On the Laplace Transform of a Continuous Additive Functional

by P.J. Fitzsimmons

The theorem below was established in [1] in the context of a stable Markov chain on {0, 1, 2, …}. We present here a simple proof based on the domination principle.

Let X = (Ω, F, F_{t}, X_{t}, θ_{t}, P^{x}) be a right Markov process with state space E, finite lifetime ζ, and cemetery state Δ. Let A = (A_{t})_{t≥0} be a continuous additive functional of X such that A_{ζ} < ∞ almost surely. Define φ(x) := P^{x}[exp(−A_{ζ})], with the understanding that φ(Δ) = 1. Observe that

(1)  1 − exp(−A_{ζ}) = ∫_{0}^{ζ} exp(−A_{ζ}^{∘}θ_{t}) dA_{t},

whence the identity

(2)  1 − φ(x) = P^{x} ∫_{0}^{ζ} φ(X_{t}) dA_{t} = U_{A}φ(x),  x ∈ E_{Δ},

where U_{A} is the potential operator associated with A: U_{A}φ(x) := P^{x} ∫_{0}^{ζ} φ(X_{t}) dA_{t}. In particular,

M_{t}^{φ} := P^{x} [∫_{0}^{ζ} φ(X_{s}) dA_{s}|F_{t}] = ∫_{0}^{t} φ(X_{s}) dA_{s} + 1 − φ(X_{t}), t ≥ 0,

is a uniformly integrable martingale under P^{x}, for each x ∈ E. The intuitive content of this statement becomes clearer when A has the simple form A_{t} = ∫_{0}^{t} a(X_{s}) ds for some bounded positive Borel function a. Writing L for the infinitesimal generator of X, we see that the martingale property of M^{φ} amounts to saying that Lφ = a · φ. The theorem stated below asserts that φ is the maximal solution of this equation with values in [0, 1].

Theorem. Let w : E_{Δ} → [0, 1] be a finely continuous function, and suppose that

M_{t} := w(X_{t}) − ∫_{0}^{t} w(X_{s}) dA_{s}

is a uniformly integrable martingale under P^{x}, for each x ∈ E. Then w ≤ φ.

Proof. Define h := w + U_{A}w. It is easy to check that h is harmonic, in the sense that P^{x}[h(X_{T})] = h(x) for each stopping time T. Observe that

U_{A}(φ − w) = 1 − h − (φ − w) ≤ 1 − h on {φ > w}.

The harmonicity of 1 − h and the domination principle now imply that

U_{A}(φ − w) ≤ 1 − h on E.

That is,

−(φ − w) ≤ 0 on E,

which means that w ≤ φ on all of E_{Δ}.

Reference

[1] P.K. Pollett and V.T. Stefanov: Path integrals for continuous-time Markov chains. J. Appl. Prob. 39 2002 901-904.