\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 130, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/130\hfil Newton's method] {Newton's method for stochastic functional differential equations} \author[M. Wrzosek \hfil EJDE-2012/130\hfilneg] {Monika Wrzosek} \address{Monika Wrzosek \newline Institute of Mathematics, University of Gda\'nsk, Wita Stwosza 57, 80-952 Gda\'nsk, Poland} \email{Monika.Wrzosek@mat.ug.edu.pl} \thanks{Submitted May 31, 2012. Published Augsut 15, 2012.} \thanks{Supported by grant BW-UG 538-5100-0964-12 from the University of Gda\'nsk} \subjclass[2000]{60H10, 60H35, 65C30} \keywords{Newton's method; stochastic functional differential equations} \begin{abstract} In this article, we apply Newton's method to stochastic functional differential equations. The first part concerns a first-order convergence. We formulate a Gronwall-type inequality which plays an important role in the proof of the convergence theorem for the Newton method. In the second part a probabilistic second-order convergence is studied. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Newton's method, known as the tangent method, was established to solve nonlinear algebraic equations of the form $F(x) = 0$ by means of the following recurrence formula $$x_{k+1} = x_k - \frac{F(x_k)}{F'(x_k)}.$$ The convergence of the sequence to the exact solution, uniqueness and local existence of the solution are stated in the Kantorovich theorem on successive approximations \cite{kant}. In \cite{chap} Chaplygin solves ordinary differential equations $$\label{ode} x' = F(t,x), \quad x(t_0) = x_0$$ by constructing convergent sequences of successive approximations $$x'_{k+1} = F(t, x_k) + F_x(t, x_k)(x_{k+1}-x_k), \quad x_{k+1}(t_0)=x_0.$$ Numerous generalizations of the method to the case of functional differential equations are also known in the literature, e.g. \cite{lak}, \cite{wang}. In particular ordinary differential equations are generalized to integro-differential, delayed, Volterra-type differential equations and problems with the Hale operator. After \cite{azb} we mention the model with an operator $\mathcal{T}$ defined on a set of absolutely continuous functions: $$x' = \mathcal{T}x, \quad x(t_0) = x_0.$$ The Newton scheme for such equations is of the form $$x'_{k+1} = \mathcal{T}x_k + (\mathcal{T}_x x_k)(x_{k+1}-x_k), \quad x_{k+1}(t_0)=x_0,$$ where $\mathcal{T}_x$ denotes the Fr\'echet derivative of an operator $\mathcal{T}$. In \cite{kaw} Kawabata and Yamada prove the convergence of Newton's method for stochastic differential equations. In \cite{amano2} Amano formulates an equivalent problem and provides a direct way to estimate the approximation error. In \cite{amano} he proposes a probabilistic second-order error estimate which has been an open problem since Kawabata and Yamada's results. The proof is based on solving the error stochastic differential equation and introducing a certain representation of the solution (see Lemma 2.5 \cite{amano}) of which components are easy to estimate. Our goal is to obtain corresponding results in the case of stochastic functional differential equations with Hale functionals. The existence and uniqueness of solutions to stochastic functional differential equations has been discussed in a large number of papers, see \cite{mao,moh,pon,turo}. The assumptions on the given functions imply the existence and uniqueness of solutions and convergence of the Newton sequence. The first part of this article concerns a first-order convergence. We formulate a Gronwall-type inequality which is a generalization of Amano's \cite{amano1}. It plays an important role in the proof of the convergence theorem for the Newton method. In the second part a probabilistic second-order convergence is studied. However, Amano's techniques \cite{amano} are not applicable to the functional case because it is not possible to construct explicit solution of linear stochastic functional differential equation using his methods. Therefore we introduce an entirely different approach and obtain a comprehensive second-order convergence theorem by simpler arguments. For this purpose we define certain sets, utilize the Gronwall-type lemma and the Chebyshev inequality. \section{Formulation of the problem} Let $(\Omega, \mathcal{F}, P)$ be a complete probability space, $(B_t)_{t\in [0,T]}$ the standard Brownian motion and $(\mathcal{F}_t)_{t\in [0,T]}$ its natural filtration. We extend this filtration as follows: $\mathcal{F}_t := \mathcal{F}_0$ for $t \in [-\tau, 0)$. By $L^2(\Omega)$ we denote the space of all random variables $Y : \Omega \to \mathbb{R}$ such that $$\|Y\|^2 = \mathbb{E}[Y^2] < \infty.$$ Let ${\mathcal{X}}_{[c,d]}$ be the space of all continuous and $\mathcal{F}_t$-adapted processes $X : [c,d] \to L^2(\Omega)$ with the norm $$\|X\|^2_{[c,d]} = \mathbb{E}[\sup_{c\leq t\leq d} |X_t|^2 ]< \infty.$$ Fix $\tau \geq 0$ and $T>0$. For a process $X \in {\mathcal{X}}_{[-\tau,T]}$ and any $t \in [0,T]$ we define the $L^2(\Omega)$-valued Hale-type operator $$X_{t+ \, \cdot}: [-\tau, 0] \to L^2(\Omega) \quad \text{by} \quad (X_{t+ \, \cdot})_s=X_{t+s} \quad \text{for} \quad s\in[-\tau,0].$$ We consider the initial value problem for stochastic functional differential equation: $$\label{prob1} \begin{gathered} dX_t = b(t, X_{t+ \, \cdot})dt + \sigma(t, X_{t+ \cdot})dB_t \text{ for } t \in [0,T], \\ X_t = \varphi_t \text{ for } t \in [-\tau,0], \end{gathered}$$ where $b, \sigma : [0,T] \times C([-\tau,0], \mathbb{R}) \to \mathbb{R}$ are deterministic Fr\'echet differentiable functions, $\varphi \in {\mathcal{X}}_{[-\tau,0]}$. Since $\mathcal{F}_t := \mathcal{F}_0$ for $t \in [-\tau, 0)$, the process $\varphi$ is deterministic thus independent of the Brownian motion on $[0,T]$. Problem \eqref{prob1} is equivalent to the stochastic functional integral equation: $$\label{prob2} \begin{gathered} X_t = \varphi_0 + \int_0^t b(s, X_{s+ \, \cdot})ds + \int_0^t \sigma(s, X_{s+ \, \cdot})dB_s \text{ for } t \in [0,T], \\ X_t = \varphi_t \text{ for } t \in [-\tau,0]. \end{gathered}$$ We formulate the Newton scheme for problem \eqref{prob1}. Let $$X^{(0)} \in {\mathcal{X}}_{[-\tau,T]}, \quad X^{(0)}_t=\varphi_t, t \in [-\tau,0]$$ and \label{prob3} \begin{aligned} dX^{(k+1)}_t & = \big[ b(t, X_{t+ \, \cdot}^{(k)}) + b_w(t, X_{t+ \, \cdot}^{(k)}) (X_{t+ \, \cdot}^{(k+1)}-X_{t+ \, \cdot}^{(k)}) \big]dt \\ &\quad + \big[ \sigma(t, X_{t+ \, \cdot}^{(k)}) + \sigma_w(t, X_{t+ \, \cdot}^{(k)})(X_{t+ \, \cdot}^{(k+1)}-X_{t+ \, \cdot}^{(k)}) \big]dB_t \quad \text{for } t \in [0,T], \\ X^{(k+1)}_t & = \varphi_t \quad \text{for } t \in [-\tau,0], \end{aligned} where $b_w(t,w), \sigma_w(t,w): C([-\tau,0], \mathbb{R}) \to \mathbb{R}$ are Fr\'echet derivatives of $b, \sigma$ with respect to the functional variable $w \in C([-\tau,0], \mathbb{R})$. We recall the functional norms of $b_w(t,w)$ and $\sigma_w(t,w)$, \begin{gather*} \|b_w(t,w)\|_F = \sup_{\sup_{s\in [-\tau,0]}|\bar{w}(s)|\leq 1} |b_w(t,w)\bar{w}|,\\ \|\sigma_w(t,w)\|_F = \sup_{\sup_{s\in [-\tau,0]}|\bar{w}(s)|\leq 1} |\sigma_w(t,w)\bar{w}|, \end{gather*} where we take the supremum over all $\bar{w} \in C([-\tau,0], \mathbb{R})$ whose uniform norms do not exceed $1$. We assume that there exists a nonnegative constant $M$ such that $$\label{zal} \|b_w(t,w)\|_F \leq M, \quad \|\sigma_w(t,w)\|_F \leq M,$$ which implies the Lipschitz condition for $b(t,w)$ and $\sigma(t,w)$: \begin{gather}\label{lip0} |b(t,w) - b(t,\bar{w})| \leq M \sup_{-\tau\leq s\leq 0}|w(s)-\bar{w}(s)|,\\ \label{lip} |\sigma(t,w) - \sigma(t,\bar{w})| \leq M\sup_{-\tau\leq s\leq 0}|w(s)-\bar{w}(s)| \end{gather} for $w, \bar{w} \in C([-\tau,0], \mathbb{R})$. Before formulating the main result we introduce the Gronwall-type inequality which will be necessary in the proof of the convergence theorem. By $(C([-\tau,0],\mathbb{R}))^*$ we denote the space of all linear and bounded functionals on $C([-\tau,0],\mathbb{R})$. \begin{lemma} \label{lem1} Suppose that $\alpha^{(1)}, \alpha^{(2)}:[0,T]\to {\mathcal{X}}_{[0,T]}$ are continuous, $A^{(1)}, A^{(2)}: [0,T] \to \left(C([-\tau,0],\mathbb{R})\right)^*$ and there exists a nonnegative constant $M$ such that $$\|A_t^{(i)}(t,w)\|_F \leq M, \quad i=1,2, \text{ for } t \in [0,T], w \in C([-\tau,0], \mathbb{R}).$$ Then for a process $X_t$ satisfying the linear stochastic functional differential equation $$\begin{gathered} dX_t = \left( \alpha^{(1)}_t + A_t^{(1)} X_{t+ \, \cdot} \right)dt + \left( \alpha^{(2)}_t + A_t^{(2)} X_{t+ \, \cdot} \right)dB_t \quad \text{for } t\in [0,T], \\ X_t = 0 \quad \text{for } t \in [-\tau,0] \end{gathered}$$ we have $$\|X\|^2_{[0,t]} \leq 4e^{4M^2(t + 4)t}\int_0^t \big[t\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]} \big]ds \quad \text{for } t\in [0,T].$$ \end{lemma} \begin{proof} Using the Schwarz inequality, It\^{o} isometry, Doob martingale inequality and the fact that $(x+y)^2 \leq 2(x^2 + y^2)$, we have \begin{align*} & \mathbb{E}\big[\sup_{0 \leq s \leq t} X^2_s\big] \\ & = \mathbb{E} \Big[\sup_{0 \leq s \leq t}\Big(\int_0^s [\alpha^{(1)}_r + A_r^{(1)}X_{r+ \, \cdot}]dr + \int_0^s[\alpha^{(2)}_r + A_r^{(2)}X_{r+ \, \cdot}]dB_r\Big)^2 \Big] \\ & \leq 2\mathbb{E} \Big[\sup_{0 \leq s \leq t}\Big(\int_0^s [\alpha^{(1)}_r + A_r^{(1)}X_{r+ \, \cdot}]dr \Big)^2\Big] \\ &\quad + 2\mathbb{E}\Big[\sup_{0 \leq s \leq t} \Big(\int_0^s [\alpha^{(2)}_r+ A_r^{(2)}X_{r+ \, \cdot}]dB_r \Big)^2\Big] \\ & \leq 2t \mathbb{E} \int_0^t[\alpha^{(1)}_s + A_s^{(1)}X_{s+ \, \cdot}]^2 ds + 2\cdot 2^2\mathbb{E} \int_0^t[\alpha^{(2)}_s+ A_s^{(2)} X_{s+ \, \cdot}]^2ds \\ & \leq 4t \int_0^t\Big(\mathbb{E}[\alpha^{(1)}_s]^2 + \mathbb{E}[A_s^{(1)} X_{s+ \, \cdot}]^2\Big) ds + 16 \int_0^t\Big(\mathbb{E}[\alpha^{(2)}_s]^2 + \mathbb{E}\big[A_s^{(2)} X_{s+ \, \cdot}\big]^2\Big)ds. \end{align*} Notice that \begin{gather*} \mathbb{E}[\alpha^{(i)}_t]^2 \leq \mathbb{E} \Big[\sup_{0 \leq s \leq t}\Big(\alpha^{(i)}_s\Big)^2\Big] = \|\alpha^{(i)}\|^2_{[0,t]}, \quad i=1,2 \\ \mathbb{E}[A_t^{(i)}X_{t+ \, \cdot}]^2 \leq \mathbb{E} \big[\|A_t^{(i)}\|^2_F \sup_{0\leq s\leq t}|X_s|^2 \big] \leq M^2 \mathbb{E} [\sup_{0\leq s\leq t}|X_s|^2] \leq M^2\|X\|^2_{[0,t]}. \end{gather*} Hence \begin{align*} &\mathbb{E}[\sup_{0 \leq s \leq t} X^2_s] \\ & \leq 4t \int_0^t [\|\alpha^{(1)}\|^2_{[0,s]} + M^2\|X\|^2_{[0,s]}] ds + 16 \int_0^t [\|\alpha^{(2)}\|^2_{[0,s]} + M^2\|X\|^2_{[0,s]} ]ds \\ & = 4\int_0^t [t\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]}]ds + 4M^2(t + 4)\int_0^t \|X\|^2_{[0,s]} ds. \end{align*} Thus $\|X\|^2_{[0,t]} \leq 4\int_0^t [t\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]} ]ds + 4M^2(t + 4)\int_0^t \|X\|^2_{[0,s]}ds.$ For a fixed $t_0$ such that $0 \leq t_0 \leq T$, we have $\|X\|^2_{[0,t]} \le 4\int_0^{t_0} \left[t_0\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]} \right]ds + 4M^2(t_0 + 4)\int_0^t \|X\|^2_{[0,s]}ds$ for $0 \leq t \leq t_0$. We apply the Gronwall inequality and obtain $\|X\|^2_{[0,t]} \leq 4e^{4M^2(t_0 + 4)t}\int_0^{t_0} \left[t_0\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]} \right]ds, \quad 0 \leq t \leq t_0.$ Since $t_0$ is fixed arbitrarily, we obtain $\|X\|^2_{[0,t]} \le 4e^{4M^2(t + 4)t}\int_0^t \left[t\|\alpha^{(1)}\|^2_{[0,s]} + 4\|\alpha^{(2)}\|^2_{[0,s]} \right]ds, \quad t\in [0,T].$ \end{proof} \section{A Convergence Theorem} For the Newton sequence $(X^{(k)})_{k \in \mathbb{N}}$, introduced in \eqref{prob3}, we denote $$\Delta X^{(k)}_t = X^{(k+1)}_t - X^{(k)}_t.$$ Observe that $\Delta X^{(k)}_t$ satisfies the following stochastic functional differential equation \label{glowny} \begin{aligned} & d\big(\Delta X^{(k+1)}_t\big) \\ & = \{b(t,X_{t+ \, \cdot}^{(k+1)})-b(t,X_{t+ \, \cdot}^{(k)})-b_w (t,X_{t+ \, \cdot}^{(k)})\Delta X^{(k)}_{t+ \, \cdot}+b_w(t, X_{t+ \, \cdot}^{(k+1)}) \Delta X^{(k+1)}_{t+ \, \cdot}\}dt \\ &\quad + \{\sigma(t,X_{t+ \, \cdot}^{(k+1)})-\sigma(t,X_{t+ \, \cdot}^{(k)}) -\sigma_w(t,X_{t+ \, \cdot}^{(k)})\Delta X^{(k)}_{t+ \, \cdot} +\sigma_w(t,X_{t+ \, \cdot}^{(k+1)})\Delta X^{(k+1)}_{t+ \, \cdot}\}dB_t \end{aligned} for $t \in [0,T]$. \begin{theorem} \label{first} Suppose \eqref{zal} holds. Then the Newton sequence $X^{(k)} = (X^{(k)})_{k \in \mathbb{N}}$ defined by \eqref{prob3} converges to the unique solution $X$ of equation \eqref{prob1} in the following sense: $$\lim_{k\to\infty} \| X^{(k)} - X \|_{[-\tau,T]} = 0.$$ \end{theorem} \begin{proof} We show that $(X^{(k)})_{k \in \mathbb{N}}$ satisfies the Cauchy condition with respect to the norm $\|\cdot\|_{[-\tau,T]}.$ From Lemma \ref{lem1} and the Doob inequality we have \begin{align*} &\|\Delta X^{(k+1)} \|^2_{[0,t]} \\ & \leq 4e^{4M^2(t+4)t}\int_0^t t \mathbb{E} \Big[\sup_{0 \leq r \leq s}| b(r,X_{r+ \, \cdot}^{(k+1)}) -b(r,X_{r+ \, \cdot}^{(k)}) - b_w(r,X_{r+ \, \cdot}^{(k)})\Delta X^{(k)}_ {r+ \, \cdot} |^2\Big] ds \\ &\quad + 4e^{4M^2(t+4)t}\int_0^t 4\mathbb{E} \Big[\sup_{0 \leq r \leq s}| \sigma(r,X_{r+ \, \cdot}^{(k+1)}) -\sigma(r,X_{r+ \, \cdot}^{(k)}) - \sigma_w(r,X_{r+ \, \cdot}^{(k)})\Delta X^{(k)}_{r+ \, \cdot} |^2 \Big]ds. \end{align*} By the Lipschitz condition for $b(t,w)$ and $\sigma(t,w)$: \begin{gather} |b(t,X_{t+ \, \cdot}^{(k+1)})-b(t,X_{t+ \, \cdot}^{(k)})| \leq M \sup_{0\leq s\leq t}|\Delta X^{(k)}_s|, \\ |\sigma(t,X_{t+ \, \cdot}^{(k+1)})-\sigma(t,X_{t+ \, \cdot}^{(k)})| \leq M \sup_{0\leq s\leq t}|\Delta X^{(k)}_s|. \end{gather} We obtain the estimate $\|\Delta X^{(k+1)} \|^2_{[0,t]} \le 16M^2(t+4)e^{4M^2(t + 4)t} \int_0^t \|\Delta X^{(k)} \|^2_{[0,s]}ds.$ Since $t \leq T$, we have $$\label{koncowe} \|\Delta X^{(k+1)}\|^2_{[0,t]} \le C \int_0^t \|\Delta X^{(k)}\|^2_{[0,s]}ds,$$ where $$C = 16M^2(T+4)e^{4M^2(T + 4)T}.$$ The recursive use of \eqref{koncowe} leads to $$\|\Delta X^{(k+1)}\|^2_{[0,t]} \leq \frac{C^{k+1}}{(k+1)!} \|\Delta X^{(0)}\|^2_{[0,t]}.$$ Thus $\|X^{(k+p)} - X^{(k)} \|^2_{[0,t]} \le \Big( \frac{C^{k+p-1}}{(k+p-1)!} + \ldots + \frac{C^{k}}{k!} \Big) \|\Delta X^{(0)}\|^2_{[0,t]}.$ We conclude that $(X^{(k)})_{k \in \mathbb{N}}$ is a Cauchy sequence in the Banach space ${\mathcal{X}}_{[-\tau,T]}$. Therefore it is convergent to $(X_t)_{t \in [-\tau,T]}$, which is a solution to equation \eqref{prob1}. This completes the proof. \end{proof} \section{Probabilistic second-order convergence} Before formulating the main result, we need the following lemma. \begin{lemma} \label{malliav} Let $(g_t)_{t\in[0,T]} \in {\mathcal{X}}_{[0,T]}$ and $\{A_t\}_{t\in[0,T]}$ be a family of $\mathcal{F}$-measurable subsets of $\Omega$ satisfying $$A_t \in \mathcal{F}_t, \quad A_t \subset A_s \ \text{ for } 0 \leq s \leq t \leq T.$$ Then we have $$\label{malliav2} \mathbb{E} \Big[\mathbf{1}_{A_{t}}\Big(\int_0^t g_sdB_s\Big)^2\Big] \leq \int_0^t \mathbb{E} [\mathbf{1}_{A_{s}} g_s^2 ]ds$$ for any $0 \leq t \leq T$. \end{lemma} \begin{proof} Since for $s \leq t$ we have the implication $$A_t \subset A_s \; \Rightarrow \; \mathbf{1}_{A_t} = \mathbf{1}_{A_t}\mathbf{1}_{A_s}$$ and $\mathbf{1}_{A_s}g_s$ is a stochastically integrable function with respect to $s$, it follows from the definition of stochastic integral that $$\mathbf{1}_{A_{t}}\Big(\int_0^t g_sdB_s\Big)^2 = \mathbf{1}_{A_{t}}\Big(\int_0^t \mathbf{1}_{A_{s}}g_sdB_s\Big)^2 \leq \Big(\int_0^t \mathbf{1}_{A_{s}}g_sdB_s\Big)^2$$ almost surely in $\Omega$ for any $0\leq t\leq T$. By It\^{o} isometry $$\mathbb{E} \Big[\mathbf{1}_{A_{t}}\Big(\int_0^t g_sdB_s\Big)^2\Big] \leq \mathbb{E} \Big[\Big(\int_0^t \mathbf{1}_{A_{s}} g_sdB_s\Big)^2\Big] = \int_0^t \mathbb{E} [\mathbf{1}_{A_{s}} g_s^2 ]ds.$$ This completes the proof. \end{proof} \begin{remark} \label{rm4.2} \rm The proof of Lemma \ref{malliav} is proposed by the Referee. Our former proof of this lemma was based on the duality property of the Malliavin calculus (see Def.1.3.1(ii) \cite{nual}), the product rule for Malliavin derivative (see Theorem 3.4 \cite{oks}), Proposition 1.2.6 \cite{nual} and Proposition 1.3.8 \cite{nual}. It also resulted in an interesting extension of the It\^{o} isometry $$\mathbb{E} \Big[\mathbf{1}_{A_{t}}\Big(\int_0^t g_sdB_s\Big)^2\Big] = \int_0^t \mathbb{E} [\mathbf{1}_{A_{t}} g_s^2 ]ds.$$ \end{remark} As in the previous sections we assume that $(X^{(k)})_{k \in \mathbb{N}}$ is the Newton sequence, where $b, \sigma : [0,T] \times C([-\tau,0], \mathbb{R}) \to \mathbb{R}$ are deterministic Fr\'echet differentiable functions, the initial function $\varphi \in {\mathcal{X}}_{[-\tau,0]}$ is a deterministic process, the Fr\'echet derivatives $b_w(t,w), \sigma_w(t,w)$ are in $\left(C([-\tau,0],\mathbb{R})\right)^*$ and condition \eqref{zal} is satisfied. \begin{theorem} \label{second} Suppose that the above assumptions are satisfied and there exists a nonnegative constant $L$ such that \begin{gather} \label{lip00} \|b_w(t,w) - b_w(t,\bar{w})\|_F \leq L \sup_{-\tau\leq s\leq 0}|w(s)-\bar{w}(s)|, \\ \label{lip2} \|\sigma_w(t,w) - \sigma_w(t,\bar{w})\|_F \leq L\sup_{-\tau\leq s\leq 0}|w(s) -\bar{w}(s)| \end{gather} for all $w, \bar{w} \in C([-\tau,0], \mathbb{R})$. Then for any $T>0$, there exists a nonnegative constant $C$ such that $$P \Big(\sup_{0\le t\le T}|\Delta X_t^{(k)}|\le\rho \quad \Rightarrow \quad \sup_{0\le t\le T}|\Delta X_t^{(k+1)}|\leq R\rho^2 \Big) \geq 1 - CTR^{-2}$$ for all $R > 0$, $0 < \rho \leq 1$, $k = 0,1,2,\dots$. \end{theorem} \begin{proof} Define the sets $$A_{\rho, t}^{(k)} = \{ \omega: \sup_{0\leq s \leq t} |\Delta X^{(k)}_s|\leq \rho \} \quad \text{for } 0 < \rho \le 1, \; 0 \leq t \leq T, \; k = 0,1,2,\dots.$$ We consider the sequence $(\Delta X^{(k)})_{k \in \mathbb{N}}$ restricted to the sets $A_{\rho, t}^{(k)}$. For this reason we multiply equation \eqref{glowny} by $\mathbf{1}_{A_{\rho, t}^{(k)}}$, the characteristic function of the set $A_{\rho, t}^{(k)}$, and obtain \begin{align*} \mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}_t &=\int_0^t \mathbf{1}_{A_{\rho, t}^{(k)}}\left[b(s,X_{s+ \,\cdot}^{(k+1)}) - b(s,X_{s+ \, \cdot}^{(k)})-b_w(s,X_{s+ \,\cdot}^{(k)})\Delta X^{(k)}_{s+ \, \cdot}\right]ds \\ &\quad +\int_0^t \mathbf{1}_{A_{\rho, t}^{(k)}}b_w(s, X_{s+ \,\cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \,\cdot}ds \\ &\quad +\mathbf{1}_{A_{\rho, t}^{(k)}}\int_0^t \left[\sigma(s,X_{s+ \,\cdot}^{(k+1)}) - \sigma(s,X_{s+ \,\cdot}^{(k)})-\sigma_w(s,X_{s+ \,\cdot}^{(k)}) \Delta X^{(k)}_{s+ \,\cdot}\right]dB_s \\ &\quad + \mathbf{1}_{A_{\rho, t}^{(k)}}\int_0^t\sigma_w(s,X_{s+ \, \cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \, \cdot}dB_s. \end{align*} Applying the Doob inequality, Lemma \ref{malliav} and the fact that $t \mapsto A_{\rho, t}^{(k)}$ is non-decreasing we obtain \begin{align*} &\|\mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}\|^2_{[0,t]} \\ & \leq 16t\int_0^t \mathbb{E}\Big[\mathbf{1}_{A_{\rho, t}^{(k)}}|b(s,X_{s+ \,\cdot}^{(k+1)}) -b(s,X_{s+ \, \cdot}^{(k)})-b_w(s,X_{s+ \,\cdot}^{(k)}) \Delta X^{(k)}_{s+ \, \cdot}|^2\Big]ds \\ &\quad + 16t \int_0^t\mathbb{E}\Big[ \mathbf{1}_{A_{\rho, t}^{(k)}} |b_w(s, X_{s+ \,\cdot}^{(k+1)})\Delta X^{(k+1)}_{s+ \,\cdot}|^2\Big]ds \\ &\quad + 16\mathbb{E}\Big[\mathbf{1}_{A_{\rho, t}^{(k)}}|\int_0^t \sigma(s,X_{s+ \,\cdot}^{(k+1)}) -\sigma(s,X_{s+ \,\cdot}^{(k)})-\sigma_w(s,X_{s+ \,\cdot}^{(k)}) \Delta X^{(k)}_{s+ \,\cdot}dB_s|^2\Big] \\ &\quad + 16\mathbb{E}\Big[\mathbf{1}_{A_{\rho, t}^{(k)}}|\int_0^t\sigma_w(s,X_{s+ \, \cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \, \cdot}dB_s|^2\Big] \\ &\leq 16t\int_0^t \mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}|b(s,X_{s+ \,\cdot}^{(k+1)}) -b(s,X_{s+ \, \cdot}^{(k)})-b_w(s,X_{s+ \,\cdot}^{(k)}) \Delta X^{(k)}_{s+ \, \cdot}|^2\Big]ds \\ &\quad + 16t \int_0^t\mathbb{E}\Big[ \mathbf{1}_{A_{\rho, s}^{(k)}}|b_w(s, X_{s+ \,\cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \,\cdot}|^2\Big]ds \\ &\quad + 16\int_0^t\mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}| \sigma(s,X_{s+ \,\cdot}^{(k+1)}) -\sigma(s,X_{s+ \,\cdot}^{(k)})-\sigma_w(s,X_{s+ \,\cdot}^{(k)}) \Delta X^{(k)}_{s+ \,\cdot}|^2\Big]ds \\ &\quad + 16\int_0^t\mathbb{E} \Big[\mathbf{1}_{A_{\rho, s}^{(k)}}|\sigma_w(s,X_{s+ \, \cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \, \cdot}|^2\Big]ds. \end{align*} From \eqref{lip00} and \eqref{lip2} we have \begin{gather} \|b_w(t,X_{t+ \, \cdot}^{(k+1)})-b_w(t,X_{t+ \, \cdot}^{(k)})\|_F \leq L \sup_{0\leq s\leq t}|\Delta X^{(k)}_s|, \\ \|\sigma_w(t,X_{t+ \, \cdot}^{(k+1)})-\sigma_w(t,X_{t+ \, \cdot}^{(k)})\|_F \leq L \sup_{0\leq s\leq t}|\Delta X^{(k)}_s|. \end{gather} From the fundamental theorem of calculus it follows that \begin{align*} &| b(r,X_{r+ \,\cdot}^{(k+1)}) - b(r,X_{r+ \, \cdot}^{(k)}) - b_w(r,X_{r+ \,\cdot}^{(k)})\Delta X^{(k)}_{r+ \, \cdot}| \\ &\leq \int_0^1 \|b_w(r,X_{r+ \,\cdot}^{(k)} + \theta \Delta X^{(k)}_{r+ \, \cdot}) - b_w(r,X_{r+ \,\cdot}^{(k)})\|_F d\theta \sup_{0\leq s\leq r}|\Delta X^{(k)}_s| \\ &\leq \frac{1}{2} L \sup_{0\leq s\leq r}|\Delta X^{(k)}_s|^2. \end{align*} Similarly, we have $$| \sigma(r,X_{r+ \,\cdot}^{(k+1)}) - \sigma(r,X_{r+ \, \cdot}^{(k)}) -\sigma_w(r,X_{r+ \,\cdot}^{(k)})\Delta X^{(k)}_{r+ \, \cdot}| \leq \frac{1}{2}L \sup_{0\leq s\leq r}|\Delta X^{(k)}_s|^2.$$ Consequently, we have the estimate \begin{align*} \|\mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}\|^2_{[0,t]} &\leq 16t\int_0^t \mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}\frac{1}{4} L^2 \sup_{0\leq r\leq s}| \Delta X^{(k)}_r|^4\Big]ds \\ &\quad+ 16t \int_0^t\mathbb{E}\Big[ \mathbf{1}_{A_{\rho, s}^{(k)}}|b_w(s, X_{s+ \,\cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \,\cdot}|^2\Big]ds \\ &\quad+ 16\int_0^t\mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}\frac{1}{4} L^2 \sup_{0\leq r\leq s}| \Delta X^{(k)}_r|^4\Big]ds \\ &\quad+ 16\int_0^t\mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}|\sigma_w(s,X_{s+ \, \cdot}^{(k+1)}) \Delta X^{(k+1)}_{s+ \, \cdot}|^2\Big]ds. \end{align*} Recall that $|\Delta X_r^{(k)}|\leq \rho$ on $A_{\rho,s}^{(k)}$ for $0\leq r\leq s$. From \eqref{zal} we have \begin{align*} &\|\mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}\|^2_{[0,t]} \\ &\leq 4(t+1)L^2t\rho^4 + 16(t+1)M^2\int_0^t \mathbb{E}\Big[\mathbf{1}_{A_{\rho, s}^{(k)}}\sup_{0\leq r\leq s}|\Delta X^{(k+1)}_r|^2\Big]ds \\ &\leq 4(T+1)tL^2\rho^4 + 16(T+1)M^2\int_0^t \|\mathbf{1}_{A_{\rho, s}^{(k)}} \Delta X^{(k+1)}\|^2_{[0,s]}ds. \end{align*} We apply the Gronwall inequality and obtain $\|\mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}\|^2_{[0,t]} \leq 4(T+1)tL^2\rho^4e^{16T(T+1)M^2}.$ The Chebyshev inequality yields \begin{align*} &P \Big(\sup_{0\le s\le t}|\Delta X_s^{(k)}|\le\rho \quad \wedge \quad \sup_{0\le s\le t}|\Delta X_s^{(k+1)}| > R\rho^2 \Big) \\ & = P\Big[\mathbf{1}_{A_{\rho, t}^{(k)}} \sup_{0\leq s\leq t}|\Delta X_s^{(k+1)}| > R\rho^2 \Big] \\ & \leq \frac{1}{R^2\rho^4} \|\mathbf{1}_{A_{\rho, t}^{(k)}}\Delta X^{(k+1)}\|^2_{[0,t]} \\ & \leq \frac{1}{R^2\rho^4} 4(T+1)tL^2\rho^4e^{16T(T+1)M^2} = CtR^{-2}. \end{align*} Hence for any $R>0$ we have \begin{align*} P \Big(\sup_{0\le s\le t}|\Delta X_s^{(k)}|\le\rho \quad \Rightarrow \quad \sup_{0\le s\le t}|\Delta X_s^{(k+1)}|\leq R\rho^2 \Big) \geq 1 - CtR^{-2}. \end{align*} Thus the assertion is true. \end{proof} \begin{remark} \label{rmk4.4} \rm Our study is devoted to the case of deterministic functionals $b$ and $\sigma$. Our result can be extended to the more complicated case of $b$ and $\sigma$ dependent on non-anticipative stochastic processes. \end{remark} \begin{remark} \label{rmk4.5} \rm If $P\big( \sup_{0\le t\le T}|\Delta X_t^{(k)}|\le\rho \big) > 0$ then the probability of the implication $$\sup_{0\le t\le T}|\Delta X_t^{(k)}|\le\rho \quad \Rightarrow \quad \sup_{0\le t\le T}|\Delta X_t^{(k+1)}|\leq R\rho^2$$ can be expressed in terms of conditional probability: $$P \Big( \sup_{0\le t\le T}|\Delta X_t^{(k+1)}|\leq R\rho^2 \; \Bigm| \; \sup_{0\le t\le T}|\Delta X_t^{(k)}|\le\rho \Big) \geq 1 - CTR^{-2},$$ which is more intuitive. \end{remark} \subsection*{An example} Let $b(t,w) = 0$ and $\sigma(t,w) = \arctan w(\frac{t}{2})$. The stochastic functional differential equation is of the form: $$\label{prz} \begin{gathered} dX_t = \arctan \big( X_{t/2} \big)dB_t \quad \text{for } t \in [0,1], \\ X_0 = 1. \end{gathered}$$ The corresponding Newton scheme is \begin{gather*} dX_t^{(k+1)} = \big[ \arctan \big( X_{t/2}^{(k)} \big) + \frac{1}{1 + \big(X_{t/2}^{(k)}\big)^2}\Delta X_{t}^{(k)} \big]dB_t \quad \text{for } t \in [0,1], \\ X_0^{(k+1)} = 1. \end{gather*} The Lipschitz condition \eqref{lip} for $\sigma(t,w)$ is satisfied with $M=1$; therefore by Theorem \ref{first} the Newton sequence is convergent to the solution of \eqref{prz}. We have the estimate \begin{align*} \|\Delta X^{(k)}\|^2_{[0,t]} &\leq \frac{(64e^{20})^k}{k!} \|\Delta X^{(0)}\|^2_{[0,t]} = \frac{(64e^{20})^k}{k!} \|\frac{\pi}{4}B_{\cdot} - 1\|^2_{[0,t]} \\ &\leq 8\frac{(64e^{20})^k}{k!}\mathbb{E} [ \frac{\pi^2}{16}B_t^2 +1 ] = \frac{(64e^{20})^k(\pi^2 + 16)}{2k!}. \end{align*} Since $\sigma_w(t,w)$ satisfies the Lipschitz condition \eqref{lip2} with $L=1$, by Theorem \ref{second} the second-order convergence is obtained. \begin{align*} P\Big[\sup_{0\le t\le 1}|\Delta X_t^{(k)}|\leq \rho \quad \Rightarrow \quad \sup_{0\leq t\leq 1}|\Delta X_t^{(k+1)}| \leq R\rho^2 \Big] \geq 1 - 8e^{32}R^{-2}. \end{align*} for all $R > 0$, $0 < \rho \le 1$, $k = 0,1,2,\dots$. \subsection*{Acknowledgements} The author is greatly indebted to the anonymous referee for the valuable comments and suggestions, which improved, especially, the main results of Section 4. The elegant and short proof of Lemma \ref{malliav} is also proposed by the referee. 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