CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-22

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The ordinary chain rule fails for Brownian motion

Statement refuted

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands, with the filtration satisfying the usual conditions, and use the $F_{0}$ -normalized everywhere-continuous representative of the standard Brownian motion.

The ordinary chain rule $d (B_{t}^{2}) = 2 B_{t} d B_{t} (claimed)$ is false for standard Brownian motion. The correct identity is $B_{t}^{2} = 2 \int_{0}^{t} B_{s} d B_{s} + t,$ and the two candidate formulas are distinguished by their expectations at every $t > 0$ : the missing term is exactly the quadratic-variation correction $t$ .

Facts & Assumptions

Given: AC, (H), the usual conditions, the $F_{0}$ -normalized everywhere-continuous adapted representative of a standard Brownian motion $B$ , and $t > 0$ .

[F1]

Correct identity. $B_{t}^{2} - t = 2 \int_{0}^{t} B_{s} d B_{s}$ up to indistinguishability, so $B_{t}^{2} = 2 \int_{0}^{t} B_{s} d B_{s} + t$ ; the integral is the localized integral of the predictable process $2 B$ , whose energy on $[0, t]$ is $E \int_{0}^{t} 4 B_{s}^{2} d s = 4 E \int_{0}^{t} B_{s}^{2} d s = 2 t^{2} < \infty$ . The Brownian square martingale Localized Ito integral Locally square-integrable predictable Brownian integrands Ito integral for square-integrable predictable processes

[F2]

Mean and second moment of the integral. A finite-energy integral $\int_{0}^{t} H d B$ is a square-integrable martingale with mean zero; in particular $E \int_{0}^{t} B_{s} d B_{s} = 0$ . The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2 Continuous-time adapted processes and martingales

[F3]

Second moment of Brownian motion. $E B_{t}^{2} = t$ : for $t > 0$ , $B_{t}$ has law $N (0, t)$ with density $(2 π t)^{- 1 / 2} e^{- x^{2} / (2 t)}$ , whose second moment is $t$ ; the value at $t = 0$ is $0$ . Standard normal and normal laws Brownian motion The standard normal density has total mass one Gaussian even moments for Brownian increments Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F4]

AC bookkeeping. Choice is declared for the ambient completeness interface. The Axiom of Choice

Counterexample

technique · direct

1.1

The alleged rule, integrated from $0$ with $B_{0} = 0$ , would give $B_{t}^{2} = 2 \int_{0}^{t} B_{s} d B_{s}$ up to indistinguishability, since the chain rule applied to $x \mapsto x^{2}$ with no quadratic correction yields exactly that identity.

F1given

2.1

Taking expectations of the alleged identity gives $E B_{t}^{2} = 2 E \int_{0}^{t} B_{s} d B_{s} = 0$ by [F2]. But the correct identity [F1] gives $E B_{t}^{2} = 2 E \int_{0}^{t} B_{s} d B_{s} + t = t$ , and [F3] confirms $E B_{t}^{2} = t$ .

F1F2F3step 1.1

3.1

Since $t > 0$ , the two values $0$ and $t$ are distinct, so the alleged chain-rule identity fails; the witness is the single process $B^{2}$ together with the two candidate formulas, and the failed conclusion is the expectation equality $E [B_{t}^{2}] = 2 E \int_{0}^{t} B_{s} d B_{s}$ for $t > 0$ .

F2F3step 2.1

4.1

Boundary and consistency cases: at $t = 0$ both candidate formulas agree, which is why the counterexample requires $t > 0$ ; the correct identity differs from the alleged one by the deterministic function $t$ , so the failure is not a null-set or version artefact; the integral in both formulas is the same object, so the discrepancy is entirely in the drift term; for $f (x) = x$ no correction appears and the ordinary rule is recovered, showing that the failure is tied to the nonvanishing second derivative; and AC enters only through [F4].

F1F2F4step 3.1∎

Source notes

Lawler, Sections 3.2--3.3, contrasts the Ito computation with the ordinary chain rule; the counterexample above isolates the discrepancy through the expectations of the two candidate formulas, using the finite-energy mean-zero property of the stochastic integral.

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Sections 3.2-3.3 (standard reference, not scraped)