1 Introduction
Stochastic evolution equations (SEE) in Hilbert spaces are generalizations of Itô stochastic equations [Reference Itô21] to the infinite dimensional case. The theory has been greatly motivated by various branches of science (e.g. physics, chemistry, biology). For example, stochastic partial differential equations of the evolutionary type are used to describe wave propagation (in random media [Reference Frisch and Bharucha-Reid14] and turbulence [Reference Chow and Bharucha-Reid9]) or the dynamics of populations [Reference Fleming, Bensoussan and Lions13]. They also appear in the study of nonlinear filtering of diffusion processes [Reference Fujisaki, Kallianpur and Kunita15], stochastic flows [Reference Kunita25] and control theory [Reference Kushner26].
Numerous papers present different approaches to the theory of SEE including the variational approach [Reference Ren, Röckner and Wang30, Reference Zhang34], the martingale (measure) approach [Reference Walsh32], the white noise approach [Reference Deck, Kruse, Potthoff, Watanabe, Da Prato and Tubaro12, Reference Holden, Øksendal, Ubøe and Zhang20], the method based on the monotonicity of the drift [Reference Pardoux29] and finally the semigroup approach [Reference Da Prato and Zabczyk10, Reference Da Prato and Zabczyk11], which we use in the paper.
The evolution models are considered in different spaces, a consequence of the choice of approach. Semigroup methods are usually applied in Hilbert spaces since real-valued stochastic integration extends directly to processes with values therein. These results by Da Prato and Zabczyk [Reference Da Prato and Zabczyk10] have been generalized to the case of a certain class of Banach spaces. Van Neerven and Veraar reformulate the integration theory for Hilbert spaces and develop the tools and techniques to study SEE in UMD spaces [Reference van Neerven, Veraar and Weis28] (which for example cover
$L^{p}$
-spaces for
$p\in$
$(1,\infty )$
) or in martingale type
$2$
spaces [Reference Brzeźniak6] that include
$L^{p}$
-spaces for
$p\in [2,\infty )$
.
The existence and uniqueness of mild solutions to stochastic functional evolution equations are considered in [Reference Taniguchi, Liu and Truman31] by using an analytic semigroup approach and fractional power operator arguments. In [Reference Jahanipur22] Jahanipur discusses the problem under assumptions of demicontinuous drift satisfying a semimonotone condition instead of Lipschitz one. In [Reference Govindan16, Reference Govindan17] Govindan studies the existence of mild solutions by the comparison theorem. See references therein for more details.
Numerical approximation of solutions to SEE has been discussed by several authors. In [Reference Gyöngy and Millet18] Gyöngy applies explicit and implicit numerical schemes to SEE with monotone operators in Banach spaces. In [Reference Hausenblas19] Hausenblas considers the Galerkin approximations for space discretization, the Euler and the Crank–Nicholson schemes for time discretization. The work of Bessaih et al [Reference Bessaih, Hausenblas, Randrianasolo and Razafimandimby4] addresses the numerical approximation of shell models of turbulence. Weak convergence of finite element approximations can be found in [Reference Kovács, Larsson and Lindgren24]. Finite-element-based space-time discretizations of the incompressible Navier–Stokes equations are studied in [Reference Brzeźniak, Carelli and Prohl7]. The splitting-up method is investigated in [Reference Bessaih, Brzeźniak and Millet3, Reference Bessaih and Millet5, Reference Carelli, Hausenblas and Prohl8, Reference Nagase27].
Newton’s methods for stochastic differential equations are studied by Kawabata and Yamada in [Reference Kawabata and Yamada23] and Amano in [Reference Amano1]. In [Reference Amano2] Amano develops techniques to prove a probabilistic second-order convergence of Newton’s methods. In [Reference Wrzosek33] we derive further non-trivial generalizations to the case of stochastic functional differential equations with Hale functionals. The aim of the present paper is to apply Newton’s method to stochastic functional evolution equations in a semigroup setting. Consider the stochastic semilinear delay heat equation
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}u(t,x)-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}u(t,x) & = & \displaystyle \int _{-r_{1}}^{0}\int _{0}^{1}f(x,y,u(t+s,y))\,dy\,ds\nonumber\\ \displaystyle & & \displaystyle +\,\int _{-r_{2}}^{0}\int _{0}^{1}b(x,y,u(t+s,y))\,dy\,ds{\dot{W}}(t)\qquad\end{eqnarray}$$
with the boundary conditions
$u(t,0)=u(t,1)=0$
,
$t\in [0,T]$
, and the initial condition
$u(t,x)=\unicode[STIX]{x1D711}(t,x)$
for
$t\in [-r,0]$
,
$x\in [0,1]$
,
$r_{1},r_{2}\geqslant 0$
, where
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}(t,\cdot )\in L^{2}([0,1])\quad \text{for }t\in [-r,0],r\geqslant r_{1},r_{2};\nonumber\\ \displaystyle & & \displaystyle \qquad \unicode[STIX]{x1D711}(\cdot ,x)\in C([-r,0],\mathbb{R})\quad \text{for }x\in [0,1]\nonumber\end{eqnarray}$$
and
$W(t)$
,
$t\in [0,T]$
is the real standard Brownian motion. Let
$H=$
$L^{2}([0,1])$
and
$A=\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}x^{2}$
be the linear operator of second-order differentation with respect to the spatial variable with the domain
$$\begin{eqnarray}{\mathcal{D}}(A)=\{u\in L^{2}([0,1]):u^{\prime },u^{\prime \prime }\in L^{2}([0,1])\text{ and }u(0)=u(1)=0\}.\end{eqnarray}$$
Then
$A:{\mathcal{D}}(A)\subseteq H\rightarrow H$
generates a strongly continuous semigroup
$S(t)$
. If we define
$F,B:C([-r,0],H)\rightarrow H$
by
$$\begin{eqnarray}\displaystyle F(\unicode[STIX]{x1D713})(x) & = & \displaystyle \int _{-r_{1}}^{0}\int _{0}^{1}f(x,y,\unicode[STIX]{x1D713}(s)(y))\,dy\,ds,\nonumber\\ \displaystyle B(\unicode[STIX]{x1D713})(x) & = & \displaystyle \int _{-r_{2}}^{0}\int _{0}^{1}b(x,y,\unicode[STIX]{x1D713}(s)(y))\,dy\,ds,\nonumber\end{eqnarray}$$
then the initial-boundary value problem (1.1) can be written as an abstract evolution equation
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}du(t)=(Au(t)+F(u_{t}))\,dt+B(u_{t})\,dW(t),\quad \\ u_{0}=\unicode[STIX]{x1D711}.\quad \end{array}\right.\end{eqnarray}$$
The paper is organized as follows. In §2 we recall some definitions and introduce basic notations. We establish a first-order convergence (§4) and a probabilistic second-order convergence (§5) of Newton’s method. The result in §4 is based on the Gronwall-type inequality presented in §3.
2 Formulation of the problem
Fix
$T>0$
. Let
$(\unicode[STIX]{x1D6FA},{\mathcal{F}},\mathbb{P})$
be a complete probability space and let
$({\mathcal{F}}_{t})_{t\in [0,T]}$
be a normal filtration. Let
$(H,\langle \cdot ,\cdot \rangle )$
and
$(U,\langle \cdot ,\cdot \rangle _{U})$
be separable Hilbert spaces with the norms denoted by
$\Vert \cdot \Vert$
,
$\Vert \cdot \Vert _{U}$
respectively and let
$L(U,H)$
be the Banach space of all linear, bounded operators from
$U$
into
$H$
with the norm
$$\begin{eqnarray}\Vert T\Vert _{L(U,H)}=\sup _{\Vert u\Vert _{U}\leqslant 1}\Vert Tu\Vert ,\quad u\in U,T\in L(U,H).\end{eqnarray}$$
Define
$L(H):=L(H,H)$
. Let
$Q$
be an
$U$
-valued linear, bounded and symmetric non-negative operator such that Tr
$Q<\infty$
. For
$U_{1}\supset U$
an
$U_{1}$
-valued stochastic process
$W(t)$
,
$t\in [0,T]$
is called a
$Q$
-Wiener process on
$[0,T]$
if:
$$\begin{eqnarray}\displaystyle & & \displaystyle W(0)=0;\nonumber\\ \displaystyle & & \displaystyle W\text{ has continuous trajectories};\nonumber\\ \displaystyle & & \displaystyle W\text{ has independent increments, and};\nonumber\\ \displaystyle & & \displaystyle W(t)-W(s)\text{ has normal distribution with mean }0\text{ and variance }(t-s)Q,\nonumber\\ \displaystyle & & \displaystyle \text{for }0\leqslant s\leqslant t\leqslant T.\nonumber\end{eqnarray}$$
Let
$U_{0}=Q^{1/2}U$
. By
$HS(U_{0},H)$
we denote the space of all Hilbert–Schmidt operators from
$U_{0}$
to
$H$
. We recall that a linear and bounded operator
$T:U_{0}\rightarrow H$
is said to be Hilbert–Schmidt if
$$\begin{eqnarray}\mathop{\sum }_{k=1}^{\infty }\Vert Te_{k}\Vert ^{2}<\infty ,\end{eqnarray}$$
where
$\{e_{k}\}\in U_{0}$
is a complete orthonormal base. The space
$HS(U_{0},H)$
equipped with the norm
$$\begin{eqnarray}\Vert T\Vert _{HS}=\biggl(\mathop{\sum }_{k=1}^{\infty }\Vert Te_{k}\Vert ^{2}\biggr)^{1/2}\end{eqnarray}$$
is a separable Hilbert space with the scalar product
$$\begin{eqnarray}\langle S,T\rangle _{HS}=\mathop{\sum }_{k=1}^{\infty }\langle Se_{k},Te_{k}\rangle ,\quad S,T\in HS(U_{0},H).\end{eqnarray}$$
Fix
$0\leqslant r<\infty$
. By
${\mathcal{C}}_{r}:=C([-r,0],H)$
denote the Banach space of all continuous functions
$v:[-r,0]\rightarrow H$
with the norm
$$\begin{eqnarray}\Vert v\Vert _{{\mathcal{C}}_{r}}=\sup _{-r\leqslant s\leqslant 0}\Vert v(s)\Vert .\end{eqnarray}$$
Set
${\mathcal{F}}_{t}:={\mathcal{F}}_{0}$
for
$t\in [-r,0]$
. For
$p>2$
,
${\mathcal{X}}_{p,T}$
is the Banach space of all continuous
${\mathcal{F}}_{t}$
-adapted processes
$X:[-r,T]\times \unicode[STIX]{x1D6FA}\rightarrow H$
such that
$$\begin{eqnarray}\Vert X\Vert _{p,t}=\biggl(\mathbb{E}\biggl[\sup _{-r\leqslant s\leqslant t}\Vert X(s)\Vert ^{p}\biggr]\biggr)^{1/p}<\infty .\end{eqnarray}$$
For a process
$X\in {\mathcal{X}}_{p,T}$
and
$t\in [0,T]$
, Hale-type operator
$X_{t}:[-r,0]\times \unicode[STIX]{x1D6FA}\rightarrow H$
is defined by
$$\begin{eqnarray}X_{t}(\unicode[STIX]{x1D70F})=X(t+\unicode[STIX]{x1D70F})\quad \text{for }\unicode[STIX]{x1D70F}\in [-r,0].\end{eqnarray}$$
We consider on
$H$
the following stochastic functional evolution equation:
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}dX(t)=(AX(t)+F(t,X_{t}))\,dt+B(t,X_{t})\,dW(t),\quad t\in [0,T]\quad \\ X(t)=\unicode[STIX]{x1D711}(t),\quad t\in [-r,0],\quad \end{array}\right.\end{eqnarray}$$
where:
(i)
$A$
is the infinitesimal generator of a strongly continuous semigroup (2.2)$$\begin{eqnarray}S(t)=\text{e}^{tA},t\geqslant 0\text{ in }H\text{ and }N=\sup _{t\in [0,T]}\Vert S(t)\Vert _{L(H)}\text{ for some }N>0;\end{eqnarray}$$
(ii)
$F:[0,T]\times \unicode[STIX]{x1D6FA}\times {\mathcal{C}}_{r}\rightarrow H$
is a measurable mapping;(iii)
$B:[0,T]\times \unicode[STIX]{x1D6FA}\times {\mathcal{C}}_{r}\rightarrow HS(U_{0},H)$
is a measurable mapping;(iv)
$W(t)$
is a
$Q$
-Wiener process on
$U_{1}\supset U$
; and(v)
$\unicode[STIX]{x1D711}\in {\mathcal{X}}_{p,0}$
.
Since
${\mathcal{F}}_{t}={\mathcal{F}}_{0}$
for
$t\in [-r,0)$
the process
$\unicode[STIX]{x1D711}$
is deterministic and thus independent of the
$Q$
-Wiener process on
$[0,T]$
. We assume that the mappings
$F$
and
$B$
satisfy the Lipschitz condition with respect to the functional variable:
$$\begin{eqnarray}\displaystyle \Vert F(t,x)-F(t,y)\Vert & {\leqslant} & \displaystyle L\Vert x-y\Vert _{{\mathcal{C}}_{r}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \Vert B(t,x)-B(t,y)\Vert _{HS} & {\leqslant} & \displaystyle L\Vert x-y\Vert _{{\mathcal{C}}_{r}}\end{eqnarray}$$
for some non-negative constant
$L$
,
$t\in [0,T]$
and
$x,y\in {\mathcal{C}}_{r}$
. A stochastic process
$X\in {\mathcal{X}}_{p,T}$
is said to be a mild solution of (2.1) if
$$\begin{eqnarray}\mathbb{P}\biggl(\int _{-r}^{T}\Vert X(s)\Vert ^{p}\,ds<\infty \biggr)=1\end{eqnarray}$$
and for arbitrary
$t\in [0,T]$
, we have
$$\begin{eqnarray}\displaystyle X(t) & = & \displaystyle S(t)\unicode[STIX]{x1D711}(0)+\int _{0}^{t}S(t-s)F(s,X_{s})\,ds\nonumber\\ \displaystyle & & \displaystyle +\int _{0}^{t}S(t-s)B(s,X_{s})\,dW(s),\quad \mathbb{P}\text{-a.s.}\end{eqnarray}$$
with
$X(t)=\unicode[STIX]{x1D711}(t)$
for
$t\in [-r,0]$
.
3 Gronwall-type inequality
By
$L({\mathcal{C}}_{r},H)$
and
$L({\mathcal{C}}_{r},HS)$
we denote the space of all linear and bounded operators from
${\mathcal{C}}_{r}$
to
$H$
and
${\mathcal{C}}_{r}$
to
$HS(U_{0},H)$
respectively with the norms
$$\begin{eqnarray}\displaystyle \Vert T_{1}\Vert _{L({\mathcal{C}}_{r},H)} & := & \displaystyle \sup _{\Vert v\Vert _{{\mathcal{C}}_{r}}\leqslant 1}\Vert T_{1}v\Vert ,\nonumber\\ \displaystyle \Vert T_{2}\Vert _{L({\mathcal{C}}_{r},HS)} & := & \displaystyle \sup _{\Vert v\Vert _{{\mathcal{C}}_{r}}\leqslant 1}\Vert T_{2}v\Vert _{HS}\nonumber\end{eqnarray}$$
for
$v\in {\mathcal{C}}_{r}$
,
$T_{1}\in L({\mathcal{C}}_{r},H)$
and
$T_{2}\in L({\mathcal{C}}_{r},HS)$
. The following lemma plays the key role in the proof of the convergence theorem in §4.
Lemma 3.1. Suppose that
$\unicode[STIX]{x1D6FC}_{1}(t):[0,T]\rightarrow H$
,
$\unicode[STIX]{x1D6FC}_{2}(t):[0,T]\rightarrow HS(U_{0},H)$
are continuous,
$T_{1}\in L({\mathcal{C}}_{r},H)$
,
$T_{2}\in L({\mathcal{C}}_{r},HS)$
and there exists a non-negative constant
$L$
such that
$$\begin{eqnarray}\Vert T_{1}(t)\Vert _{L({\mathcal{C}}_{r},H)}\leqslant L,\qquad \Vert T_{2}(t)\Vert _{L({\mathcal{C}}_{r},HS)}\leqslant L\end{eqnarray}$$
for
$t\in [0,T]$
. Then for a process
$X\in {\mathcal{X}}_{p,T}$
satisfying the stochastic functional evolution equation
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}dX(t)=(AX(t)+\unicode[STIX]{x1D6FC}_{1}(t)+T_{1}(t)X_{t})\,dt+(\unicode[STIX]{x1D6FC}_{2}(t)+T_{2}(t)X_{t})\,dW(t),\\ \quad \qquad \qquad t\in [0,T]\\ X(t)=0,\quad t\in [-r,0]\end{array}\right.\end{eqnarray}$$
we have
$$\begin{eqnarray}\Vert X\Vert _{p,t}^{p}\leqslant 4^{p-1}C_{3}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}\int _{0}^{t}(N^{p}\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}])\,ds\end{eqnarray}$$
for
$t\in [0,T]$
.
Proof. By the fact that
$$\begin{eqnarray}(|x|+|y|)^{p}\leqslant 2^{p-1}(|x|^{p}+|y|^{p})\quad \text{for }p>1\end{eqnarray}$$
we have
$$\begin{eqnarray}\displaystyle \Vert X\Vert _{p,t}^{p} & = & \displaystyle \mathbb{E}\biggl[\sup _{\tilde{t}\leqslant t}\Vert X_{\tilde{t}}\Vert ^{p}\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{p-1}\mathbb{E}\biggl[\sup _{\tilde{t}\leqslant t}\biggl\|\int _{0}^{\tilde{t}}S(\tilde{t}-s)[\unicode[STIX]{x1D6FC}_{1}(s)+T_{1}(s)X_{s}]\,ds\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & & \displaystyle +\,2^{p-1}\mathbb{E}\biggl[\sup _{\tilde{t}\leqslant t}\biggl\|\int _{0}^{\tilde{t}}S(\tilde{t}-s)[\unicode[STIX]{x1D6FC}_{2}(s)+T_{2}(s)X_{s}]\,dW_{s}\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & = & \displaystyle 2^{p-1}(I_{1}+I_{2}).\nonumber\end{eqnarray}$$
By (2.2) and the Hölder inequality for some
$C_{1}>0$
dependent on
$T$
:
$$\begin{eqnarray}\displaystyle I_{1} & {\leqslant} & \displaystyle N^{p}C_{1}\mathbb{E}\biggl[\sup _{\tilde{t}\leqslant t}\int _{0}^{\tilde{t}}\Vert \unicode[STIX]{x1D6FC}_{1}(s)+T_{1}(s)X_{s}\Vert ^{p}\,ds\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle N^{p}C_{1}\mathbb{E}\biggl[\int _{0}^{t}\Vert \unicode[STIX]{x1D6FC}_{1}(s)+T_{1}(s)X_{s}\Vert ^{p}\,ds\biggr].\nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle I_{1} & {\leqslant} & \displaystyle 2^{p-1}N^{p}C_{1}\mathbb{E}\biggl[\int _{0}^{t}\Vert \unicode[STIX]{x1D6FC}_{1}(s)\Vert ^{p}+\Vert T_{1}(s)X_{s}\Vert ^{p}\,ds\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{p-1}N^{p}C_{1}\int _{0}^{t}(\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+L^{p}\Vert X\Vert _{p,s}^{p})\,ds.\nonumber\end{eqnarray}$$
By [Reference Da Prato and Zabczyk10, Proposition 7.3] there exists
$C_{2}>0$
dependent on
$T$
such that
$$\begin{eqnarray}\displaystyle I_{2}\leqslant C_{2}\mathbb{E}\biggl[\int _{0}^{t}\Vert \unicode[STIX]{x1D6FC}_{2}(s)+T_{2}(s)X_{s}\Vert _{HS}^{p}\,ds\biggr]. & & \displaystyle \nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle I_{2}\leqslant 2^{p-1}C_{2}\int _{0}^{t}(\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}]+L^{p}\Vert X\Vert _{p,s}^{p})\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
Hence for
$C_{3}=\max \{C_{1},C_{2}\}$
:
$$\begin{eqnarray}\displaystyle \Vert X\Vert _{p,t}^{p} & {\leqslant} & \displaystyle 4^{p-1}C_{3}\int _{0}^{t}(N^{p}\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}])\,ds\nonumber\\ \displaystyle & & \displaystyle +\,4^{p-1}C_{3}L^{p}(N^{p}+1)\int _{0}^{t}\Vert X\Vert _{p,s}^{p}\,ds.\nonumber\end{eqnarray}$$
For a fixed
$t_{0}$
such that
$0\leqslant t_{0}\leqslant T$
we have
$$\begin{eqnarray}\displaystyle \Vert X\Vert _{p,t}^{p} & {\leqslant} & \displaystyle 4^{p-1}C_{3}\int _{0}^{t_{0}}(N^{p}\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}])\,ds\nonumber\\ \displaystyle & & \displaystyle +\,4^{p-1}C_{3}L^{p}(N^{p}+1)\int _{0}^{t}\Vert X\Vert _{p,s}^{p}\,ds\nonumber\end{eqnarray}$$
for
$0\leqslant t\leqslant t_{0}$
. Applying the Gronwall inequality we obtain
$$\begin{eqnarray}\displaystyle \Vert X\Vert _{p,t}^{p}\leqslant 4^{p-1}C_{3}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}\int _{0}^{t_{0}}(N^{p}\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}])\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
Since
$t_{0}$
is fixed arbitrarily we have
$$\begin{eqnarray}\displaystyle \Vert X\Vert _{p,t}^{p}\leqslant 4^{p-1}C_{3}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}\int _{0}^{t}(N^{p}\Vert \unicode[STIX]{x1D6FC}_{1}\Vert _{p,s}^{p}+\mathbb{E}[\Vert \unicode[STIX]{x1D6FC}_{2}(s)\Vert _{HS}^{p}])\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
This completes the proof. ◻
4 Convergence of Newton’s method
We formulate the Newton scheme for problem (2.1). Let
$X^{(0)}\in {\mathcal{X}}_{p,T}$
,
$X^{(0)}(t)=\unicode[STIX]{x1D711}(t)$
for
$t\in [-r,0]$
and
$$\begin{eqnarray}\displaystyle dX^{(k+1)}(t) & = & \displaystyle [AX^{(k+1)}(t)+F(t,X_{t}^{(k)})+F_{v}(t,X_{t}^{(k)})(X_{t}^{(k+1)}-X_{t}^{(k)})]\,dt\nonumber\\ \displaystyle & & \displaystyle +\,[B(t,X_{t}^{(k)})+B_{v}(t,X_{t}^{(k)})(X_{t}^{(k+1)}-X_{t}^{(k)})]\,dW(t),\nonumber\\ \displaystyle X^{(k+1)}(0) & = & \displaystyle \unicode[STIX]{x1D711}(t),\quad t\in [-r,0],\end{eqnarray}$$
where
$F_{v}\in L({\mathcal{C}}_{r},H)$
and
$B_{v}\in L({\mathcal{C}}_{r},HS)$
are Fréchet derivatives of
$F$
and
$B$
with respect to the functional variable
$v\in {\mathcal{C}}_{r}$
. Let
$\unicode[STIX]{x1D6E5}X^{(k)}(t)=X^{(k+1)}(t)-X^{(k)}(t)$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}F^{(k)}(t,X_{t}):=F(t,X_{t}^{(k+1)})-F(t,X_{t}^{(k)}),\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}B^{(k)}(t,X_{t}):=B(t,X_{t}^{(k+1)})-B(t,X_{t}^{(k)}).\nonumber\end{eqnarray}$$
We have the following differential equations for increments
$\unicode[STIX]{x1D6E5}X^{(k+1)}$
:
$$\begin{eqnarray}\displaystyle & & \displaystyle d\unicode[STIX]{x1D6E5}X^{(k+1)}(t)\nonumber\\ \displaystyle & & \displaystyle \quad = [A\unicode[STIX]{x1D6E5}X^{(k+1)}(t)+\unicode[STIX]{x1D6E5}F^{(k)}(t,X_{t})-F_{v}(t,X_{t}^{(k)})\unicode[STIX]{x1D6E5}X_{t}^{(k)}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,F_{v}(t,X_{t}^{(k+1)})\unicode[STIX]{x1D6E5}X_{t}^{(k+1)} ]\,dt\nonumber\\ \displaystyle & & \displaystyle \qquad +\,[\unicode[STIX]{x1D6E5}B^{(k)}(t,X_{t})-B_{v}(t,X_{t}^{(k)})\unicode[STIX]{x1D6E5}X_{t}^{(k)}+B_{v}(t,X_{t}^{(k+1)})\unicode[STIX]{x1D6E5}X_{t}^{(k+1)}]\,dW(t).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
Theorem 4.1. Suppose that there exists a non-negative constant
$L$
such that
$$\begin{eqnarray}\displaystyle \Vert F_{v}(t,v)\Vert _{L({\mathcal{C}}_{r},H)} & {\leqslant} & \displaystyle L,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \Vert B_{v}(t,v)\Vert _{L({\mathcal{C}}_{r},HS)} & {\leqslant} & \displaystyle L\end{eqnarray}$$
for
$t\in [0,T]$
,
$v\in {\mathcal{C}}_{r}$
which implies the Lipschitz condition
$$\begin{eqnarray}\Vert F(t,x)-F(t,y)\Vert \leqslant L\Vert x-y\Vert _{{\mathcal{C}}_{r}},\qquad \Vert B(t,x)-B(t,y)\Vert _{HS}\leqslant L\Vert x-y\Vert _{{\mathcal{C}}_{r}}\end{eqnarray}$$
for
$t\in [0,T]$
,
$x,y\in {\mathcal{C}}_{r}$
. Then the Newton sequence
$(X^{(k)})_{k\in \mathbb{N}}$
defined by (4.1) converges to the unique solution
$X$
of equation (2.1) in the following sense:
$$\begin{eqnarray}\lim _{k\rightarrow \infty }\Vert X^{(k)}-X\Vert _{p,T}=0.\end{eqnarray}$$
Proof. We show that
$(X^{(k)})_{k\in \mathbb{N}}$
satisfies the Cauchy condition with respect to the norm
$\Vert \cdot \Vert _{p,T}.$
We apply Lemma 3.1 with
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FC}_{1}(s)=\unicode[STIX]{x1D6E5}F^{(k)}(s,X_{s})-F_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)},\qquad T_{1}(s)=F_{v}(s,X_{s}^{(k+1)}),\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D6FC}_{2}(s)=\unicode[STIX]{x1D6E5}B^{(k)}(s,X_{s})-B_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)},\qquad T_{2}(s)=B_{v}(s,X_{s}^{(k+1)})\nonumber\end{eqnarray}$$
and obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 4^{p-1}C_{3}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}N^{p}\int _{0}^{t}\Vert \unicode[STIX]{x1D6E5}F^{(k)}(\cdot ,X_{\cdot })-F_{v}(\cdot ,X_{\cdot }^{(k)})\unicode[STIX]{x1D6E5}X_{\cdot }^{(k)}\Vert _{p,s}^{p}\,ds\nonumber\\ \displaystyle & & \displaystyle \qquad +\,4^{p-1}C_{3}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}\int _{0}^{t}\mathbb{E}[\Vert \unicode[STIX]{x1D6E5}B^{(k)}(s,X_{s})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,B_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{HS}^{p} ]ds.\nonumber\end{eqnarray}$$
By (3.3), (2.3), and (2.4) we have
$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D6E5}F^{(k)}(s,X_{s})-F_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert ^{p}\leqslant 2^{p}L^{p}\Vert \unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{{\mathcal{C}}_{r}}^{p} & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D6E5}B^{(k)}(s,X_{s})-B_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{HS}^{p}\leqslant 2^{p}L^{p}\Vert \unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{{\mathcal{C}}_{r}}^{p}. & & \displaystyle \nonumber\end{eqnarray}$$
Hence we have the estimate
$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\leqslant 2^{3p-1}C_{3}L^{p}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)t}(N^{p}+1)\int _{0}^{t}\Vert \unicode[STIX]{x1D6E5}X^{(k)}\Vert _{p,s}^{p}\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
Since
$t\leqslant T$
we have
$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\leqslant C\int _{0}^{t}\Vert \unicode[STIX]{x1D6E5}X^{(k)}\Vert _{p,s}^{p}\,ds, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}C=2^{3p-1}C_{3}L^{p}e^{4^{p-1}C_{3}L^{p}(N^{p}+1)T}(N^{p}+1).\end{eqnarray}$$
The recursive use of (4.5) leads to
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\leqslant \frac{C^{k+1}t^{k+1}}{(k+1)!}\Vert \unicode[STIX]{x1D6E5}X^{(0)}\Vert _{p,t}^{p}.\end{eqnarray}$$
Thus
$$\begin{eqnarray}\displaystyle \Vert X^{(k+p)}-X^{(k)}\Vert _{p,t}^{p}\leqslant \biggl(\frac{C^{k+p-1}t^{k+p-1}}{(k+p-1)!}+\cdots +\frac{C^{k}t^{k}}{k!}\biggr)\Vert \unicode[STIX]{x1D6E5}X^{(0)}\Vert _{p,t}^{p}. & & \displaystyle \nonumber\end{eqnarray}$$
We conclude that
$(X^{(k)})_{k\in \mathbb{N}}$
is a Cauchy sequence in the Banach space
${\mathcal{X}}_{p,T}$
. Therefore it is convergent to
$X$
, which is a solution to equation (2.1). This completes the proof.◻
5 Probabilistic second order convergence of Newton’s method
Theorem 5.1. Suppose that (4.3) and (4.4) hold and assume additionally that there exists a non-negative constant
$M$
such that
$$\begin{eqnarray}\displaystyle \Vert F_{v}(t,v)-F_{v}(t,\bar{v})\Vert _{L({\mathcal{C}}_{r},H)} & {\leqslant} & \displaystyle M\Vert v-\bar{v}\Vert _{{\mathcal{C}}_{r}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \Vert B_{v}(t,v)-B_{v}(t,\bar{v})\Vert _{L({\mathcal{C}}_{r},HS)} & {\leqslant} & \displaystyle M\Vert v-\bar{v}\Vert _{{\mathcal{C}}_{r}}\end{eqnarray}$$
for
$v,\bar{v}\in {\mathcal{C}}_{r}$
. Then for any
$T>0$
,
$p>2$
there exists
$\tilde{C}>0$
such that
$$\begin{eqnarray}P\biggl(\sup _{0\leqslant t\leqslant T}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(t)\Vert \leqslant \unicode[STIX]{x1D70C}\quad \Rightarrow \quad \sup _{0\leqslant t\leqslant T}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(t)\Vert \leqslant R\unicode[STIX]{x1D70C}^{2}\biggr)\geqslant 1-\tilde{C}TR^{-p}\end{eqnarray}$$
for all
$R>0$
,
$0<\unicode[STIX]{x1D70C}\leqslant 1$
,
$k=0,1,2,\ldots \,$
.
Proof. Define the sets
$$\begin{eqnarray}A_{\unicode[STIX]{x1D70C},t}^{(k)}\,=\,\bigg\{\unicode[STIX]{x1D714}:\sup _{0\leqslant s\leqslant t}\!\Vert \unicode[STIX]{x1D6E5}X^{(k)}(s)\Vert \,\leqslant \,\unicode[STIX]{x1D70C}\bigg\}\!\quad \text{for }0<\unicode[STIX]{x1D70C}\leqslant 1,0\leqslant t\leqslant T,k=0,1,2,\ldots .\end{eqnarray}$$
We consider the sequence
$(\unicode[STIX]{x1D6E5}X^{(k)})_{k\in \mathbb{N}}$
restricted to the sets
$A_{\unicode[STIX]{x1D70C},t}^{(k)}$
. For this reason we multiply equation (2.5) by
$\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}$
, the characteristic function of the set
$A_{\unicode[STIX]{x1D70C},t}^{(k)}$
:
$$\begin{eqnarray}\displaystyle \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}(t) & = & \displaystyle \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\int _{0}^{t}S(t-s)\tilde{F}^{(k,k+1)}(s,X_{s})\,ds\nonumber\\ \displaystyle & & \displaystyle +\,\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\int _{0}^{t}S(t-s)\tilde{B}^{(k,k+1)}(s,X_{s})\,dW(s),\nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle \tilde{F}^{(k,k+1)}(t,X_{t}) & := & \displaystyle \unicode[STIX]{x1D6E5}F^{(k)}(t,X_{t})-F_{v}(t,X_{t}^{(k)})\unicode[STIX]{x1D6E5}X_{t}^{(k)}+F_{v}(t,X_{t}^{(k+1)})\unicode[STIX]{x1D6E5}X_{t}^{(k+1)},\nonumber\\ \displaystyle \tilde{B}^{(k,k+1)}(t,X_{t}) & := & \displaystyle \unicode[STIX]{x1D6E5}B^{(k)}(t,X_{t})-B_{v}(t,X_{t}^{(k)})\unicode[STIX]{x1D6E5}X_{t}^{(k)}+B_{v}(t,X_{t}^{(k+1)})\unicode[STIX]{x1D6E5}X_{t}^{(k+1)}.\nonumber\end{eqnarray}$$
Applying (3.3) and by the fact that for
$s\leqslant t$
,
$$\begin{eqnarray}A_{\unicode[STIX]{x1D70C},t}^{(k)}\subset A_{\unicode[STIX]{x1D70C},s}^{(k)}\quad \Rightarrow \quad \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}=\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\end{eqnarray}$$
we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 2^{p-1}\mathbb{E}\biggl[\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\sup _{\tilde{t}\in [0,t]}\biggl\|\int _{0}^{\tilde{t}}S(\tilde{t}-s)\tilde{F}^{(k,k+1)}(s,X_{s})\,ds\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2^{p-1}\mathbb{E}\biggl[\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\sup _{\tilde{t}\in [0,t]}\biggl\|\int _{0}^{\tilde{t}}S(\tilde{t}-s)\tilde{B}^{(k,k+1)}(s,X_{s})\,dW(s)\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 2^{p-1}\mathbb{E}\biggl[\sup _{\tilde{t}\in [0,t]}\biggl\|\int _{0}^{\tilde{t}}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}S(\tilde{t}-s)\tilde{F}^{(k,k+1)}(s,X_{s})\,ds\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2^{p-1}\mathbb{E}\biggl[\sup _{\tilde{t}\in [0,t]}\biggl\|\int _{0}^{\tilde{t}}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}S(\tilde{t}-s)\tilde{B}^{(k,k+1)}(s,X_{s})\,dW(s)\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & & \displaystyle \quad =2^{p-1}(J_{1}+J_{2}).\nonumber\end{eqnarray}$$
By the Hölder inequality for some
$C_{4}>0$
dependent on
$T$
, (2.2) and (3.3) we get
$$\begin{eqnarray}\displaystyle J_{1} & {\leqslant} & \displaystyle N^{p}C_{4}\mathbb{E}\biggl[\sup _{\tilde{t}\in [0,t]}\int _{0}^{\tilde{t}}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \tilde{F}^{(k,k+1)}(s,X_{s})\Vert ^{p}\,ds\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle N^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \tilde{F}^{(k,k+1)}(s,X_{s})\Vert ^{p}\,ds\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{p-1}N^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}F^{(k)}(s,X_{s})-F_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert ^{p}\,ds\biggr]\nonumber\\ \displaystyle & & \displaystyle +\,2^{p-1}N^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert F_{v}(s,X_{s}^{(k+1)})\unicode[STIX]{x1D6E5}X_{s}^{(k+1)}\Vert ^{p}\,ds\biggr].\nonumber\end{eqnarray}$$
From the fundamental theorem of calculus and (5.1) it follows that
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \unicode[STIX]{x1D6E5}F^{(k)}(s,X_{s})-F_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert \nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \sup _{\tilde{s}\in [0,s]}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert \int _{0}^{1}\Vert F_{v}(s,X_{s}^{(k)}+\unicode[STIX]{x1D703}\unicode[STIX]{x1D6E5}X_{s}^{(k)})-F_{v}(s,X_{s}^{(k)})\Vert _{L({\mathcal{C}}_{r},H)}\,d\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{2}M\sup _{\tilde{s}\in [0,s]}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert ^{2}.\nonumber\end{eqnarray}$$
Hence by (4.3),
$$\begin{eqnarray}\displaystyle J_{1} & {\leqslant} & \displaystyle \frac{1}{2}N^{p}M^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(s)\Vert ^{2p}\,ds\biggr]\nonumber\\ \displaystyle & & \displaystyle +\,2^{p-1}N^{p}L^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(s)\Vert ^{p}\,ds\biggr].\nonumber\end{eqnarray}$$
Recall that
$\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert \leqslant \unicode[STIX]{x1D70C}$
on
$A_{\unicode[STIX]{x1D70C},s}^{(k)}$
for
$0\leqslant \tilde{s}\leqslant s$
. Thus we have
$$\begin{eqnarray}\displaystyle J_{1}\leqslant \frac{1}{2}N^{p}M^{p}C_{4}\unicode[STIX]{x1D70C}^{2p}t+2^{p-1}N^{p}L^{p}C_{4}\int _{0}^{t}\Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,s}^{p}\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
We estimate
$J_{2}$
. Since
$$\begin{eqnarray}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\tilde{B}^{(k,k+1)}(s,X_{s})\end{eqnarray}$$
is an
$HS(U_{0},H)$
-valued predictable process, we apply [Reference Da Prato and Zabczyk10, Proposition 7.3] with some constant
$C_{5}>0$
dependent on
$T$
and (3.3):
$$\begin{eqnarray}\displaystyle J_{2} & = & \displaystyle \mathbb{E}\biggl[\sup _{\tilde{t}\in [0,t]}\biggl\|\int _{0}^{\tilde{t}}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}S(\tilde{t}-s)\tilde{B}^{(k,k+1)}(s,X_{s})\,dW_{s}\biggr\|^{p}\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle C_{5}\mathbb{E}\biggl[\int _{0}^{t}\Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\tilde{B}^{(k,k+1)}(s,X_{s})\Vert _{HS}^{p}\,ds\biggr]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{p-1}C_{5}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}B^{(k)}(s,X_{s})-B_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{HS}^{p}\,ds\biggr]\nonumber\\ \displaystyle & & \displaystyle +\,2^{p-1}C_{5}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert B_{v}(s,X_{s}^{(k+1)})\unicode[STIX]{x1D6E5}X_{s}^{(k+1)}\Vert _{HS}^{p}\,ds\biggr].\nonumber\end{eqnarray}$$
From the fundamental theorem of calculus and (5.2) it follows that
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \unicode[STIX]{x1D6E5}B^{(k)}(s,X_{s})-B_{v}(s,X_{s}^{(k)})\unicode[STIX]{x1D6E5}X_{s}^{(k)}\Vert _{HS}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \sup _{\tilde{s}\in [0,s]}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert \int _{0}^{1}\Vert B_{v}(s,X_{s}^{(k)}+\unicode[STIX]{x1D703}\unicode[STIX]{x1D6E5}X_{s}^{(k)})-B_{v}(s,X_{s}^{(k)})\Vert _{L({\mathcal{C}}_{r},HS)}\,d\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{2}M\sup _{\tilde{s}\in [0,s]}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert ^{2}.\nonumber\end{eqnarray}$$
Hence
$$\begin{eqnarray}\displaystyle J_{2} & {\leqslant} & \displaystyle \frac{1}{2}M^{p}C_{4}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(s)\Vert ^{2p}\,ds\biggr]\nonumber\\ \displaystyle & & \displaystyle +\,2^{p-1}L^{p}C_{5}\mathbb{E}\biggl[\int _{0}^{t}\mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(s)\Vert ^{p}\,ds\biggr].\nonumber\end{eqnarray}$$
Recall that
$\Vert \unicode[STIX]{x1D6E5}X^{(k)}(\tilde{s})\Vert \leqslant \unicode[STIX]{x1D70C}\text{on }A_{\unicode[STIX]{x1D70C},s}^{(k)}\text{ for }0\leqslant \tilde{s}\leqslant s$
. Thus we have
$$\begin{eqnarray}\displaystyle J_{2}\leqslant \frac{1}{2}M^{p}C_{5}\unicode[STIX]{x1D70C}^{2p}t+2^{p-1}L^{p}C_{5}\int _{0}^{t}\Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,s}^{p}\,ds. & & \displaystyle \nonumber\end{eqnarray}$$
Finally, for
$C_{6}=\max \{C_{4},C_{5}\}$
we have the estimate
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\leqslant 2^{p-1}(J_{1}+J_{2})\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 2^{p-2}M^{p}C_{6}\unicode[STIX]{x1D70C}^{2p}(N^{p}+1)t+4^{p-1}L^{p}C_{6}(N^{p}+1)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\int _{0}^{t}\Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},s}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,s}^{p}\,ds.\nonumber\end{eqnarray}$$
Applying the Gronwall inequality we obtain
$$\begin{eqnarray}\displaystyle \Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\leqslant 2^{p-2}M^{p}C_{6}\unicode[STIX]{x1D70C}^{2p}(N^{p}+1)t\text{e}^{4^{p-1}L^{p}C_{6}(N^{p}+1)}. & & \displaystyle \nonumber\end{eqnarray}$$
The Chebyshev inequality yields
$$\begin{eqnarray}\displaystyle & & \displaystyle P\biggl(\sup _{0\leqslant s\leqslant t}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(s)\Vert \leqslant \unicode[STIX]{x1D70C}\wedge \sup _{0\leqslant s\leqslant t}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(s)\Vert >R\unicode[STIX]{x1D70C}^{2}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =P\biggl[\mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\sup _{0\leqslant s\leqslant t}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(s)\Vert >R\unicode[STIX]{x1D70C}^{2}\biggr]\leqslant \frac{1}{R^{p}\unicode[STIX]{x1D70C}^{2p}}\Vert \mathbf{1}_{A_{\unicode[STIX]{x1D70C},t}^{(k)}}\unicode[STIX]{x1D6E5}X^{(k+1)}\Vert _{p,t}^{p}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{R^{p}}2^{p-2}M^{p}C_{6}(N^{p}+1)t\text{e}^{4^{p-1}L^{p}C_{6}(N^{p}+1)}=\tilde{C}tR^{-p}.\nonumber\end{eqnarray}$$
Hence for any
$R>0$
we have
$$\begin{eqnarray}P\biggl(\sup _{0\leqslant s\leqslant t}\Vert \unicode[STIX]{x1D6E5}X^{(k)}(s)\Vert \leqslant \unicode[STIX]{x1D70C}\quad \Rightarrow \quad \sup _{0\leqslant s\leqslant t}\Vert \unicode[STIX]{x1D6E5}X^{(k+1)}(s)\Vert \leqslant R\unicode[STIX]{x1D70C}^{2}\biggr)\geqslant 1-\tilde{C}tR^{-p}.\end{eqnarray}$$
This completes the proof. ◻
6 Applications
Example 1
Taking
$A\equiv 0$
,
$H=\mathbb{R}$
and
${\mathcal{C}}_{r}=C([-r,0],\mathbb{R})$
in problem (2.1) we arrive at the stochastic functional differential equation of the form
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}dX(t)=F(t,X_{t})\,dt+B(t,X_{t})\,dW(t),\quad t\in [0,T]\quad \\ X(t)=\unicode[STIX]{x1D711}(t),\quad t\in [-r,0].\quad \end{array}\right.\end{eqnarray}$$
The Newton scheme for problem (6.1):
$$\begin{eqnarray}\displaystyle dX^{(k+1)}(t) & = & \displaystyle [F(t,X_{t}^{(k)})+F_{v}(t,X_{t}^{(k)})(X_{t}^{(k+1)}-X_{t}^{(k)})]\,dt\nonumber\\ \displaystyle & & \displaystyle +\,[B(t,X_{t}^{(k)})+B_{v}(t,X_{t}^{(k)})(X_{t}^{(k+1)}-X_{t}^{(k)})]\,dW(t),\quad t\in [0,T]\nonumber\\ \displaystyle X^{(k+1)}(0) & = & \displaystyle \unicode[STIX]{x1D711}(t),\quad t\in [-r,0]\nonumber\end{eqnarray}$$
is investigated in [Reference Wrzosek33]. The techniques to prove the probabilistic second-order convergence in the non-functional setting are developed in [Reference Amano2] by Amano, whose proof is based on solving an error stochastic differential equation and introducing a certain representation of the solution (see [Reference Amano2, Lemma 2.5]) with components easy to estimate. There is a significant difference between the functional and non-functional case: Amano uses Itô isometry whereas the functional case demands the Doob inequality with some tricks to overcome difficulties arising from the impossibility of constructing explicit solutions to functional SDEs.
We give examples of stochastic partial differential equations to illustrate the main assumptions implying the first-order and probabilistic second-order convergence of Newton’s method.
Example 2
Consider a special case of problem (1.1):
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}u(t,x)-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}u(t,x) & = & \displaystyle \int _{-r_{1}}^{0}\int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)u(t+s,y))\,dy\,ds\nonumber\\ \displaystyle & & \displaystyle +\,\int _{-r_{2}}^{0}\int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)u(t+s,y))\,dy\,ds{\dot{W}}(t),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FD}:[0,1]^{2}\rightarrow \mathbb{R}$
is a continuous function and thus bounded by a positive constant
$K$
. Define
$F,B:{\mathcal{C}}_{r}\rightarrow L^{2}([0,1])$
by
$$\begin{eqnarray}\displaystyle F(\unicode[STIX]{x1D713})(x) & = & \displaystyle \int _{-r_{1}}^{0}\int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)\unicode[STIX]{x1D713}(s,y))\,dy\,ds,\nonumber\\ \displaystyle B(\unicode[STIX]{x1D713})(x) & = & \displaystyle \int _{-r_{2}}^{0}\int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)\unicode[STIX]{x1D713}(s,y))\,dy\,ds.\nonumber\end{eqnarray}$$
The Fréchet derivatives
$F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713}),\hspace{2.22198pt}B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713}):{\mathcal{C}}_{r}\rightarrow L^{2}([0,1])$
,
$$\begin{eqnarray}\displaystyle (F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})h)(x) & = & \displaystyle \int _{-r_{1}}^{0}\int _{0}^{1}\cos (\unicode[STIX]{x1D6FD}(x,y)\unicode[STIX]{x1D713}(s,y))\unicode[STIX]{x1D6FD}(x,y)h(s,y)\,dy\,ds,\nonumber\\ \displaystyle (B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})h)(x) & = & \displaystyle \int _{-r_{2}}^{0}\int _{0}^{1}\cos (\unicode[STIX]{x1D6FD}(x,y)\unicode[STIX]{x1D713}(s,y))\unicode[STIX]{x1D6FD}(x,y)h(s,y)\,dy\,ds\nonumber\end{eqnarray}$$
satisfy the following conditions:
$$\begin{eqnarray}\displaystyle \Vert F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle Kr_{1},\nonumber\\ \displaystyle \Vert B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle Kr_{2}.\nonumber\end{eqnarray}$$
Hence the assumptions of Theorem 4.1 are satisfied and the Newton sequence for (6.2) converges to the solution
$u$
of equation (6.2). Moreover, the Fréchet derivatives
$F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})$
,
$B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})$
satisfy the Lipschitz condition
$$\begin{eqnarray}\displaystyle \Vert F_{\unicode[STIX]{x1D713}}(t,\unicode[STIX]{x1D713}_{1})-F_{\unicode[STIX]{x1D713}}(t,\unicode[STIX]{x1D713}_{2})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle K^{2}r_{1}\Vert \unicode[STIX]{x1D713}_{1}-\unicode[STIX]{x1D713}_{2}\Vert _{{\mathcal{C}}_{r}},\nonumber\\ \displaystyle \Vert B_{\unicode[STIX]{x1D713}}(t,\unicode[STIX]{x1D713}_{1})-B_{\unicode[STIX]{x1D713}}(t,\unicode[STIX]{x1D713}_{2})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle K^{2}r_{2}\Vert \unicode[STIX]{x1D713}_{1}-\unicode[STIX]{x1D713}_{2}\Vert _{{\mathcal{C}}_{r}}.\nonumber\end{eqnarray}$$
Thus the assumptions of Theorem 5.1 are satisfied. Therefore we obtain the probabilistic second-order convergence of Newton’s method for (6.2).
Similar results can be obtained in the modification of (6.2):
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}u(t,x)-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}u(t,x) & = & \displaystyle \int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)u(\unicode[STIX]{x1D6FE}(t),y))\,dy\nonumber\\ \displaystyle & & \displaystyle +\,\int _{0}^{1}\sin (\unicode[STIX]{x1D6FD}(x,y)u(\unicode[STIX]{x1D6FE}(t),y))\,dy{\dot{W}}(t),\nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FE}:[0,T]\rightarrow [-r,T]$
,
$\unicode[STIX]{x1D6FE}(t)\leqslant t$
.
Example 3
Consider the stochastic semilinear delay heat equation (cf. [Reference Jahanipur22, Example 1, pp. 1250–1252])
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}u(t,x)-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}u(t,x) & = & \displaystyle \int _{-r_{1}}^{0}f(u(t+s,x))\,ds\nonumber\\ \displaystyle & & \displaystyle +\,\int _{-r_{2}}^{0}b(u(t+s,x))\,ds{\dot{W}}(t)\end{eqnarray}$$
with the initial and boundary conditions as in problem (1.1). Let
$H=$
$L^{2}([0,1])$
,
$A=\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}x^{2}$
and
$f,g:\mathbb{R}\rightarrow \mathbb{R},\hspace{2.22198pt}f(u)=g(u)=\sin u$
. Define
$F,B:{\mathcal{C}}_{r}\rightarrow L^{2}([0,1])$
by
$$\begin{eqnarray}F(\unicode[STIX]{x1D713})(x)=\int _{-r_{1}}^{0}\sin (\unicode[STIX]{x1D713}(s,x))\,ds,\qquad B(\unicode[STIX]{x1D713})(x)=\int _{-r_{2}}^{0}\sin (\unicode[STIX]{x1D713}(s,x))\,ds.\end{eqnarray}$$
The Fréchet derivatives
$F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713}),B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713}):{\mathcal{C}}_{r}\rightarrow L^{2}([0,1])$
satisfy the following conditions:
$$\begin{eqnarray}\displaystyle \Vert F_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle r_{1},\nonumber\\ \displaystyle \Vert B_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D713})\Vert _{L({\mathcal{C}}_{r},L^{2}([0,1]))} & {\leqslant} & \displaystyle r_{2}.\nonumber\end{eqnarray}$$
Therefore by Theorem 4.1 the first-order convergence of Newton’s method for (6.3) is obtained. However, since the Fréchet derivatives of
$F$
and
$B$
are not Lipschitz, the assumptions of Theorem 5.1 are not satisfied. Furthermore, second-order convergence problems appear even if we reduce the equation to the following form:
$$\begin{eqnarray}du(t)=(Au(t)+F(t,u_{t}))\,dt+dW(t),\end{eqnarray}$$
where
$F(t,\unicode[STIX]{x1D713})(x)=\sin \unicode[STIX]{x1D713}(\unicode[STIX]{x1D6FE}(t)-t,x)$
,
$\unicode[STIX]{x1D6FE}:[0,T]\rightarrow [-r,T]$
,
$\unicode[STIX]{x1D6FE}(t)\leqslant t$
.
It is surprising that for the corresponding deterministic problem (
$B\equiv 0$
) one obtains the second-order convergence of Newton’s method.
