Brownian Bridge™

Setup

Mathematical context

The theory built in Modules 1–4 requires a precise language for comparing random variables and talking about sequences converging to a limit. Four distinct notions of convergence appear naturally in quantitative finance: almost sure convergence, convergence in probability, $L^p$ convergence, and convergence in distribution. These are not equivalent, and confusing them produces incorrect results — in Monte Carlo error bounds, in Central Limit Theorem applications, and in the conditions required for the Optional Stopping Theorem.

This module establishes the $L^p$ function spaces that are the natural home for random variables, derives the fundamental inequalities (Jensen, Hölder, Markov), characterises the four modes of convergence, and introduces uniform integrability — the concept that bridges $L^1$ and almost sure convergence and appears explicitly in the OST conditions of Module 4.

Stated assumptions

$(\Omega, \mathcal{F}, \mathbb{P})$ is a complete probability space (Module 1).
Random variables are real-valued measurable functions $X : \Omega \to \mathbb{R}$ (or $\mathbb{R} \cup \{+\infty\}$ where noted).
Lebesgue integration is used throughout (Module 2). The results hold under any $\sigma$ -finite measure; on a probability space the proofs simplify.
Conventions: $p \in [1, \infty)$ unless otherwise stated; the case $p = \infty$ is treated separately.

INSIGHT

Financial Insight. On a trading desk, the choice of $L^p$ space is a modelling assumption, not an abstraction. An option payoff in $L^2(\mathbb{Q})$ has finite variance under the risk-neutral measure — a necessary condition for the Black-Scholes delta hedge to be well-defined. A process in $L^1$ only has a finite price but not necessarily finite hedging error. Exotic payoffs (e.g., power options $S^{1.5}$ ) may fail to be in $L^2$ under log-normal dynamics, invalidating standard Greeks formulas. Uniform integrability appears every time you pass a limit through an expectation — which Monte Carlo methods do at every step.

Theory

1. Lp spaces

DEFINITION

Definition 5.1 ( $L^p$ space). For $p \in [1, \infty)$ , define

$\|X\|_p := \left(\mathbb{E}[|X|^p]\right)^{1/p}, \qquad L^p(\Omega, \mathcal{F}, \mathbb{P}) := \{X : \|X\|_p < \infty\}.$

For $p = \infty$ : $\|X\|_\infty := \mathrm{ess\,sup}|X|$ , the essential supremum (smallest $M$ such that $\mathbb{P}(|X| > M) = 0$ ).

Elements of $L^p$ are equivalence classes of random variables that agree $\mathbb{P}$ -almost everywhere.

The $L^p$ norm measures the average magnitude of $X$ raised to the $p$ -th power. For $p = 1$ : $\|X\|_1 = \mathbb{E}[|X|]$ — the mean absolute value. For $p = 2$ : $\|X\|_2 = \sqrt{\mathbb{E}[X^2]}$ — the root mean square, the natural norm for variance and hedging error.

THEOREM

Theorem 5.1 (Riesz-Fischer — completeness of $L^p$ ). For $p \in [1, \infty]$ , $L^p(\Omega, \mathcal{F}, \mathbb{P})$ is a Banach space (complete normed vector space). In particular, every Cauchy sequence in $L^p$ has a limit in $L^p$ .

Completeness is what guarantees that Itô integrals (defined as $L^2$ limits of simple integrands) actually exist in $L^2$ .

THEOREM

Theorem 5.2 ( $L^p$ inclusions on probability spaces). For a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with $1 \leq p \leq q \leq \infty$ :

$L^q(\Omega, \mathcal{F}, \mathbb{P}) \subseteq L^p(\Omega, \mathcal{F}, \mathbb{P}), \qquad \|X\|_p \leq \|X\|_q.$

This inclusion is specific to probability spaces (i.e., $\mathbb{P}(\Omega) = 1$ ). On a general measure space the inclusion reverses. The implication: $L^2$ is a strictly better-behaved space than $L^1$ — finite variance implies finite mean, but not vice versa.

2. Fundamental inequalities

THEOREM

Theorem 5.3 (Markov's inequality). For $X \geq 0$ and $\lambda > 0$ :

$\mathbb{P}(X \geq \lambda) \leq \frac{\mathbb{E}[X]}{\lambda}.$

Setting $X = |Y|^p$ gives Chebyshev's inequality: $\mathbb{P}(|Y| \geq \lambda) \leq \mathbb{E}[|Y|^p]/\lambda^p$ .

Markov is the simplest probabilistic bound. Its proof is a one-line application of monotonicity of integration: $\mathbb{E}[X] \geq \mathbb{E}[X \cdot \mathbf{1}_{X \geq \lambda}] \geq \lambda \, \mathbb{P}(X \geq \lambda)$ . It is tight: take $\mathbb{P}(X = \lambda) = 1/\lambda$ , $\mathbb{P}(X = 0) = 1 - 1/\lambda$ .

THEOREM

Theorem 5.4 (Jensen's inequality). Let $\varphi : \mathbb{R} \to \mathbb{R}$ be convex and $X \in L^1$ . Then:

$\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)].$

For concave $\varphi$ , the inequality reverses.

PROOF

Proof. Since $\varphi$ is convex, for any $a \in \mathbb{R}$ there exists a supporting hyperplane: $\varphi(x) \geq \varphi(a) + c(x - a)$ for some $c \in \mathbb{R}$ (the subgradient at $a$ ). Set $a = \mathbb{E}[X]$ and take expectations on both sides:

$\mathbb{E}[\varphi(X)] \geq \varphi(\mathbb{E}[X]) + c(\mathbb{E}[X] - \mathbb{E}[X]) = \varphi(\mathbb{E}[X]). \quad \square$

Jensen is omnipresent in finance. Applications: convexity of the option payoff implies the value of an option on the average is less than the average option value (Jensen's inequality in Asian pricing). The log-normal expected value: $\mathbb{E}[e^X] \geq e^{\mathbb{E}[X]}$ (convexity of $\exp$ ). The sub-additivity of $\sqrt{\cdot}$ means $\sqrt{\mathbb{E}[X^2]} \geq \mathbb{E}[|X|]$ (i.e., $\|X\|_2 \geq \|X\|_1$ ).

THEOREM

Theorem 5.5 (Hölder's inequality). For $p, q \in (1, \infty)$ with $1/p + 1/q = 1$ (Hölder conjugates):

$\mathbb{E}[|XY|] \leq \|X\|_p \|Y\|_q.$

The case $p = q = 2$ is the Cauchy-Schwarz inequality: $\mathbb{E}[|XY|] \leq \|X\|_2 \|Y\|_2$ .

Hölder's inequality is used to bound covariance terms in option pricing (e.g., proving that $\mathbb{E}[\Delta \cdot S] \leq \|\Delta\|_2 \|S\|_2$ for a self-financing portfolio with square-integrable delta), and in the theory of stochastic integration where it controls the cross-terms.

THEOREM

Theorem 5.6 (Minkowski's inequality). For $p \in [1, \infty)$ :

$\|X + Y\|_p \leq \|X\|_p + \|Y\|_p.$

This is the triangle inequality for the $L^p$ norm — what makes $L^p$ a normed space. It is the key step in verifying that $L^p$ is a vector space under the $L^p$ norm.

3. Modes of convergence

We consider a sequence $(X_n)_{n \geq 1}$ and a limit $X$ , all defined on $(\Omega, \mathcal{F}, \mathbb{P})$ .

DEFINITION

Definition 5.2 (Four modes of convergence).

(AS) Almost sure convergence: $X_n \xrightarrow{\mathrm{a.s.}} X$ if $\mathbb{P}(\omega : X_n(\omega) \to X(\omega)) = 1$ .

(P) Convergence in probability: $X_n \xrightarrow{\mathbb{P}} X$ if $\mathbb{P}(|X_n - X| > \varepsilon) \to 0$ for every $\varepsilon > 0$ .

(Lp) $L^p$ convergence: $X_n \xrightarrow{L^p} X$ if $\|X_n - X\|_p \to 0$ .

(D) Convergence in distribution: $X_n \xrightarrow{D} X$ if $\mathbb{E}[f(X_n)] \to \mathbb{E}[f(X)]$ for all bounded continuous $f$ .

The four modes are strictly ordered in strength. The implication diagram is:

$L^p \Rightarrow \mathbb{P} \Leftarrow \mathrm{a.s.}$

and both $L^p$ and a.s. convergence imply convergence in probability, which implies convergence in distribution. No other general implication holds.

EXAMPLE

Example 5.1 (Implication failures — the standard counterexamples).

(a) A.S. does not imply $L^1$ . Let $\Omega = [0,1]$ with Lebesgue measure. Set $X_n = n \cdot \mathbf{1}_{[0, 1/n]}$ . Then $X_n(\omega) \to 0$ for every $\omega > 0$ (i.e., a.s.), but $\mathbb{E}[X_n] = n \cdot (1/n) = 1 \not\to 0$ . So $X_n \to 0$ a.s. but $X_n \not\to 0$ in $L^1$ .

(b) $L^1$ does not imply a.s. The typewriter sequence: partition $[0,1]$ into intervals $[k/2^m, (k+1)/2^m)$ for $0 \leq k < 2^m$ , indexed sequentially. Set $X_n = \mathbf{1}_{I_n}$ . Then $\mathbb{E}[X_n] = 2^{-m} \to 0$ (so $X_n \to 0$ in $L^1$ ), but for every $\omega \in [0,1]$ , $X_n(\omega) = 1$ infinitely often and $= 0$ infinitely often — the sequence does not converge a.s.

(c) Convergence in probability does not imply a.s. The typewriter sequence also serves here: $X_n \to 0$ in probability (same argument as $L^1$ ) but not a.s.

REMARK

Remark (Subsequence criterion). $X_n \xrightarrow{\mathbb{P}} X$ if and only if every subsequence $(X_{n_k})$ has a further subsequence $(X_{n_{k_j}})$ with $X_{n_{k_j}} \xrightarrow{\mathrm{a.s.}} X$ . This is one of the most useful tools in stochastic analysis for promoting convergence in probability to almost sure convergence along a subsequence.

4. Uniform integrability

DEFINITION

Definition 5.3 (Uniform integrability). A family $\{X_\alpha\}$ of random variables is uniformly integrable (UI) if

$\lim_{M \to \infty} \sup_\alpha \, \mathbb{E}\!\left[|X_\alpha| \cdot \mathbf{1}_{|X_\alpha| > M}\right] = 0.$

Intuitively: the tails of all $X_\alpha$ simultaneously become negligible as the truncation level $M$ grows. Uniform integrability ensures that convergence in probability "controls the tails" well enough to imply $L^1$ convergence.

THEOREM

Theorem 5.7 (Vitali convergence theorem). Let $(X_n)$ be a UI family and $X_n \xrightarrow{\mathbb{P}} X$ . Then $X \in L^1$ and $X_n \to X$ in $L^1$ .

This is the definitive statement bridging convergence in probability and $L^1$ convergence. The Dominated Convergence Theorem (Module 2) is a special case: if $|X_n| \leq Y$ with $Y \in L^1$ , then $\{X_n\}$ is UI by $\mathbb{E}[|X_n| \cdot \mathbf{1}_{|X_n|>M}] \leq \mathbb{E}[Y \cdot \mathbf{1}_{Y > M}] \to 0$ .

REMARK

Remark (UI and martingale theory). A martingale $(M_t)$ is UI if and only if it converges a.s. and in $L^1$ to a terminal variable $M_\infty$ with $M_t = \mathbb{E}[M_\infty \mid \mathcal{F}_t]$ . This is the Doob $L^1$ martingale convergence theorem. The connection to Module 4: the OST condition "the stopped martingale is UI" is precisely saying that the stopped process has a well-behaved $L^1$ limit.

EXAMPLE

Example 5.2 (UI in option pricing). Under the risk-neutral measure $\mathbb{Q}$ , the family $\{e^{-rT} g(S_T) : T \geq 0\}$ for a bounded payoff $g$ (e.g., a call with notional cap) is UI — the payoff is dominated by a constant. For an unbounded payoff such as $g(S_T) = S_T^2$ (power option), the family is UI only if $S_T$ has sufficiently thin tails under $\mathbb{Q}$ (e.g., finite higher moments under log-normal dynamics). Failing to verify UI when applying DCT in a simulation loop is one cause of Monte Carlo bias that does not diminish with sample size.

Validation

The companion notebook verifies:

$L^p$ norms on a finite discrete probability space using exact rational arithmetic: $\|X\|_1$ , $\|X\|_2$ , $\|X\|_\infty$ , and confirms the $L^q \subseteq L^p$ inclusion numerically.
Jensen's inequality: for $\varphi(x) = e^x$ (convex) and $\varphi(x) = \log x$ (concave), verifies $\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$ and the reverse.
Hölder's and Cauchy-Schwarz inequalities: verified on explicit numerical examples.
Convergence counterexamples: simulates the $X_n = n \cdot \mathbf{1}_{[0,1/n]}$ sequence, confirms a.s. convergence to 0 while $\mathbb{E}[X_n] = 1$ for all $n$ .
Uniform integrability: empirically verifies that a bounded family is UI while an unbounded family fails the UI criterion at each truncation level.

PRACTICE

Hand exercise.

(a) Let $X$ be uniform on $\{1, 2, 3, 4\}$ with equal probability $1/4$ . Compute $\|X\|_1$ , $\|X\|_2$ , and $\|X\|_\infty$ exactly. Confirm $\|X\|_1 \leq \|X\|_2 \leq \|X\|_\infty$ .

(b) Let $\varphi(x) = x^2$ (convex). With $X$ as above, verify Jensen's inequality $\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$ by direct computation.

(c) Give an explicit example of a sequence $(X_n)$ on $[0,1]$ that converges to 0 in $L^2$ but not almost surely.

Limitations

$L^p$ inclusions reverse on infinite measure spaces. On $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$ (Lebesgue measure), $L^2 \not\subseteq L^1$ : the function $f(x) = 1/(1 + |x|)$ is in $L^2(\mathbb{R})$ but not in $L^1(\mathbb{R})$ . The inclusion $L^q \subseteq L^p$ for $q \geq p$ is specific to finite measure spaces. Applying the probability-space inclusion argument to general measure spaces is a common error when working with unnormalised Monte Carlo weights.

WARNING

Warning (confusing convergence modes in Monte Carlo). The Strong Law of Large Numbers gives a.s. convergence of sample means: $\bar{X}_n \xrightarrow{\mathrm{a.s.}} \mu$ . The Central Limit Theorem gives convergence in distribution: $\sqrt{n}(\bar{X}_n - \mu) \xrightarrow{D} N(0, \sigma^2)$ . The Monte Carlo standard error $\sigma/\sqrt{n}$ is an $L^2$ convergence rate. These three statements are about different modes of convergence and cannot be combined without justification. In particular, the CLT rate does not imply a.s. convergence at rate $1/\sqrt{n}$ ; the a.s. rate is $O(\sqrt{\log n / n})$ (law of the iterated logarithm).

Uniform integrability requires verification, not assumption. UI is often claimed without proof. A sufficient condition: $\sup_\alpha \mathbb{E}[|X_\alpha|^{1+\varepsilon}] < \infty$ for some $\varepsilon > 0$ (a bounded $L^{1+\varepsilon}$ family is UI). Checking this requires knowing the tail behaviour of $X_\alpha$ — for path-dependent payoffs under stochastic volatility models, this is not always available in closed form and must be verified numerically.

Jensen's inequality is not a pricing shortcut. The inequality $\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$ shows that the price of a convex payoff exceeds the payoff at the expected value. It does not compute the price. The gap $\mathbb{E}[\varphi(X)] - \varphi(\mathbb{E}[X])$ depends on the full distribution of $X$ , not just its mean.

Convergence in distribution does not imply anything about the random variables themselves. Two sequences $(X_n)$ and $(Y_n)$ can converge in distribution to the same limit while being defined on completely different probability spaces. In particular, $X_n \xrightarrow{D} X$ and $Y_n \xrightarrow{D} X$ does not imply $X_n - Y_n \xrightarrow{D} 0$ . This is the source of the Skorokhod representation theorem: to recover path properties from distributional convergence, one must construct a coupling on a common probability space.

Lp Spaces and Modes of Convergence

Setup

Mathematical context

Stated assumptions

Theory

1. Lp spaces

2. Fundamental inequalities

3. Modes of convergence

4. Uniform integrability

Validation

Limitations

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