{k}$ and $R_{n} : = \sum_{k} B_{t_{k + 1}} Δ_{k}$ , whose telescoping identities [F3] express them as $L_{n} = \frac{1}{2} (B_{t}^{2} - \sum_{k} Δ_{k}^{2})$ and $R_{n} = \frac{1}{2} (B_{t}^{2} + \sum_{k} Δ_{k}^{2})$ .

F3given

2.1

By the quadratic-variation limit of [F3], $L_{n} \to \frac{1}{2} (B_{t}^{2} - t)$ and $R_{n} \to \frac{1}{2} (B_{t}^{2} + t)$ almost surely; the two limits differ by $t > 0$ , so the Riemann--Stieltjes sums of $f d g$ with $g = B$ have no common limit along dyadic partitions and the pathwise Riemann--Stieltjes integral $\int_{0}^{t} B d B$ does not exist in that sense, even though the Ito integral does.

F3step 1.2

3.1

Consequently the bounded-variation construction cannot serve as the definition of the Brownian Ito integral: its central hypothesis fails almost surely on every nondegenerate interval [F1], and its conclusion fails explicitly for the witness $f (x) = x$ , $g = B$ by the limit mismatch of step 2.1. The Ito integral of the same integrand exists and equals $\frac{1}{2} (B_{t}^{2} - t)$ by the companion example page, so the failure is a failure of the pathwise construction, not of the stochastic integral. AC enters only through [F4].

F1step 2.1F4given∎

Source notes

Lawler, Section 2.8, records that Brownian paths have infinite variation on every interval and that the ordinary bounded-variation theory therefore does not apply; van der Vaart, Section 5.1, states the same boundary at the start of the stochastic-integration construction. The left/right limit mismatch is the standard quadratic-variation computation, included so that the failure is witnessed by an explicit pair of evaluation rules rather than only by the failure of a hypothesis.

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 2.8 (standard reference, not scraped)
Aad van der Vaart, Stochastic Integration and Differential Equations, Section 5.1 (standard reference, not scraped)