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 . ${\displaystyle X=(X_{t})_{t\in T}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {B}}(E)}$ ${\displaystyle T=[0,\infty )}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$ ${\displaystyle {\mathcal {A}}}$

Then the stochastic process is called progressively measurable (with respect to ), if for each the mapping ${\displaystyle \mathbb {F} }$ ${\displaystyle t\in T}$

${\displaystyle S\colon \Omega \times [0,t]\to E}$

defined by

${\displaystyle S(\omega ,s):=X_{s}(\omega )}$

always - - is measurable . ${\displaystyle {\mathcal {F}}_{t}\otimes {\mathcal {B}}([0,t])}$ ${\displaystyle {\mathcal {B}}(E)}$

In most cases it is . ${\displaystyle (E,{\mathcal {B}}(E))=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

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 . ${\displaystyle \tau }$ ${\displaystyle X_{\tau }:=X_{\tau (\omega )}(\omega )}$

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 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 574.
2. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 316.