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# adapted process

Let $\{X_{t}\mid t\in T\}$ be a stochastic process defined on a probability space $(\Omega,\mathcal{F},P)$ and $\{\mathcal{F}_{t}\mid t\in T\}$ a filtration (an increasing sequence of sigma subalgebras of $\mathcal{F}$), where $T$ is a linearly ordered subset of $\mathbb{R}$ with a minimum $t_{0}$. Then the process $\{X_{t}\}$ is said to be *adapted to* the filtration $\{\mathcal{F}_{t}\}$ if for each $t\geq t_{0}$, $X_{t}$ is $\mathcal{F}_{t}$-measurable:

$X_{t}^{{-1}}(B)\in\mathcal{F}_{t}\mbox{ for each Borel set }B\in\mathbb{R}.$ |

A stochastic process is an *adapted process* if it is adapted to some filtration.

Synonym:

adapted

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

60A99*no label found*60G07

*no label found*

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## Corrections

extra argument by Mathprof ✓

Please add by CWoo ✓

\mathcal{F} by CWoo ✓

Something is not right.... by CWoo ✓

adapted by skubeedooo ✓

Please add by CWoo ✓

\mathcal{F} by CWoo ✓

Something is not right.... by CWoo ✓

adapted by skubeedooo ✓

## Comments

## Adapted

Hello!

'A stochastic process is an adapted process if it is adapted to some filtration' what some filtration. Can I say that astochastic process is an adapted process if (X_t)_{t\geq 0}$ est (\mathcal{F}_t)_{t\geq 0}-adapted to a filtration $(\mathcal{F}_t)_{t\geq 0}$ Obviously $(X_t)_{t\geq 0}$ is adapted to the filtration \sigma(X_t, t\geq 0).