Chapter 30: What Does It Mean to Know Before It Happens?

An arrival occurs. A recorder draws a vertical mark; someone else sees the mark and then writes at the same position, “I was waiting for an arrival here.” The timestamps agree perfectly. That agreement does not mean the second person anticipated the arrival.

In a finite sequence of bets, a time step separates choosing a stake from revealing its outcome. Continuous time has no smallest step. A rule can use only information available by each present time and still exploit a jump that has just occurred at that very time. Before calculating accumulations along random paths, we must distinguish these two meanings of “already known.”

30.1 Seeing a Jump Does Not Let Us Backdate a Choice

Take a Poisson process N of rate λ>0 with its complete right-continuous natural filtration. Chapter 21 proves independent arrivals on disjoint time intervals and the martingale property of Mₜ=Nₜ−λt. No new integration theory is needed yet. On a finite interval N has finitely many jumps and λt has finite variation, so ordinary pathwise Stieltjes integrals against them are available.

Choose a bounded ξ using Fₛ before acting on (s,t]. Conditional averaging says that the compensated gain over this interval has no bias. The same remains true for a finite sum of such choices.

Now use Hₜ=ΔNₜ: take value one exactly when an arrival occurs and zero otherwise. H is adapted and can be read from the record up to t. It takes value one at every jump but zero almost everywhere for length measure. Consequently,

The calculation is correct. The mistake would be to identify “readable afterward” with “chosen beforehand.” Adaptedness alone cannot justify extending the earlier conditional-mean identity to every such rule. We need control over the boundary between the two times.

30.2 An Event Also Needs a Time Coordinate

A random variable needs measurability on Ω. A process H is a function on Ω×[0,K]; checking Hₜ separately loses their common measurable structure. We need to distinguish two kinds of rules on the product space, rather than choose an unrelated exceptional null set at every time.

The optional σ-algebra O is generated by all càdlàg adapted real processes. “Optional” does not mean permission to use arbitrary future information. It describes information readable from a path as currently displayed. Arrival counts, right-continuous prices, and first-arrival indicators belong here.

The other class places a choice before the interval on which it acts: use an Fₛ-measurable ξ on (s,t], and F₀ at the initial time. The σ-algebra generated by these rectangles is denoted P and called the predictable σ-algebra. It is equivalently generated by all left-continuous adapted processes.

The name does not express an arbitrary preference for left endpoints. The interval (s,t] specifies that a choice made at s may act on subsequent changes, but cannot be revised after seeing what happens at t. Left-continuous processes are approximated by increasingly fine left-endpoint choices; conversely, each rectangular indicator above is itself left-continuous and adapted. This proves the equivalence.

We have P⊂O, and optional processes are progressively measurable, so reading an optional process at a stopping time produces an \mathcal F_T-measurable variable. Full arguments for the generators and stopped evaluations are in the continuous-information proof. Predictability does not require continuity: an amount known in advance to be paid at noon tomorrow may enter an accumulated account by a jump.

30.3 Fixed-Time Tests Do Not Capture Knowing Just Before

For a càdlàg adapted X, the left-limit process Xₜ₋ is left-continuous and adapted on t>0, hence predictable. Xₜ can include a newly observed jump. Fixed-time tests may miss the distinction completely: ΔNₜ=0 almost surely at every deterministic t, but \Delta N_T=1 at the first arrival T.

Thus “at every fixed t it almost surely equals something known beforehand” cannot replace predictability of the whole process. Chapter 23 distinguished modifications from indistinguishable versions. Here the same issue of incompatible exceptional sets directly affects whether a rule may be used in an accumulation.

Information just before a random time T is precisely the σ-algebra below. It retains F₀ and statements of the form “B was observed at s and T had not yet arrived.” If H is predictable, H_T on {T<∞} is \mathcal F_{T-}-measurable. This follows from the rectangular generators and the monotone class theorem, without requiring T to be announceable.

30.4 Some Times Can Be Approached; Others Must Be Awaited

Suppose a continuous adapted path starts below a threshold. Before first reaching it, the path must reach slightly lower thresholds; those earlier hitting times approach the target time. A memoryless arrival is different: no sequence of observed approach signals guarantees that the next signal is the arrival.

A stopping time T with values in [0,∞] is predictable, or announceable, if stopping times Tₙ increase to T with Tₙ<T on {0<T<∞}; the initial event {T=0} is handled in F₀. One may also require Tₙ<T and Tₙ→∞ on {T=∞}, without changing the finite-time notion. Deterministic times are the simplest examples. For a continuous-path hitting time, use levels approaching from the interior; their hitting times must first be legitimate stopping times, as proved for continuous paths in Chapter 16.

Announcement gives an important measurable conclusion. The indicator of (Tₙ,T] is left-continuous and adapted; intersect these intervals over n to obtain the graph of T. The graph of an announceable time is therefore P-measurable. A prior rule can select it exactly, a stronger statement than measurable sections at each fixed time.

T is accessible if its finite graph is covered by countably many graphs of predictable stopping times. Accessibility is not predictability. For example, reveal an independent nondegenerate coin only at time 1, and let T=1 on heads and T=∞ on tails. Its finite graph is covered by deterministic time 1. If T were announceable, {T=1} would belong to F₁₋, contradicting revelation only at 1. The filtration is trivial before 1, contains the coin from 1 onward, and is given its usual augmentation.

The fact used in the last argument can be checked directly: if Tₙ announces T, then {T=1}=∩ₙ{Tₙ<1}∩{T≥1} belongs to F₁₋. In this example T takes only 1 and ∞, so the second event is the whole space.

30.5 An Unannounced Time Can Still Have a Rate

T is totally inaccessible on positive times if it coincides with any predictable S at a finite positive time with probability zero. For a first Poisson arrival, this requires proof rather than the description “the arrival is sudden.”

Starting from the simple-rule identity in 30.1, the rectangular monotone class theorem gives the following equality for every nonnegative predictable H. The argument compares the finite measures E∫H dN and λE∫H dt; monotone convergence then covers arbitrary nonnegative H.

Take H to be the indicator of the positive-time graph of a predictable S. Each graph has zero dt-length, so E[\Delta N_S;0<S\le K]=0. If the first arrival T equals S, then \Delta N_S=1; hence P(T=S\le K)=0. A union over integer K proves total inaccessibility. This does not mean arrivals cannot be predicted statistically: λ still controls conditional average counts precisely, without supplying an exact announceable time.

Accessible and totally inaccessible pieces do decompose a general stopping time. Take the supremum of the probabilities of {0<T<∞} covered by countable unions of predictable graphs. Choose covers approaching that supremum and unite them to obtain a countable cover C attaining it. Restrict T to {(ω,T)∈C} for its accessible piece and to the complement for its totally inaccessible piece. These restrictions are stopping times because coincidence events belong to the appropriate stopped information. If the complement intersected another predictable graph with positive probability, adding that graph to C would exceed the supremum. Accessible pieces of two decompositions differ only on a null set, since an accessible graph and a totally inaccessible graph have null intersection. Treat time zero separately as predictable.

30.6 A Right-Continuous Filtration Does Not Draw Paths for Us

The construction below produces conditional expectations E[Y|Fₜ]. Choosing a version at each t does not supply a càdlàg path. That version problem must be solved before the process is read at random times.

The usual conditions have two parts: F₀ contains all ambient null sets, and Fₜ is the intersection of Fₛ over s>t. Right continuity controls information obtained by approaching from the right; completeness lets us remove a common exceptional set. These are conditions on the filtration, not a promise of continuity for every adapted process.

Under the usual conditions, the closed martingale E[Y|Fₜ], Y∈L¹, has a càdlàg modification. Start with upcrossing on countable dense grids, exclude infinite oscillation and escape, and take right limits. Reverse martingale convergence for conditional expectations and right continuity of the filtration identify the fixed-time limits with the original variables. For a class D submartingale, right continuity of the means is additionally needed for this extension to be a modification. The regularization proof supplies the common null set, L¹ limits, and stopped-sampling details.

For example, the deterministic Sₜ=1{t>0} is a bounded submartingale in a trivial usual filtration but has no modification right-continuous at zero; its mean is not right-continuous there either. Conversely, any genuine martingale on a fixed interval ending at K fits the closed case with Y=M_K. Agreement at dense times and right continuity patch the finite-interval modifications, giving martingale regularization on the half-line.

30.7 Retaining Integral Mass at Randomly Selected Times

Finite expectation at a fixed time does not control times selected by observing a path. The condition needed for Doob–Meyer controls those selections directly: on [0,K], the variables Sτ over all stopping times τ≤K must form a uniformly integrable family. Such a process is of class D.

An increasing process with integrable terminal value has this property, since its stopped values are dominated by that terminal value. Closed martingales also have it: Mτ=E[Y|Fτ], and conditional averages of one integrable Y over arbitrary sub-σ-algebras form a uniformly integrable family. Prove the latter identity first on finite grids and then pass to the limit using càdlàg paths and uniform integrability, without circularly invoking a decomposition.

It prevents precisely the loss of expectation seen in the spikes of Chapter 7. Here a spike may be selected not only by a deterministic index n, but by a history-dependent time. Controlling all such selections is what retains the average accumulation as grids become finer.

30.8 On a Grid, the Growth Part Really Can Be Calculated in Advance

Let S be a càdlàg class D submartingale. Rescale time first to [0,1]. On the dyadic grid Dₙ the conditional mean of each future increase is nonnegative. This leads to the natural decomposition below: accumulate those conditional means in Cⁿ and call the remainder Mⁿ.

Cⁿ is increasing, and its jth increment is known at the preceding grid point; Mⁿ is a grid martingale. This explains why predictability enters the decomposition: we extract the prior conditional mean, not every increase seen afterward.

Cⁿ need not be monotone in n when the grid is refined, so monotone convergence does not apply. Even a fixed value of E Cⁿ₁ gives only L¹ boundedness, not an L¹-convergent subsequence. First we must show that terminal values do not transfer mass to increasingly rare huge outcomes; then we need a limiting operation that preserves conditional averaging.

30.9 Averaging Approximations Without Erasing Temporal Order

The appropriate operation is a finite forward convex combination. For uniformly integrable (fₙ), choose deterministic weights λₙ,ₖ≥0 with sum one and finite support in k≥n so that the weighted sums converge in L¹. Deterministic weights do not inspect sample outcomes; nonnegative weights preserve monotonicity; using only forward grids ensures that every fixed coarse grid is eventually contained in all participating grids.

No unestablished weak compactness theory is required: T-DM-COMPACT puts every truncation into one square-summable Hilbert space, obtains a Cauchy sequence through minimum-norm convex combinations and the parallelogram identity, then removes truncations uniformly using uniform integrability. The same proof file establishes the convex-growth criterion and the required weak L¹ subsequence result. The compactness step is actually supplied, not replaced by the name of a lemma.

30.10 Why Stopping Times Control the Compensator Tails

Subtract the closed martingale Zₜ=E[S₁|Fₜ] and put R=S−Z. The submartingale relation gives R≤0 and R₁=0. R remains càdlàg and class D. The discrete compensators Cⁿ are unchanged because every conditional mean increment of Z is zero. Write Dⁿ=Cⁿ₁. The grid martingale identity becomes

Here τ may be any Dₙ-valued stopping time. For r>0, stop at the grid point just before Cⁿ first exceeds r, calling it τₙ(r); take 1 if there is no crossing. This predecessor is still a stopping time because the next compensator increment is already measurable now. It satisfies C^n_\tau\le r and {τ<1}={Dⁿ>r}. Therefore

Markov inequality alone cannot handle the first term: it would only give a constant that does not vanish. Stop in advance again at level r/2, writing σ=τₙ(r/2). On {Dⁿ>r}, at least r/2 of compensation remains after σ; on the other paths the remaining compensation is still nonnegative. Integrating the conditional identity gives

Combine the inequalities to bound the tail expectation. The two events have probabilities at most 2c₀/r and c₀/r, uniformly tending to zero. Class D makes all these stopped values Rτ uniformly integrable, so their integrals on the small-probability events vanish uniformly. This proves uniform integrability of (Dⁿ).

Since Mⁿ₁=S₁−Dⁿ, the martingale terminal values are uniformly integrable too. The decisive point is not bounded total expectation. It is the legitimate choice of an observation time just before compensation becomes large, translating the tail back into stopped values of the original submartingale.

30.11 A Common Terminal Limit Produces Genuine Decomposition Paths

Extend Mⁿ to the closed martingale Mⁿₜ=E[Mⁿ₁|Fₜ]. Use common forward convex weights from 30.9 so that gₙ=Σλₙ,ₖMᵏ₁→g in L¹. Set Mₜ=E[g|Fₜ], take its càdlàg modification from 30.6, and put C=S−M.

At any fixed dyadic t, all participating grids contain t once n is large, so ΣλCᵏₜ=Sₜ−M̄ⁿₜ→Cₜ in L¹. Take a subsequence giving almost sure convergence simultaneously at countably many dyadic times. Each convex combination increases in time, so C increases on the dense grid; càdlàg paths extend monotonicity to every time. We have C₀=0 and integrable C₁=S₁−g. Thus we have genuine increasing paths, not merely unrelated fixed-time limiting versions.

One issue remains indispensable: adaptedness of C is not predictability. Stopping here would let the Poisson count itself masquerade as a compensator and erase the distinction discovered in 30.1.

30.12 The Proof Must Still Work at the Instant of a Jump

Extend each discrete Cᵏ to the whole interval: on ((j−1)2⁻ᵏ,j2⁻ᵏ] use its value at the next grid point, and take zero at 0. Call this Lᵏ. Although the value comes from the next grid point, it is already known at the preceding one, so Lᵏ is left-continuous and adapted. With the original convex weights form L̄ⁿ. All these processes are predictable, nonnegative and increasing.

Along the common subsequence from 30.11, convergence on the dense grid gives limsup L̄ⁿₜ≤Cₜ on every remaining path. At continuity points of C, sandwiching between grid points gives full convergence. Only the jumps of C remain. Saying that there are at most countably many jumps is not enough: their locations depend on ω and need not be a fixed sequence of times.

First prove equality at an arbitrary stopping time τ≤1. Let σₖ be the first Dₖ point not smaller than τ; then L^k_\tau=C^k_{\sigma_k}. Grid sampling gives E L^k_\tau=E S_{\sigma_k}-E S_0. Since σₖ↓τ, right continuity and class D give S_{\sigma_k}\to S_\tau in L¹. Every convex combination uses k≥n, hence E\overline L^n_\tau\to E S_\tau-E S_0=E C_\tau as well.

Also 0\le\overline L^n_\tau\le\overline L^n_1, and the latter converges in L¹ to C₁. The stopped family is therefore uniformly integrable. Apply reverse Fatou after truncation, then remove the truncation, to obtain E\limsup_n\overline L^n_\tau\ge\limsup_n E\overline L^n_\tau=E C_\tau. Together with the pathwise upper bound, this yields \limsup_n\overline L^n_\tau=C_\tau almost surely.

Only now unify the exceptional sets. For each positive rational q, set T_q=\inf\{t:C_t\ge q\}, with infinity if never reached. Since C is increasing and right-continuous, \{T_q\le t\}=\{C_t\ge q\}, so each such hitting time is a stopping time. Every positive jump crosses some rational level, hence its time belongs to this countable family of hitting times. Apply the preceding conclusion at T_q\wedge1 and discard the countable union of null sets. Equality now holds simultaneously at all jumps as well. At zero use C₀=L̄ⁿ₀=0 separately.

The right side is a predictable function, and completeness permits a common modification on the exceptional set. Thus C has an indistinguishable predictable version. No general section or début theorem was invoked: jumps were covered by rational-level hitting times of an already constructed increasing path. This step at the jump times is what actually carries “known at the preceding grid point” into continuous time.

30.13 Why Every Qualification in the Decomposition Matters

Existence has now been proved. Restoring any finite K, the result is: under the usual conditions, every càdlàg class D submartingale S has a decomposition S=M+A, unique up to indistinguishability, where M is a uniformly integrable martingale and A is a predictable càdlàg increasing process starting at zero with integrable terminal value. M₀=S₀. Here uniform integrability of the martingale means uniform integrability of all its stopped values on the interval, equivalently closure by its terminal value.

Uniqueness does not use the false unqualified assertion that every finite-variation martingale is constant. Two decompositions give a predictable finite-variation martingale D=A−A′ with integrable total variation. The proof establishes E∫H dD=0 for bounded predictable tests, then takes H=f′(D) for f(x)=√(1+x²). A convex inequality for ordinary finite-variation integration forces E f(Dₜ)≤f(0), hence Dₜ=0; right continuity upgrades this to indistinguishability.

Without predictability the Poisson process has both decompositions N=(N−λt)+λt and N=0+N into a martingale plus an increasing process. Without A₀=0 one can also move an integrable initial variable between the parts. The qualifications remove actual ambiguities; they are not formal decoration.

30.14 Compensation May Jump Without Using Newly Revealed Information

Reveal an integrable nonnegative Y at deterministic time 1, knowing only its law beforehand, and set Sₜ=Y1{t≥1}. With no extra information about Y in F₁₋, the compensator is Aₜ=EY·1{t≥1}, and the remainder is (Y−EY)1{t≥1}. Compensation jumps at 1 without using the newly revealed Y. If Y were already completely known before 1, the compensator would instead be S itself.

More generally, at an announceable bounded positive stopping time T, a uniformly integrable martingale satisfies E[\Delta M_T\mid\mathcal F_{T-}]=0. The announcing sequence approaches from the left, and uniform integrability of stopped values permits L¹ passage to the left limit. The full conditional argument checks the required generators. Consequently,

This is not an unconditional identity at every random time. At the first Poisson arrival T, \Delta N_T=1 while \Delta(\lambda t)_T=0; precisely this T cannot be announced. Failing to distinguish types of times would turn a correct conditional identity into a false pathwise formula.

30.15 The Same Arrival Path Can Carry Different Prior Information

Observe only the first Poisson arrival, with time T and Jₜ=1{T≤t}. The rule 1{s≤T} is left-continuous and adapted, hence predictable. Using the identity in 30.5, or first testing rectangles and then stopping, gives compensator λ(t∧T). Both terminal values are integrable on finite intervals, and their difference is a martingale, also verified directly by conditional interval tests against Fₛ.

If the observer is told T completely at the start, enlarge the initial information to contain T. Then (T−1/n)⁺ announces T, J itself is predictable, its compensator is J, and its martingale part is zero. Neither the law of T nor any arrival path has changed. What changes is the information allowed for prior choices. The old λ(t∧T) is no longer the compensator in this enlarged information structure.

Thus “unpredictable” is not a label attached to an arrival independently of observation, just as Markov behavior in Chapter 17 was not memorylessness independent of the chosen state. The filtration participates in the mathematical conclusion; changing it arbitrarily does not preserve old martingale identities.

30.16 Even an Unadapted Accumulation Has a Prior Projection

Let B be a nonnegative increasing right-continuous process with left limits, B₀=0 and E B_K<\infty. Assume joint measurability for the ambient σ-algebra and time, but not adaptedness. We seek a predictable increasing A such that every nonnegative predictable rule has the same expected accumulation against A as against B. This is the finite-variation foundation of dual predictable projection.

Bₜ itself need not be a submartingale because it may not be adapted. Instead set U_t=E[B_K-B_t\mid\mathcal F_t]. It is a nonnegative supermartingale dominated by E[B_K\mid\mathcal F_t], so its stopped family is uniformly integrable; right continuity of B and dominated convergence make its mean right-continuous. Regularization from 30.6 applied to −U gives a càdlàg modification. Therefore,

is a càdlàg class D submartingale with Vₜ=E[Bₜ|Fₜ] at each deterministic t. For U, first control the stopped family on arbitrary grids by the conditional upper bound; after regularization, approach arbitrary stops from the right. This avoids sampling paths before they have been constructed. V₀=0. Apply the proved Doob–Meyer decomposition to V=L+A.

For bounded ξ∈Fₛ, the martingale increment has zero conditional mean, and the conditional expectations at both deterministic endpoints can be replaced by B. This proves the rectangular identity below. Both sides define finite positive measures on the predictable σ-algebra, so the monotone class theorem and monotone convergence give the identity for all required nonnegative tests.

If A′ also satisfies the identity, testing H=ξ1(s,t] makes A−A′ a predictable finite-variation martingale; uniqueness from 30.13 forces indistinguishability. For a signed raw process starting at zero with integrable total variation, project the positive and negative pathwise Jordan variations separately and subtract. Joint measurability of those variations follows from the supremum over rational partitions and right continuity.

This projection is not completed merely by replacing Bₜ with its conditional expectation at each time. V gives the currently visible average level; A extracts an accumulated measure consistent with every prior rule. The operations act on different objects: one averages values conditionally, the other identifies mass in time through dual tests.

30.17 Extending Observation in Time While Retaining the Conditions

If S is class D on every finite [0,K], decompose on each interval. Uniqueness makes restrictions from longer to shorter intervals agree, so the pieces combine into M and A on the half-line. M is uniformly integrable on each finite interval; this does not mean that all its stopped values over the entire half-line are uniformly integrable. That global conclusion requires additional tail control.

A local version also needs actual localization assumptions: if stopping times ρₙ↑∞ make every stopped S^{\rho_n} satisfy the preceding conditions on finite intervals, their decompositions patch consistently into a local martingale and a locally integrable predictable increasing process. Stopping a predictable càdlàg process preserves predictability: A_{t\wedge\rho}=A_t\mathbf1_{\{t\le\rho\}}+A_\rho\mathbf1_{\{\rho<t\}}. The first term is a product of predictable processes; the second is left-continuous and adapted. This permits uniqueness on overlapping stopped decompositions. Merely saying “localize” does not ensure these conditions.

30.18 Bounded Expectations Can Still Lose Mass at the Endpoint

Class D cannot be replaced by supₜE|Sₜ|<∞. Put tₙ=1−2⁻ⁿ and reveal fair coins successively at these times. Capital is 2ⁿ after n heads, and becomes zero permanently at the first tail. Keep the nth capital on [tₙ,tₙ₊₁), call it Wₜ, and set W₀=1 and W₁=0. Almost every path has already encountered a tail before 1, so W has càdlàg paths; modify the all-heads null event jointly if needed.

Before 1, W is a martingale, but it is not a martingale when the terminal time 1 is included. Conversely S=−W is a submartingale on [0,1]: for s<1 the conditional mean of its endpoint is zero, indeed at least Sₛ. Almost every path is already zero beforehand; the loss of expectation is not a common jump at the endpoint. It has E|Sₜ|=1 for t<1 and S₁=0, while along tₙ it tends to zero almost surely but not in L¹. A decomposition into a uniformly integrable martingale and an increasing process with integrable terminal value would make both stopped families uniformly integrable and therefore make S class D, a contradiction.

No finite discrete conditional-mean calculation fails here. Every finite grid can be decomposed. What fails is transporting those averages intact to the accumulation time 1. The stopped-tail control in 30.10 was precisely what excluded this failure.

30.19 The Boundary of Time Becomes Part of Calculation

Return to the vertical mark of the opening arrival. We can now distinguish a rule available before it appears, a record readable only when it appears, and a time that can have a definite rate without being announceable. Predictability has not removed randomness. It has expressed “not borrowing the next change” as a mathematical structure that survives passage to a limit.

The decomposition is more than bookkeeping. We began with conditional means over small steps. Uniform integrability, convex combinations and identification at jump times have assembled them into a unique predictable accumulating path. The remainder is still a martingale. Looking only at a completed path, both parts may seem readable afterward. Requiring every value to respect the order of information is what actually determines the decomposition.

We have not yet constructed integration against a general rough martingale. Finite-variation sums used for Poisson counts cannot simply be transferred to Brownian motion. We have determined which value rules can be chosen before a change. The next difficulty is to prove when multiplying such rules by increasingly fine random increments yields a stable accumulation. Temporal order is now clear; path roughness still requires its own control.

Sources and Mathematical Scope

Meyer published his original continuous-time decomposition work in 1962 and 1963; the predictable formulation is connected with Doléans-Dade’s work on natural increasing processes. Dates and bibliographic entries can be checked in the authors’ paper and references. The grids, forward convex combinations and jump identification here are a modern mathematical reconstruction, not a reenactment of the historical discovery sequence. The proof structure can be compared with the author-hosted version by Beiglböck, Schachermayer and Veliyev; the chapter and companion files supply the actual derivations.

The proved scope is Doob–Meyer for class D càdlàg submartingales on finite intervals under the usual conditions, and dual predictable projection of raw càdlàg processes with integrable total variation. General random-measure compensation additionally requires marks, σ-finiteness and localization, to be handled with jump integration. Measurable projection, section and general début theorems for arbitrary random sets do not follow automatically from the level-hitting argument here; the decomposition proof does not assume those general theorems.