Progressive, optional and predictable measurability

Let $\left\{ \mathcal{F}_{t}\right\}$ be a filtration satisfying the usual conditions and let $\mathcal{B}[0, t]$ be the Borel $\sigma$-field on $[0, t]$. A real-valued stochastic process $X$ is progressively measurable if for each $t \geq 0$, the map $(s, \omega ) \rightarrow X(s, \omega )$ from $[0, t] \times \Omega$ to $\mathbb {R}$ is measurable with respect to the product $\sigma$-field $\mathcal{B}[0, t] \times \mathcal{F}_{t}$.

Give an example of an adapted process that is not progressively measurable.

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Prove that, equivalently, the optional $\sigma$-algebra $\mathscr {O}$ can be defined as the $\sigma$-algebra of subsets of $\mathbb {R}_{+} \times \Omega$ generated by the stochastic intervals of the form

where $\tau (\omega )$ are Markov times (with respect to the filtration $\mathbb {F}$ ).

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Prove that, equivalently, the predictable $\sigma$-algebra $\mathscr {P}$ can be defined as the $\sigma$-algebra generated by all adapted processes whose sample paths are merely left-continuous on $(0, \infty )$.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Verify that the $\sigma$-algebra $\mathscr {P}$ is generated by each of the following classes of subsets:

Definition 7. A subset $A$ of $\mathbb {R}_{+} \times \Omega$ is called progressively measurable (or simply progressive) if the process $X = \mathbf{1}_{A}$ (i.e., $X_{t}(\omega ) = \mathbf{1}_{A}(t, \omega )$, $t \geqslant 0, \omega \in \Omega$) is progressively measurable.

The collection of progressive sets $A$ forms a $\sigma$-algebra, denoted $\operatorname {Prog}$; here $\mathscr {F}_{\infty } = \bigvee_{t \geqslant 0} \mathscr {F}_{t}$, where $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ is the filtration under consideration.

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Verify that the following inclusions hold among the $\sigma$-algebras introduced above:

Let the process $X$ be progressively measurable and let $\tau$ be a finite Markov time. Show that the random variable $X_{\tau }$ is $\mathscr {F}_{\tau }$-measurable (this property is very nearly what makes it worthwhile to introduce the notion of progressive measurability at all).

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Verify that simple (piecewise right-continuous) functions $f = f(t, \omega )$ of the form

where the variables $\xi_{k}(\omega )$ are $\mathscr {F}_{t_{k}}$-measurable and $0 = t_{0} < \cdots < t_{m} = T$, are optional.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Show that simple (piecewise left-continuous for $t \in (0, T]$ ) functions $f = f(t, \omega )$ of the form

where the variable $\eta (\omega )$ is $\mathscr {F}_{0}$-measurable, the $\eta_{k}(\omega )$ are $\mathscr {F}_{t_{k}}$-measurable, and $0 = t_{0} < \cdots < t_{m} = T$, are predictable.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Wiener process, $\mathscr {F}_{t}^{W}$ the $\sigma$-algebra generated by the random variables $W_{s}$, $0 \leqslant s \leqslant t$, $\mathbb {F}_{T}^{W} = \left(\mathscr {F}_{t}^{W}\right)_{t \in [0, T]}$ and $\mathscr {F}^{W} = \sigma \left\{ W_{t}, t \geqslant 0\right\}$; finally, $L_{T}^{2} = L^{2}\left(\Omega \times [0, T], \mathscr {F}^{W} \otimes \mathscr {B}([0, T]), \mathbb {P} \otimes \operatorname {mes}\right)$.

Definition 1. We say that a random function $f = \left\{ f_{t}, t \in [0, T]\right\}$ is nonanticipating with respect to the filtration $\mathbb {F}_{T}^{W}$ if it is adapted to $\mathbb {F}_{T}^{W}$ (i.e., for each $t \in [0, T]$ the random variable $f_{t}$ is $\mathscr {F}_{t}^{W}$-measurable) and is $\mathscr {B}([0, T]) \otimes \mathscr {F}^{W}$-measurable in the pair of variables $(t, \omega ) \in [0, T] \times \Omega$. The class of such random functions is denoted by $\mathscr {A}_{T}$.

Carry out the construction of the Itô integral presented in § 2 and § 3, starting from the class of optional simple functions appearing in Exercise 7 (to which the same value of the Itô integral is assigned as to the predictable simple functions of Exercise 8). Can one assert that the closure in $L_{T}^{2}$ of this collection of simple functions belonging to $L_{T}^{2}$ yields the space $\mathscr {A}_{T} \cap L_{T}^{2}$ ?

The idea of the construction of the Itô integral in § 2 and § 3 is as follows: (1) first the integrals are defined (in a quite natural way) for piecewise constant "nonanticipating" random functions, and then (2) for more general functions $f$, by approximating them by simple functions $f^{(n)}$ and relying on isometry ideas, the stochastic integrals $I_{T}(f)$ are defined as the limits (in the mean-square sense) of the variables $I_{T}\left(f^{(n)}\right)$, $n \rightarrow \infty$.

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a one - dimensional Brownian motion and $\tau$ a stopping time. Show that $f(s, \omega ): = \; \mathbb {1}_{[0, T \wedge \tau (\omega ))}(s), 0 \leqslant s \leqslant T < \infty$ is progressively measurable. Use the very definition of the Itô integral to calculate $\int_{0}^{T} f(s) d B_{s}$ and compare the result with the localization principle of Theorem 15.15.

Show that

is a $\sigma$ - algebra.

Show that every right or left continuous adapted process is $\mathscr {P}$ measurable.

Let $\tau$ be a stopping time. Show that $\; \mathbb {1}_{[0, \tau )}$ is $\mathscr {P}$ measurable.

Show that the process $f^{2} \bullet \left\langle M\right\rangle_{t}: = \int_{0}^{t}\left|f(s)\right|^{2} d\left\langle M\right\rangle_{s}$ appearing in Theorem 17.9.b) is adapted.