Stochastic processes filtrations and stopping times

Definition Fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a time set $\mathbb{T}$, either the dates $\{0, 1, \dots, N\}$ or the interval $[0, \infty)$. A filtration is a family $(\mathcal{F}_t)_{t \in \mathbb{T}}$ of sigma algebras with

where an event belongs to $\mathcal{F}_t$ when the information available at time $t$ decides whether it has occurred. A stochastic process $X = (X_t)_{t \in \mathbb{T}}$ is a family of random variables on $\Omega$, one for each time. It is adapted to the filtration when

for every $t \in \mathbb{T}$. The natural filtration $\mathcal{F}^X_t = \sigma(X_u : u \le t)$ is the smallest filtration to which $X$ is adapted. Adaptedness is a statement about information only. It says nothing about continuity or smoothness, or about how well future values can be forecast, and a process adapted to one filtration need not be adapted to a smaller one.

Definition In discrete time, let $H_k$ be the number of shares held from date $k$ to date $k + 1$, for $k = 0, \dots, N - 1$. The strategy is predictable when each $H_k$ is $\mathcal{F}_k$-measurable, so every holding is fixed with the information available when the position is opened. Many texts index the same holding by the right endpoint of its period, and then predictability says that the holding over the period from $k - 1$ to $k$ is $\mathcal{F}_{k-1}$-measurable. The gain process, in currency units, is

Suppose $S$ is a martingale under $\mathbb{P}$ for the filtration, so each $S_k$ is integrable and $\mathbb{E}[\Delta S_{k+1} \mid \mathcal{F}_k] = 0$ with $\Delta S_{k+1} = S_{k+1} - S_k$, and suppose each $H_k$ is bounded. Then the known factor comes out of the conditional expectation,

so $G$ is a martingale and $\mathbb{E}[G_n] = 0$ for every $n$. Boundedness of $H_k$ can be weakened to integrability of each product $H_k \Delta S_{k+1}$. The argument uses that $H_k$ is $\mathcal{F}_k$-measurable and fails for a holding that depends on $\Delta S_{k+1}$.

Definition In continuous time with a filtration $(\mathcal{F}_t)_{t \ge 0}$, the predictable sigma algebra $\mathcal{P}$ is the sigma algebra on $[0, \infty) \times \Omega$ generated by all adapted processes with left-continuous paths, each viewed as the map $(t, \omega) \mapsto X_t(\omega)$. Equivalently, it is generated by the sets $\{0\} \times A$ with $A \in \mathcal{F}_0$ and $(s, t] \times A$ with $s < t$ and $A \in \mathcal{F}_s$. A process is predictable when it is measurable with respect to $\mathcal{P}$. A simple predictable process has the form

with times $0 = t_0 < t_1 < \dots < t_n$ and each $a_i$ bounded and $\mathcal{F}_{t_i}$-measurable, and its gain against a price $S$ up to time $t_n$ is $\sum_{i} a_i (S_{t_{i+1}} - S_{t_i})$. Every left-continuous adapted process, in particular every continuous adapted process, is predictable, and every predictable process is adapted. An adapted process with jumps need not be predictable.

Definition A random time $\tau$ with values in $\mathbb{T} \cup \{\infty\}$ is a stopping time for the filtration when

for every $t \in \mathbb{T}$. In discrete time this is equivalent to $\{\tau = k\} \in \mathcal{F}_k$ for every date $k$. The value $\infty$ means that the event never happens. Constant times are stopping times, and so are the minimum and the maximum of two stopping times. For a continuous adapted price $S$ and a lower barrier $b$, the first hitting time

with $\inf \varnothing = \infty$, is a stopping time. The set of levels at most $b$ is closed, so $\tau_b \le t$ exactly when the minimum of $S$ over $[0, t]$ is at most $b$, and by continuity that minimum equals the infimum over the rational times in $[0, t]$ together with $t$, which is $\mathcal{F}_t$-measurable. For the strict condition $S_t < b$ the first hitting time is a stopping time once the filtration is right-continuous, which is one reason for the usual conditions below.

Definition A filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ satisfies the usual conditions when the probability space is complete, $\mathcal{F}_0$ contains every $\mathbb{P}$-null set of $\mathcal{F}$, and the filtration is right-continuous, which means $\mathcal{F}_t = \mathcal{F}_{t+}$ for every $t \ge 0$, where

Every filtration can be enlarged to a smallest filtration that satisfies both conditions, its usual augmentation. Completeness ensures that a process equal at each time, almost surely, to an adapted process is itself adapted. Right-continuity ensures that the first time a right-continuous adapted process enters an open set is a stopping time. A standard result, stated here without proof, is that the completed natural filtration of a Brownian motion is already right-continuous and that the process is still a Brownian motion for it. This lesson defines the usual conditions and uses them only as hypotheses.

Definition A trading strategy is anticipatory when some holding is not measurable with respect to the information available when the position is opened, so in discrete time some $H_k$ is not $\mathcal{F}_k$-measurable. Its gains are outside the predictable-gain argument above, and an exercise or stopping rule that is not a stopping time anticipates in the same sense. Revealing a future move early, by replacing $\mathcal{F}_k$ with a larger filtration $\mathcal{G}_k$ for which $\Delta S_{k+1}$ is $\mathcal{G}_k$-measurable, changes the model. Then $\mathbb{E}[\Delta S_{k+1} \mid \mathcal{G}_k] = \Delta S_{k+1}$, so the increment no longer has conditional mean zero unless it is zero almost surely, and $S$ is not a martingale for $(\mathcal{G}_k)$.