一类带局部时的随机微分方程和双壁斜布朗运动

A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH LOCAL TIME AND SKEW BROWNIAN MOTION WITH TWO BOUNDARIES

  • 摘要: Itô 和 Mckean ^1 (Problem 1, Seotion 4.2) 首先给出了斜布朗运动的直观描述. Walsh ^2, Harrison 和 Shepp ^3 分别从扩散过程和随机微分方程角度,进一步对单辟斜布朗运动进行了讨论. 本文讨论带两个局部时的随机微分方程:d X(t)=\sigma\left(X_t\right) d B_i+b\left(X_t\right) d t+\beta_1 d \hatL_i^0(X)+\beta_2 d \hatL_i^1(X) \cdots \cdots其中 \beta_1 、 \beta_2 为常数, \hatL_i^z(X) 表示 X 在 x 处的对称局部时,即:\hatL_i(X)=\frac12\left(L_i^*(X)+L_t^-*(-X)\right),B 为标准布朗运动。反复应用凸函数的 Itô 公式,我们得到了 (0.1) 解存在的充分条件.接着,讨论了双壁斜布朗运动;d X_t=d B(t)+\beta_1 d \hatL_t^0(X)+\beta_2 d \hatL_i^1(X) \cdots的构造,并给出它的鞅刻划。在本文中,将经常利用如下基本事实; d \mathcalL_i(X) 以集合 \left\s_1, X_\mathrmH-=x\right\ 为支擈。定理1 考虑方程 (0.1) ,若下列条件满足:(1) \left|\beta_1\right|< 1,\left|\beta_2\right|< 1;(2) \sigma: R \rightarrow R 为 Borel 可测, \forall x \in R, \sigma(x) \neq 0, 且对 \forall u, v:-\infty< u< v< \infty, 有: \int_u^0 \fracd x\sigma_a(x)< \infty, \quad \int_v^0 \frac|b(x)|\sigma^2(x) d x< \infty ;(3) \int_-\infty^0 \exp \left\\int_y^0 \frac2 b(x)\sigma^2(x) d x\right\ d y=\int_0^\infty \exp \left\-\int_0^y \frac2 b(x)\sigma^2(x) d x\right\ d y< \infty.则方程 (0.1) 有唯一解。

     

    Abstract: Consider the following stochastic differential equation: d X(t)=\sigma\left(X_t\right) d B(t)+b\left(X_t\right) d t+\beta_1 d \hatL_t^0(X)+\beta_2 d \hatL_t^1(X) where B(t) is a normalized Brownian motion, \beta_1 and \beta_2 are constants, \hatL_t^*(\boldsymbolX) is the symmetric local time at x of X, the following theorem is established: Theorem 1: If the following conditions are satisfied: (1) \left|\beta_1\right|< 1,\left|\beta_2\right|< 1; (2) \sigma: R \rightarrow R, \sigma(x) is Borel measurable, \forall x \in R_7 \sigma(x) \neq 0, and for any u, v_;-\infty< u< v< \infty, we hewe: \int_u^0 \frac1\sigma^2(x) d x< \infty, \int_u^v \frac|b(x)|\sigma^2(x) d x< \infty; (3) \int_-\infty^0 \exp \left\\int_y^0 \frac2 b(z)\sigma^2(z) d z\right\ d y=\int_0^\infty \exp \left\-\int_0^\nu \frac2 b(z)\sigma^2(z) d z\right\ d y=\infty, then there exists a unique solution X of equation (1) X is called a skew Brownian motion with two boundaries, if d X(t)=d B(t)+\beta_1 d \hatL_i^0(X)+\beta_2 d \hatL_t^1(X). In this paper, the construction and martingale charaoterization of skew Brownian motions with two boundaries are established, too.

     

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