T-LEVY-PATHS: Jump Measures, Compensated Limits and Decomposition on the Original Space
This completes the path proofs of Chapter 27. We first construct paths from a finite-dimensional Lévy triplet, then recover the actual jump measure and independent Gaussian remainder of any given càdlàg Lévy process. Only the rational-coordinate σ-algebra is needed, not the Polish or compactness theorems for the J₁ topology. Random predictable integration is not used.
1. Construct a counting measure from independent bands
Partition the punctured mark space into countably many Borel bands of finite ν-mass. On each band use a Chapter 21 Poisson clock with rate equal to its mass and independent marks distributed according to normalized ν there; zero-mass bands are empty. Chapters 8 and 10 put all bands and an independent Gaussian process on a common product space. Let N count their atoms in Borel time-mark sets, allowing infinity. Each realization is a countable sum of Dirac measures, hence countably additive.
Given the count in a finite window, Chapter 21 identifies arrival times as ordered independent uniforms. With independent marks, conditional expectation for a nonnegative measurable f is a single-point integral raised to the count. The Poisson generating function gives the next identity. Start with finitely many bands and a finite window, then apply MCT; integrability of f is not required.
\begin{gathered}
\mu(ds,dx)=ds\,\nu(dx),\qquad
Ee^{-N(f)}=\exp\!\left[-\int(1-e^{-f})\,d\mu\right],\\
E N(f)=\int f\,d\mu\quad(f\ge0).
\end{gathered}The mean identity also follows from simple functions, single-band count means and MCT. For disjoint finite-intensity sets, take f to be a nonnegative linear combination of indicators. The first formula factors into Poisson count transforms, and the counts are nonnegative integers, so successive one-variable uniqueness from Chapter 13 identifies every coefficient of the joint PGF. The joint law is the product Poisson law, proving independence. Infinite-intensity sets have infinite count almost surely because the expectation of the exponential of minus their count is zero. Actual counts and independence have been established, not merely the name PRM.
2. The compensated isometry for deterministic functions
For a simple function on disjoint finite-intensity sets, define I(f) by weighting count minus intensity. Poisson variance equals its mean and disjoint counts are independent; multiplication gives the isometry. Such simple functions are dense in L² by σ-finiteness, and T-LP completeness uniquely extends I.
EI(f)=0,\qquad E[I(f)I(g)]=\int fg\,d\mu,\qquad
E|I(f)-I(g)|^2=\|f-g\|_{L^2(\mu)}^2.Here f,g are deterministic real functions. Define vector functions coordinatewise and sum squared norms. This is not integration of not-yet-developed predictable random integrands. For functions in both L¹ and L², the ordinary count integral minus the intensity integral agrees with this extension, by simple approximation in both norms.
3. From fixed-time L² to a common path limit
\begin{gathered}
M^\varepsilon_t=\int_{(0,t]\times\{\varepsilon<\|x\|\le1\}}
x\,[N(ds,dx)-ds\,\nu(dx)],\qquad
v(\varepsilon)=\int_{0<\|x\|\le\varepsilon}\|x\|^2\nu(dx)\downarrow0,\\
E\sup_{t\le T}\|M^\delta_t-M^\varepsilon_t\|^2
\le4T v(\varepsilon)\quad(0<\delta<\varepsilon\le1).
\end{gathered}Use the completion of the natural filtration generated by all past counts and Gaussian coordinates: add every null set from the total σ-field at every time, without requiring right continuity. Finite truncations are finite jump sums minus linear functions and are càdlàg. Independent time-window counts give the martingale property relative to all past counts and an independent Gaussian past. Apply Chapter 15’s scalar L² Doob inequality on dyadic grids and sum coordinate suprema. Càdlàg paths make the supremum over dense grids equal the full supremum; MCT/Fatou yields the vector bound. Section 2 computes terminal variance, without a general continuous-time maximal theorem.
\begin{gathered}
v(\varepsilon_j)\le2^{-4j},\qquad \varepsilon_j\downarrow0,\\
P\!\left(\sup_{t\le j}\|M^{\varepsilon_{j+1}}_t-
M^{\varepsilon_j}_t\|>2^{-j}\right)\le4j\,2^{-2j}.
\end{gathered}Borel–Cantelli eventually bounds the differences by 2⁻ʲ. Telescoping gives uniform Cauchy convergence on every compact interval on one probability-one event. Because the filtration is completed, setting the process to zero on the exceptional null set preserves adaptedness. The uniform limit is càdlàg: fix an accurate truncation, use its right continuity and left limits, and use uniform Cauchy control for the left-limit values themselves. Denote the limit by M and set it to zero on the exceptional set.
E\sup_{t\le T}\|M_t-M^\varepsilon_t\|^2
\le4T v(\varepsilon)\longrightarrow0.Fatou gives the estimate for the whole truncation family, not just the selected subsequence. Fixed-time L² convergence and conditional expectation contraction transfer the martingale property to M; increment variance is duration times the small-jump second integral. Finite truncations have factored joint increment characteristic functions, and convergence in probability preserves factorization. Thus M has independent stationary increments and is stochastically continuous.
Fix any 0<ε≤1. Eventually εⱼ<ε. The common uniform limit also gives convergence of supremum differences, and Fatou yields
E\sup_{t\le T}\|M_t-M^\varepsilon_t\|^2
\le\liminf_{j\to\infty}4T
\int_{\varepsilon_j<\|x\|\le\varepsilon}\|x\|^2\nu(dx)
=4T v(\varepsilon).This treats every cutoff ε, not only the selected sequence. Intersect probability-one events first over positive integer T; every finite window is contained in one of them. Pass the fixed-time conditional-expectation identities through L² contraction to retain the martingale property. Joint characteristic functions of increments on disjoint time windows pass through probability limits, retaining stationary independence.
4. The model atoms really are its jumps
Y_t=bt+G_t+
\int_{(0,t]\times\{\|x\|>1\}}x\,N(ds,dx)+M_t,\qquad
E[G_sG_t^\top]=(s\wedge t)A.Construct G from Chapter 23 Brownian motion and a matrix square root, independently of N. Large jumps are finite in compact windows even if their marks lack first moments. The model is càdlàg, has independent stationary increments and is stochastically continuous. Multiply finite-truncation characteristic functions and use second-order domination as ε decreases to zero, obtaining the unit-ball-truncated LK exponent.
Two arrival times in different bands coincide with probability zero because their laws are continuous and independent; a countable union excludes every cross-band coincidence. Within each band arrivals are distinct too. On the common uniform-convergence event, the jump of a remainder is at most twice its uniform norm. Thus each actual atom is eventually visible and recovered with its mark, while at a nonatom all truncation jumps are zero and so is the limit jump. This controls all times simultaneously rather than joining fixed-time null sets.
\|\Delta(M-M^{\varepsilon_j})_t\|
\le2\sup_{s\le T}\|M_s-M^{\varepsilon_j}_s\|,
\qquad N_Y=\sum_{t>0:\Delta Y_t\ne0}\delta_{(t,\Delta Y_t)}=N.5. Read a path’s jump measure from rational coordinates
First a deterministic lemma. Every càdlàg vector path z on [0,T] admits finite-step uniform approximations. With infimum of the empty set equal to infinity, use the recursion below, stopping at the first T. On [tⱼ,tⱼ₊₁) take value z(tⱼ), and give T its separate value z(T).
t_0=0,\qquad
t_{j+1}=\inf\{s\in(t_j,T]:\|z(s)-z(t_j)\|>\eta\}\wedge T .Right continuity gives positive segment lengths and error at most η before the next segment. If a new time is less than T, a sequence in the trigger set decreasing to it and right continuity give separation at least η between adjacent anchors. Infinitely many segment times accumulating at a point would make those anchors Cauchy by its left limit, a contradiction. There are finitely many segments; assigning T separately handles its terminal jump. Since η is arbitrary, this proves uniform approximation. It is a deterministic argument and uses no stochastic first-hitting measurability theorem.
Take a continuous compactly supported mark function f vanishing within radius η and a continuous time weight a. Approximate the path by a finite step function within η/4. On a fine dyadic mesh, cells containing no step boundary have increments smaller than η and contribute nothing. Every actual jump that can contribute to f also lies in that finite boundary set: away from a step boundary its norm is at most twice the approximation error, less than η. A sufficiently fine mesh has at most one boundary in each cell. Only these finitely many cells remain; their endpoints approach the boundary’s left and right values, and time weights converge. Hence
\sum_i a(t_i)f(x(t_i)-x(t_{i-1}))
\longrightarrow\sum_{0<s\le T}a(s)f(\Delta x_s).Use cells (tᵢ₋₁,tᵢ]: a dyadic boundary is captured by the left cell, the jump at T by the last cell, and time zero is excluded. Integer T and dyadic endpoints make every left side depend only on rational coordinates. For relatively open time-mark rectangles, approximate their indicators increasingly by nonnegative continuous product functions, with marks restricted to bounded annuli away from zero. Write the path as z and use the increasing windows below.
\begin{gathered}
E_m=(0,m)\times\{x:1/m<\|x\|<m\},\qquad m\ge2,\\
N_z(E_m)<\infty,\qquad
N_z(B)=\lim_{m\to\infty}N_z(B\cap E_m).
\end{gathered}Fix m. The Borel subsets B of Eₘ for which z↦N_z(B) is measurable form a λ-system: it contains Eₘ; complements use the finite difference N_z(E_m)-N_z(B), never infinity minus infinity; disjoint countable unions use sums. Relative open rectangles form a generating π-system, so π–λ gives all Borel subsets of Eₘ. Increasing windows then give every Borel count, possibly infinite. Finitely many jumps in each Eₘ, guaranteed by the step lemma, are essential for the complement argument.
Rational coordinates of the large-jump sum J^{>1} and finite compensated truncations are measurable integrals. The common good set has the following countable form, with all indices positive integers.
\begin{gathered}
\mathcal H=\bigcap_{m,r\ge1}\ \bigcup_{J\ge1}\
\bigcap_{j,k\ge J}
\left\{z:\sup_{q\in\mathbb Q\cap[0,m]}
\|M_q^{\varepsilon_j}(z)-M_q^{\varepsilon_k}(z)\|\le1/r\right\},\\
C_t(z)=z_t-bt-J_t^{>1}(z)-M_t(z).
\end{gathered}Each cutoff difference is càdlàg, so its rational-time supremum equals its full compact-window supremum. Thus H gives exactly locally uniform Cauchy convergence. Define M as that uniform limit on H and zero off H. The limit and residual C are càdlàg, with measurable rational coordinates. Only the rational-coordinate σ-algebra is used; D-space Polishness or topological compactness is not needed.
6. Return the decomposition to the original probability space
Now X is a given càdlàg, stochastically continuous process with stationary independent increments and X₀=0. Its unit-time law is infinitely divisible. Obtain its unique parameters from T-LEVY-CHARACTERISTICS first, without using this path decomposition. If its unit exponent is Ψ, the characteristic function at time 1/n times exp(−Ψ/n) is continuous, takes values among nth roots of unity and equals one at frequency zero, hence is identically one. Independence gives all rational-time exponents, and stochastic continuity extends to every time.
Thus X and the model Y have the same rational finite-dimensional laws. By π–λ they have the same law on the rational-coordinate σ-algebra, and all joint measurable functionals from Section 5 have the same law. On the model these recover N,M and G. Transfer that joint identity back to X to obtain the following result on its original space.
\begin{gathered}
N_X=\sum_{s>0:\Delta X_s\ne0}\delta_{(s,\Delta X_s)}
\text{ is PRM with intensity }ds\,\nu(dx),\\
X_t=bt+C_t+\int_{(0,t]\times\{\|x\|>1\}}x\,N_X(ds,dx)
+\lim_j\int_{(0,t]\times\{\varepsilon_j<\|x\|\le1\}}x\,[N_X-ds\,\nu],\\
C\text{ is continuous Gaussian with covariance }(s\wedge t)A,\qquad
C\ \perp\!\!\!\perp\ N_X .
\end{gathered}The identity holds for all times on one probability-one event. The selected compensated limit is locally uniform and the full truncation family converges ucp. If A is singular, only the stated continuous Gaussian remainder is asserted; the original space need not contain extra independent standard Brownian coordinates for zero-variance directions. If A is zero, C is zero.
Continuity of the remainder must itself be included in the measurable transfer, rather than transferring separate fixed times. A càdlàg C is continuous on every compact window exactly when its rational coordinates satisfy the uniform-modulus condition below. Sufficiency follows by uniformly continuous extension of the rational values and identification by right continuity; necessity is uniform continuity on compact intervals.
\mathcal C=\bigcap_{m,r\ge1}\ \bigcup_{n\ge1}\
\bigcap_{\substack{p,q\in\mathbb Q\cap[0,m]\\|p-q|<1/n}}
\{z:\|C_p(z)-C_q(z)\|\le1/r\}.Both H and this continuity event belong to the rational-coordinate σ-algebra and have probability one in the model, hence also for X. Transfer Gaussian finite-dimensional laws first at rational times. For finite families of these coordinates and Borel counting events, model independence gives product probabilities. Apply π–λ on both sides to obtain independence of the generated σ-fields. Continuity extends the rational coordinates to all times. Once the decomposition holds at rational times, càdlàg paths of every term extend that same-event identity to all times; no uncountable union of null sets is used.
The decomposition also respects the completion of X’s natural filtration. For fixed t, restrict Section 5’s mesh to [0,t]. Even for irrational t it uses only coordinates at times at most t. Past jump counts, finite compensated sums and their limits are measurable with respect to the completed natural past; the residual definition gives the same for C. Conversely the path decomposition recovers X’s past from C and N, so
\overline{\mathcal F_t^X}=
\overline{\sigma\!\left(C_s,N_X((0,s]\times B):
s\le t,\ B\text{ Borel},\ \nu(B)<\infty\right)}.The bars add null sets of the original total σ-field, without requiring right-continuous augmentation. In the model, future increments of (C,N) are independent of this past; finite-cylinder transfer and π–λ give the same fact on the original space. The continuous remainder thus has Brownian increment structure with covariance A relative to this natural information. No such assertion is made for an arbitrary enlarged filtration revealing the future.
7. Finite activity and finite variation criteria
Finite jump activity is equivalent to finite total ν-mass. In the finite case Poisson counts are finite on compact windows; in the infinite case the Laplace formula gives infinite count in every positive rational window almost surely, and a countable intersection suffices. This counts jumps without excluding a separate rough Gaussian part.
X\text{ has finite variation on compact intervals a.s.}
\quad\Longleftrightarrow\quad
A=0,\qquad\int_{0<\|x\|\le1}\|x\|\nu(dx)<\infty.If the integral diverges, the Laplace formula makes the expected exponential of minus the absolute small-jump sum zero, so that sum is infinite almost surely. A deterministic càdlàg path’s variation is at least the sum of its jump norms: select finitely many jumps, choose disjoint small left neighborhoods, shrink them, then take the supremum over finite jump sets. Finite variation is impossible.
If A is nonzero, choose a projection of positive variance. The continuous Gaussian part is independent of the full jump path. For a fixed jump path j, if Gaussian plus j has finite variation with probability p(j)>0, take two independent Gaussians sharing j; both have finite variation with probability p(j)². Their difference is nonzero scaled Brownian of finite variation, contradicting Chapter 26. Fubini shows that positive total probability would give positive integral of p(j)². Finite variation is measurable through suprema over rational partitions. No general quadratic variation is needed.
Conversely Campbell’s mean formula gives finite absolute small-jump sum on compact windows. Rewrite the compensated term as an absolutely convergent ordinary jump sum plus a finite linear drift. Large jumps are locally finite, proving finite variation.
X_t=dt+\sum_{s\le t}\Delta X_s,\qquad
d=b-\int_{\|x\|\le1}x\,\nu(dx)\quad\text{in the finite-variation case}.Complete two limits in the variation criterion. For necessity, use 1−e^{−r}≥(1−e^{−1})r for 0≤r≤1 in the Poisson Laplace formula. An infinite first integral makes the expectation of the negative exponential of the absolute jump sum zero, hence that sum infinite almost surely. For sufficiency use the integrable random total and pathwise tail bound below.
\begin{gathered}
A_T^{\rm jump}=\sum_{\substack{s\le T\\0<\|\Delta X_s\|\le1}}
\|\Delta X_s\|,\qquad
EA_T^{\rm jump}=T\int_{0<\|x\|\le1}\|x\|\nu(dx)<\infty,\\
\sup_{t\le T}\left\|\sum_{\substack{s\le t\\0<\|\Delta X_s\|\le\varepsilon}}\Delta X_s\right\|
\le\sum_{\substack{s\le T\\0<\|\Delta X_s\|\le\varepsilon}}
\|\Delta X_s\|\longrightarrow0 .
\end{gathered}On the common event of finite absolute sums on integer windows, the last bound holds for every t simultaneously. The deterministic compensation tail is bounded by T times the small-ball first integral and also vanishes. Thus the ordinary absolute jump sum minus its linear integral equals the previously constructed M, not merely a variable with the same fixed-time law. The variation event used in Gaussian necessity is measurable: variation on a compact window is the supremum of increment sums over rational partitions. Treat Gaussian and jump paths as independent rational-coordinate random elements; Fubini gives the two-Gaussian-copies-with-one-jump-path argument without an unproved conditional path distribution.
8. The precise boundary for increasing paths
For real, non-killed processes of infinite lifetime, X is a subordinator from zero exactly when A=0, ν has no negative mass, small jumps have integrable first magnitude, and the actual drift d in Section 7 is nonnegative. Positive ordinary jump sums prove sufficiency. Necessity follows from finite variation and nonnegative jumps, except that the sign of d still needs proof.
Let S be the positive jump sum and split a short interval h at mark size ηh. The chance of a larger jump is at most hν((ηh,∞)), which tends to zero by domination with x/η on (0,1] and DCT; on (1,∞) it is h times a finite mass, also tending to zero. The mean of the remaining sum divided by h is the first integral below that threshold, also tending to zero. Thus S_h/h tends to zero in probability and X_h/h tends to d. Nonnegative X_h forces d≥0.
Ee^{-qX_t}=
\exp\left[-t\left(dq+\int_{(0,\infty)}(1-e^{-qx})\nu(dx)\right)\right],
\qquad q\ge0.Density 1/x on (0,1) gives infinite activity with finite variation. Density 1/x² with compensation still has only positive jumps, but its absolute small-jump sum diverges; it is not a subordinator. This compensated process has mean zero and variance t and is nondegenerate. If it were nonnegative, its zero mean would force degeneracy, another direct reason why absence of negative jumps does not imply monotonicity.