Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 130, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NEWTON’S METHOD FOR STOCHASTIC FUNCTIONAL
DIFFERENTIAL EQUATIONS
MONIKA WRZOSEK
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.
1. 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
F (xk )
.
xk+1 = xk − 0
F (xk )
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 [7]. In [5] Chaplygin solves ordinary differential equations
x0 = F (t, x),
x(t0 ) = x0
(1.1)
by constructing convergent sequences of successive approximations
x0k+1 = F (t, xk ) + Fx (t, xk )(xk+1 − xk ),
xk+1 (t0 ) = x0 .
Numerous generalizations of the method to the case of functional differential equations are also known in the literature, e.g. [9], [15]. In particular ordinary differential equations are generalized to integro-differential, delayed, Volterra-type differential equations and problems with the Hale operator. After [4] we mention the
model with an operator T defined on a set of absolutely continuous functions:
x0 = T x,
x(t0 ) = x0 .
The Newton scheme for such equations is of the form
x0k+1 = T xk + (Tx xk )(xk+1 − xk ),
xk+1 (t0 ) = x0 ,
2000 Mathematics Subject Classification. 60H10, 60H35, 65C30.
Key words and phrases. Newton’s method; stochastic functional differential equations.
c 2012 Texas State University - San Marcos.
Submitted May 31, 2012. Published Augsut 15, 2012.
Supported by grant BW-UG 538-5100-0964-12 from the University of Gdańsk.
1
(1.2)
2
M. WRZOSEK
EJDE-2012/130
where Tx denotes the Fréchet derivative of an operator T . In [8] Kawabata and
Yamada prove the convergence of Newton’s method for stochastic differential equations. In [2] Amano formulates an equivalent problem and provides a direct way to
estimate the approximation error. In [3] 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 [3]) 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 [10, 11, 13, 14]. 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 [1]. 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 [3] 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.
2. Formulation of the problem
Let (Ω, F, P ) be a complete probability space, (Bt )t∈[0,T ] the standard Brownian
motion and (Ft )t∈[0,T ] its natural filtration. We extend this filtration as follows:
Ft := F0 for t ∈ [−τ, 0). By L2 (Ω) we denote the space of all random variables
Y : Ω → R such that
kY k2 = E[Y 2 ] < ∞.
Let X[c,d] be the space of all continuous and Ft -adapted processes X : [c, d] → L2 (Ω)
with the norm
kXk2[c,d] = E[ sup |Xt |2 ] < ∞.
c≤t≤d
Fix τ ≥ 0 and T > 0. For a process X ∈ X[−τ,T ] and any t ∈ [0, T ] we define the
L2 (Ω)-valued Hale-type operator
Xt+ · : [−τ, 0] → L2 (Ω)
by
(Xt+ · )s = Xt+s
for s ∈ [−τ, 0].
We consider the initial value problem for stochastic functional differential equation:
dXt = b(t, Xt+ · )dt + σ(t, Xt+· )dBt for t ∈ [0, T ],
Xt = ϕt for t ∈ [−τ, 0],
(2.1)
where b, σ : [0, T ] × C([−τ, 0], R) → R are deterministic Fréchet differentiable functions, ϕ ∈ X[−τ,0] . Since Ft := F0 for t ∈ [−τ, 0), the process ϕ is deterministic
thus independent of the Brownian motion on [0, T ]. Problem (2.1) is equivalent to
EJDE-2012/130
NEWTON’S METHOD
3
the stochastic functional integral equation:
Z t
Z t
X t = ϕ0 +
b(s, Xs+ · )ds +
σ(s, Xs+ · )dBs for t ∈ [0, T ],
0
0
(2.2)
Xt = ϕt for t ∈ [−τ, 0].
We formulate the Newton scheme for problem (2.1). Let
X (0) ∈ X[−τ,T ] ,
(0)
Xt
= ϕt , t ∈ [−τ, 0]
and
(k+1)
dXt
(k+1)
Xt
(k)
(k+1)
(k) (k)
= b(t, Xt+ · ) + bw (t, Xt+ · )(Xt+ · − Xt+ · ) dt
(k)
(k)
(k+1)
(k) + σ(t, Xt+ · ) + σw (t, Xt+ · )(Xt+ · − Xt+ · ) dBt
= ϕt
for t ∈ [0, T ], (2.3)
for t ∈ [−τ, 0],
where bw (t, w), σw (t, w) : C([−τ, 0], R) → R are Fréchet derivatives of b, σ with
respect to the functional variable w ∈ C([−τ, 0], R). We recall the functional norms
of bw (t, w) and σw (t, w),
kbw (t, w)kF =
|bw (t, w)w̄|,
sup
sups∈[−τ,0] |w̄(s)|≤1
kσw (t, w)kF =
|σw (t, w)w̄|,
sup
sups∈[−τ,0] |w̄(s)|≤1
where we take the supremum over all w̄ ∈ C([−τ, 0], R) whose uniform norms do
not exceed 1. We assume that there exists a nonnegative constant M such that
kbw (t, w)kF ≤ M,
kσw (t, w)kF ≤ M,
(2.4)
which implies the Lipschitz condition for b(t, w) and σ(t, w):
|b(t, w) − b(t, w̄)| ≤ M
sup |w(s) − w̄(s)|,
(2.5)
−τ ≤s≤0
|σ(t, w) − σ(t, w̄)| ≤ M
sup |w(s) − w̄(s)|
(2.6)
−τ ≤s≤0
for w, w̄ ∈ C([−τ, 0], 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([−τ, 0], R))∗
we denote the space of all linear and bounded functionals on C([−τ, 0], R).
Lemma 2.1. Suppose that α(1) , α(2) : [0, T ] → X[0,T ] are continuous, A(1) , A(2) :
∗
[0, T ] → (C([−τ, 0], R)) and there exists a nonnegative constant M such that
(i)
kAt (t, w)kF ≤ M,
i = 1, 2, for t ∈ [0, T ], w ∈ C([−τ, 0], R).
Then for a process Xt satisfying the linear stochastic functional differential equation
(1)
(1)
(2)
(2)
dXt = αt + At Xt+ · dt + αt + At Xt+ · dBt for t ∈ [0, T ],
(2.7)
Xt = 0 for t ∈ [−τ, 0]
we have
kXk2[0,t]
4M 2 (t+4)t
Z
≤ 4e
0
t
tkα(1) k2[0,s] + 4kα(2) k2[0,s] ds
for t ∈ [0, T ].
4
M. WRZOSEK
EJDE-2012/130
Proof. Using the Schwarz inequality, Itô isometry, Doob martingale inequality and
the fact that (x + y)2 ≤ 2(x2 + y 2 ), we have
E sup Xs2
0≤s≤t
Z s
h
Z s
2 i
(2)
(2)
= E sup
[αr(1) + A(1)
X
]dr
+
[α
+
A
X
]dB
r+
·
r+
·
r
r
r
r
0≤s≤t
0
0
h
Z s
2 i
≤ 2E sup
[αr(1) + A(1)
X
]dr
r+ ·
r
0≤s≤t
0
h
Z s
2 i
+ 2E sup
[αr(2) + A(2)
r Xr+ · ]dBr
0≤s≤t
0
Z t
2
2
2
≤ 2tE
[αs(1) + A(1)
X
]
ds
+
2
·
2
E
[αs(2) + A(2)
s+
·
s
s Xs+ · ] ds
0
0
Z t
Z t
2 (2)
2
(2) 2
≤ 4t
E[αs(1) ]2 + E[A(1)
X
]
ds
+
16
E[α
]
+
E
A
X
ds.
s+
·
s+
·
s
s
s
Z
t
0
0
Notice that
h
2 i
(i)
E[αt ]2 ≤ E sup αs(i)
= kα(i) k2[0,t] ,
i = 1, 2
0≤s≤t
(i)
(i)
E[At Xt+ · ]2 ≤ E kAt k2F sup |Xs |2 ≤ M 2 E[ sup |Xs |2 ] ≤ M 2 kXk2[0,t] .
0≤s≤t
0≤s≤t
Hence
E[ sup Xs2 ]
0≤s≤t
Z
t
[kα(1) k2[0,s]
2
kXk2[0,s] ]ds
Z
t
≤ 4t
+M
+ 16
[kα(2) k2[0,s] + M 2 kXk2[0,s] ]ds
0
0
Z t
Z t
(1) 2
(2) 2
2
=4
[tkα k[0,s] + 4kα k[0,s] ]ds + 4M (t + 4)
kXk2[0,s] ds.
0
0
Thus
kXk2[0,t]
Z
≤4
t
[tkα(1) k2[0,s]
+
4kα(2) k2[0,s] ]ds
2
Z
+ 4M (t + 4)
0
t
kXk2[0,s] ds.
0
For a fixed t0 such that 0 ≤ t0 ≤ T , we have
Z t0 h
Z t
i
kXk2[0,t] ≤ 4
t0 kα(1) k2[0,s] + 4kα(2) k2[0,s] ds + 4M 2 (t0 + 4)
kXk2[0,s] ds
0
0
for 0 ≤ t ≤ t0 . We apply the Gronwall inequality and obtain
Z t0 h
i
2
4M 2 (t0 +4)t
kXk[0,t] ≤ 4e
t0 kα(1) k2[0,s] + 4kα(2) k2[0,s] ds,
0 ≤ t ≤ t0 .
0
Since t0 is fixed arbitrarily, we obtain
Z th
i
2
tkα(1) k2[0,s] + 4kα(2) k2[0,s] ds,
kXk2[0,t] ≤ 4e4M (t+4)t
t ∈ [0, T ].
0
EJDE-2012/130
NEWTON’S METHOD
5
3. A Convergence Theorem
For the Newton sequence (X (k) )k∈N , introduced in (2.3), we denote
(k)
∆Xt
(k)
Observe that ∆Xt
(k+1) d ∆Xt
(k+1)
= Xt
(k)
− Xt .
satisfies the following stochastic functional differential equation
(k+1)
(k)
(k)
(k)
(k+1)
(k+1)
= {b(t, Xt+ · ) − b(t, Xt+ · ) − bw (t, Xt+ · )∆Xt+ · + bw (t, Xt+ · )∆Xt+ · }dt
(k+1)
(k)
(k)
(k)
(k+1)
(k+1)
+ {σ(t, Xt+ · ) − σ(t, Xt+ · ) − σw (t, Xt+ · )∆Xt+ · + σw (t, Xt+ · )∆Xt+ · }dBt
(3.1)
for t ∈ [0, T ].
Theorem 3.1. Suppose (2.4) holds. Then the Newton sequence X (k) = (X (k) )k∈N
defined by (2.3) converges to the unique solution X of equation (2.1) in the following
sense:
lim kX (k) − Xk[−τ,T ] = 0.
k→∞
Proof. We show that (X (k) )k∈N satisfies the Cauchy condition with respect to the
norm k · k[−τ,T ] . From Lemma 2.1 and the Doob inequality we have
k∆X (k+1) k2[0,t]
≤ 4e4M
2
(t+4)t
t
Z
h
i
(k+1)
(k)
(k)
(k)
tE sup |b(r, Xr+ · ) − b(r, Xr+ · ) − bw (r, Xr+ · )∆Xr+ · |2 ds
0≤r≤s
0
+ 4e4M
2
(t+4)t
t
Z
h
i
(k+1)
(k)
(k)
(k)
4E sup |σ(r, Xr+ · ) − σ(r, Xr+ · ) − σw (r, Xr+ · )∆Xr+ · |2 ds.
0≤r≤s
0
By the Lipschitz condition for b(t, w) and σ(t, w):
(k+1)
(k)
|b(t, Xt+ · ) − b(t, Xt+ · )| ≤ M sup |∆Xs(k) |,
(3.2)
0≤s≤t
(k+1)
(k)
|σ(t, Xt+ · ) − σ(t, Xt+ · )| ≤ M sup |∆Xs(k) |.
(3.3)
0≤s≤t
We obtain the estimate
k∆X (k+1) k2[0,t]
4M 2 (t+4)t
2
Z
t
≤ 16M (t + 4)e
k∆X (k) k2[0,s] ds.
0
Since t ≤ T , we have
k∆X (k+1) k2[0,t]
Z
≤C
t
k∆X (k) k2[0,s] ds,
0
where
C = 16M 2 (T + 4)e4M
The recursive use of (3.4) leads to
k∆X (k+1) k2[0,t] ≤
Thus
kX (k+p) − X (k) k2[0,t] ≤
2
(T +4)T
.
C k+1
k∆X (0) k2[0,t] .
(k + 1)!
C k+p−1
Ck + ... +
k∆X (0) k2[0,t] .
(k + p − 1)!
k!
(3.4)
6
M. WRZOSEK
EJDE-2012/130
We conclude that (X (k) )k∈N is a Cauchy sequence in the Banach space X[−τ,T ] .
Therefore it is convergent to (Xt )t∈[−τ,T ] , which is a solution to equation (2.1).
This completes the proof.
4. Probabilistic second-order convergence
Before formulating the main result, we need the following lemma.
Lemma 4.1. Let (gt )t∈[0,T ] ∈ X[0,T ] and {At }t∈[0,T ] be a family of F-measurable
subsets of Ω satisfying
At ∈ F t ,
At ⊂ As for 0 ≤ s ≤ t ≤ T.
Then we have
h
Z t
2 i Z t
E 1A t
gs dBs
≤
E[1As gs2 ]ds
0
(4.1)
0
for any 0 ≤ t ≤ T .
Proof. Since for s ≤ t we have the implication
At ⊂ As ⇒ 1At = 1At 1As
and 1As gs is a stochastically integrable function with respect to s, it follows from
the definition of stochastic integral that
2
2 Z t
2
Z t
Z t
1As gs dBs
1As gs dBs ≤
gs dBs = 1At
1A t
0
0
0
almost surely in Ω for any 0 ≤ t ≤ T . By Itô isometry
Z t
h
2 i Z t
2 i
h Z t
1As gs dBs
=
E[1As gs2 ]ds.
gs dBs
≤E
E 1A t
0
0
0
This completes the proof.
Remark 4.2. The proof of Lemma 4.1 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) [12]), the product rule for Malliavin derivative (see Theorem 3.4 [6]),
Proposition 1.2.6 [12] and Proposition 1.3.8 [12]. It also resulted in an interesting
extension of the Itô isometry
h
Z t
2 i Z t
E 1A t
gs dBs
=
E[1At gs2 ]ds.
0
0
As in the previous sections we assume that (X (k) )k∈N is the Newton sequence,
where b, σ : [0, T ] × C([−τ, 0], R) → R are deterministic Fréchet differentiable functions, the initial function ϕ ∈ X[−τ,0] is a deterministic process, the Fréchet deriva∗
tives bw (t, w), σw (t, w) are in (C([−τ, 0], R)) and condition (2.4) is satisfied.
Theorem 4.3. Suppose that the above assumptions are satisfied and there exists a
nonnegative constant L such that
kbw (t, w) − bw (t, w̄)kF ≤ L sup |w(s) − w̄(s)|,
(4.2)
−τ ≤s≤0
kσw (t, w) − σw (t, w̄)kF ≤ L sup |w(s) − w̄(s)|
−τ ≤s≤0
(4.3)
EJDE-2012/130
NEWTON’S METHOD
7
for all w, w̄ ∈ C([−τ, 0], R). Then for any T > 0, there exists a nonnegative constant
C such that
(k)
(k+1)
P sup |∆Xt | ≤ ρ ⇒
sup |∆Xt
| ≤ Rρ2 ≥ 1 − CT R−2
0≤t≤T
0≤t≤T
for all R > 0, 0 < ρ ≤ 1, k = 0, 1, 2, . . . .
Proof. Define the sets
(k)
Aρ,t = {ω : sup |∆Xs(k) | ≤ ρ}
for 0 < ρ ≤ 1, 0 ≤ t ≤ T, k = 0, 1, 2, . . . .
0≤s≤t
(k)
We consider the sequence (∆X (k) )k∈N restricted to the sets Aρ,t . For this reason
(k)
we multiply equation (3.1) by 1A(k) , the characteristic function of the set Aρ,t , and
ρ,t
obtain
Z t
h
i
(k+1)
(k+1)
(k)
(k)
(k)
1A(k) ∆Xt
=
1A(k) b(s, Xs+ · ) − b(s, Xs+ · ) − bw (s, Xs+ · )∆Xs+ · ds
ρ,t
ρ,t
0
Z t
(k+1)
(k+1)
+
1A(k) bw (s, Xs+ · )∆Xs+ · ds
ρ,t
0
Z th
i
(k+1)
(k)
(k)
(k)
+ 1A(k)
σ(s, Xs+ · ) − σ(s, Xs+ · ) − σw (s, Xs+ · )∆Xs+ · dBs
ρ,t
0
Z t
(k+1)
(k+1)
+ 1A(k)
σw (s, Xs+ · )∆Xs+ · dBs .
ρ,t
0
(k)
Applying the Doob inequality, Lemma 4.1 and the fact that t 7→ Aρ,t is nondecreasing we obtain
k1A(k) ∆X (k+1) k2[0,t]
ρ,t
Z t h
i
(k+1)
(k)
(k)
(k)
≤ 16t
E 1A(k) |b(s, Xs+ · ) − b(s, Xs+ · ) − bw (s, Xs+ · )∆Xs+ · |2 ds
ρ,t
0
Z t h
i
(k+1)
(k+1)
+ 16t
E 1A(k) |bw (s, Xs+ · )∆Xs+ · |2 ds
ρ,t
0
Z t
h
i
(k+1)
(k)
(k)
(k)
+ 16E 1A(k) |
σ(s, Xs+ · ) − σ(s, Xs+ · ) − σw (s, Xs+ · )∆Xs+ · dBs |2
ρ,t
0
Z t
h
i
(k+1)
(k+1)
+ 16E 1A(k) |
σw (s, Xs+ · )∆Xs+ · dBs |2
ρ,t
0
Z t h
i
(k+1)
(k)
(k)
(k)
≤ 16t
E 1A(k) |b(s, Xs+ · ) − b(s, Xs+ · ) − bw (s, Xs+ · )∆Xs+ · |2 ds
ρ,s
0
Z t h
i
(k+1)
(k+1)
E 1A(k) |bw (s, Xs+ · )∆Xs+ · |2 ds
+ 16t
ρ,s
0
Z t h
i
(k+1)
(k)
(k)
(k)
+ 16
E 1A(k) |σ(s, Xs+ · ) − σ(s, Xs+ · ) − σw (s, Xs+ · )∆Xs+ · |2 ds
ρ,s
0
Z t h
i
(k+1)
(k+1)
E 1A(k) |σw (s, Xs+ · )∆Xs+ · |2 ds.
+ 16
0
ρ,s
From (4.2) and (4.3) we have
(k+1)
(k)
kbw (t, Xt+ · ) − bw (t, Xt+ · )kF ≤ L sup |∆Xs(k) |,
0≤s≤t
(4.4)
8
M. WRZOSEK
(k+1)
EJDE-2012/130
(k)
kσw (t, Xt+ · ) − σw (t, Xt+ · )kF ≤ L sup |∆Xs(k) |.
0≤s≤t
From the fundamental theorem of calculus it follows that
(k+1)
(k)
(k)
(k)
|b(r, Xr+ · ) − b(r, Xr+ · ) − bw (r, Xr+ · )∆Xr+ · |
Z 1
(k)
(k)
(k)
≤
kbw (r, Xr+ · + θ∆Xr+ · ) − bw (r, Xr+ · )kF dθ sup |∆Xs(k) |
0≤s≤r
0
1
≤ L sup |∆Xs(k) |2 .
2 0≤s≤r
Similarly, we have
(k+1)
(k)
(k)
(k)
|σ(r, Xr+ · ) − σ(r, Xr+ · ) − σw (r, Xr+ · )∆Xr+ · | ≤
1
L sup |∆Xs(k) |2 .
2 0≤s≤r
Consequently, we have the estimate
Z t h
i
1
k1A(k) ∆X (k+1) k2[0,t] ≤ 16t
E 1A(k) L2 sup |∆Xr(k) |4 ds
ρ,s 4
ρ,t
0≤r≤s
0
Z t h
i
(k+1)
(k+1)
+ 16t
E 1A(k) |bw (s, Xs+ · )∆Xs+ · |2 ds
ρ,s
0
Z t h
i
1
+ 16
E 1A(k) L2 sup |∆Xr(k) |4 ds
ρ,s 4
0≤r≤s
0
Z t h
i
(k+1)
(k+1)
+ 16
E 1A(k) |σw (s, Xs+ · )∆Xs+ · |2 ds.
ρ,s
0
Recall that
(k)
|∆Xr |
≤ ρ on
(k)
Aρ,s
for 0 ≤ r ≤ s. From (2.4) we have
k1A(k) ∆X (k+1) k2[0,t]
ρ,t
≤ 4(t + 1)L2 tρ4 + 16(t + 1)M 2
t
Z
ρ,s
0
≤ 4(T + 1)tL2 ρ4 + 16(T + 1)M 2
h
i
E 1A(k) sup |∆Xr(k+1) |2 ds
0≤r≤s
t
Z
0
k1A(k) ∆X (k+1) k2[0,s] ds.
ρ,s
We apply the Gronwall inequality and obtain
2
k1A(k) ∆X (k+1) k2[0,t] ≤ 4(T + 1)tL2 ρ4 e16T (T +1)M .
ρ,t
The Chebyshev inequality yields
P sup |∆Xs(k) | ≤ ρ
∧
sup |∆Xs(k+1) | > Rρ2
0≤s≤t
0≤s≤t
h
= P 1A(k) sup |∆Xs(k+1) | > Rρ2
ρ,t
i
0≤s≤t
1
≤ 2 4 k1A(k) ∆X (k+1) k2[0,t]
ρ,t
R ρ
2
1
≤ 2 4 4(T + 1)tL2 ρ4 e16T (T +1)M = CtR−2 .
R ρ
Hence for any R > 0 we have
P sup |∆Xs(k) | ≤ ρ ⇒
sup |∆Xs(k+1) | ≤ Rρ2 ≥ 1 − CtR−2 .
0≤s≤t
0≤s≤t
(4.5)
EJDE-2012/130
NEWTON’S METHOD
9
Thus the assertion is true.
Remark 4.4. Our study is devoted to the case of deterministic functionals b and
σ. Our result can be extended to the more complicated case of b and σ dependent
on non-anticipative stochastic processes.
(k)
Remark 4.5. If P sup0≤t≤T |∆Xt | ≤ ρ > 0 then the probability of the implication
(k)
sup |∆Xt | ≤ ρ
(k+1)
⇒
sup |∆Xt
| ≤ Rρ2
0≤t≤T
0≤t≤T
can be expressed in terms of conditional probability:
(k)
(k+1)
| ≤ Rρ2
sup |∆Xt | ≤ ρ ≥ 1 − CT R−2 ,
P sup |∆Xt
0≤t≤T
0≤t≤T
which is more intuitive.
An example. Let b(t, w) = 0 and σ(t, w) = arctan w( 2t ). The stochastic functional
differential equation is of the form:
dXt = arctan Xt/2 dBt for t ∈ [0, 1],
(4.6)
X0 = 1.
The corresponding Newton scheme is
(k+1)
dXt
(k) = arctan Xt/2 +
1
1+
(k) 2
Xt/2
(k+1)
X0
(k) ∆Xt
dBt
for t ∈ [0, 1],
= 1.
The Lipschitz condition (2.6) for σ(t, w) is satisfied with M = 1; therefore by
Theorem 3.1 the Newton sequence is convergent to the solution of (4.6). We have
the estimate
(64e20 )k
(64e20 )k π
k∆X (0) k2[0,t] =
k B· − 1k2[0,t]
k!
k!
4
(64e20 )k π 2 2
(64e20 )k (π 2 + 16)
≤8
E[ Bt + 1] =
.
k!
16
2k!
k∆X (k) k2[0,t] ≤
Since σw (t, w) satisfies the Lipschitz condition (4.3) with L = 1, by Theorem 4.3
the second-order convergence is obtained.
h
i
(k)
(k+1)
P sup |∆Xt | ≤ ρ ⇒
sup |∆Xt
| ≤ Rρ2 ≥ 1 − 8e32 R−2 .
0≤t≤1
0≤t≤1
for all R > 0, 0 < ρ ≤ 1, k = 0, 1, 2, . . . .
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 4.1 is also proposed by
the referee. The original proof was one page long.
10
M. WRZOSEK
EJDE-2012/130
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Monika Wrzosek
Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk,
Poland
E-mail address: Monika.Wrzosek@mat.ug.edu.pl