Strong Markov property of the stable process on local càdlàg paths #
This module proves the strong Markov property of stableLocalLaw at stopping times of the
right-continuous natural filtration. Countably valued upper grid approximations reduce the
argument to deterministic times, and càdlàg sample paths give convergence of finite-dimensional
future laws (J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter I,
Proposition 6; O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002,
Theorem 13.11).
Strong Markov property, integral form. For A in the stopped σ-algebra of τ with
A ⊆ {τ < ⊤} and bounded Borel F on DLoc,
∫_A F(shiftAt τ f) dP = P(A) · ∫ F dP. (A = ∅ and τ ≡ 0 allowed.)
Independence form. Under the conditional probability given {τ < ⊤} (when that event has
positive probability), the stopped σ-algebra restricted to the event is independent of the
post-τ path. Stated with the restricted (non-normalised) measure, so that the null case is
vacuous-true and no division is needed: for stopped events A ⊆ {τ < ⊤} and Borel B,
P(A ∩ shiftAt τ ⁻¹ B) = P(A) · P(B).
Law of the post-τ path. Conditionally on {τ < ⊤} (i.e. under the restriction of P
to it), shiftAt τ has the law stableLocalLaw θ, scaled by the mass of {τ < ⊤}.
Compact-window form: the post-τ path cut at horizon U.
Almost surely finite stopping time: when τ < ⊤ a.s., the post-τ path has law
stableLocalLaw θ and is independent of the whole stopped σ-algebra.
Strong Markov, conditional-expectation form (bounded test, on {τ < ⊤}): relative to the
stopped σ-algebra, E[1_{τ<⊤} F(shiftAt τ) | F_τ] = 1_{τ<⊤} E[F] a.s.
Version without indicators when τ < ⊤ a.s.