properties of X-integrable processes

Let X be a semimartingale. Then a predictable process ξ is X-integrable if the stochastic integral ξ𝑑X is defined, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the set

{0tα𝑑X:|α||ξ| is predictable}

being bounded in probability, for each t>0. We list some properties of X-integrable processes.

  1. 1.

    Every locally bounded predictable process is X-integrable.

  2. 2.

    The X-integrable processes are closed under linear combinationsMathworldPlanetmath. That is, if α,β are X-integrable and λ,μ, then λα+μβ is X-integrable.

  3. 3.

    If |α||β| are predictable processes and β is X-integrable, then so is α.

  4. 4.

    A process is X-integrable if it is locally X-integrable. That is, if there are stopping times τn almost surely increasing to infinityMathworldPlanetmath and such that 1{tτn}ξt is X-integrable, then ξ is X-integrable.

Title properties of X-integrable processes
Canonical name PropertiesOfXintegrableProcesses
Date of creation 2013-03-22 18:40:59
Last modified on 2013-03-22 18:40:59
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 60H10
Classification msc 60G07
Classification msc 60H05
Related topic StochasticIntegration