Introduction

In stochastic modeling, we often work with a filtered probability space $(\Omega, \mathcal{F},\mathbb{F},\mathbb{P})$ where the probability measure $\mathbb{P}$ encodes our “beliefs”, and the filtration—collection of $\sigma$-algebras— \(\mathbb{F}=(\mathcal{F}_t)_{t\ge0}\) represents the information that’s available to us at any time. In Bayesian inference, we typically update our beliefs by changing the probability measure. However, there are many situations in engineering and finance where it’s more natural to change the filtration while holding the probability measure fixed.

For instance, imagine an investor who gains insider information about the outcome of a future random variable (e.g., default price, terminal price). This extra information alters the structure of “fairness" in the underlying stochastic system, i.e., processes that were martingales under the original information may no longer be so under the new one.

The mathematical framework that studies how stochastic processes behave under such changes in information is called Enlargement of Filtrations, which has been studied extensively in the probability literature; see (Grigorian & Jarrow, 2023; Jacod, 2006; Jeanblanc et al., 2009).

Foundations

Before diving into filtration enlargement, it’s worth reviewing a few basic concepts in probability theory. A probability space is a triple $(\Omega, \mathcal{F},\mathbb{P})$, consisting of:

• $\Omega$: a set of possible outcomes,

• $\mathcal{F}$: a set of measurable (observable) events,

• $\mathbb{P}$: a probability measure assigning likelihoods to events in $\mathcal{F}$.

In the presence of a temporal dimension (e.g., when working with stochastic processes), it’s helpful to specify not only what events are measurable (according to $\mathcal{F}$) but also at what time they become measurable. This leads to the concept of a filtration:

\[\mathbb{F}=(\mathcal{F}_t)_{t\ge 0}\]

where each \(\mathcal{F}_t\) encodes the information available at time $t$. A filtered probability space is simply the quadruple $(\Omega, \mathcal{F},\mathbb{F},\mathbb{P})$. A stochastic process $(X_t)_{t\ge0}$ is said to be adapted to $\mathbb{F}$ if $X_t$ is $\mathcal{F}_t$-measurable for all $t$. That is, the value of $X_t$ can be determined exactly by the information available at time $t$.

Martingales and Semi-martingales

Let \((\Omega, \mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) be a filtered probability space, and let $M_t$ be an $\mathbb{F}$-adapted stochastic process with $\mathbb{E}\left|M_t\right|<\infty$ for all $t$. If \(\mathbb{E} \!\left[M_t|\mathcal{F}_s\right]=M_s\) for all $0\le s< t$, then $M_t$ is said to be an $\mathbb{F}$-martingale. Intuitively, a martingale represents a “fair game": given the current information, its expected future value is just the present value. A process $X_t$ is a semimartingale if it can be decomposed as \(X_t=M_t+A_t\) where $M_t$ is a local martingale (which behaves like a martingale up to random stopping times)¹ and $A_t$ is an $\mathbb{F}$-adapted finite-variation process. Semimartingales form the largest class of stochastic processes for which we can talk about Itô calculus.

Enlargement of filtrations

A martingale represents a fair game, but fairness is only relative to the filtration (relative to the given information). A fair game in the eyes of one player may not be a fair game in the eyes of another.

Consider two students who are taking bets based on a one-dimensional Brownian motion $X_t$. One has access only to the natural filtration $\mathbb{F}$ (consisting of the values of the Brownian motion up to current time), while the other secretly knows the value of some random variable $\zeta$ (e.g. value of the Brownian motion at some later time $T$). From the perspective of the better-informed student, an $\mathbb{F}$-Brownian motion $X_t$ may no longer be a martingale considering the information from $\zeta$. Describing $X_t$ in the presence of extra information is the goal of Enlargement of Filtrations.

Setup

There are two main types of “enlargement":

• Initial enlargement: all secret information is available at time $0$.

• Progressive enlargement: information is gradually revealed over time.

We will focus on the first case. Let \(\mathbb{F} = (\mathcal{F}_t)_{t \ge 0}\) be the original filtration, and let $\zeta$ be an $\mathcal{F}$-measurable random variable taking values in some measurable space $E$. We define the enlarged filtration \(\mathbb{G} = (\mathcal{G}_t)_{t \ge 0}\) by

\[\mathcal{G}_t = \mathcal{F}_t \vee \sigma(\zeta),\]

where $\vee$ denotes the smallest $\sigma$-algebra containing elements from $\mathcal{F}_t$ and $\sigma(\zeta)$. Intuitively, $\mathbb{G}$ describes a world in which the value of $\zeta$ is known from the start—one has “insider” information about $\zeta$ at time $0$.

The $\mathcal{H}$ and $\mathcal{H}’$ Hypotheses

The central question is how martingales behave under a filtration enlargement.

If every $\mathbb{F}$-martingale remains a $\mathbb{G}$-martingale, we say that $\mathbb{F}$ is immersed in $\mathbb{G}$, or that the pair $(\mathbb{F}, \mathbb{G})$ satisfies the $\mathcal{H}$-hypothesis. This is quite a strong condition—informally, it means that the extra information carried by $\zeta$ does not interfere with the “fairness” of any $\mathbb{F}$-martingale.

A weaker but more flexible condition is the $\mathcal{H}’$-hypothesis: every $\mathbb{F}$-martingale remains a $\mathbb{G}$-semimartingale. Under this assumption, Itô calculus remains valid, but an additional drift term may appear when expressing the $\mathbb{F}$-martingale in the $\mathbb{G}$ filtration.

Jacod’s condition

Among several sufficient conditions ensuring the $\mathcal{H}’$-hypothesis, Jacod’s condition is often the most convenient in practice. The following statements (theorem and lemma) are adapted from [@Grigorian2023].

::: theorem Theorem 1 (Jacod’s condition). *Let $\zeta$ be a random element in a standard Borel space $(E,\mathcal{E})$, and let $Q_t(\omega, dx)$ be the regular conditional distribution of $\zeta$ given $\mathcal{F}_t$ for all $t\ge0$. Suppose that there exists a positive $\sigma$-finite² measure $\eta$ on $(E,\mathcal{E})$ such that

\[Q_t(\omega, dx)\ll\eta(dx)\text{ a.s., }\forall t\ge0.\]

Then $\mathcal{H}’$ holds.* :::

Just knowing that a process remains a semimartingale is usually not enough, we typically would like the semimartingale decomposition in the enlarged filtration³ $\mathbb{F}^{\sigma(\zeta)}$. To do so, we need the following lemma and theorem.

Lemma 1. Under $\mathcal{J}$, there exists a nonnegative $\mathcal{O}(\mathbb{F})\times\mathcal{B}(\bar{\mathbb{R}})$-measurable function \(\Omega\times \mathbb{R}_+\times\bar{\mathbb{R}}\ni (\omega, t,x)\mapsto p_t(\omega, x)\) cadlag in $t$ such that*

• $Q_t(\omega,dx)=p_t(\omega,x)\eta(dx)$ for every $t\ge0$,

• for each $x\in\bar{\mathbb{R}}$, the process $(p_t(x))_{t\ge0}$ is an $\mathbb{F}$-martingale,

• *$p(x)>0$ and $p_-(x)>0$ on⁴ \(⟦0,\zeta^x ⟦\) and $p(x)=0$ on \(⟦\zeta^x,\infty⟦\), where $\zeta^x:=\inf{t:p_{-}(x)=0}$. :::

Theorem 2. *Suppose that the random element $\zeta$ satisfies $\mathcal{J}$. If $X$ is an $\mathbb{F}$-local martingale, the process $\tilde{X}$ defined as

\[\tilde{X}_t:=X_t-\int_0^t\frac{1}{p_{s-}(\zeta)}d\langle{X,p(u)\rangle}^\mathbb{F}_s|_{u=\zeta}, \quad t\le T\quad\]

is an $\mathbb{F}^{\sigma(\zeta)}$-local martingale.* :::

Example: Brownian bridge via initial enlargement

Consider a standard Brownian motion $B=(B_t)_{t\ge 0}$ on $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ with its natural filtration ($\mathcal{F}_t=\sigma(X_s,0\le s\le t)$). Suppose we enlarge $\mathbb{F}$ with the information given by $\zeta:=B_T$, i.e.,

\[\mathcal{G}_t = \mathcal{F}_t \vee \sigma(B_T), \qquad \mathbb{G}=(\mathcal{G}_t)_{t\ge 0}.\]

Again, $\mathbb{G}$ is assumed to be right-continuous by taking

\[\mathcal{G}^+_t=\bigcap_{s>t}\mathcal{G}_s.\]

Our goals are:

1. Verify Jacod’s condition;

2. Compute the semimartingale decomposition of $B$ w.r.t. $\mathbb{G}$;

Step 1: Verifying Jacod’s condition.

For $t<T$, the conditional law of $B_T$ given $\mathcal{F}_t$ is Gaussian:

\[B_T \mid \mathcal{F}_t \sim \mathcal{N}\big(B_t,\; T-t\big).\]

Hence, there exists a dominating measure (the Lebesgue measure) $\eta(dx)=dx$. $Q_t(\omega,dx)$ has a density with respect to Lebesgue measure: $Q_t(\omega,dx) \;=\; p_t(\omega,x)\,dx$, with

\[p_t(x) = \frac{1}{\sqrt{2\pi (T-t)}}\exp\!\Big(-\frac{(x-B_t)^2}{2(T-t)}\Big),\quad 0\le t<T.\]

Thus $Q_t(\cdot,dx)\ll \eta (dx)$ a.s. for each $t<T$, so Jacod’s condition holds on $[0, t)$. (On $[T,\infty)$, the conditional law is degenerate at $B_T$.)

Step 2: Computing the semimartingale decomposition.

We apply Theorem 2 with $X=B$. To do so, we first compute the $\mathbb{F}$-predictable covariation $\langle B, p(x)\rangle^\mathbb{F}$. By Itô’s formula, we can get

\[dp_t(x) = \partial_{B_t} p_t(x)\, dB_t \;=\; \frac{x-B_t}{T-t}\, p_t(x)\, dB_t, \qquad t<T.\]

Hence the predictable covariation with $B$ is just

\[d\big\langle B, p(x)\big\rangle_t^{\mathbb{F}} = \frac{x-B_t}{T-t}\, p_t(x)\, dt, \qquad 0\le t<T.\]

Plugging this into $(*)$ with $x=\zeta=B_T$, we obtain from Theorem [2] that the process

\[\tilde{B}_t:=B_t-\int_0^t\frac{B_T-B_s}{T-s}\,ds, \qquad0\le t<T\]

is a $\mathbb{G}$-local martingale. Since $B$ is continuous with quadratic variation $[B]_t=t$, it follows that $[\tilde{B}]_t=t$ as well, and by Lévy’s characterization $\tilde{B}$ is a $\mathbb{G}$-Brownian motion on $[0,T)$. Equivalently, $B$ admits the $\mathbb{G}$-semimartingale (indeed, SDE) representation

\[dB_t = d\tilde{B}_t \;+\; \frac{B_T - B_t}{T-t}\, dt, \qquad 0\le t<T,\]

where $\tilde{B}$ is a $\mathbb{G}$-Brownian motion. This is precisely the Brownian bridge drift toward the terminal pinning value $B_T$.

Notes

## References

1. Enlargement of Filtrations: An Exposition of Core Ideas with Financial Examples

   Karen Grigorian and Robert A. Jarrow

   2023
2. Grossissement initial, Hypothese (H’) et theoreme de Girsanov

   Jean Jacod

   In Grossissements de filtrations: exemples et applications: Séminaire de Calcul Stochastique 1982/83 Université Paris VI, 2006
3. Mathematical Methods for Financial Markets

   Monique Jeanblanc, Marc Yor, and Marc Chesney

   2009