The submartingale problem for a class of degenerate elliptic operators

• 成果类型: Article
• 署名作者: Bass, Richard F.; Lavrentiev, Alexander
• 署名单位: University of Connecticut; University of Wisconsin System
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-006-0047-9
• 发表日期: 2007
• 页码: 415-449
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; boundary-conditions; chains

摘要

We consider the degenerate elliptic operator acting on C-b(2) functions on [0,infinity) (d): [GRAPHIC] where the a(i) are continuous functions that are bounded above and below by positive constants, the bi are bounded and measurable, and the alpha(i) is an element of (0, 1). We impose Neumann boundary conditions on the boundary of [0,infinity)(d). There will not be uniqueness for the submartingale problem corresponding to L. If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for L holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations dX(t)(i) = root 2a(i)(X-t)(X-t(i))(alpha i/2)dW(t)(i) + b(i)(X-t)d(t) + dL(t)(Xi), X-t(i) >= 0, where W-t(i) are independent Brownian motions and L-t(Xi) is a local time at 0 for X-i.