A càdlàg path is right-continuous and has a left limit
Objective: Without looking back, explain why “On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold.” follows, identify the decisive convention and estimate, and name the first failed line under “Left-continuity without right-continuity does not satisfy the convention and moves the recorded jump to a different side of the event time.”
Unit 1, chapter 1 starts with an observable question: “A càdlàg path is right-continuous and has a left limit.” A jump is a discontinuous change at a stated time, not a decorative spike on a plot. Before calculating, record the sample path convention, left limit, jump size, time interval, filtration, probability law, and whether the object is raw, predictable, compensated, or integrated.
Read the core definition in ordinary language before reading its symbols: A path x on [0,T] is càdlàg when x_t approaches x_s as t decreases to s and the finite left limit x_{s-} exists for every s>0. The path layer records when and by how much a state changes. The random-measure layer records how many marked jumps land in a time–mark set. The compensator records the predictable average rate. These are related objects, but they are not interchangeable.
The bounded theorem target is: On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold. Every conclusion belongs to the declared truncation, filtration, topology, and integrability range. Equality in distribution, pathwise equality, indistinguishability, convergence in probability, and convergence in a Skorokhod topology are different claims and are never silently substituted.
The proof route is to begin from “A path x on [0,T] is càdlàg when x_t approaches x_s as t decreases to s and the finite left limit x_{s-} exists for every s>0.”; retain the filtration, jump time, mark, truncation convention, and integrability class; apply only the estimate named in “On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold.”; and stop before the boundary “Left-continuity without right-continuity does not satisfy the convention and moves the recorded jump to a different side of the event time.”. A valid arrow may use independent increments, a Poisson generating function, monotone convergence, compensation, an isometry, localization, Itô’s formula with jumps, a Gronwall estimate, or a martingale-problem argument. Naming one of these tools without checking its hypotheses does not justify the arrow.
The exact finite checkpoint is: Let x_t=0 for t<1 and x_t=3 for t≥1 on [0,2]; give x_{1-}, x_1, and the right limit at 1. Keep the event rows, half-open time cells, jump marks, signs, units, probabilities, and rational arithmetic visible. The audited endpoint is They are 0, 3, and 3, so the path is right-continuous at 1 and has a left limit there. A finite table verifies one identity or estimator; it does not prove a process-level limit, a tail asymptotic, or an application claim.
The first invalid step after changing assumptions is part of the lesson: Left-continuity without right-continuity does not satisfy the convention and moves the recorded jump to a different side of the event time. Locate the first missing measurability, predictability, integrability, independence, finite-activity, Lipschitz, or topology condition. Preserve earlier valid lines and state the extra condition needed for repair instead of forcing the original result.
Keep five evidence layers separate. Path bookkeeping identifies jumps and left limits. Measure theory defines intensity and compensation. Martingale calculus controls centered fluctuations. Analytic estimates support existence, uniqueness, and limits. Modeling choices specify data, loss, interventions, or physical meaning. Passing one layer never certifies the other four automatically.
The chapter closes with reconstruction rather than recognition. Starting from raw event rows, rebuild the definition, justify the decisive theorem line, reproduce the exact endpoint, activate the stated mutation, and report the strongest conclusion that survives. A familiar formula, one matching simulation, or the word “Lévy” is not completion.
Bounded theorem for this chapter: On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold. The claim has exactly the pathwise, probabilistic, and topological meaning stated here; no stronger frontier or application conclusion is included.
Type every object in “A path x on [0,T] is càdlàg when x_t approaches x_s as t decreases to s and the finite left limit x_{s-} exists for every s>0..” State whether it is a càdlàg path, stopping time, jump size, counting measure, intensity measure, predictable compensator, local martingale, finite-variation process, semimartingale, integrand, generator, or probability law. Record its domain and sigma-field.
Fix conventions before algebra. Use half-open time intervals consistently, choose one truncation function, distinguish X_t from X_{t-}, define Delta X_t=X_t-X_{t-}, and state whether small jumps are raw or compensated. A convention change must also change drift and formula terms.
Verify the finite event algebra. Count each recorded jump once, add its mark with the declared sign, separate disjoint cells, and recompute the same total both from the path increments and from the associated counting measure.
Establish the analytic budget required by the theorem. Integrate the named function against the intensity or Lévy measure, check square or absolute integrability on the correct region, localize if only local control is available, and state every finite-activity or moment threshold.
Apply the decisive arrow: begin from “A path x on [0,T] is càdlàg when x_t approaches x_s as t decreases to s and the finite left limit x_{s-} exists for every s>0.”; retain the filtration, jump time, mark, truncation convention, and integrability class; apply only the estimate named in “On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold.”; and stop before the boundary “Left-continuity without right-continuity does not satisfy the convention and moves the recorded jump to a different side of the event time.”. If compensation is used, subtract the predictable mean; if an isometry is used, match both measures and integrability classes; if an SDE estimate is used, compare the same stopped solutions and include jump terms.
Reconstruct the exact checkpoint independently: Let x_t=0 for t<1 and x_t=3 for t≥1 on [0,2]; give x_{1-}, x_1, and the right limit at 1. Verify They are 0, 3, and 3, so the path is right-continuous at 1 and has a left limit there. Then repeat by a second route such as conditioning on the jump count, summing event rows, applying the characteristic exponent, or evaluating the generator on the test function.
Activate the mutation “Left-continuity without right-continuity does not satisfy the convention and moves the recorded jump to a different side of the event time..” Report the first line that loses meaning or its bound. These steps prove exactly “On a compact time interval, a càdlàg real path is bounded and has only finitely many jumps larger than any fixed positive threshold.”; they do not prove unlisted fluctuation identities, sharp tails, infinite-dimensional equations, singular SPDE, production simulation accuracy, causal validity, or domain performance.
Worked jump model to reconstruct from exact inputs: Let x_t=0 for t<1 and x_t=3 for t≥1 on [0,2]; give x_{1-}, x_1, and the right limit at 1. Preserve event order, cell endpoints, marks, compensators, and exact arithmetic until the final comparison.
- Write the complete finite input for “Let x_t=0 for t<1 and x_t=3 for t≥1 on [0,2]; give x_{1-}, x_1, and the right limit at 1.”: observation window, half-open cells, event times, marks, intensity or Lévy measure, truncation, integrand, and every parameter. Reject unordered or dimensionally ambiguous rows.
- Build the path and measure views separately. Compute left limits and jumps from the path; then place the same events into the counting measure and verify that no endpoint is counted twice.
- Apply the chapter construction: expectation, compensation, characteristic exponent, jump integral, Itô correction, generator action, recursion, or likelihood contribution. Keep exact values until the last comparison.
- Check the decisive measurability and integrability condition, then recompute the endpoint by an independent route. Compare object types and conventions as well as numerical values.
- Both valid routes meet at They are 0, 3, and 3, so the path is right-continuous at 1 and has a left limit there. State the hypothesis that made them agree, activate the boundary mutation, and record the first missing estimate or changed drift term.
Result: Audited answer: They are 0, 3, and 3, so the path is right-continuous at 1 and has a left limit there. The calculation certifies this finite checkpoint under the stated convention and assumptions; it does not certify every path law, asymptotic regime, numerical implementation, or application.