Quasimartingale

An adapted stochastic process of finite mean variation; with càdlàg paths it coincides with a semimartingale.

Version

v1 · 2026-09-28 · History

Domain-specific #

11629

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Stochastic Processes, Stochastic Integration → Mathematics

Aliases

Quasi-martingale, Finite Mean Variation Process

Core Idea

A quasimartingale is an adapted integrable stochastic process with finite mean variation, defined as a supremum over partitions of conditional expected increments plus the terminal expectation.

Partition sums of conditional expected increments remain uniformly bounded. A càdlàg quasimartingale falls into the semimartingale class.

Structural Signature

Sig role-phrases:

• Filtered space — Supplies time-indexed information. It is context. Counterfactual: No filtration means no conditional increments.
• Adapted process — Makes each value observable at its time. It is object. Counterfactual: Anticipating values violate the definition.
• Partitions — Discretize time for variation sums. It is test. Counterfactual: One partition cannot establish the supremum.
• Conditional increments — Measure predictable mean change. It is term. Counterfactual: Raw path variation is different.
• Finite supremum — Defines quasimartingale status. It is bound. Counterfactual: Infinite mean variation fails.
• Càdlàg regularity — Connects to semimartingales. It is special case. Counterfactual: Without it the classes need not coincide.

What It Is Not

• It is not finite pathwise variation.
• It is not necessarily càdlàg.
• It is not nonadapted.
• It is not automatically a martingale.
• Closest near-miss. A semimartingale is a close neighbor; equivalence requires the path-regularity convention stated by the source.

Scope of Application

• Stochastic processes. Classifies integrator-like processes.
• Semimartingale theory. Uses the càdlàg equivalence.
• Probability. Studies conditional variation.
• Finance. Clarifies admissible process classes.

Clarity

Include adapted integrable processes whose conditional mean-variation supremum is finite. Exclude finite pathwise variation alone, nonadapted processes, and semimartingale synonymy without a càdlàg convention.

Manages Complexity

The definition controls conditional drift, not total sample oscillation. Dropping càdlàg broadens the class.

Abstract Reasoning

1. Filtered space — Supplies time-indexed information. No filtration means no conditional increments.
2. Adapted process — Makes each value observable at its time. Anticipating values violate the definition.
3. Partitions — Discretize time for variation sums. One partition cannot establish the supremum.
4. Conditional increments — Measure predictable mean change. Raw path variation is different.
5. Finite supremum — Defines quasimartingale status. Infinite mean variation fails.
6. Càdlàg regularity — Connects to semimartingales. Without it the classes need not coincide.

Knowledge Transfer

Finite-mean-variation analysis transfers among filtered processes only with integrability, adaptation, and partition conventions fixed; semimartingale conclusions require the stated path regularity.

Examples

Applied / In Practice

Partition sums of conditional expected increments remain uniformly bounded.

Mapped back: supremum → finite.

Applied / In Practice

A càdlàg quasimartingale falls into the semimartingale class.

Mapped back: paths → cadlag.

Structural Tensions

T1 — Mean Variation versus Path Variation. The definition controls conditional drift, not total sample oscillation.

Diagnostic: Which variation is measured?

T2 — General Paths versus Semimartingale Regularity. Dropping càdlàg broadens the class.

Diagnostic: Which convention is used?

Structural–Framed Character

Supplies time-indexed information. Makes each value observable at its time. The definition controls conditional drift, not total sample oscillation.

Structural Core vs. Domain Accent

Defines quasimartingale status. Connects to semimartingales. The class ends when mean variation diverges.

Instantiates / Related Primes

This entry is a kind of Stochastic Process.

• Approved root. The frozen graph retains quasimartingale without a parent edge.

• Related — Martingale and Semimartingale. Has zero conditional drift. Coincides under càdlàg regularity.

Relationships to Other Abstractions

Current abstraction Quasimartingale Domain-specific

Parents (1) — more general patterns this builds on

• Quasimartingale is a kind of Stochastic Process Prime

  Quasimartingale is a strict kind of Stochastic Process: it is an adapted stochastic process satisfying finite mean variation.

Hierarchy path (1) — routes to 1 parentless root

• Quasimartingale → Stochastic Process

Neighborhood in Abstraction Space

Quasimartingale sits in a crowded region of the domain-specific corpus (40^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Market & Economic Indices (5 abstractions)

Nearest neighbors

• Concurrent Estimation — 0.88
• Etemadi's Inequality — 0.88
• First-Hitting-Time Model — 0.88
• Maximising measure — 0.87
• Probability Density Function — 0.87

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

• Martingale. Tell: Has zero conditional drift.
• Semimartingale. Tell: Coincides under càdlàg regularity.
• Finite-variation process. Tell: Uses pathwise variation.
• Submartingale. Tell: Has one-sided conditional drift.

References

• Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Quasimartingale (revision 1329653434).
• Preserved source candidate: https://www.ams.org/tran/1965-120-03/S0002-9947-1965-0192542-5

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.