Itô’s formula for Walsh’s Brownian motion and applications

1. Itô’s formula for Walsh’s Brownian motion Let E = R2; we will use polar co-ordinates (r, α) to denote points in E. We denote by C(E) the space of all continuous functions on E. For f ∈ C(E), we define fα(r) = f (r, α), r > 0, α ∈ [0, 2π [. Throughout this paper we fix μ a probability measure on [0, 2π [, also we define f (r) =  2π 0 f (r, α) μ(dα), r > 0. Let (P t )t≥0 be the semigroup of a reflecting Brownian motion on [0, ∞[ and let (P0 t )t≥0 be the semigroup of a Brownian motion on [0, ∞[ killed at 0. Then for t ≥ 0, define Pt to act on f ∈ C0(E) as follows: Pt f (0, α) = P t f (0), Pt f (r, α) = P t f (r) + P 0 t (fα − f )(r) for r > 0 and α ∈ [0, 2π [. We recall the following Theorem 1.1 (Barlow et al., 1989). (Pt)t≥0 is a Feller semigroup on C0(E), the associated process is by definition the Walsh’s Brownian motion (Zt)t≥0 (with angles distributed as μ). We can write Zt = (Rt , Θt) where the radial part Rt = |Zt | is a reflected Brownian motion and (Θt)t≥0 is called the angular process of Z . To well define Θt , when Zt = 0, we set by convention Θt = 0. ∗ Corresponding author. E-mail addresses: [email protected] (H. Hajri), [email protected] (W. Touhami). 0167-7152/$ – see front matter© 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.12.021 H. Hajri, W. Touhami / Statistics and Probability Letters 87 (2014) 48–53 49 Notations: We denote by Eμ, the state space of Walsh’s Brownian motion, i.e. Eμ = {(r, α) : r ≥ 0, α ∈ supp(μ)}. We define the tree-metric ρ on R2 as follows: for z1 = (r1, α1), z2 = (r2, α2) ∈ R2, ρ(z1, z2) = (r1 + r2)1{α1≠α2} + |r1 − r2|1{α1=α2}. Note that the sample paths of Walsh’s Brownian motion are continuous with respect to the tree-topology, since the process cannot jump from one ray to another one without passing through the origin. These paths are also continuous with respect to the relative topology induced by the Euclidean metric on R2. Now denote by D the set of all functions f : Eμ → R such that (i) f is continuous for the tree-topology. (ii) For all α ∈ supp(μ), fα is C2 on ]0, ∞] and f ′ α(0+), f ′′ α (0+) exist and are finite. (iii) For all K > 0, sup 0 0 and for ε > 0, define τ ε 0 = 0 and for n ≥ 0 σ ε n = inf{r ≥ τ ε n : |Zr | = ε}, τ ε n+1 = inf{r ≥ σ ε n : Zr = 0}. Set g(z) = f (z) − f ′ α(0+)|z|1{z≠0} where α = arg(z). Note that g ∈ D and g ′ α(0 ) = 0. We will prove our formula first for g . We have g(Zt) − g(0) = lim ε→0 ∞  n=0  g(Zσ ε n+1∧t) − g(Zσ ε n ∧t)  = lim ε→0  Aεt + D ε t 