Brownian Bridge™

Setup

Mathematical context

A probability model that never changes is insufficient for finance. Asset prices, interest rates, and positions all evolve in time, and the information available to an agent at time $t$ determines which trades are feasible, which prices are observable, and which expectations are well-defined. The purpose of this module is to formalise the notion of evolving information and the processes that respect it.

We work on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ , extending the probability space of Module 1 with a time-indexed structure.

Stated assumptions

$(\Omega, \mathcal{F}, \mathbb{P})$ is a complete probability space (all $\mathbb{P}$ -null sets of $\mathcal{F}$ belong to $\mathcal{F}$ ; see Module 1).
The filtration $(\mathcal{F}_t)_{t \geq 0}$ $(F_{t})_{t \geq 0}$ satisfies the usual conditions:
- (UC1) Completeness: each $\mathcal{F}_t$ contains all $\mathbb{P}$ -null sets of $\mathcal{F}$ .
- (UC2) Right-continuity: $\mathcal{F}_t = \mathcal{F}_{t^+} := \bigcap_{s > t} \mathcal{F}_s$ for all $t \geq 0$ .
Unless stated otherwise, all processes are real-valued and indexed by $t \in [0, T]$ or $[0, \infty)$ .
Conventions: time is continuous unless the discrete-time subscript notation $M_n$ is used explicitly.

INSIGHT

Financial Insight. On a trading desk, the filtration $(\mathcal{F}_t)$ models the information set available at time $t$ : price histories, rates, volatility surfaces. The usual conditions are technical hygiene with practical consequences. Right-continuity (UC2) ensures that stopping times can always be approximated from above, which underpins the optional stopping theorem for continuous-time processes. Completeness (UC1) prevents measure-zero events from creating pathological measurability failures in stopped processes. Most references impose these without comment; understanding why they are needed separates a practitioner from a student.

Theory

1. Filtrations

DEFINITION

Definition 4.1 (Filtration). A filtration on $(\Omega, \mathcal{F}, \mathbb{P})$ is a family $(\mathcal{F}_t)_{t \geq 0}$ of sub-σ-algebras of $\mathcal{F}$ satisfying:

$\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} \quad \text{for all } 0 \leq s \leq t.$

The terminal σ-algebra is $\mathcal{F}_\infty := \sigma\!\left(\bigcup_{t \geq 0} \mathcal{F}_t\right)$ .

Intuitively: $\mathcal{F}_t$ is the collection of all events whose occurrence is determined by time $t$ . The inclusion $\mathcal{F}_s \subseteq \mathcal{F}_t$ for $s \leq t$ formalises the fact that information is never forgotten.

EXAMPLE

Example 4.1 (Two-flip coin-toss filtration). Let $\Omega = \{HH, HT, TH, TT\}$ with $\mathbb{P}$ uniform. Define:

$\mathcal{F}_0 = \{\emptyset, \Omega\}$ — no information.
$\mathcal{F}_1 = \sigma\!\left(\{HH, HT\},\, \{TH, TT\}\right)$ — outcome of flip 1 is known.
$\mathcal{F}_2 = 2^\Omega$ — both flips revealed; full information.

Then $\mathcal{F}_0 \subsetneq \mathcal{F}_1 \subsetneq \mathcal{F}_2$ , a strictly increasing chain.

2. Adapted processes and the natural filtration

DEFINITION

Definition 4.2 (Adapted process). A stochastic process $(X_t)_{t \geq 0}$ is $(\mathcal{F}_t)$ -adapted if $X_t$ is $\mathcal{F}_t$ -measurable for every $t \geq 0$ .

Adaptation says: the value of $X$ at time $t$ is determined by the information available at time $t$ — no peeking at the future. A non-adapted process would require knowledge of events that have not yet occurred.

DEFINITION

Definition 4.3 (Natural filtration). For a process $(X_t)_{t \geq 0}$ , the natural filtration is

$\mathcal{F}_t^X := \sigma(X_s : 0 \leq s \leq t).$

This is the smallest filtration to which $(X_t)$ is adapted.

REMARK

Remark. The natural filtration is the coarsest (most parsimonious) filtration making $(X_t)$ adapted. In continuous time one typically augments $\mathcal{F}_t^X$ with $\mathbb{P}$ -null sets to satisfy the usual conditions — the result is the augmented natural filtration and is the standard choice for Brownian motion.

3. Stopping times

DEFINITION

Definition 4.4 (Stopping time). A random variable $\tau : \Omega \to [0, \infty]$ is a stopping time with respect to $(\mathcal{F}_t)$ if

$\{\tau \leq t\} \in \mathcal{F}_t \quad \text{for all } t \geq 0.$

Economically: the decision to stop at time $\tau$ depends only on information available at $\tau$ — not on future events. A portfolio manager who decides to exit a position when the asset first hits a barrier is executing a stopping time strategy.

EXAMPLE

Example 4.2 (First hitting time). Let $(B_t)$ be a standard Brownian motion adapted to its augmented natural filtration $(\mathcal{F}_t)$ . For level $a > 0$ , define $\tau_a := \inf\{t \geq 0 : B_t = a\}$ . Then $\tau_a$ is a stopping time: $\{\tau_a \leq t\} = \{\sup_{0 \leq s \leq t} B_s \geq a\} \in \mathcal{F}_t$ .

REMARK

Remark (Predictable vs. stopping time). A process $(H_t)$ is predictable if it is $(\mathcal{F}_{t^-})$ -measurable — it is determined by information strictly before time $t$ . Predictability is the correct integrability condition for the Itô integral: the integrand must not anticipate the future. For continuous-time processes with continuous paths, adapted and predictable coincide. For jump processes, they differ: using $H_t$ rather than $H_{t^-}$ as the integrand changes the stochastic integral by a sum over jump times. This distinction is not pedantic — a trading strategy requiring knowledge of today's closing price to determine today's position is not implementable in continuous time.

4. Martingales

DEFINITION

Definition 4.5 (Martingale / sub- / supermartingale). An adapted process $(M_t)_{t \geq 0}$ with $\mathbb{E}[|M_t|] < \infty$ for all $t$ is a:

Martingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s$ a.s. for all $0 \leq s \leq t$ .
Submartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \geq M_s$ a.s. for all $0 \leq s \leq t$ .
Supermartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \leq M_s$ a.s. for all $0 \leq s \leq t$ .

The martingale condition says: your best prediction of the future value, given all current information, is exactly the current value. There is no predictable drift. It formalises the notion of a fair game.

EXAMPLE

Example 4.3 (Canonical martingales in quantitative finance).

(a) Standard Brownian motion. $(B_t)$ under $\mathbb{P}$ is a martingale. Proof: $\mathbb{E}[B_t \mid \mathcal{F}_s] = \mathbb{E}[B_t - B_s \mid \mathcal{F}_s] + B_s = \mathbb{E}[B_t - B_s] + B_s = 0 + B_s = B_s$ , using that $B_t - B_s$ is independent of $\mathcal{F}_s$ and has mean zero.

(b) Stochastic exponential (Girsanov density). $\mathcal{E}_t(\sigma B) := \exp\!\left(\sigma B_t - \tfrac{1}{2}\sigma^2 t\right)$ is a martingale under $\mathbb{P}$ , provided $\sigma$ is a constant. This is the Radon-Nikodym derivative $d\mathbb{Q}/d\mathbb{P}\big|_{\mathcal{F}_t}$ of the risk-neutral measure — the subject of Module 3 in Stochastic Calculus.

(c) Discounted asset price under $\mathbb{Q}$ . If $dS_t = \mu S_t \, dt + \sigma S_t \, dB_t^{\mathbb{P}}$ , the Girsanov theorem provides a $\mathbb{Q}$ under which $d(e^{-rt} S_t)$ has no $dt$ term. Under $\mathbb{Q}$ : $d(e^{-rt}S_t) = e^{-rt}S_t \sigma \, dB_t^{\mathbb{Q}}$ , a local martingale. The fundamental theorem of asset pricing states: no arbitrage $\Longleftrightarrow$ there exists $\mathbb{Q}$ under which discounted prices are local martingales.

(d) Compensated Poisson process. If $N_t \sim \mathrm{Poisson}(\lambda t)$ , then $M_t = N_t - \lambda t$ is a martingale. The term $\lambda t$ is the compensator that removes the drift of $N_t$ .

5. Key martingale theorems

THEOREM

Theorem 4.1 (Doob's Optional Stopping Theorem — OST). Let $(M_n)_{n=0}^N$ be a discrete-time martingale and $\tau$ a stopping time with $\tau \leq N$ a.s. Then:

$\mathbb{E}[M_\tau] = \mathbb{E}[M_0].$

More generally, if $\tau$ is a.s. finite and either (i) $(M_n)$ is uniformly integrable, or (ii) $\mathbb{E}[\tau] < \infty$ and $|M_{n+1} - M_n| \leq C$ a.s. for some constant $C$ , then $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$ .

PROOF

Proof sketch (bounded $\tau$ case). Write the stopped process as a telescoping sum using the martingale difference:

$M_{\tau \wedge n} = M_0 + \sum_{k=0}^{n-1} \mathbf{1}_{\{\tau > k\}} (M_{k+1} - M_k).$

Since $\tau$ is a stopping time, $\{\tau > k\} = \{\tau \leq k\}^c \in \mathcal{F}_k$ . So $\mathbf{1}_{\{\tau > k\}}$ is $\mathcal{F}_k$ -measurable, and each term $\mathbf{1}_{\{\tau > k\}} (M_{k+1} - M_k)$ is a martingale difference. Taking expectations: $\mathbb{E}[M_{\tau \wedge n}] = \mathbb{E}[M_0]$ for all $n$ . At $n = N$ , since $\tau \leq N$ a.s., we get $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$ . $\square$

WARNING

Warning (OST failure without integrability). The integrability conditions in the OST cannot be dropped. Consider a symmetric random walk $(S_n)$ and $\tau = \inf\{n : S_n = 1\}$ . Then $\tau$ is a.s. finite, but $\mathbb{E}[\tau] = \infty$ . Since $S_\tau = 1$ a.s., we would need $\mathbb{E}[S_\tau] = 1$ , but $\mathbb{E}[S_0] = 0$ — a contradiction. The OST does not apply here. This is the mathematical formalisation of why doubling strategies (Martingale betting) fail in the presence of any budget constraint.

THEOREM

Theorem 4.2 (Doob's Maximal Inequality). Let $(M_n)_{0 \leq n \leq N}$ be a non-negative submartingale. For $\lambda > 0$ :

$\mathbb{P}\!\left(\max_{0 \leq k \leq N} M_k \geq \lambda\right) \leq \frac{\mathbb{E}[M_N]}{\lambda}.$

For a martingale and $p > 1$ :

$\mathbb{E}\!\left[\max_{0 \leq k \leq N} |M_k|^p\right] \leq \left(\frac{p}{p-1}\right)^p \mathbb{E}\!\left[|M_N|^p\right].$

The maximal inequality controls the running maximum of a martingale in terms of its terminal distribution. It underlies the tightness arguments in weak convergence of stochastic processes and the $L^p$ theory of martingale integrals.

THEOREM

Theorem 4.3 (Martingale Convergence Theorem). Let $(M_n)_{n \geq 0}$ be a martingale bounded in $L^1$ (i.e., $\sup_{n} \mathbb{E}[|M_n|] < \infty$ ). Then there exists an integrable random variable $M_\infty$ such that $M_n \to M_\infty$ a.s.

6. Doob-Meyer decomposition

THEOREM

Theorem 4.4 (Doob-Meyer decomposition). Every submartingale $(X_n)$ with $\sup_n \mathbb{E}[X_n^+] < \infty$ can be written uniquely as

$X_n = M_n + A_n,$

where $(M_n)$ is a martingale and $(A_n)$ is a predictable, non-decreasing process with $A_0 = 0$ .

The process $(A_n)$ is the compensator of $(X_n)$ . It absorbs all the drift of $X$ , leaving a pure martingale $M_n$ . In continuous time, applying this decomposition to $M_t^2$ (when $(M_t)$ is a square-integrable martingale) yields the quadratic variation process $\langle M \rangle_t$ , defined as the unique predictable increasing process such that $M_t^2 - \langle M \rangle_t$ is a martingale. For Brownian motion, $\langle B \rangle_t = t$ . This is the bridge from martingale theory to Itô calculus.

7. Martingale representation theorem (preview)

REMARK

Remark (Brownian martingale representation). On the augmented natural filtration of a standard Brownian motion, every $L^2$ martingale $(M_t)_{t \in [0,T]}$ has the representation

$M_t = M_0 + \int_0^t H_s \, dB_s$

for a unique predictable process $(H_s)$ with $\mathbb{E}\!\left[\int_0^T H_s^2 \, ds\right] < \infty$ . The financial interpretation is market completeness: every $L^2(\mathcal{F}_T, \mathbb{Q})$ contingent claim is replicable by the hedging strategy $(H_t)$ . The integrand $(H_t)$ is the delta of the claim. The uniqueness of $(H_t)$ is what makes the Black-Scholes hedge unique.

Validation

The companion notebook verifies:

The filtration axioms on the two-flip coin-toss example (Example 4.1): each $\mathcal{F}_n$ is confirmed as a valid σ-algebra and the inclusion $\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2$ is checked by direct set comparison.
Adaptation of the symmetric random walk $S_n$ to its natural filtration: the σ-algebra generated by $(S_0, \ldots, S_n)$ is computed and $S_n$ -measurability verified.
The discrete martingale property for $(S_n)$ : exact rational computation of $\mathbb{E}[S_{n+1} \mid \mathcal{F}_n] = S_n$ for $n = 0, 1, 2$ over all eight three-step paths.
The optional stopping theorem: for the bounded stopping time $\tau = \min\{n : S_n = 1 \text{ or } n = 3\}$ , the notebook computes $\mathbb{E}[S_\tau]$ exactly and confirms it equals $\mathbb{E}[S_0] = 0$ .
Doob's maximal inequality: the empirical maximum over 1 000 simulated paths is compared with the theoretical bound $\mathbb{E}[|S_N|]/\lambda$ .

PRACTICE

Hand exercise. Let $\Omega = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$ with $\mathbb{P}$ uniform (three fair flips). Set $S_n = (\text{number of H in first } n \text{ flips}) - n/2$ for $n = 0, 1, 2, 3$ .

(a) Write down $\mathcal{F}_1$ and $\mathcal{F}_2$ explicitly as sets of events.

(b) Show $(S_n)_{n=0}^3$ is a martingale by verifying $\mathbb{E}[S_{n+1} \mid \mathcal{F}_n] = S_n$ for $n = 0, 1, 2$ using the atom-averaging formula from Module 3.

(c) Let $\tau = \min\{n \geq 0 : S_n = 1\}$ . Is $\tau$ a stopping time? Compute $\mathbb{E}[\tau]$ and verify $\mathbb{E}[S_{\tau \wedge 3}] = 0$ directly.

Limitations

Path regularity in continuous time. The martingale property $\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s$ holds for each fixed pair $(s, t)$ . It does not, by itself, constrain the paths of $M$ . Under the usual conditions, every martingale has a càdlàg modification (right-continuous with left limits), but this must be proved, not assumed. Working with a version of $M$ without confirmed path regularity makes Itô integration ill-defined.

OST: integrability is binding. In continuous time, applying OST to hitting times of barriers requires verifying that the stopped process is uniformly integrable. For the stochastic exponential $\mathcal{E}_t(\theta B)$ , the OST holds at a stopping time $\tau$ if and only if $\mathbb{E}[\mathcal{E}_\tau(\theta B)] = 1$ , which requires a Novikov-type condition $\mathbb{E}[e^{\frac{1}{2}\theta^2 \tau}] < \infty$ . Failing to check this when computing barrier option prices produces a systematic bias that does not disappear with more simulation paths.

WARNING

Warning (wrong measure for the martingale check). The discounted asset price $e^{-rt} S_t$ is a martingale under the risk-neutral measure $\mathbb{Q}$ , not under the real-world measure $\mathbb{P}$ . Under $\mathbb{P}$ , the process has drift $(\mu - r)$ . Checking the martingale property under $\mathbb{P}$ and concluding no-arbitrage is a fundamental error. The entire content of the Girsanov theorem is the passage from $\mathbb{P}$ (where the drift is $\mu$ ) to $\mathbb{Q}$ (where the drift is $r$ , absorbed into the risk-free discounting).

Discrete-time results do not transfer automatically. The discrete OST proof uses finite sums that terminate. The continuous-time version requires an additional approximation: replace $\tau$ by $\tau_n = \lceil 2^n \tau \rceil / 2^n$ and pass to the limit. Bounded stopping times in continuous time require no further justification; unbounded ones require the integrability conditions.

Predictability matters for jumps. For pure-diffusion processes, adapted and predictable coincide (since the paths are continuous, $H_{t^-} = H_t$ a.s.). For processes with jumps, the distinction is binding: an integrand that uses $H_t$ instead of $H_{t^-}$ changes the Itô formula by a sum over jump times and leads to different pricing formulas.

Interview Angle

PRACTICE

L1 — Junior. Expected: intuition and basic implementation.

"What does it mean for a process to be adapted to a filtration? Give an example of a process that is NOT adapted." Expected answer: $X_t$ adapted means $X_t$ is $\mathcal{F}_t$ -measurable — its value at time $t$ depends only on information available up to $t$ . Non-adapted example: $X_t = B_T$ for $t < T$ (the terminal value of Brownian motion, which requires knowledge of the future).
"Is the sum of two martingales a martingale?" Yes, by linearity of conditional expectation. Common mistake: asserting the same for products (false in general) or for the maximum of two martingales (also false).
"State the optional stopping theorem for a symmetric random walk with a bounded stopping time." Must state the condition $\tau \leq N$ a.s. and the conclusion $\mathbb{E}[S_\tau] = \mathbb{E}[S_0]$ . The candidate should note the condition cannot be dropped.

PRACTICE

L2 — Senior. Expected: derivation and edge cases.

"Prove that standard Brownian motion is a martingale." Use independent increments: $\mathbb{E}[B_t \mid \mathcal{F}_s] = \mathbb{E}[B_t - B_s \mid \mathcal{F}_s] + \mathbb{E}[B_s \mid \mathcal{F}_s] = \mathbb{E}[B_t - B_s] + B_s = 0 + B_s = B_s$ , where the first equality is linearity of CE, the second uses $B_t - B_s \perp \mathcal{F}_s$ , and the last uses $\mathbb{E}[B_t - B_s] = 0$ .
"Why is the discounted stock price a martingale under the risk-neutral measure but not under the real-world measure?" Under $\mathbb{P}$ : $d(e^{-rt}S_t) = e^{-rt}S_t(\mu - r) \, dt + e^{-rt}S_t \sigma \, dB_t^{\mathbb{P}}$ — non-zero drift unless $\mu = r$ . The Girsanov theorem subtracts the market price of risk $\theta = (\mu - r)/\sigma$ from $B_t^{\mathbb{P}}$ to produce $B_t^{\mathbb{Q}} = B_t^{\mathbb{P}} + \theta t$ , under which the drift term vanishes.
"When does the optional stopping theorem fail? Give a financial example." OST fails when $\mathbb{E}[\tau] = \infty$ and the martingale is not uniformly integrable. The doubling strategy: place $2^n$ on each losing round and stop at first win. The stopping time is a.s. finite but $\mathbb{E}[\tau] = \infty$ . The stopped process is not UI. Result: $S_\tau = 1$ a.s. yet $\mathbb{E}[S_0] = 0$ — OST inapplicable. This is why budget constraints (infinite borrowing = impossible) are essential to ruling out doubling strategies.

PRACTICE

L3 — Researcher. Expected: original reasoning and model critique.

"Explain the Doob-Meyer decomposition and its role in defining quadratic variation. How does this connect to the Itô isometry?" Every submartingale $X = M + A$ (martingale + predictable compensator). Applying this to $M_t^2$ (a submartingale when $M$ is a martingale): the compensator is the quadratic variation $\langle M \rangle_t$ , uniquely characterised by $M_t^2 - \langle M \rangle_t$ being a martingale. The Itô isometry $\mathbb{E}\!\left[\left(\int_0^T H_s \, dM_s\right)^2\right] = \mathbb{E}\!\left[\int_0^T H_s^2 \, d\langle M \rangle_s\right]$ is the $L^2$ norm of the stochastic integral expressed through the quadratic variation bracket — direct consequence of the Doob-Meyer decomposition applied to the square of the integral.
"What is the martingale representation theorem, and what does market completeness mean in its terms?" On the Brownian filtration: every $L^2(\mathbb{Q})$ variable $X$ decomposes as $X = \mathbb{E}^\mathbb{Q}[X] + \int_0^T H_s \, dB_s^\mathbb{Q}$ for a unique predictable square-integrable $(H_t)$ . Market completeness = uniqueness of $(H_t)$ = uniqueness of the equivalent martingale measure $\mathbb{Q}$ . Stochastic volatility models (Heston, SABR) are incomplete: the filtration is generated by $B^S$ and $B^\sigma$ , two independent Brownians. The MRT then requires two integrands, and the second hedge ratio for vol risk cannot be determined without adding a traded variance instrument.
"In a jump-diffusion model, why is predictability of the integrand more restrictive than in the continuous case?" For continuous semimartingales, $\mathcal{F}_{t^-} = \mathcal{F}_t$ a.s. (paths are continuous, so the information just before $t$ and at $t$ coincide). For jump processes, $\Delta M_t = M_t - M_{t^-}$ is non-zero at jump times: using $H_t$ vs. $H_{t^-}$ as the integrand changes the integral by $\sum_{s \leq t} H_s \Delta M_s - \sum_{s \leq t} H_{s^-} \Delta M_s = \sum_{s \leq t} \Delta H_s \Delta M_s$ . This correction term appears explicitly in the jump-diffusion Itô formula and changes the replication cost of claims with discontinuous payoffs.