dx(t)=a1(x(t),y(t),t)dt+b1(x(t),y(t),t)dw1(t)+c1(x(t),y(t),t)dw2(t),dy(t)=a2(x(t),y(t),t)dt+b2(x(t),y(t),t)dw1(t)+c2(x(t),y(t),t)dw2(t),w1(t)和w2(t)是维纳过程且统计独立;将其改写为stratonovich型随机微分方程
时间: 2023-08-08 07:08:18 浏览: 44
根据Ito到Stratonovich的转换公式,可以将原始随机微分方程转化为Stratonovich型随机微分方程:
$$
\begin{aligned}
dx(t)&=a_1(x(t),y(t),t)dt+b_1(x(t),y(t),t)\circ dw_1(t)+c_1(x(t),y(t),t)\circ dw_2(t)\\
&+\frac{1}{2}\left(\frac{\partial b_1}{\partial x}(x(t),y(t),t)c_1(x(t),y(t),t)-\frac{\partial c_1}{\partial y}(x(t),y(t),t)b_1(x(t),y(t),t)\right)dt\\
dy(t)&=a_2(x(t),y(t),t)dt+b_2(x(t),y(t),t)\circ dw_1(t)+c_2(x(t),y(t),t)\circ dw_2(t)\\
&+\frac{1}{2}\left(\frac{\partial b_2}{\partial x}(x(t),y(t),t)c_2(x(t),y(t),t)-\frac{\partial c_2}{\partial y}(x(t),y(t),t)b_2(x(t),y(t),t)\right)dt
\end{aligned}
$$
其中,$\circ$表示Stratonovich积分符号,$\frac{\partial b_1}{\partial x}$表示$b_1$对$x$的偏导数,$\frac{\partial c_1}{\partial y}$表示$c_1$对$y$的偏导数,$\frac{\partial b_2}{\partial x}$表示$b_2$对$x$的偏导数,$\frac{\partial c_2}{\partial y}$表示$c_2$对$y$的偏导数。