Optional stopping theorem
In probability theory, the optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its in
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of
Minimal-entropy martingale measure
In probability theory, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the entropy difference between the objective probability measure, , and the
Martingale representation theorem
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of a
Azuma's inequality
In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppos
Martingale (probability theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equ
Forward measure
In finance, a T-forward measure is a pricing measure absolutely continuous with respect to a risk-neutral measure, but rather than using the money market as numeraire, it uses a bond with maturity T.
Kazamaki's condition
In mathematics Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's the
Fleming–Viot process
In probability theory, a Fleming–Viot process (F–V process) is a member of a particular subset of probability measure-valued Markov processes on compact metric spaces, as defined in the 1979 paper by
Doob's martingale inequality
In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartin
Doob martingale
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the
McDiarmid's inequality
In probability theory and theoretical computer science, McDiarmid's inequality is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain func
Doob's martingale convergence theorems
In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American ma
Doob decomposition theorem
In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and stochastic pr
Martingale difference sequence
In probability theory, a martingale difference sequence (MDS) is related to the concept of the martingale. A stochastic series X is an MDS if its expectation with respect to the past is zero. Formally
Wald's martingale
In probability theory Wald's martingale, named after Abraham Wald and more commonlyknown as the geometric Brownian motion, is a stochastic process of the form for any real value λ where Wt is a Wiener
Tanaka's formula
In the stochastic calculus, Tanaka's formula states that where Bt is the standard Brownian motion, sgn denotes the sign function and Lt is its local time at 0 (the local time spent by B at 0 before ti
Uniform integrability
In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.
Local martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale
Semimartingale
In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimarting
Martingale central limit theorem
In probability theory, the central limit theorem says that, under certain conditions, the sum of many independent identically-distributed random variables, when scaled appropriately, converges in dist
Doob–Meyer decomposition theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasi
Novikov's condition
In probability theory, Novikov's condition is the sufficient condition for a stochastic process which takes the form of the Radon–Nikodym derivative in Girsanov's theorem to be a martingale. If satisf
Sigma-martingale
In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 19
Doléans-Dade exponential
In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation where denotes the process of