Planar diffusions with rank-based characteristics and perturbed Tanaka equations

• 成果类型: Article
• 署名作者: Fernholz, E. Robert; Ichiba, Tomoyuki; Karatzas, Ioannis; Prokaj, Vilmos
• 署名单位: University of California System; University of California Santa Barbara; Columbia University; Eotvos Lorand University; Hungarian Academy of Sciences; HUN-REN; HUN-REN Institute for Computer Science & Control
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0430-7
• 发表日期: 2013
• 页码: 343-374
• 关键词: uniqueness

摘要

For given nonnegative constants g, h, rho, sigma with rho(2)+sigma(2) = 1 and g+h > 0, we construct a diffusion process (X-1(center dot), X-2(center dot)) with values in the plane and infinitesimal generator L=1({x1>x2})(rho(2)/2 alpha(2)/alpha x12+sigma(2)/2 alpha(2)/alpha x22-h alpha/alpha x1+g alpha/alpha x2) +1({x1>x2})(sigma(2)/2 alpha(2)/alpha x12+rho(2)/2 alpha(2)/alpha x22 + g alpha/alpha x1-h alpha/alpha x2), (0.1) 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)=xi driven by suitable continuous, orthogonal semimartingales M(center dot) and N(center dot) and with f(center dot) of bounded variation, which we study here in detail; and those of a one-dimensional diffusion Y(center dot) with bang-bang drift dU(t)=-lambda sign(Y(t))dt+dW(t), Y(0)=y driven by a standard Brownian motion W(center dot). We also show that the planar diffusion (X-1(center dot), X-2(center dot)) can be represented in terms of this process Y(center dot), its local time L-Y (center dot) at the origin, and an independent standard Brownian motion Q(center dot), in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.