SDE | 数值方法介绍
2026-06-12 11:27:54 星期五
本节介绍常见的几种数值方法，收敛阶等问题在下一节分析.
Euler - Mavuyama method.
考虑一维的SDE:
\[dX_t=f(t,X_t)dt+\sigma(t,X_t)dB_t, \quad t\in [0,T]
\]
在ODE中，我们对\([0,T]\)进行剖分：
在 \([t_n, t_{n+1}]\) 上
\[X(t_{n+1}) = X(t_n) + \int_{t_n}^{t_{n+1}} f(t, X(t)) dt + \int_{t_n}^{t_{n+1}} \sigma(t, X(t)) dB(t)
\]
用左端点逼近
\[\approx X(t_n) + \int_{t_n}^{t_{n+1}} f(t_n, X(t_n)) dt + \int_{t_n}^{t_{n+1}} \sigma(t_n, X(t_n)) dB(t)
\]
记 \(\overline{X}_n = X(t_n)\)，则数值格式为
\[\overline{X}_{n+1} = \overline{X}_n + f(t_n, \overline{X}_n) \Delta t + \sigma(t_n, \overline{X}_n) \bigl( B(t_{n+1}) - B(t_n) \bigr)
\]
Milstein method
记 \(\Delta_{n+1} B = B(t_{n+1}) - B(t_n) \sim N(0, \Delta t)\).
下面处理 \(B(t_{n+1}) - B(t_n)\): 用 Itô 公式
\[d\sigma(t, X_t) = \left[ \frac{\partial \sigma}{\partial t}(t, X_t) + \frac{\partial \sigma}{\partial x}(t, X_t) f(t, X_t) + \frac{1}{2} \frac{\partial^2 \sigma}{\partial x^2}(t, X_t) \sigma^2(t, X_t) \right] dt + \left( \frac{\partial \sigma}{\partial x} \sigma \right)(t, X_t) dB_t
\]
\[\sigma(t, X_t) = \sigma(t_n, X_{t_n}) + \int_{t_n}^t \left( \frac{\partial \sigma}{\partial t} + \frac{\partial \sigma}{\partial x} f + \frac{1}{2} \frac{\partial^2 \sigma}{\partial x^2} \sigma^2 \right)(s, X_s) ds + \int_{t_n}^t \left( \frac{\partial \sigma}{\partial x} \sigma \right)(s, X_s) dB_s
\]
\[\begin{aligned}
\int_{t_n}^{t_{n+1}} \sigma(t, X_t) dB_t &= \sigma(t_n, X_{t_n}) \Delta_{n+1} B \\
&\quad + \int_{t_n}^{t_{n+1}} \int_{t_n}^t \left( \frac{\partial \sigma}{\partial x} \sigma \right)(s, X_s) dB_s dB_t \\
&\quad + \int_{t_n}^{t_{n+1}} \int_{t_n}^t \left( \frac{\partial \sigma}{\partial t} + \frac{\partial \sigma}{\partial x} f + \frac{1}{2} \frac{\partial^2 \sigma}{\partial x^2} \sigma^2 \right)(s, X_s) ds dB_t \\
&\approx \sigma(t_n, X_{t_n}) \Delta_{n+1} B + \left( \frac{\partial \sigma}{\partial x} \sigma \right)(t_n, X_{t_n}) \int_{t_n}^{t_{n+1}} \int_{t_n}^t dB_s dB_t
\end{aligned}
\]
所以数值格式为：
\[\overline{X}_{n+1} = \overline{X}_n + f(t_n, \overline{X}_n) \Delta t + \sigma(t_n, \overline{X}_n) \Delta_{n+1} B + \left( \frac{\partial \sigma}{\partial x} \sigma \right)(t_n, \overline{X}_n) \int_{t_n}^{t_{n+1}} \int_{t_n}^{t} dB_s dB_t
\]
可以发现milstein格式要比EM格式要多一项. 对于第三项，可以进一步转化：
\[\begin{aligned}
d(B_t - B_{t_n})^2 &= 2(B_t - B_{t_n}) dB_t + \frac{1}{2} \cdot 2 dt \\
(B_{t_{n+1}} - B_{t_n})^2 &= 2 \int_{t_n}^{t_{n+1}} (B_t - B_{t_n}) dB_t + \int_{t_n}^{t_{n+1}} dt \\
&= 2 \int_{t_n}^{t_{n+1}} \int_{t_n}^t dB_s dB_t + \Delta t \\
\Rightarrow \int_{t_n}^{t_{n+1}} \int_{t_n}^t dB_s dB_t &= \frac{1}{2} \left( (\Delta_{n+1} B)^2 - \Delta t \right)
\end{aligned}
\]
\(\theta -EM\) method
\[X(t_{n+1}) = X(t_n) + \underbrace{\int_{t_n}^{t_{n+1}} f(t, X(t)) dt}_{\text{可用 } [t_n, t_{n+1}] \text{ 中任意点逼近}} + \underbrace{\int_{t_n}^{t_{n+1}} \sigma(t, X(t)) dB(t)}_{\text{只能用左端点}}
\]
\[\int_{t_n}^{t_{n+1}} f(t, X(t)) dt \approx \int_{t_n}^{t_{n+1}} \big[ (1-\theta) f(t_n, X(t_n)) + \theta f(t_{n+1}, X(t_{n+1})) \big] dt, \quad \theta \in [0,1]
\]
\(\theta\)-EM method:
\[\overline{X}_{n+1} = \overline{X}_n + (1-\theta)f(t_n, \overline{X}_n)\Delta t + \theta f(t_{n+1}, \overline{X}_{n+1})\Delta t + \sigma(t_n, \overline{X}_n)\Delta_{n+1}B
\]
-
\(\theta=0\) 即为 EM.
-
\(\theta\in(0,1]\) 时，\(\theta f(t_{n+1}, \overline{X}_{n+1})\Delta t\) 中也包含 \(\overline{X}_{n+1}\)，是隐格式，用压缩映射定理：
\[\Phi(z) = \overline{X}_n + (1-\theta)f(t_n, \overline{X}_n)\Delta t + \theta f(t_{n+1}, z)\Delta t + \sigma(t_n, \overline{X}_n)\Delta_{n+1}B \]压缩性:
\[|\Phi(z_1) - \Phi(z_2)| = \big| \theta\Delta t \big[ f(t_{n+1}, z_1) - f(t_{n+1}, z_2) \big] \big| \leq L\theta\Delta t |z_1 - z_2| \]选取 \(\Delta t^*\) 使得 \(L\theta\Delta t^* < 1\)，则 \(\forall \Delta t \in (0, \Delta t^*]\)，\(\Phi\) 是压缩映射.
\(\theta-Milstein\) method
\[\overline{X}_{n+1} = \overline{X}_n + (1-\theta) f(t_n, \overline{X}_n) \Delta t + \theta f(t_{n+1}, \overline{X}_{n+1}) \Delta t + \sigma(t_n, \overline{X}_n) \Delta_{n+1}B + \left( \frac{\partial \sigma}{\partial x} \sigma \right)(t_n, \overline{X}_n) \int_{t_n}^{t_{n+1}} \int_{t_n}^t dB_s dB_t
\]
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