Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes.^{[clarification needed ]}

Given a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )}${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )}, then a stochastic process ${\displaystyle (X_{n})_{n\in \mathbb {N} }}${\displaystyle (X_{n})_{n\in \mathbb {N} }} is predictable if ${\displaystyle X_{n+1}}${\displaystyle X_{n+1}} is measurable with respect to the σ-algebra ${\displaystyle {\mathcal {F}}_{n}}${\displaystyle {\mathcal {F}}_{n}} for each n.^[1]

Given a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}, then a continuous-time stochastic process ${\displaystyle (X_{t})_{t\geq 0}}${\displaystyle (X_{t})_{t\geq 0}} is predictable if ${\displaystyle X}${\displaystyle X}, considered as a mapping from ${\displaystyle \Omega \times \mathbb {R} _{+}}${\displaystyle \Omega \times \mathbb {R} _{+}}, is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.^[2] This σ-algebra is also called the predictable σ-algebra.

• Every deterministic process is a predictable process.^{[citation needed ]}
• Every continuous-time adapted process that is left continuous is a predictable process.^{[citation needed ]}

• Adapted process
• Martingale

• Stochastic processes

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