Stochastic calculus for symmetric Markov processes

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Chen, Z.-Q. ; Fitzsimmons, P. J. ; Kuwae, K. ; Zhang, T.-S. (2008)
  • Publisher: The Institute of Mathematical Statistics
  • Journal: (issn: 0091-1798)
  • Related identifiers: doi: 10.1214/07-AOP347
  • Subject: 60J55 | 60J57 | 60H05 | additive functional | 31C25 | generalized Itô formula | Symmetric Markov process | dual additive functional | Mathematics - Probability | 31C25 (Primary) 60J57, 60J55, 60H05 (Secondary) | stochastic integral | Revuz measure | time reversal | martingale additive functional | dual predictable projection

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained.
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    Lemma 2.17. Let A be a finite rcll PrAF with defining sets {Ξ, Ξt, t ≥ 0}. There then exists a real-valued Borel function ϕ on EΔ × EΔ with ϕ(x, x) = 0 for x ∈ EΔ such that A with defining sets

    Proof. Let Ξ ∈ F∞, Ξt ∈ Ftm, t > 0 be the defining sets of A admitting m-null set. We easily see rt−1(Ξt) ∩ {t < ζ} ⊂ rs−1(Ξs) ∩ {s < ζ} for s ∈ ]0, t[ by use of Lemma 2.15(i) and θt−sΞt ⊂ Ξs.

    Set Ξbt := rt−1(Ξt) for t > 0 and Ξb := Tt>0 Ξbt. We then see that Ξb = Tt>0,t∈Q Ξbt by use of rt−1(Ξt) ∩ {t ≥ ζ} = {t ≥ ζ} and the monotonicity of Lemma 3.1.

    Theorem 3.6. For an MAF M of finite energy, Λ(M ) defined above coincides on [[0, ζ[[ with Γ(M ) defined in (1.5), Pm-a.e. for all t > 0 by way of its subadditivity. Hence, we obtain (3.12).

    (iii) Suppose that f ∈ F . Let Kt be a purely discontinuous local MAF on [[0, ζ[[ with Kt − Kt− = −ϕ(Xt−, Xt) − ϕ(Xt, Xt−) on ]0, ζ[. Then, 2 i,j=1 s [2] Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T.-S. (2008). Perturba-

    tion of symmetric Markov processes. Probab. Theory Related Fields 140 239-275. [3] Chen, Z.-Q., Fitzsimmons, P. J., Takeda, M., Ying, J. and Zhang, T.-S. (2004).

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