Quasimartingale

A quasimartingale is a concept from stochastic processes and refers to a stochastic process that has finite mean variation. Quasimartingales are generalizing semimartingales in the sense as they do not have to be càdlàg, and they are exactly semimartingales if they are càdlàg. Quasimartingales were introduced by the American mathematician Donald Fisk in 1965.{{Cite journal | author = Donald L. Fisk | title = Quasi-martingales | journal = Transactions of the American Mathematical Society | year = 1965 | volume = 120 | issue = 3 | pages = 369–389 | doi = 10.1090/S0002-9947-1965-0192542-5 | url = https://www.ams.org/tran/1965-120-03/S0002-9947-1965-0192542-5| url-access = subscription }}

Some authors use the term as a synonym for semimartingale and assume the process is càdlàg.

== Quasimartingale ==

Let $(\backslash Omega,\; \backslash mathcal\{F\},\; (\backslash mathcal\{F\}\_t)\_\{t\; \backslash geq\; 0\},\; P)$ be a filtred probability space and let $\backslash tau$ be a partition of the interval $[0,\; \backslash infty]$. Further, let $X\; =\; (X\_t)\{t\; \backslash geq\; 0\}$ be an adapted stochastic process. The (mean) variation of $X$ is defined as

:$\backslash operatorname\{Var\}(X)\; :=\; \backslash sup\backslash limits\_\{\backslash tau\}\; \backslash mathbb\{E\}\backslash left[\backslash sum\backslash limits\_\{i=0\}^n\; \backslash left|\backslash mathbb\{E\}[X\_\{t\_i\}\; -\; X\_\{t\_\{i+1\}\}\; |\; \backslash mathcal\{F\}\_\{t\_i\}]\backslash right|\backslash right]$

The process $X$ is a quasimartingale if for all $t$ and the process has finite variation:

:{{Cite book | author = Philip E. Protter | editor = Springer | title = Stochastic Integration and Differential Equations | year = 2004 | isbn = 3-540-00313-4 | pages = 116}}