wordpress站点地址和,网站列表页内容,电商系统网站建设,深圳深圳网站开发本文的目的在于提供0.5#xff0c;1和1.5阶强SDE数值格式的推导和内容#xff0c;所有推导基于 I t o − T a y l o r Ito-Taylor Ito−Taylor展开#xff0c;由于国内外网站缺少关于强SDE数值阶的总结#xff0c;笔者在此特作总结#xff0c;为使用SDE数值格式的读者提供…本文的目的在于提供0.51和1.5阶强SDE数值格式的推导和内容所有推导基于 I t o − T a y l o r Ito-Taylor Ito−Taylor展开由于国内外网站缺少关于强SDE数值阶的总结笔者在此特作总结为使用SDE数值格式的读者提供帮助本文需要读者预先已经知道了有关Brown-motion的基本性质和随机微分的基本性质否则格式推导会看不懂。如果想要实际应用直接看数值格式即可。
1. SDE强收敛阶判断定理
本文考虑如下SDE: d X b ( x ) d t σ ( x ) d W . dXb(x)dt\sigma(x)dW. dXb(x)dtσ(x)dW. 使用数值SDE在时间 t t t下以 x x x为初值的近似解定义为 X ‾ t , x ( t h ) \overline{X}_{t,x}(th) Xt,x(th),其中 t h ≥ t th\geq t th≥t.真解为: X t , x ( t h ) X_{t,x}(th) Xt,x(th).若满足如下定理: 定理1: 若存在 p 1 , p 2 p_1,p_2 p1,p2,满足: E ∣ X t , x ( t h ) − X ‾ t , x ( t h ) ∣ ≤ k ( 1 ∣ x ∣ 2 ) h p 1 . E |X_{t,x}(th)-\overline{X}_{t,x}(th)|\leq k(1|x|^2)h^{p_1}. E∣Xt,x(th)−Xt,x(th)∣≤k(1∣x∣2)hp1. [ E ∣ X t , x ( t h ) − X ‾ t , x ( t h ) ∣ 2 ] 1 2 ≤ k ( 1 ∣ x ∣ 2 ) 1 2 h p 2 . [E |X_{t,x}(th)-\overline{X}_{t,x}(th)|^2]^{\frac{1}{2}}\leq k(1|x|^2)^{\frac{1}{2}}h^{p_2}. [E∣Xt,x(th)−Xt,x(th)∣2]21≤k(1∣x∣2)21hp2. 且满足 p 1 ≥ p 2 1 2 p_1\geq p_2\frac{1}{2} p1≥p221, p 2 ≥ 1 2 p_2\geq\frac{1}{2} p2≥21.则可认为强收敛阶为 p 2 − 1 2 p_2-\frac{1}{2} p2−21. 该定理的证明笔者在此不做推导感兴趣的读者自行阅读其他文献或教材这一般都是有的这里我们直接利用它进行以下的 I t o − T a y l o r Ito-Taylor Ito−Taylor展开。
2. I t o − T a y l o r Ito-Taylor Ito−Taylor展开
我们考虑: d x b ( x ) d t σ ( x ) d W . dxb(x)dt\sigma(x)dW. dxb(x)dtσ(x)dW. 若有一个非常光滑的 f ( x ) f(x) f(x),根据 I t o Ito Ito公式会有: d f ( x ) [ b ( x ) f x ( x ) 1 2 σ 2 f x x ( x ) ] d t [ σ ( x ) f x ( x ) ] d W . df(x)[b(x)f_x(x)\frac{1}{2} \sigma^2 f_{xx}(x)]dt[\sigma(x)f_x(x)]dW. df(x)[b(x)fx(x)21σ2fxx(x)]dt[σ(x)fx(x)]dW. 这两个微分算子可以被定义为: L 0 b ( x ) d d x 1 2 σ 2 ( x ) d 2 d x 2 L^0b(x)\frac{d}{dx}\frac{1}{2}\sigma^2(x)\frac{d^2}{dx^2} L0b(x)dxd21σ2(x)dx2d2 L 1 σ ( x ) d d x L^1\sigma(x)\frac{d}{dx} L1σ(x)dxd 则对两边取积分会有: f ( x t ) f ( x t 0 ) ∫ t 0 t L 0 f ( x s ) d s ∫ t 0 t L 1 f ( x s ) d W s . f(x_t)f(x_{t0})\int_{t_0}^t L^0f(x_s)ds\int_{t_0}^tL^1f(x_s)dW_s. f(xt)f(xt0)∫t0tL0f(xs)ds∫t0tL1f(xs)dWs. 注意到: ∫ t 0 t L 0 f ( x s ) d s L 0 f ( x t 0 ) ( t − t 0 ) ∫ t 0 t [ L 0 f ( x s ) − L 0 f ( x t 0 ) ] . \int_{t_0}^t L^0f(x_s)dsL^0 f(x_{t_0})(t-t_0)\int_{t_0}^t[L^0f(x_s)-L^0f(x_{t_0})]. ∫t0tL0f(xs)dsL0f(xt0)(t−t0)∫t0t[L0f(xs)−L0f(xt0)]. ∫ t 0 t L 1 f ( x s ) d W s L 1 f ( x t 0 ) ( W t − W t 0 ) ∫ t 0 t [ L 1 f ( x s ) − L 1 f ( x t 0 ) ] d W s . \int_{t_0}^t L^1f(x_s)dWsL^1 f(x_{t_0})(W_t-W_{t_0})\int_{t_0}^t[L^1f(x_s)-L^1f(x_{t_0})]dWs. ∫t0tL1f(xs)dWsL1f(xt0)(Wt−Wt0)∫t0t[L1f(xs)−L1f(xt0)]dWs. 注意到: ∫ t 0 t [ L 0 f ( x s ) − L 0 f ( x t 0 ) ] ∫ t 0 t ∫ t 0 s d [ L 0 f ( x s 1 ) ] d s \int_{t_0}^t[L^0f(x_s)-L^0f(x_{t_0})]\int_{t_0}^t\int_{t_0}^s d[L^0f(x_{s_1})]ds ∫t0t[L0f(xs)−L0f(xt0)]∫t0t∫t0sd[L0f(xs1)]ds ∫ t 0 t [ L 1 f ( x s ) − L 1 f ( x t 0 ) ] ∫ t 0 t ∫ t 0 s d [ L 1 f ( x s 1 ) ] d W s \int_{t_0}^t[L^1f(x_s)-L^1f(x_{t_0})]\int_{t_0}^t\int_{t_0}^s d[L^1f(x_{s_1})]dWs ∫t0t[L1f(xs)−L1f(xt0)]∫t0t∫t0sd[L1f(xs1)]dWs 使用 I t o Ito Ito公式我们可以知道: d [ L 0 f ( x ) ] [ L 0 L 0 f ] d t [ L 1 L 0 f ] d W . d[L^0f(x)][L^0L^0f]dt[L^1L^0f]dW. d[L0f(x)][L0L0f]dt[L1L0f]dW. d [ L 1 f ( x ) ] [ L 0 L 1 f ] d t [ L 1 L 1 f ] d W . d[L^1f(x)][L^0L^1f]dt[L^1L^1f]dW. d[L1f(x)][L0L1f]dt[L1L1f]dW. 我们便可以得到: f ( x t ) f ( x t 0 ) L 0 f ( x t 0 ) ( t − t 0 ) L 1 f ( x t 0 ) ( W t − W t 0 ) R ‾ f(x_t)f(x_{t0})L^0 f(x_{t_0})(t-t_0)L^1 f(x_{t_0})(W_t-W_{t_0})\overline{R} f(xt)f(xt0)L0f(xt0)(t−t0)L1f(xt0)(Wt−Wt0)R 其中: R ‾ ∫ t 0 t ∫ t 0 s L 0 L 0 f ( s 1 ) d s 1 d s ∫ t 0 t ∫ t 0 s L 1 L 0 f ( s 1 ) d W s 1 d s ∫ t 0 t ∫ t 0 s L 0 L 1 f ( s 1 ) d s 1 d W s ∫ t 0 t ∫ t 0 s L 1 L 1 f ( s 1 ) d W s 1 d W s \overline{R}\int_{t_0}^t\int_{t_0}^s L^0L^0f(s_1)ds_1ds\int_{t_0}^t\int_{t_0}^s L^1L^0f(s_1)dWs_1ds\int_{t_0}^t\int_{t_0}^s L^0L^1f(s_1)ds_1dWs\int_{t_0}^t\int_{t_0}^s L^1L^1f(s_1)dWs_1dWs R∫t0t∫t0sL0L0f(s1)ds1ds∫t0t∫t0sL1L0f(s1)dWs1ds∫t0t∫t0sL0L1f(s1)ds1dWs∫t0t∫t0sL1L1f(s1)dWs1dWs 当然可以继续展开下去。 我们在这里只关心 f ( x ) x f(x)x f(x)x的情况因此这会导出本文的主题0.5(Euler-Maruyama), 1(Milstein), 和1.5 阶强Stochastic Differential Equation格式。
3、强格式1: 0.5阶强SDE (Euler-Maruyama)
令 f ( x ) x f(x)x f(x)x,显然地,我们会得到: L 0 f L 0 x b ( x ) L^0fL^0xb(x) L0fL0xb(x), L 1 f L 1 x σ ( x ) L^1fL^1x\sigma(x) L1fL1xσ(x). 则根据 f ( x t ) f ( x t 0 ) L 0 f ( x t 0 ) ( t − t 0 ) L 1 f ( x t 0 ) ( W t − W t 0 ) R ‾ f(x_t)f(x_{t0})L^0 f(x_{t_0})(t-t_0)L^1 f(x_{t_0})(W_t-W_{t_0})\overline{R} f(xt)f(xt0)L0f(xt0)(t−t0)L1f(xt0)(Wt−Wt0)R 代入 f ( x ) x f(x)x f(x)x可得: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) R ‾ x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0})\overline{R} xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)R 此时 R ‾ \overline{R} R为: R ‾ ∫ t 0 t ∫ t 0 s L 0 b ( x s 1 ) d s 1 d s ∫ t 0 t ∫ t 0 s L 1 b ( x s 1 ) d W s 1 d s ∫ t 0 t ∫ t 0 s L 0 σ ( x s 1 ) d s 1 d W s ∫ t 0 t ∫ t 0 s L 1 σ ( x s 1 ) d W s 1 d W s \overline{R}\int_{t_0}^t\int_{t_0}^s L^0b(x_{s_1})ds_1ds\int_{t_0}^t\int_{t_0}^s L^1b(x_{s_1})dWs_1ds\int_{t_0}^t\int_{t_0}^s L^0\sigma(x_{s_1})ds_1dWs\int_{t_0}^t\int_{t_0}^s L^1\sigma(x_{s_1})dWs_1dWs R∫t0t∫t0sL0b(xs1)ds1ds∫t0t∫t0sL1b(xs1)dWs1ds∫t0t∫t0sL0σ(xs1)ds1dWs∫t0t∫t0sL1σ(xs1)dWs1dWs 称下式为Euler-Maruyama格式: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0}) xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)需要用到如下随机微分性质: E [ d W s 2 ] d s E[dWs^2]ds E[dWs2]ds 根据定理和随机微分的性质可知余项 R ‾ \overline{R} R中满足: p 1 2 p_12 p12 (具有 d W s dWs dWs会使得阶数为0), p 2 1 p_21 p21(一个 d s ds ds贡献一个阶,一个 d W s dWs dWs贡献0.5个阶), 满足定理格式此时强收敛阶为 p 2 − 0.5 0.5 p_2-0.50.5 p2−0.50.5阶,则该格式具有0.5阶强收敛性。
4、强格式2/3: 1阶强SDE (Milstein)和1.5阶强SDE.
注意到余项: R ‾ ∫ t 0 t ∫ t 0 s L 0 b ( x s 1 ) d s 1 d s ∫ t 0 t ∫ t 0 s L 1 b ( x s 1 ) d W s 1 d s ∫ t 0 t ∫ t 0 s L 0 σ ( x s 1 ) d s 1 d W s ∫ t 0 t ∫ t 0 s L 1 σ ( x s 1 ) d W s 1 d W s \overline{R}\int_{t_0}^t\int_{t_0}^s L^0b(x_{s_1})ds_1ds\int_{t_0}^t\int_{t_0}^s L^1b(x_{s_1})dWs_1ds\int_{t_0}^t\int_{t_0}^s L^0\sigma(x_{s_1})ds_1dWs\int_{t_0}^t\int_{t_0}^s L^1\sigma(x_{s_1})dWs_1dWs R∫t0t∫t0sL0b(xs1)ds1ds∫t0t∫t0sL1b(xs1)dWs1ds∫t0t∫t0sL0σ(xs1)ds1dWs∫t0t∫t0sL1σ(xs1)dWs1dWs 我们想要提升强格式阶那么显然此时若能够提升 p 2 p_2 p2的阶就可以了注意到限制了 p 2 p_2 p2的阶为 R ‾ \overline{R} R中这一项: G ∫ t 0 t ∫ t 0 s L 1 σ ( x s 1 ) d W s 1 d W s G\int_{t_0}^t\int_{t_0}^s L^1\sigma(x_{s_1})dWs_1dWs G∫t0t∫t0sL1σ(xs1)dWs1dWs那么我们需要对其进行再一次的展开此时根据 I t o Ito Ito公式我们可以将它展开为: G L 1 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s d W s 1 d W s ∫ t 0 t ∫ t 0 s ∫ t 0 s 1 L 0 L 1 σ ( x s 2 ) d s 2 d W s 1 d W s ∫ t 0 t ∫ t 0 s ∫ t 0 s 1 L 1 L 1 σ ( x s 2 ) d W s 2 d W s 1 d W s GL^1\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s dWs_1dWs\int_{t_0}^t\int_{t_0}^s\int _{t_0}^{s_1}L^0L^1\sigma(x_{s_2})d{s_2}dWs_1dWs\int_{t_0}^t\int_{t_0}^s\int _{t_0}^{s_1}L^1L^1\sigma(x_{s_2})d{Ws_2}dWs_1dWs GL1σ(xt0)∫t0t∫t0sdWs1dWs∫t0t∫t0s∫t0s1L0L1σ(xs2)ds2dWs1dWs∫t0t∫t0s∫t0s1L1L1σ(xs2)dWs2dWs1dWs 此时可以得到Milstein格式为: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) L 1 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s d W s 1 d W s x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0})L^1\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s dWs_1dWs xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)L1σ(xt0)∫t0t∫t0sdWs1dWs这即: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) σ ′ ( x t 0 ) σ ( x t 0 ) ∫ t 0 t ∫ t 0 s d W s 1 d W s x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0})\sigma^{}(x_{t_0})\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s dWs_1dWs xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)σ′(xt0)σ(xt0)∫t0t∫t0sdWs1dWs其中: ∫ t 0 t ∫ t 0 s d W s 1 d W s 1 2 ( W t − W t 0 ) 2 − 1 2 ( t − t 0 ) \int_{t_0}^t\int_{t_0}^s dWs_1dWs\frac{1}{2}(W_t-W_{t_0})^2-\frac{1}{2}(t-t_0) ∫t0t∫t0sdWs1dWs21(Wt−Wt0)2−21(t−t0)这即可以得到最终格式为: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) 1 2 σ ′ ( x t 0 ) σ ( x t 0 ) [ ( W t − W t 0 ) 2 − ( t − t 0 ) ] x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0})\frac{1}{2}\sigma^{}(x_{t_0})\sigma(x_{t_0})[(W_t-W_{t_0})^2-(t-t_0)] xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)21σ′(xt0)σ(xt0)[(Wt−Wt0)2−(t−t0)] 此时注意到 p 1 2 p_12 p12不变因为提升了 p 2 3 2 p_2\frac{3}{2} p223满足定理要求因此该格式收敛阶为1.下面继续提升注意到此时若想提升强收敛阶很明显提升 p 2 p_2 p2是不够的了因为此时 p 1 p_1 p1也需要被提升但是很明显的是在 R ‾ \overline{R} R中有 p 2 3 2 p_2\frac{3}{2} p223的项两个: ∫ t 0 t ∫ t 0 s L 0 σ ( x s 1 ) d s 1 d W s \int_{t_0}^t\int_{t_0}^s L^0\sigma(x_{s_1})ds_1dWs ∫t0t∫t0sL0σ(xs1)ds1dWs和 ∫ t 0 t ∫ t 0 s L 1 b ( x s 1 ) d W s 1 d s \int_{t_0}^t\int_{t_0}^s L^1b(x_{s_1})dWs_1ds ∫t0t∫t0sL1b(xs1)dWs1ds则他们也需要被包含在内才可以提升该格式的收敛阶因此在 M i l s t e i n Milstein Milstein的基础上我们继续延拓格式 T T T为: T M i l s t e i n K TMilsteinK TMilsteinK K L 0 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d W s L 1 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d W s 1 d s L 0 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d s ( L 1 ) 2 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s ∫ t 0 s 1 d W s 2 d W s 1 d W s KL^0\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^sds_1dWsL^1b(x_{t_0})\int_{t_0}^t\int_{t_0}^s dWs_1dsL^0b(x_{t_0})\int_{t_0}^t\int_{t_0}^s ds_1ds(L^1)^2\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s\int _{t_0}^{s_1}d{Ws_2}dWs_1dWs KL0σ(xt0)∫t0t∫t0sds1dWsL1b(xt0)∫t0t∫t0sdWs1dsL0b(xt0)∫t0t∫t0sds1ds(L1)2σ(xt0)∫t0t∫t0s∫t0s1dWs2dWs1dWs我们仅考虑从 n n n到 n 1 n1 n1会发生什么令 △ t t n 1 − t n \triangle tt_{n1}-t_n △ttn1−tn, △ W W t n 1 − W t n \triangle WW_{t_{n1}}-W_{t_n} △WWtn1−Wtn ( L 1 ) 2 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s ∫ t 0 s 1 d W s 2 d W s 1 d W s ( L 1 ) 2 σ ( x t 0 ) [ 1 6 △ W 2 − 1 2 △ t ] △ W (L^1)^2\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s\int _{t_0}^{s_1}d{Ws_2}dWs_1dWs(L^1)^2\sigma(x_{t_0})[\frac{1}{6}\triangle W^2-\frac{1}{2}\triangle t]\triangle W (L1)2σ(xt0)∫t0t∫t0s∫t0s1dWs2dWs1dWs(L1)2σ(xt0)[61△W2−21△t]△W这一点使用 I t o Ito Ito公式很容易证明。 另一方面令: △ Z ∫ t n t n 1 ∫ t n s d W s 1 d s ∫ t n t n 1 [ t n 1 − s ] d W s \triangle Z\int_{t_n}^{t_{n1}}\int_{t_n}^s dWs_1ds\int_{t_n}^{t_{n1}}[t_{n1}-s] dW_s △Z∫tntn1∫tnsdWs1ds∫tntn1[tn1−s]dWs根据随机微分的性质,显然 △ Z ∼ N ( 0 , ∫ t n t n 1 [ t n 1 − s ] 2 d s ) N ( 0 , 1 3 △ t 3 ) \triangle Z \sim N(0,\int_{t_n}^{t_{n1}}[t_{n1}-s]^2 ds)N(0,\frac{1}{3} \triangle t^3) △Z∼N(0,∫tntn1[tn1−s]2ds)N(0,31△t3)注意到: E [ △ Z △ W ] ∫ t n t n 1 [ t n 1 − s ] d s 1 2 △ t 2 E[\triangle Z \triangle W]\int_{t_n}^{t_{n1}}[t_{n1}-s] ds\frac{1}{2} \triangle t^2 E[△Z△W]∫tntn1[tn1−s]ds21△t2因此: E [ ( △ Z − 1 2 △ W ) △ W ] 0 E[(\triangle Z-\frac{1}{2} \triangle W)\triangle W]0 E[(△Z−21△W)△W]0这样找到了独立的样本且注意到此时根据随机微分的性质可得到: ( △ Z − 1 2 △ W ) N ( 0 , 1 12 △ t 3 ) (\triangle Z-\frac{1}{2} \triangle W)N(0,\frac{1}{12}\triangle t^3) (△Z−21△W)N(0,121△t3)注意到: ∫ t 0 t ∫ t 0 s d s 1 d W s △ t △ W − △ Z \int_{t_0}^t\int_{t_0}^sds_1dWs\triangle t \triangle W-\triangle Z ∫t0t∫t0sds1dWs△t△W−△Z那么显然地,我们可以根据如下方法来给出1.5阶格式,若我们有 G 1 G_1 G1, G 2 G_2 G2两个独立的同分布正态分布变量 G 1 N ( 0 , 1 ) , G 2 N ( 0 , 1 ) G_1N(0,1),G_2N(0,1) G1N(0,1),G2N(0,1)那么根据我们的推导 K L 0 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d W s L 1 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d W s 1 d s L 0 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d s ( L 1 ) 2 σ ( x t 0 ) ∫ t 0 t ∫ t 0 s ∫ t 0 s 1 d W s 2 d W s 1 d W s KL^0\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^sds_1dWsL^1b(x_{t_0})\int_{t_0}^t\int_{t_0}^s dWs_1dsL^0b(x_{t_0})\int_{t_0}^t\int_{t_0}^s ds_1ds(L^1)^2\sigma(x_{t_0})\int_{t_0}^t\int_{t_0}^s\int _{t_0}^{s_1}d{Ws_2}dWs_1dWs KL0σ(xt0)∫t0t∫t0sds1dWsL1b(xt0)∫t0t∫t0sdWs1dsL0b(xt0)∫t0t∫t0sds1ds(L1)2σ(xt0)∫t0t∫t0s∫t0s1dWs2dWs1dWs K L 0 σ ( x t 0 ) [ △ t △ W − △ Z ] L 1 b ( x t 0 ) △ Z L 0 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d s ( L 1 ) 2 σ ( x t 0 ) [ 1 6 △ W 2 − 1 2 △ t ] △ W KL^0\sigma(x_{t_0})[\triangle t \triangle W-\triangle Z]L^1b(x_{t_0})\triangle ZL^0b(x_{t_0})\int_{t_0}^t\int_{t_0}^s ds_1ds(L^1)^2\sigma(x_{t_0})[\frac{1}{6}\triangle W^2-\frac{1}{2}\triangle t]\triangle W KL0σ(xt0)[△t△W−△Z]L1b(xt0)△ZL0b(xt0)∫t0t∫t0sds1ds(L1)2σ(xt0)[61△W2−21△t]△W其中: △ W △ t G 1 . \triangle W \sqrt{\triangle t} G_1. △W△t G1. △ Z 1 2 △ t △ W 1 2 3 △ t 3 2 G 2 \triangle Z \frac{1}{2} \triangle t\triangle W\frac{1}{2 \sqrt{3}} \triangle t^{\frac{3}{2}} G_2 △Z21△t△W23 1△t23G2此时 K K K中所有成分可求这样的1.5阶格式为: x t x t 0 b ( x t 0 ) ( t − t 0 ) σ ( x t 0 ) ( W t − W t 0 ) 1 2 σ ′ ( x t 0 ) σ ( x t 0 ) [ ( W t − W t 0 ) 2 − ( t − t 0 ) ] K x_tx_{t_0}b(x_{t_0})(t-t_0)\sigma(x_{t_0})(W_t-W_{t_0})\frac{1}{2}\sigma^{}(x_{t_0})\sigma(x_{t_0})[(W_t-W_{t_0})^2-(t-t_0)]K xtxt0b(xt0)(t−t0)σ(xt0)(Wt−Wt0)21σ′(xt0)σ(xt0)[(Wt−Wt0)2−(t−t0)]K K L 0 σ ( x t 0 ) [ △ t △ W − △ Z ] L 1 b ( x t 0 ) △ Z L 0 b ( x t 0 ) ∫ t 0 t ∫ t 0 s d s 1 d s ( L 1 ) 2 σ ( x t 0 ) [ 1 6 △ W 2 − 1 2 △ t ] △ W KL^0\sigma(x_{t_0})[\triangle t \triangle W-\triangle Z]L^1b(x_{t_0})\triangle ZL^0b(x_{t_0})\int_{t_0}^t\int_{t_0}^s ds_1ds(L^1)^2\sigma(x_{t_0})[\frac{1}{6}\triangle W^2-\frac{1}{2}\triangle t]\triangle W KL0σ(xt0)[△t△W−△Z]L1b(xt0)△ZL0b(xt0)∫t0t∫t0sds1ds(L1)2σ(xt0)[61△W2−21△t]△W当然读者可以自行推导更高阶的 I t o − T a y l o r Ito-Taylor Ito−Taylor展开笔者在这里不过多介绍。
