Progressively measurable stochastic process
A progressively measurable stochastic process is a stochastic process in probability theory that still meets additional measurability criteria. Processes that can be measured progressively are a tightening of adapted processes and occur, for example, when examining stop times . They also play a central role in the construction of the Itō integral in stochastic analysis .
definition
A stochastic process is given with values in a Polish space , provided with Borel's σ-algebra and index set as well as a filtration in .
Then the stochastic process is called progressively measurable (with respect to ), if for each the mapping
defined by
always - - is measurable .
In most cases it is .
properties
- Every progressively measurable process is an adapted process and product measurable . Conversely, it can be shown that an adapted product-measurable process always has a progressively measurable modification .
- If a process is adapted and left continuous or right continuous , it can be measured progressively. Thus, due to the definition of the predictable σ-algebra , every predictable process is also progressively measurable.
- If, on the other hand, the process is almost certainly left-continuous or right-continuous, then there is a modification of the process that can be measured progressively.
- The random variable assigned to a real-valued stochastic process and a stop time is always measurable for progressively measurable stochastic processes with respect to the σ-algebra of the τ-past .
literature
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
- David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .