Modified equations for weakly convergent stochastic symplectic schemes via their generating functions

Abstract

In this paper, an approach of constructing modified equations of weak $k+k'$ order ($k'\ge 1$) apart from the k-th order weakly convergent stochastic symplectic methods, i.e., stochastic symplectic methods with respect to weak convergence and of weak order k, is given using the underlying generating functions of them. This approach is valid for stochastic Hamiltonian systems with additive noises, and those with multiplicative noises but for which the Hamiltonian functions $H_r(p,q),\,\,r\ge 1$ associated to the diffusion parts depend only on p or only on q. In such cases, we find that the modified equations of the weakly convergent stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original systems.

Appendices

Appendix 1: An illustrating explanation regarding influence of the approximation error of $\bar{I}_{\alpha }^h$

For example, given a two-dimensional ($d=2$) stochastic Hamiltonian system with additive noise, or the $H_r(p,q),\,\,r\ge 1$ depend only on p or only on q, and its symplectic scheme of weak order 1, i.e. $k=1$. Suppose the generating function of the scheme is

$$\begin{aligned} S^1(P,q,h)=\mathop {\mathop {\sum }\limits _{\alpha }}\limits _{l(\alpha )\le 1}\left( \sum _{\beta }C_{\alpha }^{\beta }G_{\beta }^1(P,q)\right) \bar{I}_{\alpha }^h, \end{aligned}$$

and the generating function for the modified equation of the numerical scheme is

$$\begin{aligned} \tilde{S}^1(\tilde{P},q,h)=\sum _{\alpha }\left( \sum _{\beta }C_{\alpha }^{\beta }\bar{G}_{\beta }^1(\tilde{P},q)\right) I_{\alpha }^h, \end{aligned}$$

where, for convenience of comparison, we have transformed the $J_{\beta }^h$ in $\tilde{S}^1$ into combination of some $I_{\alpha }^h$ according to (4.16). By definition of $\bar{G}_{\beta }^1$ given in (4.11), we have

$$\begin{aligned} \bar{G}_{\beta }^1(\tilde{P},q)=G_{\beta }^1(\tilde{P},q)+M_{\beta }^1(\tilde{P},q), \end{aligned}$$

where $M_{\beta }^1(\tilde{P},q)$ denote all the other terms except for the first term $G_{\beta }^1(\tilde{P},q)$ in $\bar{G}_{\beta }^1(\tilde{P},q)$. A straightforward calculation gives

$$\begin{aligned} \mathbf {E}\left( \frac{\partial S^1}{\partial q}\right) ^2-\mathbf {E}\left( \frac{\partial \tilde{S}^1}{\partial q}\right) ^2=\mathbf {E}(\triangle _1^2)-\mathbf {E}(\triangle _2+\triangle _3)^2, \end{aligned}$$

where

$$\begin{aligned} \triangle _1= & {} \frac{\partial }{\partial q}\left[ (G_{(0)}^1+\frac{1}{2}G_{(1,1)}^1)h+G_{(1)}^1\bar{\xi }\sqrt{h}\right] \\ \triangle _2= & {} \frac{\partial }{\partial q}\left[ (G_{(0)}^{1[0]}+\frac{1}{2}G_{(1,1)}^{1[0]})h+G_{(1)}^{1[0]}\xi \sqrt{h}\right] \\ \triangle _3= & {} \frac{\partial }{\partial q}\left[ G_{(1,1)}^{1[0]}I_{(1,1)}+(G_{(1,0)}^{1[0]}+H_{1}^{[1]})I_{(1,0)}\right. \\&\left. +(G_{(0,1)}^{1[0]}+H_{1}^{[1]})I_{(0,1)} +G_{(1,1,1)}^{1[0]}I_{(1,1,1)}+\cdots \right] \end{aligned}$$

with all the appearing $G_{\alpha }^1$ taking values at (P, q), and the functions coming from $\bar{G}_{\alpha }^1$ taking values at $(\tilde{P},q)$. Thus,

$$\begin{aligned} \mathbf {E}\left( \frac{\partial S^1}{\partial q}\right) ^2-\mathbf {E}\left( \frac{\partial \tilde{S}^1}{\partial q}\right) ^2=\mathbf {E}(\triangle _1^2)-\mathbf {E}(\triangle _2^2)-\mathbf {E}(\triangle _3^2)-2\mathbf {E}(\triangle _2\triangle _3). \end{aligned}$$

Performing Taylor expansion of $G_{\alpha }^1(P,q)$ and those functions from $\bar{G}_{\alpha }^1(\tilde{P},q)$ at (p, q), we get

$$\begin{aligned}&\mathbf {E}(\triangle _1^2)-\mathbf {E}(\triangle _2^2)=h\left[ \frac{\partial G_{(1)}^1}{\partial q}(p,q)\right] ^2\mathbf {E}(\bar{\xi }^2-\xi ^2)\\&\quad +h^2\left[ \varPsi _1\left( \mathbf {E}(\bar{\xi }^2-\xi ^2),\mathbf {E}(\bar{\xi }^4-\xi ^4),\ldots ,\mathbf {E}(\bar{\xi }^{2(k+k')}-\xi ^{2(k+k')})\right) \right] \\&\quad +h^2\left[ \varPsi _2\left( \frac{\partial H_{(1)}^{[1]}}{\partial q}(p,q),\frac{\partial H_{(0)}^{[1]}}{\partial q}(p,q),\frac{\partial H_{(1)}^{[2]}}{\partial q}(p,q),\frac{\partial H_{(0)}^{[2]}}{\partial q}(p,q),\ldots \right) \right] , \end{aligned}$$

where [20]

$$\begin{aligned} \bar{\xi }=\left\{ \begin{array}{ll}-A_h,&{}\xi <-A_h,\\ \xi ,&{}|\xi |\le A_h,\\ A_h,&{}\xi > A_h,\end{array}\right. \end{aligned}$$

with $\xi \sim \mathscr {N}(0,1)$ and $A_h=\sqrt{2\mu \ln |h|}$ with $\mu \ge 1$. $\varPsi _1$ is a linear function of $\mathbf {E}(\bar{\xi }^{2l}-\xi ^{2l})$ $(1\le l\le k+k')$, and $\varPsi _2$ is a function of partial derivatives of the unknown functions $H_j^{[i]}$. Then,

$$\begin{aligned} \begin{aligned}&\mathbf {E}\left( \frac{\partial S^1}{\partial q}\right) ^2-\mathbf {E}\left( \frac{\partial \tilde{S}^1}{\partial q}\right) ^2\\&\quad =\left[ \left( \frac{\partial G_{(1)}^1}{\partial q}(p,q)\right) ^2\mathbf {E}(\bar{\xi }^2-\xi ^2)h\right. \\&\qquad \left. +h^2\left( \varPsi _1\left( \mathbf {E}(\bar{\xi }^2-\xi ^2),\mathbf {E}(\bar{\xi }^4-\xi ^4),\ldots ,\mathbf {E}(\bar{\xi }^{2(k+k')}-\xi ^{2(k+k')})\right) \right) \right] \\&\qquad +\left[ h^2\left( \varPsi _2\left( \frac{\partial H_{(1)}^{[1]}}{\partial q},\frac{\partial H_{(0)}^{[1]}}{\partial q},\frac{\partial H_{(1)}^{[2]}}{\partial q},\frac{\partial H_{(0)}^{[2]}}{\partial q},\ldots \right) \right) -\mathbf {E}(\triangle _3^2)-2\mathbf {E}(\triangle _2\triangle _3)\right] \\&\quad :=F_1\left( \mathbf {E}(\bar{\xi }^2-\xi ^2),\mathbf {E}(\bar{\xi }^4-\xi ^4),\ldots ,\mathbf {E}(\bar{\xi }^{2(k+k')}-\xi ^{2(k+k')})\right) \\&\qquad +F_2\left( \frac{\partial H_{(1)}^{[1]}}{\partial q},\frac{\partial H_{(0)}^{[1]}}{\partial q},\frac{\partial H_{(1)}^{[2]}}{\partial q},\frac{\partial H_{(0)}^{[2]}}{\partial q},\ldots \right) , \end{aligned} \end{aligned}$$

where $F_2$ is a function including partial derivatives of the unknown functions $H_j^{[i]}$ but without $\bar{\xi }$.

To make $\left| \mathbf {E}\left( \frac{\partial S^1}{\partial q}\right) ^2-\mathbf {E}\left( \frac{\partial \tilde{S}^1}{\partial q}\right) ^2\right| \le O(h^{k+k'+1})$, we only need $|F_1|\le O(h^{k+k'+1})$ which implies

$$\begin{aligned} \left\{ \begin{array}{l}\mathbf{{E}}[\bar{\xi }^{(2k+2k')}-{\xi }^{(2k+2k')}]\le O(h),\,\,\,\,\,\,\\ \mathbf{{E}}[\bar{\xi }^{(2k+2k'-2)}-{\xi }^{(2k+2k'-2)}]\le O(h^2),\,\,\,\,\,\,\\ \vdots \\ \mathbf{{E}}[\bar{\xi }^{2}-{\xi }^{2}]\le O(h^{(k+k')}),\,\,\,\,\,\,\end{array}\right. \end{aligned}$$

and $|F_2|\le O(h^{k+k'+1})$, from which we can derive some partial derivatives of the unknown functions such as $\frac{\partial H_j^{[i]}}{\partial q}$ for some i and j. Similar to the analysis in [20], we can derive that $\mu \ge k+k'+1$ solves the group of inequalities above. In other words, as long as we choose $\mu \ge k+k'+1$ in the approximation $\bar{I}_1^h=\bar{\xi }\sqrt{h}$ in which $\bar{\xi }$ has a boundary containing the parameter $\mu$, the approximation error totally contained in $|F_1|$ will be merged into the desired error order of the modified equation $O(h^{k+k'+1})$, and it will not affect the determination of the unknown functions $H_j^{[i]}$ which are all included in the other term $F_2$.

It can be derived similarly that, to guarantee the weakly convergent order k of the numerical method not being affected by the truncation of the random variable, one needs to choose $\mu \ge k+1$. Therefore, considering both the matching between the numerical method and its modified equation, and the weakly convergent order of the numerical method, we choose $\mu \ge \max \{k+k'+1,k+1\}=k+k'+1$.

As long as the approximation error $\bar{I}_{\alpha }^h$ can be at last due to truncations of the Gaussian random variables, it can be merged into the desired error order of the modified equations via choosing sufficiently large values of $\mu$, similar to the analysis given by Milstein, Repin and Tretyakov, which implies the possibility of adapting the truncation method to constructing implicit schemes of arbitrary desired root mean-square orders s by choosing sufficiently large $\mu \ge 2s$ [20].

Appendix 2: Modified equation with $k=1$ and $k'=2$

Here we derive the modified equation that is globally weakly 2 order closer to the numerical method (5.5) than the true solution of the stochastic system (5.1) does, i.e. k=1, $k'$=2. We start by presenting the calculations for finding the functions $\bar{G}_{\alpha }^{1}$ in addition to those in (5.6),

$$\begin{aligned}&\bar{G}_{(1,0,1)}^1=G_{(1,0,1)}^{1[0]}+G_{(1,1)}^{1[1]}=G_{(1,1)}^{1[1]},\\&\bar{G}_{(0,0,1)}^1=G_{(0,0,1)}^{1[0]}+2G_{(0,1)}^{1[1]}+2G_{(1)}^{1[2]}=-\sigma q+2G_{(0,1)}^{1[1]}+2H_1^{[2]},\\&\bar{G}_{(0,1,0)}^1=G_{(0,1,0)}^{1[0]}+G_{(0,1)}^{1[1]}+G_{(1,0)}^{1[1]}+2G_{(1)}^{1[2]}=-\sigma q+G_{(0,1)}^{1[1]}+G_{(1,0)}^{1[1]}+2H_1^{[2]},\\&\bar{G}_{(1,0,0)}^1=G_{(1,0,0)}^{1[0]}+2G_{(0,1)}^{1[1]}+2G_{(1)}^{1[2]}=2G_{(1,0)}^{1[1]}+2H_1^{[2]},\\&\bar{G}_{(0,1,1,1)}^1=G_{(0,1,1,1)}^{1[0]}+G_{(1,1,1)}^{1[1]}=G_{(1,1,1)}^{1[1]},\\&\bar{G}_{(1,0,1,1)}^1=G_{(1,0,1,1)}^{1[0]}+G_{(1,1,1)}^{1[1]}=G_{(1,1,1)}^{1[1]},\\&\bar{G}_{(1,1,0,1)}^1=G_{(1,1,0,1)}^{1[0]}+G_{(1,1,1)}^{1[1]}=G_{(1,1,1)}^{1[1]},\\&\bar{G}_{(1,1,1,0)}^1=G_{(1,1,1,0)}^{1[0]}+G_{(1,1,1)}^{1[1]}=G_{(1,1,1)}^{1[1]},\\&\bar{G}_{(0,0,0)}^1=G_{(0,0,0)}^{1[0]}+3G_{(0,0)}^{1[1]}+6G_{(0)}^{1[2]}={\tilde{P}}^{2}+q^2+3G_{(0,0)}^{1[1]}+6H_0^{[2]},\\&\bar{G}_{(1,1,1,1,0)}^1=\bar{G}_{(1,1,1,0,1)}^1=\bar{G}_{(1,1,0,1,1)}^1=\bar{G}_{(1,0,1,1,1)}^{1}=\bar{G}_{(0,1,1,1,1)}^{1}=G_{(1,1,1,1)}^{1[1]},\\&\bar{G}_{(0,0,1,1)}^1=G_{(0,0,1,1)}^{1[0]}+2G_{(0,1,1)}^{1[1]}+2G_{(1,1)}^{1[2]}=2G_{(0,1,1)}^{1[1]}+2G_{(1,1)}^{1[2]},\\&\bar{G}_{(0,1,0,1)}^1=G_{(0,1,0,1)}^{1[0]}+G_{(0,1,1)}^{1[1]}+G_{(1,0,1)}^{1[1]}+2G_{(1,1)}^{1[2]}=G_{(0,1,1)}^{1[1]}+G_{(1,0,1)}^{1[1]}+2G_{(1,1)}^{1[2]},\\&\bar{G}_{(0,1,1,0)}^1=G_{(0,1,1,0)}^{1[0]}+G_{(0,1,1)}^{1[1]}+G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]}=G_{(0,1,1)}^{1[1]}+G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]},\\&\bar{G}_{(1,1,0,0)}^1=G_{(1,1,0,0)}^{1[0]}+2G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]}=2G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]},\\&\bar{G}_{(1,0,1,0)}^1=G_{(1,0,1,0)}^{1[0]}+G_{(1,0,1)}^{1[1]}+G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]}=G_{(1,0,1)}^{1[1]}+G_{(1,1,0)}^{1[1]}+2G_{(1,1)}^{1[2]},\\&\bar{G}_{(1,0,0,1)}^1=G_{(1,0,0,1)}^{1[0]}+2G_{(1,0,1)}^{1[1]}+2G_{(1,1)}^{1[2]}=2G_{(1,0,1)}^{1[1]}+2G_{(1,1)}^{1[2]}.\\&\vdots \end{aligned}$$

Based on the functions $\bar{G}_{\alpha }^{1}$ listed above, we can reattain the equations (5.7), (5.8), (5.9) and (5.10) when we compare coefficients of h and $h^2$,

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\sigma \frac{\partial ^2H_1^{[1]}}{\partial p\partial q}-\frac{\partial H_0^{[1]}}{\partial q}=\frac{1}{2}p,\quad \frac{1}{2}\sigma \frac{\partial ^2H_1^{[1]}}{\partial p^2}-\frac{\partial H_0^{[1]}}{\partial p} =\frac{1}{2}q,\\&\frac{\partial H_1^{[1]}}{\partial q}=0,\quad \frac{\partial H_1^{[1]}}{\partial p}=\frac{1}{2}\sigma . \end{aligned} \end{aligned}$$

Substituting (5.9) into (5.7) and (5.8), and comparing the coefficients of $h^3$, we get the following

$$\begin{aligned}&\frac{\partial H_0^{[1]}}{\partial p}=-\frac{1}{2}q,\quad \frac{\partial H_0^{[1]}}{\partial q}=-\frac{1}{2}p,\\&G_{(1,1)}^{1[1]}=\frac{\partial H_1^{[1]}}{\partial q}\frac{\partial G_1^{[0]}}{\partial {\tilde{P}}}+\frac{\partial H_1^{[0]}}{\partial q}\frac{\partial G_1^{[1]}}{\partial {\tilde{P}}}=-\frac{1}{2}{\sigma }^2,\\&G_{(0,1)}^{1[1]}=\frac{\partial H_1^{[1]}}{\partial q}\frac{\partial G_0^{[0]}}{\partial {\tilde{P}}}+\frac{\partial H_1^{[0]}}{\partial q}\frac{\partial G_0^{[1]}}{\partial {\tilde{P}}}=\frac{1}{2}\sigma q,\\&G_{(1,0)}^{1[1]}=\frac{\partial H_0^{[1]}}{\partial q}\frac{\partial G_1^{[0]}}{\partial {\tilde{P}}}+\frac{\partial H_0^{[0]}}{\partial q}\frac{\partial G_1^{[1]}}{\partial {\tilde{P}}}=\frac{1}{2}\sigma q,\\&G_{(1,1,1)}^{1[1]}=0,\quad {\bar{G}}_{(0,0,1)}^1={\bar{G}}_{(0,1,0)}^1=2H_1^{[2]},\\&{\bar{G}}_{(1,0,0)}^1=\sigma q+2H_1^{[2]},\\&\frac{\partial H_1^{[2]}}{\partial p}=0,\quad \frac{\partial H_1^{[2]}}{\partial q}=-\frac{1}{6}\sigma ,\quad G_{(1,1,1,1)}^{1[1]}=0,\\&G_{(0,1,1)}^{1[1]}=G_{(1,0,1)}^{1[1]}=G_{(1,1,0)}^{1[1]}=G_{(1,1,1,1)}^{1[1]}=G_{(1,1)}^{1[2]}=0,\\&G_{(0,0)}^{1[1]}=\frac{\partial H_0^{[1]}}{\partial q}\frac{\partial G_0^{[0]}}{\partial {\tilde{P}}}+\frac{\partial H_0^{[0]}}{\partial q}\frac{\partial G_0^{[1]}}{\partial {\tilde{P}}}=-\frac{1}{2}({\tilde{P}}^2+q^2),\\&\bar{G}_{(0,0,0)}^1=-\frac{1}{2}({\tilde{P}}^{2}+q^2)+6H_0^{[2]},\\&\frac{\partial H_0^{[2]}}{\partial p}=\frac{1}{6}p,\quad \frac{\partial H_0^{[2]}}{\partial q}=\frac{1}{6}q. \end{aligned}$$

According to the definition of the modified equation in Theorem (4.1), the modified equation of weak third order apart from the numerical method (5.5) is

$$\begin{aligned} dp= & {} \left( -q+h\frac{p}{2}-h^2\frac{q}{6}\right) dt+ \left( \sigma +h^2\frac{\sigma }{6}\right) dW(t),\nonumber \\ dq= & {} \left( p-h\frac{q}{2}+h^2\frac{p}{6}\right) dt+h\frac{\sigma }{2} dW(t). \end{aligned}$$