Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

An interesting property of a stochastic differential equation (SDE) or a stochastic partial differential equation (SPDE) is the qualitative behaviour of its second moment for large times. Both types of equations can be interpreted as SDEs on a (here separable) Hilbert space \((H, \left\langle \cdot , \cdot \right\rangle _{H} )\). More specifically, let us consider a complete filtered probability space \((\varOmega , \mathscr {A}, (\mathscr {F}_t,{t\ge 0}), P)\) satisfying the “usual conditions” and the model problemwith \(\mathscr {F}_0\)-measurable, square-integrable initial condition \(X(0) = X_0\). Here, the operator \(A: \mathscr {D}(A) \rightarrow H\) is the generator of a \(C_0\)-semigroup \(S=(S(t), t \ge 0)\) on H and F is a linear and bounded operator on H, i.e., \(F \in L(H)\). Furthermore, L denotes a U-valued Q-Lévy process that is assumed to be a square-integrable martingale as considered in [27] on the real separable Hilbert space \((U, \left\langle \cdot , \cdot \right\rangle _{U} )\) with covariance \(Q \in L(U)\) of trace class and let \(G \in L(H;L(U;H))\).

We recall from [24] that an equilibrium (solution) of (1.1) is the zero solution \((X_\mathrm {e}(t) = 0, t\ge 0)\). It is called mean-square stable if, for every \(\varepsilon > 0\), there exists \(\delta > 0\) such that \({{\mathrm{\mathbb {E}}}}[\Vert X(t) \Vert _H^2] < \varepsilon \) for all \(t\ge 0\) whenever \({{\mathrm{\mathbb {E}}}}[\Vert X_0\Vert _H^2] < \delta \). It is further asymptotically mean-square stable if it is mean-square stable and there exists \(\delta > 0\) such that \(\mathbb {E}[\Vert X_0\Vert _H^2] < \delta \) implies \(\lim _{t \rightarrow \infty } {{\mathrm{\mathbb {E}}}}[ \Vert X(t) \Vert _H^2] = 0\). A lot of effort has been dedicated to the asymptotic mean-square stability analysis in finite and infinite dimensions, see e.g., [2, 17, 24, 26].

Since analytical solutions to SDEs are rarely available, approximations in time and possibly in space by numerical methods have to be considered. The main focus of research in recent years has been on strong and weak convergence when the discretization parameters \(\Delta t\) in time and h in space tend to zero. However, this property does not guarantee that the approximation shares the same (asymptotic) mean-square stability properties as the analytical solution. For finite-dimensional SDEs it is known that the specific choice of \(\Delta t\) is essential. The goal of this manuscript is to generalize the theory of asymptotic mean-square stability analysis to a Hilbert space setting. We develop a theory for approximation schemes that has apriori no relation to the original Eq. (1.1) and its properties. Later on, we will discuss which conditions on (1.1) and its approximation lead to similar behaviour. An important application of mean-square stability is in multilevel Monte Carlo methods, where combinations of approximations on different space and time grids are computed. If the solution is mean-square unstable on any of the included levels, this is enough for the estimator not to behave as it should, see, e.g., [1].

The mean-square stability analysis of numerical approximations of SDEs started by considering approximations of the one-dimensional geometric Brownian motion, see e.g., [14, 15, 29]. As it has been pointed out in [10, 11], the analysis of higher-dimensional systems and their approximations is also necessary, since the asymptotic behaviour of the corresponding mean-square processes of systems with commuting and non-commuting matrices often differs. The tools to perform mean-square stability analysis of SDE approximations presented in [11] could in principle be used for approximations of infinite-dimensional SDEs by a method of lines approach: After projection on an \(N_h\)-dimensional space the mean-square stability properties of the resulting finite-dimensional SDEs and their approximations can be determined by considering the eigenvalues of \(N_h^2 \times N_h^2\)-dimensional matrices. However, due to the computational complexity as \(N_h \rightarrow \infty \), neither the symbolic nor the numerical computation of these eigenvalues can be done for arbitrarily large systems. For this reason, we use an approach based on tensor-product-space-valued processes and properties of tensorized linear operators.

The outline of this article is as follows: Sect. 2 sets up a theory of mean-square stability analysis for discrete stochastic processes derived from recursions as they appear in approximations of infinite-dimensional SDEs. In the main result, necessary and sufficient conditions for asymptotic mean-square stability are shown. These results are then applied in Sect. 3 to numerical approximations of (1.1) based on spatial Galerkin discretization schemes and time discretizations with Euler–Maruyama and Milstein methods using backward/forward Euler and Crank–Nicolson as rational semigroup approximations. We conclude this work presenting simulations of stochastic heat equations with spectral Galerkin and finite element methods in Sect. 4 that illustrate the theory.

This section is devoted to the setup of asymptotic mean-square stability for families of stochastic processes in discrete time given by recursion schemes as they typically show up in approximations of (1.1). We derive necessary and sufficient conditions ensuring asymptotic mean-square stability that can be checked in practice as it is shown later in Sect. 3.

Let \((V_h, h \in (0,1])\) be a family of finite-dimensional subspaces \(V_h \subset H\) with \({\text {dim}}(V_h) = N_h\in \mathbb {N}\) indexed by a refinement parameter h. With an inner product induced by \( \left\langle \cdot , \cdot \right\rangle _{H} \), \(V_h\) becomes a Hilbert space with norm \(\Vert \cdot \Vert _{H}\). For a linear operator \(D: V_h \rightarrow V_h\), the operator norm \(\Vert D \Vert _{L(V_h)}\) is therefore given by \(\sup _{v \in V_h} \Vert D v \Vert _{H}/\Vert v \Vert _{H}\) and can be seen to coincide with \(\Vert D P_h \Vert _{L(H)}\), where \(P_h\) denotes the orthogonal projection onto \(V_h\).

Let us further consider the time interval \([0,\infty )\) and for convenience equidistant time steps \(t_j = j \Delta t\), \(j \in \mathbb {N}_0\), with fixed step size \(\Delta t > 0\). Hence, \(t \rightarrow \infty \) is equivalent to \(j\rightarrow \infty \). Assume that we are given a sequence of \(V_h\)-valued random variables \({(X_h^j, j \in \mathbb {N}_0)}\) determined by the linear recursion schemewith \(\mathscr {F}_0\)-measurable initial condition \(X_h^0 \in L^2(\varOmega ;V_h)\), i.e., \({{\mathrm{\mathbb {E}}}}[\Vert X_h^0 \Vert _{V_h}^2]< \infty \). Here \({D_{\Delta t, h}^{{\text {det}}}\in L(V_h)}\) and \(D_{\Delta t, h}^{{\text {stoch}},j}\) is an \(L(V_h)\)-valued random variable for all j.

In terms of SDE (1.1), one can think of \(D_{\Delta t, h}^{{\text {det}}}\) as the approximation of the solution operator of the deterministic partand \(D_{\Delta t, h}^{{\text {stoch}},j}\) approximates the stochastic partAlthough, in general, any not necessarily equidistant time discretization \((t_j, j \in \mathbb {N}_0)\) that satisfies \(t_j \rightarrow \infty \) if \(j \rightarrow \infty \) would be sufficient for the following theory, we see in the given SDE example that \(D_{\Delta t, h}^{{\text {det}}}\) would be j-dependent in this case, which we want to omit for the sake of readability.

Inspired by properties of standard approximation schemes for (1.1), we put the following assumptions on the family \((D_{\Delta t, h}^{{\text {stoch}},j}, j \in \mathbb {N}_0)\).

For the recursion scheme (2.1) an equilibrium (solution) is given by the zero solution, which is defined as \(X_{h,\mathrm {e}}^j = 0\) for all \(j\in \mathbb {N}_0\). We define mean-square stability of the zero solution of (2.1) in what follows.

It is called asymptotically mean-square stable if it is mean-square stable and there exists \(\delta > 0\) such that \(\mathbb {E}[\Vert X_h^0\Vert _H^2] < \delta \) implies \(\lim _{j \rightarrow \infty } {{\mathrm{\mathbb {E}}}}[ \Vert X_h^j \Vert _H^2]=0\). Furthermore, it is called asymptotically mean-square unstable if it is not asymptotically mean-square stable.

For convenience, the abbreviation (asymptotic) mean-square stability is used for the (asymptotic) mean-square stability of the zero solution of (2.1) or (1.1) if it is clear from the context.

When applied to \(Y_j = X^j_h\), the following lemma provides an equivalent condition for mean-square stability in terms of the tensor-product-space-valued process \({X_h^j \otimes X_h^j \in V_h^{(2)}}\). Here, for a general Hilbert space H, the abbreviation \({H^{(2)} = H \otimes H}\) is used and \(H^{(2)}\) is defined as the completion of the algebraic tensor product with respect to the norm induced bywhere \(v = \sum ^N_{i=1} v_{1,i} \otimes v_{2,i}\) and \(w = \sum ^M_{j=1} w_{1,j} \otimes w_{2,j}\) are representations of elements v and w in the algebraic tensor product.

This lemma enables us to show the following sufficient condition for asymptotic mean-square stability.

In many examples the operators \((D_{\Delta t, h}^{{\text {stoch}},j},j\in \mathbb {N}_0)\) have a constant covariance, i.e., they satisfy for all \(j \in \mathbb {N}_0\) Often as in the following example they are even independent and identically distributed, which implies (2.2).

Having this example in mind, we are able to give a necessary and sufficient condition for asymptotic mean-square stability when assuming (2.2) and therefore to specify Theorem 2.1. The condition relies on the spectrum of a single linear operator \(\mathscr {S}\in L(V^{(2)}_h)\).

In the framework of SDE approximations, note that this corollary is an SPDE version formulated with operators of the results for finite-dimensional linear systems in [11]. There, the proposed method relies on a matrix eigenvalue problem. For SPDE approximations, this approach is not suitable, since the dimension of the considered eigenvalue problem increases heavily with space refinement. More precisely, for \(h>0\), the spectral radius of an \((N_h^2 \times N_h^2)\)-matrix has to be computed. To overcome this problem, we perform, in what follows, mean-square stability analysis of SPDE approximations based on operators as introduced above.

We continue by applying the previous results to the analysis of some classical numerical approximations of (1.1) which admits by results in [27, Chapter 9] an up to modification unique mild càdlàg solution and is for \(t \ge 0\) given byWe assume further that the operator \(-A: \mathscr {D}(-A) \subset H \rightarrow H\) of (1.1) is densely defined, self-adjoint, and positive definite with compact inverse. This implies that \(-A\) has a non-decreasing sequence of positive eigenvalues \({(\lambda _i, i \in \mathbb {N})}\) for an orthonormal basis of eigenfunctions \((e_i, i \in \mathbb {N})\) in H and fractional powers of \(-A\) are provided byfor all \(i \in \mathbb {N}\) and \(r > 0\). For each \(r>0\), \(\dot{H}^r = \mathscr {D}((-A)^{r/2})\) with inner product \( \left\langle \cdot , \cdot \right\rangle _{r} = \left\langle (-A)^{r/2}\cdot , (-A)^{r/2}\cdot \right\rangle _{H} \) defines a separable Hilbert space (see, e.g., [20, Appendix B]).

We assume that the sequence \((V_h, h \in (0,1])\) of finite-dimensional subspaces fulfil \(V_h\subset \dot{H}^1\subset H\) and define the discrete operator \(-A_h : V_h \rightarrow V_h\) byfor all \(v_h, w_h \in V_h\). This definition implies that \(-A_h\) is self-adjoint and positive definite on \(V_h\) and therefore has a sequence of orthonormal eigenfunctions \((e_{h,i}, i = 1,\ldots ,N_h)\) and positive non-decreasing eigenvalues \((\lambda _{h,i}, i = 1,\ldots ,N_h)\) (see e.g., [20, Chapter 3]). By using basic properties of the Rayleigh quotient, we bound the smallest eigenvalue \(\lambda _{h,1}\) of \(-A_h\) from below by the smallest eigenvalue \(\lambda _1\) of \(-A\) throughsince \(V_h \subset H\), cf. [9]. This estimate turns out to be useful when comparing asymptotic mean-square stability of (1.1) and its approximation later in this section.

Let the covariance of the Lévy process L be self-adjoint, positive semidefinite, and of trace class. Then results in [27, Chapter 4] imply the existence of an orthonormal basis \((f_i, i \in \mathbb {N})\) of U and a non-increasing sequence of non-negative real numbers \((\mu _i, i \in \mathbb {N})\) such that for all \(i \in \mathbb {N}\), \(Q f_i = \mu _i f_i\) with \({{\mathrm{Tr}}}(Q) = \sum _{i=1}^\infty \mu _i < \infty \) and L admits a Karhunen–Loève expansionwhere \((L_i, i \in \mathbb {N})\) is a family of real-valued, square-integrable, uncorrelated Lévy processes satisfying \({{\mathrm{\mathbb {E}}}}[(L_i(t))^2]=t\) for all \(t \ge 0\). Note that due to the martingale property of L, the real-valued Lévy processes satisfy \({{\mathrm{\mathbb {E}}}}[L_i(t)] = 0\) for all \(t \ge 0\) and \(i \in \mathbb {N}\). This implies, together with the stationarity of the Lévy increments \(\Delta L^j_i = L_i(t_{j+1})-L_i(t_j)\), that for all \(i \in \mathbb {N}\) and \(j \in \mathbb {N}_0\),Since the series representation of L can be infinite, an approximation of L might be required to implement a fully discrete approximation scheme, which is typically done by truncation of the Karhunen–Loève expansion, i.e., for \(\kappa \in \mathbb {N}\), set the truncated process \({L^\kappa (t) = \sum _{i=1}^\kappa \sqrt{\mu _i} L_i(t) f_i}\). Note that the choice of \(\kappa \) is essential and should be coupled with the overall convergence of the numerical scheme as is discussed in, e.g., [3, 5, 25]. Within this work, we consider numerical methods based on the original Karhunen–Loève expansion (3.3) of L. However, this does not restrict the applicability of the results since \(L^\kappa \) fits in the framework by setting \(\mu _i = 0\) for all \(i > \kappa \).

As standard example in this context we consider the stochastic heat equation which is used for simulations in Sect. 4.

In this section we adopt the setting of Example 3.1 and use numerical simulations to illustrate our theoretical results. More specifically, we consider the stochastic heat equationwith \(X_0(x) = \sqrt{30} x(1-x)\), then \({{\mathrm{\mathbb {E}}}}[\Vert X_0 \Vert _H^2] = 1\). We consider a Q-Wiener process \(W(t) = \sum _{i=1}^\infty \sqrt{\mu _i} \beta _i(t) e_i\), where \((\beta _i,i\in \mathbb {N})\) is a sequence of independent, real-valued Brownian motions, and assume \(\mu _i = C_\mu i^{-\alpha }\) with \(C_\mu > 0\) and \(\alpha > 1\). Here, \(C_\mu \) scales the noise intensity and \(\alpha \) controls the space regularity of W, see, e.g., [23, 25].