Reflection, restarts and excursion windows #
Deterministic adapters for the Skorokhod reflection of a local càdlàg path: restart formulas, strict downward passage, and generalized minima on windows around excursion starts (J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapters IV and VII, for the reflected process and its excursions away from the infimum).
The reflected path t ↦ y + X_t - min 0 (y + I_t) is càdlàg: I is monotone, bounded on
compacts and right-continuous.
Reflection at zero of y + X: t ↦ y + X_t - min 0 (y + I_t).
Equations
- BGW.StableTrees.reflectedFrom y f = ⟨fun (t : NNReal) => ↑y + ↑f t - min 0 (↑y + BGW.StableTrees.runningInfLoc f t), ⋯⟩
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Reflection from zero is strongly progressive for the local filtration.
No-negative-jumps hypothesis, as used in runningInfLoc_continuous.
Equations
- BGW.StableTrees.NoNegJumps f = ∀ (s : NNReal), Function.leftLim (↑f) s ≤ ↑f s
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Excursion candidate of the reflected path from 0: positive on (g,d), R(g-) = 0,
R(d) = 0; R(g) = 0 is NOT required (jump starts allowed).
Equations
- One or more equations did not get rendered due to their size.
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Excursion interval of the reflected path from 0: an excursion candidate with
R(g) = 0 (continuous start; jump starts are excluded by design).
Equations
- One or more equations did not get rendered due to their size.
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(i) Under NoNegJumps, R g = 0 implies R(g-) = 0 (R ≥ 0, I continuous, so
R(g-) = X(g-) - min 0 I_g ≤ X g - min 0 I_g = R g).
(ii) An excursion candidate with R g = 0 is an excursion interval.
An excursion interval is a candidate.
Reflection is nonnegative.
For X_0 = 0 the reflection from 0 is X - I.
Deterministic restart. Reflection from y at time s + u is the reflection, from the
value R_s reached at s, of the centered shifted path at time u. No hypothesis on f.
Restart at a random time (everywhere form; on {τ f < ⊤}; junk shift otherwise).
Joint measurability of the reflection in (y, f).
Joint measurability of (y, f, t) ↦ (reflectedFrom y f) t.
First passage as the generalized inverse of the running infimum. Everywhere (no
hypothesis): T_x = inf {t | I_t < -x}.
Reflection vanishes at the passage time (creeping): for a spectrally positive path with
X_0 = 0 and T_x finite, I_{T_x} = -x = X_{T_x}, hence R_{T_x} = 0.
Restart at the passage time: the reflected path after T_x is the reflection from 0
of the centered post-T_x path.
The running infimum is constant on an excursion interval (spectrally positive,
X_0 = 0): I_t = I_g for g ≤ t ≤ d.
The left limit of X at the start of an excursion is the running infimum there.
Rational windows around the start g. Every excursion interval with 0 < g has
nonnegative rationals r < g < q < d (for g = 0 no r exists, hence 0 < g).
Candidate version of the window infimum: for an excursion candidate (g,d) (no R g = 0
needed) under NoNegJumps, the infimum of X over [r,q], r < g < q < d, equals X(g-).
The window infimum is the left limit at the start. If (g,d) is an excursion interval and
r < g < q < d, the infimum of X over [r,q] equals X(g-) = I_g.
Window minimizer adapter (feeds IsStableLevyLaw.ae_continuousAt_of_isGenMinimizer):
for an excursion interval (g,d) and r < g < q < d, g - r is a generalized minimizer of the
centered shift at r, restricted to the horizon [0, q - r]. (Here g is generalized, through
its left limit.)
Window minimizer adapter for candidates: the same statement for an excursion candidate
(g,d) (jump start allowed); the generalized minimum over the rational window is attained at g
as a left limit. Sealed continuity of minimizers on countably many rational windows then gives
R g = 0 (a.s., stochastic step).
End of an excursion = first zero of the reflection after any interior rational time:
d = inf {t ≥ r | R_t = 0} for r ∈ (g,d).
Excursion intervals are disjoint: a later start cannot lie inside an earlier interval.