DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pathwise and reparameterized simulation-gradient estimators

Definition

Use Simulation objectives and unbiased payoff oracles and write $J (θ) = E [H (θ, Z)]$ using one fixed random input $Z$ whose law does not depend on $θ$ . If, for almost every input, the map $θ \mapsto H (θ, Z)$ is differentiable, its pathwise (or reparameterized) gradient estimator is $G_{p w} (θ, Z) : = D_{θ} H (θ, Z) .$ When $H (θ, Z) = h (T (θ, Z), θ)$ , with differentiable $T$ and $h$ , the chain rule gives $G_{p w} = D_{x} h (T (θ, Z), θ) D_{θ} T (θ, Z) + \partial_{θ} h (T (θ, Z), θ) .$ The estimator differentiates the simulated outcome while holding the same random input fixed. This definition alone does not assert that its expectation equals $\nabla J (θ)$ .

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael C. Fu, Stochastic Gradient Estimation, preprint of Handbook of Simulation Optimization, §3.1 (standard reference, not scraped)
Paul Glasserman, Monte Carlo Methods in Financial Engineering, §7.2 (standard reference, not scraped)