Keywords

1 Introduction

Numerical simulations [5, 9, 11] show that for stochastic Hamiltonian systems (SHS) symplectic schemes give more accurate results for long term simulation that non-symplectic schemes, but, to the best of our knowledge, no theoretical proof was done in the stochastic case. For a SHS and a first weak order symplectic scheme, in [2] we present an expansion of the global approximation error in powers of the discretization step size. Comparing this expansion with the global error expansion obtained in [13] for the Euler scheme (which has also weak order one), we justify the superior performance of the symplectic scheme for the simple linear SHS corresponding to the Kobo oscillator [2]. However, this justification can not be easily extended for general non-linear SHSs. Here we use backward error analysis to find an expansion of error for the symplectic scheme in terms of the powers of the discretization step size and the solutions of the modified Kolmogorov equation [3].

Backward error analysis was successfully applied to study long term behavior of deterministic Hamiltonian systems [4]. Recently, backward error analysis was extended to stochastic differential equations (SDE). Modified SDEs associated with various numerical schemes are presented in [1, 10, 14]. A SDE defined on the n-dimensional torus and its approximation by the explicit Euler scheme are studied using backward error analysis in [3].

We follow the same approach as in [3], and we construct the modified equation not at the level of the SDE, but at the level of the associated Kolmogorov equation. Compared with [3] we consider a fully implicit scheme instead of an explicit one, and we consider a SHS with additive or multiplicative noise defined on \(\mathrm {R}^{2n}\) instead of the compact n dimensional torus. Implicit numerical schemes are also considered in [6, 7], but for Langevin SDEs on \(\mathrm {R}^n\) with additive noise. Studying the multiplicative noise case is more difficult, especially for a fully implicit numerical scheme.

In the next section we present some preliminary results regarding the solution of the SHS and the approximate solution given by the numerical scheme. The steps followed for the backward error analysis are included in Sect. 3. The last section contains the conclusions.

2 Assumptions and Preliminary Results

We introduce a few definitions and notations. We denote \(\mathrm {N}=\{1,2,\ldots \}\), \(N^*=\{1,2,\ldots \}\) and for any \(x=(x_1,\ldots ,x_n)^T\in \mathrm {R}^n\), |x| represents the Euclidean norm.

For any multi-index \(\alpha =(\alpha _1,\ldots ,\alpha _r)\in \mathrm {N}^{r}\) with length \(|\alpha |=\alpha _1+\cdots +\alpha _r\), let \(\partial _{\alpha }=\frac{\partial ^{|\alpha |}}{\partial _1^{\alpha _1}\cdots \partial _r^{\alpha _r}}\) denote the partial derivative of order \(|\alpha |\).

We define the following space of functions with polynomial growth:

$$ C_{pol}^\infty (\mathrm {R}^{2n})=\left\{ f \in C^\infty (\mathrm {R}^{2n})\text { such that } f \text { and all its derivatives have polynomial growth}\right\} $$

For any \(k, l\in \mathrm {N}\), we denote

$$\begin{aligned}&C_k^l(\mathrm {R}^{2n})=\biggl \{f\in C^l(\mathrm {R}^{2n}): \text { there exists } C_{l,k}>0 \text { such that for all } x\in \mathrm {R}^{2n}\text { and any index } \\&\alpha \in \mathrm {N}^{2n}, |\alpha |\le l, |\partial _\alpha f(x)|\le C_{l,k} (1+|x|^{2k})\biggl \}. \end{aligned}$$

On \(C_k^l(\mathrm {R}^{2n})\) we define [7] the norm \(\Vert \cdot \Vert _{l,k}\) and the semi norm \(|\cdot |_{l,k}\):

$$\begin{aligned} \Vert f\Vert _{l,k}= \sup _{\alpha ,|\alpha |\le l}\frac{|\partial _\alpha f(x)|}{1+|x|^{2k}},\quad |f|_{l,k}= \sup _{\alpha ,1\le |\alpha |\le l}\frac{|\partial _\alpha f(x)|}{1+|x|^{2k}}. \end{aligned}$$
(1)

Notice that if \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\), then for all \(d\in \mathrm {N}\), there exists \(r_d\in \mathrm {N}\) such that \(\phi \in C_{r_d}^d(\mathrm {R}^{2n})\).

We consider the following stochastic Hamiltonian system

$$\begin{aligned}&dP = -\partial _Q H_0(P,Q)dt-\sum _{r =1}^{m} \partial _Q H_r(P,Q) \circ dw_t^r, \quad P(0) = p\nonumber \\&dQ = \partial _P H_0(P,Q)dt+\sum _{r =1}^{m} \partial _P H_r(P,Q)\circ dw_t^r, \quad Q(0) = q , \end{aligned}$$
(2)

where P, Q, p, q are n-dimensional column vectors, \(w_t^r, r= 1,\ldots ,m\) are independent standard Wiener processes, and for any function f defined on \(\mathrm {R}^n\times \mathrm {R}^n\), \(\partial _P f\) and \(\partial _Q f\) denote the column vectors with components \((\partial f/\partial P_i), 1\le i\le n\) and \((\partial f/\partial Q_i), 1\le i\le n\), respectively. The stochastic flow \((p,q) \longrightarrow (P,Q)\) of the SHS (2) preserves the symplectic structure [9]: \(dP \wedge dQ = dp \wedge dq\), where the differential 2-form \(dp \wedge dq = dp_1 \wedge dq_1 + \dots + dp_n \wedge dq_n\).

The system (2) can be re-written in the Ito formulation:

$$\begin{aligned} dP&=a(P,Q)dt+\sum _{r=1}^m\sigma ^{r}(P,Q) dw_t^r,\quad P(0)=p \end{aligned}$$
(3)
$$\begin{aligned} dQ&=b(P,Q)dt+\sum _{r=1}^m\gamma ^{r}(P,Q)dw_t^r,\quad Q(0)=q, \end{aligned}$$
(4)

where

$$\begin{aligned} a&=-\partial _Q H_0+\frac{1}{2}\sum _{r=1}^m\sum _{j=1}^n \biggl (\frac{\partial H_r}{\partial Q_j}\partial _Q\left( \frac{\partial H_r}{\partial P_j}\right) -\frac{\partial H_r}{\partial P_j}\partial _Q\left( \frac{\partial H_r}{\partial Q_j}\right) \biggl ) \\ b&=\partial _P H_0+\frac{1}{2}\sum _{r=1}^m\sum _{j=1}^n \biggl (-\frac{\partial H_r}{\partial Q_j}\partial _P\left( \frac{\partial H_r}{\partial P_j}\right) +\frac{\partial H_r}{\partial P_j}\partial _P\left( \frac{\partial H_r}{\partial Q_j}\right) \biggl ) \\ \sigma ^r&=-\partial _Q H_r,\quad \gamma ^r=\partial _P H_r. \end{aligned}$$

Here everywhere the arguments are (P, Q), and a, b, \(\sigma ^r\), \(\gamma ^r\), \(r=1,\ldots ,m\) are \(n-\)dimensional column vectors.

The Kolmogorov generator \(L(p,q,\partial _p, \partial _q)\) associated with the SHS (3)–(4) has the following form [12]

$$\begin{aligned} {\begin{matrix} &{}L(p,q,\partial _p, \partial _q)\phi (p,q)=\sum _{j=1}^n \left( a_j\frac{\partial }{\partial p_j}\phi (p,q)+b_j\frac{\partial }{\partial q_j}\phi (p,q)\right) +\frac{1}{2}\sum _{r=1}^m\sum _{i,j=1}^n \biggl (\sigma _i^r\sigma _j^r\\ &{} \frac{\partial ^2 }{\partial p_ip_j}\phi (p,q)+\gamma _i^r\gamma _j^r \frac{\partial ^2 }{\partial q_iq_j}\phi (p,q)+2\sigma _i^r\gamma _j^r \frac{\partial ^2 }{\partial p_iq_j}\phi (p,q)\biggl ), \phi \in C^\infty (\mathrm {R}^{2n}) \end{matrix}} \end{aligned}$$

Throughout the paper we make the same assumptions as in [12, 13]:

  1. A1.

    The derivatives of any order of \(H_i\in C^{\infty }\), \(i=1,\ldots , m\) are bounded, and the derivative of any order \(k\ge 2\) of \(H_0\in C^{\infty }\) are bounded.

  2. A2.

    The operator L is uniformly elliptic: there exists a constant \(\alpha >0\) such that for all \(x=(p,q)^T\in \mathrm {R}^{2n}\) we have

    $$\begin{aligned} \alpha |x|^2\le \sum _{r=1}^m\sum _{i,j=1}^n \left( \sigma _i^r\sigma _j^r p_ip_j+\gamma _i^r\gamma _j^r q_iq_j+2\sigma _i^r\gamma _j^r p_iq_j\right) \end{aligned}$$
    (5)
  3. A3.

    There exists a constant \(\beta >0\) and a compact set K such that for all \(x=(p,q)^T\) \(\in \mathrm {R}^{2n}-K\) we have \(p\cdot a(x)+q\cdot b(x)\le -\beta |x|^2\).

Notice that assumption A1 implies that we have a Lipschitz condition, i.e. there exists \(L_1>0\) such that for all \(X=(P,Q)^T\), \(x=(p,q)^T\in \mathrm {R}^{2n}\) we have

$$\begin{aligned} \sum _{j=0}^m\biggl | \left( \partial _P H_{j},\partial _Q H_{j}\right) ^T(X)-\left( \partial _p H_{j},\partial _q H_{j}\right) ^T(x)\biggl | \le L_1|X-x|. \end{aligned}$$
(6)

2.1 Results Regarding the Solution of the Stochastic Hamiltonian System

Proceeding as in Proposition 3.1 in [12], under the assumptions A1-A3 we can prove the following result regarding the solution \(\left( X^{0,x_0}(t)\right) =\left( (P(t,p_0,q_0),Q(t,p_0,q_0))^T\right) \) of the SHS (2) with the initial condition \(x_0=(p_0,q_0)^T\in \mathrm {R}^{2n}\).

Lemma 1

The Markov process \(\left( X^{0,x_0}(t)\right) \) is ergodic. The unique invariant probability measure \(\mu \) has finite moments of any order and a density \(\rho \ge 0\). Moreover, for any \(k\in \mathrm {N}\) there exist \(C_k,\gamma _k>0\) such that for any \(x_0=(p_0,q_0)^T\in \mathrm {R}^{2n}\), and any \(t\ge 0\) we have:

$$\begin{aligned} E(|X^{0,x_0}(t)|^k)\le C_k\left( 1+|x_0|^k \exp (-\gamma _k t)\right) . \end{aligned}$$
(7)

We consider any function \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\), and for all \(x=(p,q)^T \in \mathrm {R}^{2n}\) and all \(t>0\) we define \(u(t,p,q):=E[\phi (X^{0,x}(t)]\). Notice that Lemma 1 implies that u is well defined. It is well known [12] that u(t, p, q) is a classical solution of the Kolmogorov equation

$$\begin{aligned} \frac{d u}{d t}(t,p,q)=Lu(t,p,q),\quad u(0,p,q)=\phi (p,q), (p,q)^T\in \mathrm {R}^{2n}, t>0. \end{aligned}$$
(8)

For any function \(f\in C_{pol}^\infty (\mathrm {R}^{2n})\) we denote the average

$$\begin{aligned} <f>:=\int f(x)d\mu (x) \end{aligned}$$

The results included in the following lemma show the exponential convergence of u and its derivatives and are essential for the backward error analysis presented in this paper. The proof is an extension of the proof of Theorem 3.4 in [12], based on Theorem 2.5 in [8].

Lemma 2

Let \(k\in \mathrm {N}\), \(k\ge 1\), and \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n}) \cap C_{r_{k+n+1}}^{k+n+1}(\mathrm {R}^{2n})\), \( r_{k+n+1} \in \mathrm {N.}\) Then there exist \(\gamma _k>0\), \(C_k>0\) and \(l_k\in \mathrm {N}\) such that \(l_k>r_{k+n+1}\) and for any \(0<\gamma <\gamma _k\) and all \(t\ge 0\) we have

$$\begin{aligned}&|u(t,x)|_{k,l_k}\le C_k\left\| \phi -<\phi >\right\| _{k+n+1,r_{k+n+1}}\exp (-\gamma t).\end{aligned}$$
(9)
$$\begin{aligned}&\Vert u(t,x)-<\phi>\Vert _{0,l_0}\le C_0\left\| \phi -<\phi >\right\| _{n+1,r_{n+1}}\exp (-\gamma t). \end{aligned}$$
(10)

2.2 Results Regarding the Symplectic Scheme

We consider the following one-step approximation [9] for the system (2):

$$\begin{aligned} {P}_{k+1}={P}_k-h\left( \partial _Q H_0+\frac{1}{2}\sum _{r=1}^m\partial _Q G_{(r,r)}\right) -\sqrt{h}\sum _{r=1}^m\varsigma _{rk}\partial _Q H_r, \quad P_0=p_0\end{aligned}$$
(11)
$$\begin{aligned} {Q}_{k+1}={Q}_k+h\left( \partial _P H_0+\frac{1}{2}\sum _{r=1}^m\partial _P G_{(r,r)}\right) +\sqrt{h}\sum _{r=1}^m\varsigma _{rk}\partial _P H_r\quad Q_0=q_0 \end{aligned}$$
(12)

where \(G_{(r,r)}=\sum _{i=1}^n \frac{\partial H_r}{\partial Q_i}\frac{\partial H_r}{\partial P_i}\), the random variables \(\varsigma _{rk}\) are mutually independent identically distributed according to the law, \(\mathrm {P}(\varsigma _{rk}=\pm 1)=1/2\), and everywhere the arguments are \(({P}_{k+1},{Q}_k)\).

Notice that the first equation (11) is implicit. Let denote \(\delta :=\sqrt{h}\) and \(F(p,q)=\left( H_0(p,q)+\frac{1}{2}\sum _{r=1}^mG_{(r,r)}(p,q)\right) \). Then we can reformulate the scheme (11)–(12) as follows:

$$\begin{aligned} P_{k+1}&=P_k-\delta ^2\partial _Q F(P_{k+1},Q_k)-\delta \sum _{r=1}^m\varsigma _{rk}\partial _Q H_r (P_{k+1},Q_k)\end{aligned}$$
(13)
$$\begin{aligned} Q_{k+1}&=Q_k+\delta ^2\partial _P F(P_{k+1},Q_k)+\delta \sum _{r=1}^m\varsigma _{rk}\partial _P H_r (P_{k+1},Q_k) \end{aligned}$$
(14)

Using the Lipschitz condition (6) and proceeding as in the proof of Theorem 4.6.1 in [9] we can show that the scheme (13)–(14) is well defined:

Lemma 3

There exist \(h_{01}>0\), \(C>0\) such that for any \(0<h\le h_{01}\) and any \((p,q)^t\in \mathrm {R}^{2n} \) there exists a unique \(z\in \mathrm {R}^{n} \) such that \( z=p-h\partial _q F(z,q)-\sqrt{h}\sum _{r=1}^m\varsigma _{rk}\partial _q H_r (z,q)\) which satisfies \(|z-p|\le C(1+|p|)\sqrt{h}.\)

Moreover, Theorem 4.6.1 in [9] shows that implicit method (13)–(14) is symplectic and of first weak order: for any \(T>0\), and any \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\) we have

$$\begin{aligned} |E (\phi (P_k,Q_k))-E(\phi (X^{0,x_0}(kh)))|\le c(\phi , T)h, k=0,\ldots ,\lfloor T/h \rfloor , c(\phi , T)>0. \end{aligned}$$
(15)

We define the function \(\phi _\delta \) which associate to \((q,p)\in \mathrm {R}^{2n}\) the solution \(z=(z_1,z_2)^T\in \mathrm {R}^{2n}\) of \(f(\delta ,q,p,z_1,z_2)=0\), where

$$\begin{aligned} f(\delta ,q,p,z)=\begin{bmatrix} z_1-p+\delta ^2\partial _q F(z_1,q)+\delta \sum _{r=1}^m\varsigma _{r}\partial _q H_r (z_1,q)\\ z_2-q-\delta ^2\partial _p F(z_1,q)-\delta \sum _{r=1}^m\varsigma _{r}\partial _p H_r (z_1,q) \end{bmatrix} \end{aligned}$$
(16)

where the random variables \(\varsigma _{r}\) are mutually independent identically distributed according to the law, \(\mathrm {P}(\varsigma _{r}=\pm 1)=1/2\), Since the scheme (13)–(14) is well defined, the function \(\phi _\delta \) is also well defined for any \(\delta \in (0, \sqrt{h_{01}})\). Using A1 it is easy to show that there exists \(h_{03}\le h_{01}\) such that \(\partial _z f(\delta ,q,p,z)=I-B(\delta ,p,q,z)\) where \(\Vert B(\delta ,p,q,z)\Vert <1\) for any \((\delta , p,q,z)\in (0, \sqrt{h_{03}})\times \mathrm {R}^{2n}\,{\times }\, \mathrm {R}^{2n}\). Thus, \(\partial _z f(\delta ,q,p,z)\) is invertible, and from the Implicit Functions Theorem we obtain that the function defined by \((\delta ,p,q)\rightarrow \phi _\delta (p,q)\) is \(C^\infty \) on a neighborhood of each point of \((0, \sqrt{h_{03}})\times \mathrm {R}^{2n}\).

Following the same approach as in the proof of Proposition 7.1 in [12] we can show that the moments of the approximating process \((P_k,Q_k)\) satisfy a similar property with (7):

Lemma 4

There exist \(0<h_{02}\le h_{01}\) such that the symplectic scheme (11)–(12) with any initial condition \((p,q)^t\in \mathrm {R}^{2n}\) and any \(0<h\le h_{02}\) satisfies for any \(l\in \mathrm {N}^*\)

$$\begin{aligned} E_{p,q}(|P_k|^{2l}+|Q_k|^{2l})\le C_l \left( 1+(|p|^{2l}+|q|^{2l})\exp (-\alpha _l kh)\right) ,\quad C_l>0, \alpha _l>0. \end{aligned}$$
(17)

3 Asymptotic Expansion of the Weak Error

Using a Taylor expansion and the fact that u is a solution of the Kolmogorov equation (8) we obtain the following expansion.

Proposition 1

Let consider any \(N\in \mathrm {N}\) and any \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n}) \cap C_{r_{2N+n+3}}^{2N+n+3}(\mathrm {R}^{2n})\), \( r_{2N+n+3} \in \mathrm {N}\). There exist \(c(N)>0\) and \( l_N\in \mathrm {N}\), \(l_N>r_{2N+n+3}\) such that for all \(h>0\) and \((p,q)^T\in \mathrm {R^{2n}}\) we have

$$\begin{aligned}&|u(h,p,q)-\sum _{k=0}^N \frac{h^k}{k!}L^k\phi (p,q)|\le c(N)h^{N+1}\Vert \phi -<\phi >\Vert _{2N+3+n,r_{2N+3+n}} \nonumber \\&(1+|p|^{2l_N}+|q|^{2l_N}) \end{aligned}$$
(18)

Let \(h_0=\min \{h_{02},h_{03}\}\). We study the first step of the approximating process \((P_k,Q_k)\), and later we will use the Markov property to extend the results at all steps. The following result gives an expansion for the symplectic scheme, similar with the expansion (18).

Proposition 2

For any \(k\in \mathrm {N}\) there exists an operator \(A_k\) of order 2k with coefficients in \( C_{pol}^\infty (\mathrm {R^{2n}})\) such that for any \(N\in \mathrm {N}\) and any \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n}) \cap C_{r_{2N+2}}^{2N+2}(\mathrm {R}^{2n})\), \( r_{2N+2} \in \mathrm {N}\), there exist \(C_N>0\) and \(l_N\in \mathrm {N}\) such that for all \(0<h\le h_0\) and \((p,q)^T\in \mathrm {R^{2n}}\) we have \(A_0=I\), \(A_1=L\), and

$$\begin{aligned} |E(\phi (Q_1,P_1))-\sum _{k=0}^N h^k A_k(p,q)\phi (p,q)|\le C_N h^{N+1}(1+|p|^{2l_N}+|q|^{2l_N})|\phi |_{2N+2, r_{2N+2}} \end{aligned}$$

Proof

Firstly we use Taylor expansions to obtain expansions for \(P_1\) and \(Q_1\) (see also the proof of Lemma 3.4 in [6]). Then the proof can be done using the same approach as in the proof of Proposition 3.2 in [6].

3.1 The Modified Generator

Following the same approach as in [3], we want to construct a formal series \(\mathscr {L}=L+hL_1+\cdots +h^kL_k+\cdots \) such that formally the solution v(h, p, q) of the equation

$$ \partial _t v(t,p,q)=\mathscr {L}v(t,p,q), t>0,\quad v(0,p,q)=\phi (p,q), (p,q)^T\in \mathrm {R^{2n}}, $$

coincides in the sense of asymptotic expansion with the transition semigroup \(E(\phi (P_1\), \(Q_1))\) studied in Proposition 2. In order to have

$$ \exp (h\mathscr {L})\phi =\phi +\sum _{k\ge 1}h^kA_k\phi $$

we define the \(L_k\) operator as

$$\begin{aligned} L_k=A_{k+1}+\sum _{l=1}^k\frac{B_l}{l!}\sum _{k_1+\cdots +k_{l+1}=k-l}L_{k_1}\ldots L_{k_l}A_{k_{l+1}+1} \end{aligned}$$
(19)

\(B_l\) are the Bernoulli numbers and \(L_k\) is an operator of order \(2k+2\) with coefficients in \(C_{pol}^\infty (\mathrm {R}^{2n})\) and \(L_k 1=0\). We also have

$$\begin{aligned} A_k=\sum _{l=1}^k\frac{1}{l!}\sum _{k_1+\cdots +k_{l}=k-l}L_{k_1}\ldots L_{k_l}. \end{aligned}$$
(20)

We define the modified generator

$$\begin{aligned} L^{(N)}=L+\sum _{k=1}^Nh^kL_k, \quad N\in \mathrm {N}^*. \end{aligned}$$
(21)

Since we do not know if the modified equation

$$ \partial _t v^{(N)}(t,p,q)=L^{(N)}v^{(N)}(t,p,q), t>0,\quad v^{(N)}(0,p,q)=\phi (p,q), (p,q)^T\in \mathrm {R^{2n}}, $$

has a solution, we construct an approximate solution associated to (21).

Proposition 3

Let \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\). For all \(k\in \mathrm {N}\) there exist functions \(v_k(t,\cdot )\in C_{pol}^\infty (\mathrm {R}^{2n})\) defined for all \(t\ge 0\) such that \(v_0(0,\cdot )=\phi (\cdot )\), \(v_k(0,\cdot )=0\), \(k\ge 1\), and

$$\begin{aligned} \partial _t v_k(t,p,q)-Lv_k(t,p,q)=\sum _{l=1}^k L_lv_{k-l}(t,p,q), \quad t\ge 0. \end{aligned}$$
(22)

Moreover, for all \(k\in \mathrm {N}\), \(j\in \mathrm {N}^*\) there exist \(\gamma _{k,j}>0\) and positive integers \(\alpha _{k,j}\) and \(l_{k,0}\) such that for all \(t\ge 0\) we have

$$\begin{aligned}&|v_k(t)|_{j,\alpha _{k,j}}\le Q_{k,j}(t)e^{-\gamma _{k,j} t}\Vert \phi -<\phi >\Vert _{j+(n+1)(k+1)+4k,r_{j+(n+1)(k+1)+4k}}, \end{aligned}$$
(23)
$$\begin{aligned}&\Vert v_k(t)\Vert _{0,l_{k,0}}\le C_{0,k}\Vert \phi -<\phi >\Vert _{(n+1)(k+1)+4k,r_{(n+1)(k+1)+4k}}, \end{aligned}$$
(24)

Here \(Q_{k,j}:[0,\infty )\rightarrow [0,\infty )\) are polynomial functions with positive coefficients and the constants \(C_{0,k}\) do not depend on t.

Proof

The proof is similar with the proof of Theorem 4.1 in [6]. Inequalities (23)–(24) are a consequence of the results presented in Lemma 2.

For any \(N\ge 0\), we define the approximate solution of the modified flow as:

$$\begin{aligned} v^{(N)}(t,p,q)=\sum _{k=0}^N h^k v_k(t,p,q). \end{aligned}$$
(25)

We can easily show that for all \(t\ge 0\) we have

$$\begin{aligned} \partial _t v^{(N)}(t,p,q)=L^{(N)}v^{(N)}(t,p,q)-R^{(N)}(t,p,q), \quad v^{(N)}(0,p,q)=\phi (p,q), \end{aligned}$$
(26)

where

$$\begin{aligned} R^{(N)}(t,p,q)=\sum _{i=N+1}^{2N} h^i \sum _{k=i-N}^N L_k v_{i-k} \end{aligned}$$
(27)

is of order \(O(h^{N+1})\). The following result can be proved similarly with Theorem 4.1 in [3].

Proposition 4

Let \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\). For any \(N\in \mathrm {N}^*\) there exist \(C_N>0\) and \(l_N\), \(k_{2N+2} \in \mathrm {N}\) such that for all \(t\ge 0\), \(0<h\le h_0\), \((p,q)\in \mathrm {R}^{2n}\) we have

$$\begin{aligned}&\biggl | E(v^{(N)}(t,P_1,Q_1)-v^{(N)}(t+h, p,q))\biggl |\nonumber \\&\quad \le h^{N+1}C_N(1+|p|^{2l_N}+|q|^{2l_N})\sup _{\begin{array}{c} s\in [0,h]\\ k=0,\ldots ,N \end{array}}|v_k(t+s,\cdot )|_{2N+2,k_{2N+2}}. \end{aligned}$$
(28)

3.2 Main Result

We now study the long time behavior of the numerical solution. We obtain an expansion similar with the one for the exact solution, given in Proposition 1.

Theorem 1

Let \(N\in \mathrm {N}\) be fixed, and let \((P_k,Q_k)\) be the discrete process defined by the symplectic scheme. Let \(0<h\le h_0\), \(\alpha _N=6N+8+(n+1)(N+2)\) and \(\phi \in C_{pol}^\infty (\mathrm {R}^{2n})\cap C_{r_{\alpha _N}}^{\alpha _N}\) . Then there exist \(C_N>0\) and \(l_N\in \mathrm {N}\) such that for all \(k\in \mathrm {N}\)

$$\begin{aligned} |E(\phi (P_k,Q_k))-v^{(N)}(kh,p,q)|\le h^{N+1}C_N(1+|p|^{2l_N}+|q|^{2l_N})\Vert \phi -<\phi >\Vert _{\alpha _N,r_{\alpha _N}}. \end{aligned}$$

Proof

Let \(t_k=kh\). By the Markov property of \((P_k,Q_k)\) we have

$$\begin{aligned}&|E(\phi (P_k,Q_k)-v^{(N+1)}(t_k,p,q)|=|E(v^{(N+1)}(0,P_k,Q_k))-v^{(N+1)}(t_k,p,q)|= \\&\left| E\left( \sum _{j=0}^{k-1}E\left( v^{(N+1)}(t_j,P_{k-j},Q_{k-j})-v^{(N+1)}(t_{j+1},P_{k-j-1},Q_{k-j-1})\biggl |P_{k-j-1},Q_{k-j-1}\right) \right) \right| \\&\le \sum _{j=0}^{k-1}\biggl |E\biggl (E\biggl (v^{(N+1)}\left( t_j,P_1(P_{k-j-1},Q_{k-j-1}), Q_1(P_{k-j-1},Q_{k-j-1})\right) -v^{(N+1)}(t_{j+1}, \\&P_{k-j-1},Q_{k-j-1})\biggl |P_{k-j-1},Q_{k-j-1}\biggl )\biggl )\biggl |, \end{aligned}$$

where \(\left( P_1(p,q), Q_1(p,q)\right) \) is the first step of the scheme (11)–(12) when the initial condition is (p, q). Using inequalities (17), (23), and (28), with \(t=t_j\), \(j=0,\ldots , k-1\), we deduce that there exist positive integers \(l_N\), \(k_N\) such that

$$\begin{aligned}&\Vert E(v^{(N+1)}(0,P_k,Q_k)-v^{(N+1)}(t_k,p,q)\Vert _{0,l_N}\le h^{N+2}c\sum _{j=0}^{k-1}\sup _{\begin{array}{c} s\in [0,h]\\ i=0,\ldots ,N+1 \end{array}} |v_i(t_j+s,\cdot )|_{2N+4,k_N} \\&\le h^{N+2}c\Vert \phi -<\phi>\Vert _{\alpha _N, r_{\alpha _N}}\sum _{j=0}^{k-1}Q_{2N+4}(t_j) e^{-\lambda _{2N+4} t_j} \\&\le h^{N+2}c\Vert \phi -<\phi >\Vert _{\alpha _N, r_{\alpha _N}}\sum _{j=0}^{k-1} e^{-\tilde{\lambda }_{2N+4} t_j}, \end{aligned}$$

where \(c>0\), \(0<\tilde{\lambda }_{2N+4}<\lambda _{2N+4}\) and \(Q_{2N+4}\) is a polynomial function with positive coefficients. Notice that for a fixed constant \(\lambda >0\) we have

$$ \sum _{j=0}^{k-1} e^{-\lambda t_j}\le \frac{1}{1-e^{-\lambda h}}\le \frac{c_1}{h}, $$

where the constant \(c_1\) depends on \(\lambda \) and \(h_0\). Hence, using the previous inequality and (24) we get

$$\begin{aligned}&\Vert E(\phi (P_k,Q_k)-v^{(N)}(t_k,p,q)\Vert _{0,l_N}=\Vert E(v^{(N+1)}(0,P_k,Q_k)-v^{(N+1)}(t_k,p,q) \\&+h^{N+1}v_{N+1}(t_k,p,q)\Vert _{0,l_N} \le h^{N+1}c_2\Vert \phi -<\phi>\Vert _{\alpha _N, r_{\alpha _N}}+h^{N+1}\Vert v_{N+1}(t_k,p,q)\Vert _{0,l_N} \\&\le h^{N+1}c_2\Vert \phi -<\phi>\Vert _{\alpha _N, r_{\alpha _N}}+h^{N+1}C_{0,N+1}\Vert \phi -<\phi>\Vert _{(n+1)(N+2)+4(N+1),r_{\alpha _N}} \\&\le h^{N+1}C_N\Vert \phi -<\phi >\Vert _{\alpha _N, r_{\alpha _N}} \end{aligned}$$

4 Conclusions and Future Work

We have presented a weak backward error analysis for a SHS system and a symplectic scheme of first weak order. The main tools are the exponential convergence to equilibrium of the solution of the Kolmogorov equation, and the uniform ellipticity of the associated operator. We plan to do a backward error analysis under less restrictive assumptions. The main difficulty is that the symplectic schemes are fully implicit, and for SDEs with multiplicative noise and unbounded coefficients, methods from Malliavin calculus are needed.