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Martingales

Filtrations

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space.

A filtration $(\mathcal F_n)_{n \ge 0}$ is a non-decreasing sequence of $\sigma$-algebras: \(\mathcal F_0 \subseteq \mathcal F_1 \subseteq \cdots \subseteq \mathcal F.\)

Interpretation: $\mathcal F_n$ represents all information available up to time $n$.

Canonical example (discrete time):

  • $\Omega = {0,1}^{\mathbb N}$
  • $\mathcal F_n$ generated by the first $n$ coordinates
  • First $n$ coordinates fixed, rest free

Stopping Times

A random variable $\tau : \Omega \to \mathbb N \cup {\infty}$ is a stopping time w.r.t. $(\mathcal F_n)$ if \(\{\tau \le n\} \in \mathcal F_n \quad \text{for all } n.\)

Equivalent (discrete time): \(\{\tau = n\} \in \mathcal F_n \quad \forall n.\)

Stopping Time Properties

For stopping times $S$ and $T$, and constant $c$ the following is always true,

  • $min(S,T)$ and $max(S,T)$ is a stopping time.
  • $c$ is a stopping time therefore $min(S,c)$ and $max(S,c)$ are a stopping times.
  • S + c is a stopping time.

Not a Stopping Time

  • S+T in general (see composite stopping times.)

Examples

  • Constants: $\tau \equiv c$
  • First hitting time: $\tau = \inf{n : X_n \in A}$

Closure Properties

If $\tau, \sigma$ are stopping times, then:

  • $\min(\tau,\sigma)$, $\max(\tau,\sigma)$ are stopping times
  • $\tau + c$ (constant $c$) is a stopping time

Composite Stopping Times

If $\tau$ is a stopping time w.r.t. $(\mathcal F_n)$, and $\sigma$ is a stopping time w.r.t. the shifted filtration $(\mathcal F_{\tau+n})$, then \(\tau + \sigma\) is a stopping time w.r.t. $(\mathcal F_n)$.

(Justified via Strong Markov Property in BM settings.)

Martingales

Let $(X_n, \mathcal F_n)$ be a stochastic process.

Definition

$(X_n)$ is a martingale if:

  1. Adapted: $X_n$ is $\mathcal F_n$-measurable
  2. Integrable: $\mathbb E|X_n| < \infty, \forall n=1,2,…$
  3. Fair game:
    \(\mathbb E[X_{n+1} \mid \mathcal F_n] = X_n\)

Variants

  • Submartingale:
    \(\mathbb E[X_{n+1} \mid \mathcal F_n] \ge X_n\)
  • Supermartingale:
    \(\mathbb E[X_{n+1} \mid \mathcal F_n] \le X_n\)

**Martingale via increments (discrete time).

Let $(\mathcal F_n)$ be a filtration and $(Y_n)$ be adapted with $\mathbb E|Y_n|<\infty$ for all $n$. If the increments $\Delta Y_k := Y_k-Y_{k-1}$ satisfy \(\mathbb E(\Delta Y_k \mid \mathcal F_{k-1})=0 \quad \text{a.s. for all } k\ge 1,\) then $(Y_n,\mathcal F_n)$ is a martingale, since \(\mathbb E(Y_k\mid \mathcal F_{k-1}) =\mathbb E(Y_{k-1}+\Delta Y_k\mid \mathcal F_{k-1}) =Y_{k-1}.\)

Basic Properties

  • Constant process is a martingale
  • If $X_n$ is a martingale, then so is $\mathbb E[X_n \mid \mathcal G_n]$ for sub-filtration $\mathcal G_n \subseteq \mathcal F_n$
  • Linear combinations of martingales are martingales

Martingale Algebra (Variance, Covariance, Equality)

  • Var(M_n) = sum of squared increments
  • Cov(M_s, M_t) = Var(M_s)
  • Orthogonality of increments
  • Equality requires a.s. or L^2

Stopped Martingales

If $(X_n)$ is a martingale and $\tau$ is a stopping time, define \(X_n^\tau := X_{n \wedge \tau}.\)

Theorem:
$(X_n^\tau)$ is a martingale w.r.t. $(\mathcal F_n)$.

Shift Operators (Time Shifts)

Shift operators formalize the idea of restarting a process after a stopping time. They are essential for:

  • composite stopping times
  • multi-stage hitting times
  • gambling systems that restart
  • Strong Markov–type arguments

Definition (Pathwise Shift)

Let $\omega = (\omega_1, \omega_2, \ldots)$ be a sample path.

The shift operator $\theta_n$ acts on paths by \((\theta_n \omega)_k := \omega_{n+k}, \quad k \ge 1.\)

Interpretation:

  • $\theta_n$ discards the first $n$ steps
  • time is “reset” to zero after step $n$

Shifted Process

Given a stochastic process $(X_n)$, define the shifted process \((X \circ \theta_n)_k := X_{n+k}.\)

This represents the process as seen after time $n$.

Random Shifts (Stopping Times)

If $\tau$ is a stopping time, define the random shift \((\theta_\tau \omega)_k := \omega_{\tau(\omega)+k}.\)

Then: \((X \circ \theta_\tau)_k = X_{\tau+k}.\)

This is how we formalize “restart the process at time $\tau$”.

Shifted Filtration

Given a filtration $(\mathcal F_n)$ and a stopping time $\tau$, define the shifted filtration \(\mathcal F^{(\tau)}_n := \mathcal F_{\tau+n}.\)

A random time $\sigma$ is a stopping time after $\tau$ if \(\{\sigma \le n\} \in \mathcal F_{\tau+n}.\)

Composite Stopping Times

If:

  • $\tau$ is a stopping time w.r.t. $(\mathcal F_n)$
  • $\sigma$ is a stopping time w.r.t. $(\mathcal F^{(\tau)}_n)$

then: \(\tau + \sigma\) is a stopping time w.r.t. $(\mathcal F_n)$.

This is the correct way to form finite sums of stopping times.

Martingales and Shift Operators

If $(M_n)$ is a martingale, then the shifted process \((M_{\tau+n} - M_\tau)_{n \ge 0}\) is a martingale w.r.t. the shifted filtration $(\mathcal F_{\tau+n})$.

(This is the discrete-time version of the Strong Markov Property.)

How Shift Operators Appear on Prelims

Look for phrases like:

  • “after time $\tau$”
  • “restart the process”
  • “wait until the next event”
  • “define another stopping time after hitting $a$”

These are signals that you need: \(X_{\tau+n} = X \circ \theta_\tau \quad \text{and} \quad \mathcal F_{\tau+n}.\)

Exam Heuristic

If you see two stages in time, think: \(\text{shift operator} \;\Rightarrow\; \text{shifted filtration} \;\Rightarrow\; \tau + \sigma.\)

Predictable Processes

A process $H_n$ is predictable w.r.t. $(\mathcal F_n)$ if \(H_n \text{ is } \mathcal F_{n-1}\text{-measurable for all } n \ge 1.\)

If $X_n$ is a martingale and $H_n$ is bounded and predictable, then \(Y_n := \sum_{k=1}^n H_k (X_k - X_{k-1})\) is a martingale.

Doob Decomposition

If $(X_n)$ is a submartingale, then there exists a unique decomposition \(X_n = M_n + A_n\) where:

  • $M_n$ is a martingale
  • $A_n$ is predictable, increasing, and $A_0 = 0$

Explicitly, \(A_n = \sum_{k=1}^n \mathbb E[X_k - X_{k-1} \mid \mathcal F_{k-1}].\)

Convex Transformations

If $(X_n)$ is a martingale and $\varphi$ is convex with $\mathbb E|\varphi(X_n)|<\infty$, then \(\varphi(X_n) \text{ is a submartingale}.\)

If $\varphi$ is increasing and convex:

  • subMG → subMG
  • superMG → superMG

Orthogonality of Martingale Increments

Let $(M_n)$ be a martingale in $L^2$.

If $i < j$, then \(\mathbb E[(M_j - M_{j-1})(M_i - M_{i-1})] = 0.\)

Conditional Variance Formula

If $(M_n)$ is a martingale in $L^2$, then \(\mathrm{Var}(M_n) = \sum_{k=1}^n \mathbb E\left[\mathrm{Var}(M_k \mid \mathcal F_{k-1})\right].\)

Martingale Convergence Theorem (MGCT)

Let $(X_n)$ be a submartingale.

Version 1 (Boundedness)

If $(X_n)$ is bounded in $L^1$ and either:

  • bounded above (subMG), or
  • bounded below (superMG),

then \(X_n \to X_\infty \quad \text{a.s.}\)

Version 2 (Uniform Integrability)

If $(X_n)$ is a uniformly integrable martingale, then \(X_n \to X_\infty \quad \text{a.s. and in } L^1.\)

Uniform Integrability (UI)

A family ${X_n}$ is uniformly integrable if \(\lim_{K \to \infty} \sup_n \mathbb E\left[|X_n| \mathbf 1_{\{|X_n|>K\}}\right] = 0.\)

Equivalent Conditions (TFAE)

For integrable $X_n$:

  1. $(X_n)$ is UI
  2. $\sup_n \mathbb E|X_n| < \infty$ and tails vanish
  3. $X_n \to X$ in $L^1$ implies UI

Useful Facts

  • $L^p$-bounded ($p>1$) ⇒ UI
  • Bounded martingale ⇒ UI
  • UI preserved under stopping

Doob’s Inequalities

Doob Maximal Inequality (Submartingale)

If $(X_n)$ is a nonnegative submartingale, then for $\lambda > 0$, \(\mathbb P\left(\max_{k \le n} X_k \ge \lambda\right) \le \frac{\mathbb E[X_n]}{\lambda}.\)

$L^p$ Inequality

If $X_n$ is a martingale and $p>1$, \(\mathbb E\left[\max_{k \le n} |X_k|^p\right] \le \left(\frac{p}{p-1}\right)^p \mathbb E|X_n|^p.\)

Consequences:

  • $L^p$-bounded MG ⇒ UI
  • Almost sure convergence

Upcrossing Inequality

Let $(X_n)$ be a submartingale and $a < b$. Let $U_n(a,b)$ be the number of upcrossings of $[a,b]$ by time $n$.

Then \((b-a)\mathbb E[U_n(a,b)] \le \mathbb E[(X_n - a)^+].\)

Consequence: Finite upcrossings ⇒ almost sure convergence.

Quadratic Variation (Discrete Time)

For a martingale $(M_n)$, \([M]_n := \sum_{k=1}^n (M_k - M_{k-1})^2.\)

Then:

  • $M_n^2 - [M]_n$ is a martingale
  • Used for variance tracking and CLTs

Maximal Inequality (Supermartingale)

If $(X_n)$ is a nonnegative supermartingale, then \(\lambda \mathbb P\left(\sup_n X_n \ge \lambda\right) \le \mathbb E[X_0].\)

Martingale Convergence Summary

Assumption Conclusion
UI martingale a.s. + $L^1$ convergence
$L^p$-bounded, $p>1$ UI ⇒ convergence
Bounded MG UI ⇒ convergence
SubMG bounded above a.s. convergence

Optional Sampling Theorem (OST)

Let $(X_n)$ be a martingale and $\sigma \le \tau$ stopping times.

If any one of the following holds:

  1. $\sigma,\tau$ bounded a.s.
  2. $(X_n)$ is UI
  3. Bounded increments + integrable stopping times

Then \(\mathbb E[X_\tau \mid \mathcal F_\sigma] = X_\sigma, \quad \mathbb E[X_\tau] = \mathbb E[X_\sigma].\)

OST Tricks

  • Compute expectations at hitting times
  • Prove fairness of gambling systems
  • Reduce problems to stopped martingales

Optional Stopping: Common Failure Modes

OST may fail if:

  • Stopping time is unbounded
  • Martingale is not UI
  • Expected stopping time is infinite
  • Martingale has heavy tails

Classic counterexample:

  • Simple symmetric random walk
  • Hitting time of a single point

Martingale Difference Sequences (MD)

A sequence $(D_n)$ is a martingale difference if: \(\mathbb E[D_{n+1} \mid \mathcal F_n] = 0, \quad X_n := \sum_{k=1}^n D_k \text{ is a martingale}.\)

CLT for MD Arrays (Sketch)

If:

  1. Conditional variances converge
  2. Lindeberg-type condition holds

Then partial sums satisfy a CLT.

Brownian Motion and Martingales

Standard Brownian Motion $(B_t)$:

  • Continuous paths
  • Independent increments
  • $B_t - B_s \sim N(0, t-s)$

Common Martingales

  • $B_t$
  • $B_t^2 - t$
  • Exponentials via Girsanov

Strong Markov Property

If $\tau$ is a stopping time, then \((B_{\tau+t} - B_\tau)_{t \ge 0}\) is independent of $\mathcal F_\tau$ and is standard Brownian motion.

Used to justify:

  • Composite stopping times
  • Multi-stage hitting times

Key Takeaways

  • Always check adapted + integrable + conditional expectation
  • Stopped martingales preserve structure
  • UI is the cleanest path to convergence and OST
  • Doob inequalities are your main technical engine
  • You can always create your own stopping time

Exam Heuristics

  • See conditional expectation = current value → martingale
  • See stopping time → try OST
  • See maximum → Doob inequality
  • See convergence → check UI or boundedness
  • See gambling system → predictable process
  • See Brownian hitting time → Strong Markov

Gambling Systems (Martingale Betting Strategies)

Gambling systems formalize adaptive betting strategies applied to martingales and are best viewed as predictable transformations of martingales.

They encode the principle that no fair game can be turned into a winning one using only past information.

Setup

Let $(X_n, \mathcal F_n)$ be a martingale.

Interpretation:

  • $X_n$: wealth after round $n$
  • $X_n - X_{n-1}$: payoff of round $n$

Predictable Betting Strategy

A process $(H_n)$ is a gambling system if \(H_n \text{ is } \mathcal F_{n-1}\text{-measurable for all } n \ge 1.\)

Interpretation:

  • $H_n$ is the amount wagered before round $n$
  • Strategy depends only on past information

Such processes are also called predictable processes.

Gains Process

Given a gambling system $(H_n)$, define the cumulative gains \(Y_n := \sum_{k=1}^n H_k (X_k - X_{k-1}), \quad Y_0 = 0.\)

  • $Y_n$ represents the wealth generated purely by betting
  • Often written as a discrete stochastic integral

Fair Gambling Theorem

If:

  • $(X_n)$ is a martingale
  • $(H_n)$ is predictable and bounded

then \((Y_n) \text{ is a martingale.}\)

Proof sketch: \(\mathbb E[Y_{n+1} - Y_n \mid \mathcal F_n] = H_{n+1} \mathbb E[X_{n+1} - X_n \mid \mathcal F_n] = 0.\)

Interpretation:
No bounded betting strategy can create expected profit from a fair game.

Submartingale and Supermartingale Cases

  • If $X_n$ is a submartingale and $H_n \ge 0$, then $Y_n$ is a submartingale
  • If $X_n$ is a supermartingale and $H_n \ge 0$, then $Y_n$ is a supermartingale

Stopped Gambling Systems

Let $\tau$ be a stopping time.

Define the stopped gains: \(Y_n^\tau := \sum_{k=1}^{n \wedge \tau} H_k (X_k - X_{k-1}).\)

Then:

  • $Y_n^\tau$ is a martingale
  • Allows strategies such as “stop when rich” or “stop when ruined”

Optional Sampling for Gambling Systems

If:

  • $(Y_n)$ is uniformly integrable, or
  • $\tau$ is bounded a.s.

then \(\mathbb E[Y_\tau] = 0.\)

Interpretation:
Even with adaptive betting and stopping rules, expected profit remains zero.

Failure Modes (Important for Prelims)

Optional Sampling may fail if:

  • The betting strategy $(H_n)$ is unbounded
  • The stopping time $\tau$ is unbounded
  • Expected values are infinite

Classic counterexample (Doubling Strategy):

  • Bet $H_n = 2^n$ after consecutive losses
  • Predictable but unbounded
  • Expected wealth undefined
  • OST does not apply

Exam Heuristics (Gambling Language)

  • “Bet”, “strategy”, “capital”, “winnings” → gambling system
  • “Depends only on the past” → predictability
  • “Stop when…” → stopping time
  • “Expected gain” → martingale + OST
  • “Doubling” → unbounded strategy, OST failure
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