Brownian Motion and Stochastic Calculus

Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer.

Type

Book

Intellectual base

Review or monograph

Year

1991

DOI

10.1007/978-1-4612-0949-2

ISBN

9780387976556

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-1-4612-0949-2

Cited from

mathematics

Cited by

7 citations across 7 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Stochastic Process
  • The index set is continuous time \(t \ge 0\); the state space is the real line; at each index \(W_t\) is a random variable (Gaussian with mean $0$ and variance \(t\)); and the single joint law is specified by the finite-dimensional distributions: for any finite set of times \(0 \le t_1 < \cdots < t_n\), the vector \((W_{t_1}, \dots, W_{t_n})\) is jointly Gaussian with covariance \(\operatorname{Cov}(W_s, W_t) = \min(s,t)\).

    This sourceStandard reference for the Wiener process, its finite-dimensional Gaussian law with min(s,t) covariance, and its Markov, martingale, and Lévy structure.

    Supported in partVerified against the publisher's abstract

    The publisher abstract confirms this is a graduate text built around Brownian motion, but it says nothing about Gaussian finite-dimensional distributions or the min(s,t) covariance, so the quantified half of the claim is unread.

    “The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths.”

    Read by an automated reader; not a human review. How support was checked

Domain-specific

• Diffusion Process
  • A mathematical model must state which solution concept and boundary behavior it uses.
• Generalized Wiener process
  • … identity and a convenient observable transfers from Geometric Brownian motion uses drift proportional to the current state and volatility proportional to that state. to A modeler distinguishes informal white-noise notation from a well-defined SDE and verifies existence, boundary and calibration assumptions..
• Itô isometry
  • … or martingale, predictable or adapted integrand, square-integrability, integration interval, stochastic-integral construction, exact expectation equality and extension argument are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Natural filtration
  • … transfers from A canonical instance directly demonstrates that each time-indexed sigma-algebra is generated precisely by the process history through that time and the family is increasing. to An applied instance preserves the same invariant under a changed notation, scale, implementation, or empirical setting..
• Progressively measurable process
  • Progressive measurability is stronger than adaptedness in general and supports measurable stopping and Itô integration; path regularity can imply it under standard filtration conditions.
• Reflection principle (Wiener process)
  • The reflection principle states that reflecting a standard Wiener path about level \(a\) after its first hitting time \(\tau_a=\inf\{t:W_t=a\}\) produces another Wiener process with the same distribution.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 1 of 7 citations: 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Registry ID ref:1d4d30016e89 · see in the full table