Absolute Continuity of Diffusion Bridges
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Absolute Continuity of Diffusion Bridges Paul McGill Universit´e Claude Bernard Lyon 1, Villeurbanne, France Abstract Given a Brownian motion (bt ) and φ : R → R of finite variation on compacts, the SDE z = b − φ(z) determines a unique regular diffusion. We establish equivalence of their bridge laws on C([0, 1]) via absolute continuity for an approximate version – endpoint in a bounded interval. The periodic case facilitates manipulations with circular diffusion measures. AMS (2000) subject classification. Primary 60J65, Secondary 60H10, 28C20. Keywords and phrases. Approximate bridge, Girsanov formula, Circular measure
NOTATION Let Px [zt = y] denote the semigroup density of (zt ) and write T = R/N. We reserve F for a generic bounded Borel functional on C([0, ∞)), C([0, 1]) or C(T) as appropriate. Here (bt ) signifies Brownian motion and C > 0 is constant. Given φ : R → R, locally bounded and Borel, we know from Zvonkin (1974) that the Itˆo SDE dzt = dbt − φ(zt )dt (0.1) enjoys the pathwise uniqueness property. Thus for each z0 = x it has a unique strong solution (zt ) on the interval of non-explosion [0, ζ). These processes define regular diffusions on R in the sense of Itˆo and McKean (1974). We consider a subfamily: write (z, φ) ∈ Z if (0.1) holds with φ of finite variation on compact intervals. Our main result says that as (z, φ) varies over Z, with (x, y) ∈ R2 held fixed, on C([0, 1]). the bridge laws F → Ex [F(z)|z1 = y] 1define equivalent measures . 1 In the statement we use 0 φ(z) := 0 φ(zs )ds and Φ := 0 φ. Proposition 1. Given (z, φ), (w, θ) ∈ Z then 1 1 Ex [F (z)|z1 = y]h(z, φ; x, y) = Ex F (w)e− 2 0 Mφ,θ (w) |w1 = y h(w, θ; x, y) where h(z, φ; x, y) = eΦ(y)−Φ(x) Px [z1 = y] and Mφ,θ = φ2 − θ2 − (φ − θ) .
2
P. McGill (1) Recall from Rudin (1966) that Φ = φ. . (2) For existence of 0 Mφ,θ (ws )ds, where φ and θ define Radon measures, we invoke the occupation density formula (e.g. Revuz and Yor (1999), Ch.VI). a.e.
Remark 1.
(3) At fixed F our Radon-Nikodym formula holds y almost everywhere, with equality everywhere if the conditional expectations have continuous versions. Our proof incorporates a determining set of such functionals. When (w, θ) = (b, 0) the formula simplifies and especially so on periodic law paths. Using the standard bridge (ξt ) = (bt − tb1 ) and Miura’s map Mφ = φ2 − φ , we find 1 1 1 (0.2) Ex [F (z)|z1 = x]Px [z1 = x] = E F (x + ξ)e− 2 0 Mφ (x+ξ) √ . 2π One can use this Brownian representation to investigate properties of periodic z-bridges. Compare (Cambronero and McKean, 1999). Starting with a regular diffusion (zt ) and Radon measure μ ≥ 0 on R2 one can define a measure on C([0, 1]) by mixing the bridge laws as in F → Ex [F (z)|z1 = y]μ(dx, dy) for F ≥ 0. The z-reciprocal processes of Jamison (1974) arise from letting μ range over probability measures. Here we select an intrinsic mixture of periodic bridges, specifically the unique measure zˇ on C(T) such that for all F ≥0 F (γ)ˇ z (dγ) = Ex [F (z)|z1 = x]Px [z1 = x]dx. (0.3) C(T)
R
Known as the circular measu
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