Predictable Processes and Stopping Times

In a stochastic integral the position multiplying each price increment must be committed before that increment is revealed. That non-anticipation requirement is encoded by a single object, the predictable $\sigma$ -algebra, and the integrands of stochastic calculus are exactly its measurable processes.

#Filtrations and stopping times

A filtration $(\F_t)_{t\ge 0}$ on $(\Omega,\F,\P)$ is an increasing family of sub- $\sigma$ -algebras, $\F_s\subseteq\F_t$ for $s\le t$ . It satisfies the usual conditions when $\F_0$ contains every $\P$ -null set and the filtration is right-continuous, $\F_t=\bigcap_{u>t}\F_u$ . A process $X$ is adapted when each $X_t$ is $\F_t$ -measurable.

Definition1

A stopping time is a map $\tau:\Omega\to[0,\infty]$ with $\{\tau\le t\}\in\F_t$ for every $t\ge 0$ . Its stopping-time $\sigma$ -algebra is

\F_\tau=\big\{A\in\F : A\cap\{\tau\le t\}\in\F_t\ \text{for all } t\ge 0\big\}. \tag{1}

Proposition2

The collection $\F_\tau$ is a $\sigma$ -algebra, $\tau$ is $\F_\tau$ -measurable, and for stopping times $\sigma\le\tau$ one has $\F_\sigma\subseteq\F_\tau$ . Moreover $\sigma\wedge\tau$ and $\sigma\vee\tau$ are stopping times.

Proof

That $\F_\tau$ is a $\sigma$ -algebra follows because the defining condition in Equation (1) is preserved under complementation and countable unions. Countable unions are immediate from $(\bigcup_n A_n)\cap\{\tau\le t\}=\bigcup_n(A_n\cap\{\tau\le t\})\in\F_t$ . The complement step is the only nontrivial one and uses $\{\tau\le t\}\in\F_t$ . If $A\in\F_\tau$ then $A^c\cap\{\tau\le t\}=\{\tau\le t\}\setminus(A\cap\{\tau\le t\})\in\F_t$ as a difference of two sets in $\F_t$ , so $A^c\in\F_\tau$ . For measurability of $\tau$ , the event $\{\tau\le s\}\cap\{\tau\le t\}=\{\tau\le\min(s,t)\}\in\F_{\min(s,t)}\subseteq\F_t$ , so $\{\tau\le s\}\in\F_\tau$ for every $s$ and these generate. If $\sigma\le\tau$ and $A\in\F_\sigma$ , then $A\cap\{\tau\le t\}=\big(A\cap\{\sigma\le t\}\big)\cap\{\tau\le t\}\in\F_t$ , giving $A\in\F_\tau$ . Finally $\{\sigma\wedge\tau\le t\}=\{\sigma\le t\}\cup\{\tau\le t\}\in\F_t$ and $\{\sigma\vee\tau\le t\}=\{\sigma\le t\}\cap\{\tau\le t\}\in\F_t$ .

A process $X$ is progressively measurable when, for each $t$ , the map $(s,\omega)\mapsto X_s(\omega)$ on $[0,t]\times\Omega$ is $\mathcal B([0,t])\otimes\F_t$ -measurable. Every right- or left-continuous adapted process is progressive, since it is the pointwise limit of adapted step processes that are product-measurable [1].

#The predictable sigma-algebra

Work on $\Omega\times(0,\infty)$ . A simple predictable process has the form

H_t(\omega)=\sum_{i=1}^{m} h_i(\omega)\,\ind_{(t_i,\,t_{i+1}]}(t), \tag{2}

with $0\le t_1<\cdots<t_{m+1}$ and each $h_i$ bounded and $\F_{t_i}$ -measurable. The value on $(t_i,t_{i+1}]$ is settled by $\F_{t_i}$ , the information available at the left endpoint, which is the algebraic form of non-anticipation.

Definition3

The predictable $\sigma$ -algebra $\Pred$ on $\Omega\times(0,\infty)$ is generated by the left-continuous adapted processes, viewed as maps $(\omega,t)\mapsto X_t(\omega)$ . A process is predictable when it is $\Pred$ -measurable.

Proposition4

The predictable $\sigma$ -algebra is generated by the simple predictable processes, equivalently by the rectangles $(s,t]\times A$ with $A\in\F_s$ . Every left-continuous adapted process is predictable.

Proof

The simple predictable processes and the rectangles generate the same $\sigma$ -algebra. Each rectangle indicator $\ind_{(s,t]\times A}=\ind_A\,\ind_{(s,t]}$ is simple predictable, with $\ind_A$ bounded and $\F_s$ -measurable. Conversely a single term $h\,\ind_{(a,b]}$ with $h$ bounded and $\F_a$ -measurable is rectangle-measurable, since for Borel $B$ the set $\{h\,\ind_{(a,b]}\in B\}$ is built from $\{h\in B'\}\times(a,b]$ with $\{h\in B'\}\in\F_a$ and its time-complement, both rectangle-generated. So $\sigma(\text{simple})=\sigma(\text{rectangles})$ .

Each rectangle indicator is also left-continuous in time, so the $\sigma$ -algebra generated by the rectangles is contained in $\Pred$ . For the reverse inclusion, let $X$ be left-continuous and adapted and set

X^n_t=\sum_{i\ge 0} X_{i/2^n}\,\ind_{(i/2^n,\,(i+1)/2^n]}(t). \tag{3}

Each $X^n$ is a countable sum of terms $X_{i/2^n}\,\ind_{(i/2^n,(i+1)/2^n]}$ , each rectangle-measurable since $\{X_{i/2^n}\in B\}\in\F_{i/2^n}$ , so $X^n$ is measurable with respect to the rectangle-generated $\sigma$ -algebra. Fix $t>0$ ; it lies in the unique dyadic interval $(i_n/2^n,(i_n+1)/2^n]$ with $i_n=\lceil t2^n\rceil-1$ , so $X^n_t=X_{i_n/2^n}$ with $i_n/2^n\uparrow t$ strictly from the left and $t-i_n/2^n\le 2^{-n}\to 0$ . Left-continuity then gives $X^n_t(\omega)=X_{i_n/2^n}(\omega)\to X_{t^-}(\omega)=X_t(\omega)$ for every $(\omega,t)$ ; the left-open intervals are what place the approximating times below $t$ . A pointwise limit of measurable functions is measurable, so $X$ lies in the rectangle-generated $\sigma$ -algebra. The two $\sigma$ -algebras therefore coincide, and both equal $\Pred$ .

Remark

Predictability is the correct measurability for an integrand because the increment $dM_t$ of a martingale is unforecastable from $\F_{t^-}$ , while a predictable $H_t$ is determined by $\F_{t^-}$ . Pairing the two in $\sum_i h_i\,(M_{t_{i+1}}-M_{t_i})$ keeps each term a martingale increment, which is what makes the stochastic integral itself a martingale. An adapted but not predictable integrand, by contrast, could peek at the increment it multiplies.

The graph of a stopping time and the stochastic interval $[\![0,\tau]\!]$ are predictable when $\tau$ is predictable, that is, announced by an increasing sequence of stopping times, the predictable stopping times of the general theory [2]. This is the structure on which the stochastic integral is built.

[1]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, 1991.

[2]

P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed. Springer, 2005.

Explore connections

see in the atlas

cite

@misc{predictable-processes,
  author = {Zac Kienzle},
  title  = {Predictable Processes and Stopping Times},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/predictable-processes}
}

#Filtrations and stopping times

Definition1

A stopping time is a map $\tau:\Omega\to[0,\infty]$ with $\{\tau\le t\}\in\F_t$ for every $t\ge 0$ . Its stopping-time $\sigma$ -algebra is

\F_\tau=\big\{A\in\F : A\cap\{\tau\le t\}\in\F_t\ \text{for all } t\ge 0\big\}. \tag{1}

Proposition2

Proof

#The predictable sigma-algebra

Work on $\Omega\times(0,\infty)$ . A simple predictable process has the form

H_t(\omega)=\sum_{i=1}^{m} h_i(\omega)\,\ind_{(t_i,\,t_{i+1}]}(t), \tag{2}

Definition3

Proposition4

Proof

X^n_t=\sum_{i\ge 0} X_{i/2^n}\,\ind_{(i/2^n,\,(i+1)/2^n]}(t). \tag{3}

Remark

[1]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, 1991.

[2]

P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed. Springer, 2005.

Explore connections

see in the atlas

cite

@misc{predictable-processes,
  author = {Zac Kienzle},
  title  = {Predictable Processes and Stopping Times},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/predictable-processes}
}