Stochastic Calculus : Discrete Time Processes

This is a review of chapter 1 of Capinski’s book, however the overview will be slightly different.

definition: A measurable space is a pair $(\Omega,S)$ where $\Omega$ is a set and $S$ is a sigma-algebra on $X$ .

The role of the $S$ is to model the information available through observation. It is a sigma algebra to enable attachment of a measure. An element of $S$ is called a measurable set.

If $\Omega = \mathbb{R}^n$ , then $S = \mathcal{B}^n$ the set of measurable sets generated from the open sets of the standard metric induced by norm.

definition: A measurable map is a function on measurable spaces $f : (X, S) \rightarrow (Y,T)$ such that $f^{-1} T \subseteq S$

The composition of measurable maps is a measurable map. A map is measurable with respect to information $U$ or $U$ -measurable if $f = f | (X,U)$ . IN other words, the information $f$ provides is contained in $U$ .

definition: A random variable is a real-valued measurable map.

definition: A probability space $X$ is a measurable space equipped with a probability measure defined on the measurable sets of the space.

definition: A real discrete-time process is a real-valued sequence. A random discrete-time process is a sequence of random variables.

For a given outcome $\omega \in \Omega$ of a measurable space, a random process $f$ is realized as a real sequence $f(\omega)$ . The value of the process at time $n$ is $f(n)$ .

definition: A filtration is a non-decreasing sequence of collections of measurable sets.

If $\mathcal{F}$ is a filtration, then $\mathcal{F}_n$ is the information that can be measured at time $n$ .

definition: The natural filtration of a random variable $f$ is the filtration $\mathcal{F}^f$ generated from the $f^{-1}(i) \mathcal{B}$ for $i = 1,\ldots, n$ : $\mathcal{F}^f(n) = \sigma \cup_{i=1..n} f^{-1}(n) \mathcal{B}$ .

definition: A random process $f$ is adapted to a filtration $\mathcal{F}$ if for all $n \ge 1$ , $f^{-1} \mathcal{B}(n) \subseteq \mathcal{F}(n)$

definition: A filtered measurable space is a measurable space equipped with a filtration of the measurable sets of the space.

This structure models the causal information flow. Here information will be taken to be synomous with sigma algebra, and information flow with a filtration.

definition: The conditional expectation of a random variable $X : (\Omega, \mathcal{F}) \rightarrow (\mathbb{R},\mathcal{B})$ with respect to information $\mathcal{G}$ is an integrable random variable $Y : (\Omega, \mathcal{F}) \rightarrow (\mathbb{R},\mathcal{B})$ such that $Y$ is $\mathcal{G}$ -measurable, and for any $\mathcal{G}$ -measurable set $A, \mathbb{E} X \mathbb{I}_A = \mathbb{E} Y \mathbb{I}_A$ .

Since it can be shown that the collection of all such conditional expectations are identical to each other almost everywhere, it is customary to ignore the difference and identify the conditional expectation by $\mathbb{E} [X | \mathcal{G}]$ or $\mathbb{E}_\mathcal{G} X$ .

A consequence of the definition is the sequence of conditional expectations defined on a filtration satisfy the so-called tower law:

lemma: when $\mathcal{F}$ is a filtration, $\mathbb{E}_{\mathcal{F}_n} X = \mathbb{E}_{\mathcal{F}_n} \mathbb{E}_{\mathcal{F}_{n+1}} X$

It is convenient to simplify notation and write $\mathbb{E}_n = \mathbb{E}_{{\mathcal F}_n}$ when it is clear the conditional expectation is with respect to the filtration of a filtered space. Also instead of saying $f$ is $\mathcal{F}_n$ -measurable, it is convenient to say that $f$ is measurable at time $n$ .

definition: A random process $f$ is adapated to a filtered space if for any time $n$ , $f^{-1}(n)$ is measurable at time $n$ .

definition: A [sub | super | ] martingale of a filtration is a random process $M$ such that for any $n \ge 0, \mathbb{E}_n M_{n+1} [\ge | \le | =] M_n$ .

definition: a process $f$ is predictable if for any time $n, f(n+1)$ is adapted at time $n$ .

In other words, the value of the process at the next time instant is determined by the information available at the current time.

proposition: a predictable martingale is constant

Since $\mathbb{E}_n X(n+1) = X(n)$ but since $X_{n+1}$ is measurable at time $n, \mathbb{E}_n X(n+1) = X(n+1)$ , and so $X(n) = X(n+1)$ . By induction, for any time $n, X(n) = X(0)$ . And so for any outcome $\omega, X(\omega)$ is a constant.

theorem: Let $M$ be a martingale and $H$ be a predictable process. Provided $H$ is bounded or $H,M$ are both square-integrable, then the process defined by $X(0) = 0$ and for $n > 0$ ,

$X(n) = (H \cdot M)(n) := \sum_{j=1..n} H(j) \Delta^- M(j)$

(where $\Delta^- q(j) = q(j) - q(j-1)$ ) defines a martingale.

Note that

$\mathbb{E}_n X(n+1) = \sum_{j=1..n} [ H(j) \Delta^- M(j)] + \mathbb{E}_n [ H(n+1) \Delta^- M(n+1)]$

$\mathbb{E}_n [ H(n+1) \Delta^- M(n+1)] = H(n+1) \mathbb{E}_n \Delta^- M(n+1) = 0$

with the added technical conditions ensuring integrability of $X$ .

lemma: (Jensen’s inequality) Let $\phi$ be a convex real operator such that given a random variable $X$ defined on a probability space $\Omega$ , $\phi(X) \in L^1(\Omega)$ . Then for any information structure $\mathcal G$ of $\Omega$ , $\mathbb \phi(\mathbb{E}_\mathcal{G} X) \le \mathbb{E}_\mathcal{G} \phi(X)$ .

lemma: if $M$ is a square-integrable martingale, then $M^2$ is a submartingale.

By Jensen, $\mathbb{E}_n M(n+1)^2 \ge (\mathbb{E}_n M(n+1))^2 = M_n^2$

definition: a process $X$ is non-decreasing if for any $\omega$ , $X(\omega)$ is a non-decreasing sequence.

proposition: an increasing martingale is constant.

Since $M$ is increasing $\Delta^- M \ge 0$ , and for all $n, \mathbb{E}_n \Delta^- M(n+1) = 0$ implies $\Delta^- M = 0$ . and so $M$ is constant.

theorem: (Doob) If $Y$ is a submartingale of a filtered space $\Omega$ , then there is a martingale $M$ and a non-decreasing predictable process $A$ defined on $\Omega$ such that $M(0) = A(0) = 0$ and $Y = Y(0) + M + A$ . Furthermore, the representation is (a.e.) unique.

Doob provide the proof necessary to demonstrate one can always convert a submartingale into a martingale by adding its compensator. Of particular interest is the square of a martingale.

proposition: If $M$ is a square integrable martingale such that $M(0) = 0$ , then when $n > m$ , $\mathbb{E}_m [M(n) - M(m)]^2 = \mathbb{E}_m M(n)^2 - M(m)^2$

This is a key result for discrete Ito integrals

proposition: (Ito Isometry) If $H \cdot M$ is square integrable then $\mathbb{E} (H \cdot M)^2 = \mathbb{E} (H^2 \cdot A)$ where $A$ is the compensator of $M^2$ .

This replaces the stochastic integrator $M$ with a predictable integrator $A$ under appropriate technical conditions.

Stopping Times

definition: A stopping time is a random variable $T$ whose range is $0..$ defined on a filtered space $\Omega$ such that such that $\mathbb{I}_{T \le n}$ is $n$ -measurable.

The role of a stopping time is to model an adapted process that announces when something occurs at a given time (or if infinite, never occurs) .

proposition: Constant stopping times, max/min of stopping times, and the limit of a convergent sequence of stopping are stopping times.

definition: Let $X$ be an adapted process defined on a filtered space and $\tau$ be a stopping time that is (almost surely) finite. Then the random variable $X(\tau) \mathbb{I}(\tau < \infty)$ is the value of the stopped process at the stopping time.

definition: for a finite stopping time $\tau$ , the information known at time $\tau$ is $\mathcal{F}_\tau = \sigma \{A \in \mathcal{F} : A \cap [\tau = m] \in \mathcal{F}_m \}$ .

It can be shown that $\mathcal{F}_\tau$ is sub-information of $\mathcal{F}$ , and that when $\tau,\nu$ are stopping times such that $\tau \le \nu$ almost surely then $\mathcal{F}_\tau$ is coarser than $\mathcal{F}_\nu$ , and that any stopping time is measurable with respect to its information.

definition: the process $X_\tau(n)=X(n \wedge \tau)$ is called the process $X$ stopped at $\tau$ .

proposition: $X_\tau$ is adapated.

theorem: if $M$ is a martingale and $\tau$ is a stopping time, then $M_\tau$ is a martingale.

The next result is a generalization of the martingale definition that applies to ordered stopping times.

theorem: (Doob’s optional sampling theorem) If $\tau,\nu$ are stopping times such that $\tau \le \nu$ , then for any martingale $M$ , $\mathbb{E}_\tau M_\nu = M_\tau$ .

Doob’s Inequalities and Martingale Convergence

The first inequality relates the probability of a submartingale surmounting a barrier over some time interval to the terminal expectation of the submartingale. The second inequality related the expectation of the square of the maximum of a martingale over some time interval to the terminal expectation of the square of the martingale. These are technical results that can be used to simplify analysis in the proof of the Martingale convergence theorem.

theorem: If $M$ is a non-negative submartingale, then for any time $n > 0$ and positive value $\lambda > 0$ ,

$\mathbb{P} [\max {M(i) : i=0..n} \ge \lambda] \le \lambda^{-1} \mathbb{P} [M(n) \mathbb{I}([\max {M(i) : i=0..n} \ge \lambda] )] \le \lambda^{-1} \mathbb{P} M(n)$

nd: interpret $P$ as a measure on functions where appropriate.

theorem: if $M$ is a square-integrable non-negative submartingale then $\mathbb{E} \max_{i=0..n} M(i)^2 \le 4 \mathbb{E} M(n)^2$ . Furthermore, if $\mathbb{E} M^2$ is uniformly bounded then $\mathbb{E} \sup_{i \ge 0} M(i)^2 \le 4 \sup_{i \ge 0} \mathbb{E} M(n)^2$

The main result of the section is a Martingale convergence theorem, which is an assertion that under appropriate technical conditions, the limit of a martingale is a well-defined random variable such that the martingale can be equivalently represented by the sequence of conditional expectations of the random variable.

theorem: let $M$ be $L^2$ -bounded martingale (uniformly bounded expectation of $M^2$ ). Then there is a random variable $Y$ such that $\mathbb{E} Y^2 < \infty$ and

$P[\lim_n M_n = Y] = 1$
$\lim_n \mathbb{E} (M_n - Y)^2 = 0$
for all $n$ , $M(n) = \mathbb{E}_n Y$

the limitation that stopping times be finite can be relaxed to extend the optional sampling theorem.

theorem: Let $M$ be an $L^2$ bounded martingale and $Y$ be a random variable such that $M(n) =\mathbb{E}_n Y$ . Then for any pair of stopping times $\tau,\nu$ such that $\tau \le \nu$ , $\mathbb{E}_\tau M(\nu) = M(\tau)$

Markov Process

A Markov process is a process whose next value depends only on the information determined from the current value.

definition: A random process $X$ is a markov process given a space equipped with the filter generated from $X$ , if for any bounded borel operator $f, \mathbb{E}_n fX(n+1) = \mathbb{E}_{X(n)} fX(n+1)$ .

nd: the information determined by $X(n)$ is the information $\sigma X(n)^{-1} \mathcal{B}$ .

[to be completed]

Stochastic Calculus : Discrete Time Processes

Stopping Times

Doob’s Inequalities and Martingale Convergence

Markov Process

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