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

Link

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

Cited from

mathematics

Cited by

6 citations across 6 artifacts.

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

Primes

• Stochastic Process
  • … 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.

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..
• 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

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