Malliavin calculus for stochastic differential equations with jumps

带跳跃的随机微分方程的 Malliavin 微积分

• 批准号: 13640132
• 负责人: KOMATSU Takashi
• 金额: $ 2.05万
• 依托单位: Osaka City University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640132/
• 关键词: Hormander condition; Malliavin calculus; filtering equation; jump type process; hypoellipticity; Hilbert space; stochastic flow; stochastic differential equation; 飛躍型確立過程; フィルタリング方程式; セミマルチンゲール; ヘビィ過程; 確率密度関数

The first result is on the regularity of solution to the filtering equation for a certain system of jump type processes. Applying a new key lemma in the Malliavin calculus, the existence of smooth density of the solution was proved under a generalized Hormander condition. The second result is the following :Let a_0(x,x^^-) be an R^N - valued smooth function on R^N × R^N , v(dθ) be a finite measure on a discrete space θ, β_t = (β^θ_t) be a Wiener process, and let μ(dv) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u ∈ R^d, 【numerical formula】Assume that x^u(0) is smooth in u and the process x(t) = (x^u(t)) takes its values in the Hilbert space H = (L^2(R^d,B(R^d),μ))^N. Let π: H → R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H : 【numerical formula】The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H with the help of a new key lemma. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.

第一个结果是针对某个跳跃类型过程的过滤方程的解决方案的规律性。在Malliavin计算中应用新的钥匙引理时，溶液的平滑密度的存在被证明是在广义的荷尔曼德条件下。第二个结果是：让A_0（X，X ^^ - ）成为R^n×r^n，v（dθ）上的R^n - 在离散空间θ，β_T=（β^θ_t）上是有限的测量，是一个wiener过程，让μ（dv）是一个强大的集成条件。考虑一个特定类型的SDE系统：对于u∈R^d，[数值公式]假设X^u（0）在u中是平滑的，并且过程x（t）=（x^u（t））在Hilbert Space H =（l^2（l^d，b（r^d），r^d），μ），μ）^n中采用其值。令π：h→r^m为有界的线性映射。随机变量π（x（t））定律的平滑密度的存在称为SDE的部分低纤维化。介绍了H：[数值公式]的矢量场的部分荷尔曼德条件。部分荷尔曼德定理可以应用于关于由SDE系统定义的随机流诱导的措施的绝对连续性传播的问题。

项目成果

A.Takeuchi: "Malliavin calculus for SDE with jumps and the partially hypoelliptic problem"Osaka J. Math.. 39. 523-559 (2002)

A.Takeuchi：“带跳跃的 SDE 的 Malliavin 演算和部分亚椭圆问题”Osaka J. Math.. 39. 523-559 (2002)

T. Kamae: "Stochastic analysis based on deterministic Brownian motion"Israel J. Math.. 125. 317-346 (2001)

T. Kamae：“基于确定性布朗运动的随机分析”Israel J. Math.. 125. 317-346 (2001)

M. Yoshida: "Gap invariance of a symmetric invariant lamination"Tokyo J. Math.. 24. 1-12 (2001)

M. Yoshida：“对称不变层压的间隙不变性”Tokyo J. Math.. 24. 1-12 (2001)

M.Yoshida: "On the pinched circle model and the……topological polynomial"Indagationes Math.. (to appear).

M.Yoshida：“关于收缩圆模型和......拓扑多项式”Indagationes Math..（即将出现）。

T.Kamae 他3名: "Automata, algebraicity and distribution of sequences of powers"Ann. Inst. Fourier, Grenoble. 51・No.3. 687-705 (2001)

T.Kamae 和其他 3 人：“自动机、代数性和幂序列的分布”Ann. Fourier，Grenoble。 51·No.3（2001）