Planar diffusions with rank-based characteristics and perturbed Tanaka equations

Abstract

For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1(·), X 2(·)) with values in the plane and infinitesimal generator and discuss its realization in terms of appropriate systems of stochastic differential equations. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation dZ(t) = f (Z (t))dM(t) + dN(t), Z(0) = ξ driven by suitable continuous, orthogonal semimartingales M(·) and N(·) and with f(·) of bounded variation, which we study here in detail; and those of a one-dimensional diffusion Y(·) with bang-bang drift dY(t) = -λsign(Y(t))dt + dW (t), Y(0)=y driven by a standard Brownian motion W(·). We also show that the planar diffusion (X 1(·), X 2(·)) can be represented in terms of this process Y(·), its local time L Y (·) at the origin, and an independent standard Brownian motion Q(·), in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion. © 2012 Springer-Verlag.