Appl Math Optim 45:23–44 (2002) DOI: 10.1007/s00245-001-0024-8

© 2002 Springer-Verlag New York Inc.

An Approximation for the Zakai Equation Y. Hu,1 G. Kallianpur,2 and J. Xiong3 1 Department

of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142, USA [email protected]

2 Department

of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA [email protected]

3 Department

of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA [email protected]

Abstract. In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. Key Words. Filtering, Zakai equation, Stochastic differential equation, Weak convergence. AMS Classification.

1.

Primary 60G35, 93E11, Secondary 60F25, 60H10.

Introduction

On the stochastic basis (Ä, F, Ft , P), let X be the d-dimensional signal process governed by the following stochastic differential equation (SDE): Z X t = x0 + 0

t

Z b(s, X s ) ds + 0

t

σ (s, X s ) d Bs ,

(1.1)

24

Y. Hu, G. Kallianpur, and J. Xiong

where b: [0, T ] × Rd → Rd and σ : [0, T ] × Rd → Rd×d are two measurable maps, B is a d-dimensional Wiener process. Let Y be the m-dimensional observation process given by Z t h(s, X s ) ds + Wt , (1.2) Yt = 0

where h: [0, T ] × Rd → Rm is a measurable map such that Z T |h(s, X s )|2 ds < ∞ E

(1.3)

0

and W is an m-dimensional Wiener process independent of B. Let Cb (Rd ) be the collection of all bounded continuous functions on Rd . Let ½ ¾ ∂ f (x) ∂ 2 f (x) 2 d d d , ∈ Cb (R ), 1 ≤ i, j ≤ d . Cb (R ) = f ∈ Cb (R ): ∂ xi ∂ xi ∂ x j For any f ∈ Cb2 (Rd ), let d d ∂ 2 f (x) X ∂ f (x) 1 X ai j (t, x) + bi (t, x) , 2 i, j=1 ∂ xi ∂ x j ∂ xi i=1

L(t) f (x) = where ai j (t, x) =

d X

σik (t, x)σk j (t, x).

k=1

For f ∈ Cb (Rd ), let πt f = E[ f (X t ) | FtY ],

(1.4)

where FtY = σ {Ys : 0 ≤ s ≤ t} is the information available up to time t. Let M+ (Rd ) be the collection of all finite positive Borel measures on Rd . Let { f j } be a dense subset of { f ∈ Cb (Rd ): k f k ≤ 1} and X 2− j (|µf j − ν f j | ∧ 1). (1.5) d(µ, ν) = j

Then (M+ (Rd ), d) is a Polish space. It is well known that πt f =

σt f , σt 1

∀ f ∈ Cb (Rd ),

t ≥ 0,

where σt , the unnormalized conditional measure, satisfies the Zakai equation: Z t σs (L(s) f ) ds σt f = f (x0 ) + 0

Z

t

+ 0

hσs (h(s, ·) f ), dYs i,

∀ f ∈ Cb2 (Rd ).

(1.6)

An Approximation for the Zakai Equation

25

The uniqueness of the solution to (1.6) has been discussed by various authors. We make the following assumption on the coefficients a and b: (B1) ai j (t, x), bi (t, x), 1 ≤ i, j ≤ d, are bounded continuous functions on [0, T ] × Rd . Further, ∀t ∈ [0, T ], ai j (t, ·), bi (t, ·) are Lipschitz continuous functions on Rd . Under Assumption (B1) and (1.3), it is easy to verify the conditions of Theorem 4.1 of [1]. Therefore, (1.6) has a unique solution in the class of the M+ (Rd )-valued continuous Ft -adapted process with Z T σt (Rd ) dt < ∞. E 0

Assumption (B1) can be relaxed by the theorem we mentioned above. However, we need this condition also for an approximation to the solution of the Zakai equation in the next section. Let P1 ∼ P be the probability measure such that ¶ µ Z T Z 1 T d P1 hh(s, X s ), d Ws i − |h(s, X s )|2 ds . (ω) = exp − dP 2 0 0 It follows from Girsanov’s formula (see [7]) that, under P1 , Y is an m-dimensional Wiener process. For simplicity, we consider the solution of the SDE (1.6) based on the stochastic basis (Ä, F, Ft , P1 ). As the driving process Y has no smooth path, we approximate it by processes with piecewise smooth sample paths. This method of solving the SDEs was initiated by Wong and Zakai in [12]. Since then, various authors have studied this problem under different setups. We refer the reader to [7] for more references. In Section 2 we introduce an approximation to the (measure-valued) solution to (1.6) and estimate the convergence rate for the distance between measures given by (1.5). Under certain hypoellipticity conditions (see Assumption (L) in Section 3) on the diffusion (1.1), the measure πt or σt is absolutely continuous with respect to the Lebesgue measure, i.e., Z u t (x) f (x) d x, σt f = Rd

and u t is the solution of the following Zakai equation: du t = L(t)∗ u t dt + h(t, ·)u t dwt , where

(1.7)

! Ã d d d X ∂ai j (t, x) 1 X ∂ 2 u(x) X ∂u(x) ai j (t, x) + − bi (t, x) L(t) u(x) = 2 i, j=1 ∂ xi ∂ x j ∂ xj ∂ xi i=1 j=1 Ã ! d d X ∂ 2 ai j (t, x) ∂bi (t, x) 1 X u(x). + − 2 j,k=1 ∂ x j ∂ xk ∂ xk i=1 ∗

26

Y. Hu, G. Kallianpur, and J. Xiong

In Section 3 we study an approximation to the solution of the density-valued equation (1.7) in L p space and estimate the convergence rate. The approximation of the Wong–Zakai type studied in this paper was also investigated by other authors, e.g., by Chaleyat-Maurel and Michel [4]. Now we compare our results with theirs. First, they fixed f and considered σt f as a random field with time t and initial state x0 as parameters. In this paper we fix the initial x0 and consider σt as a measure-valued process. Secondly, Chaleyat-Maurel and Michel derived their results by making use of the Zakai equation and the method of Moulinier [10], [11]. In this paper we apply the Kallianpur–Striebel formula to the filtering and apply the Feynman–Kac formula to the approximating PDEs. We work with the solutions given by the above mentioned two formulae directly. The main contribution of this article is the rate of convergence. It seems that there is not much work about the rate of convergence of the Wong–Zakai type of approximation. However, in [4] it was shown that the rate of convergence is 1/20. Namely, the difference between the true solution and the approximate one is less than or equal to C|5|1/20 for some constant C. This paper greatly improves the estimate and concludes that the difference between the true solution and the approximate one is less than or equal to C|5|α for any α < 12 . See Theorems 2.7 and 3.1 below. It is easy to see that even for the Brownian motion Yt and its polygonal approximation Yt5 , (E sup0≤t≤T |Yt5 − Yt |2 )1/2 ≥ C|5|1/2 for some positive constant. Thus our rate estimate is approximately exact in some sense.

2.

Polygonal Approximation

In this section we first define the Wong–Zakai type approximation σ 5 by replacing Y by a sequence of piecewise linear functions. Then we show that σ 5 converges to a process which is the solution of an SDE obtained by modifying the Zakai equation (1.6). Then we modify the approximating sequence such that the limit is the solution to the Zakai equation. An estimate of the convergence rate is also given. Let 5: 0 = t0 < t1 < t2 < · · · < tn = T be a partition of [0, T ]. Denote |5| = max1≤k≤n (tk − tk−1 ). For t ∈ [tk−1 , tk ), k = 1, 2, . . . , n, let Yt − Ytk−1 (t − tk−1 ). Yt5 = Ytk−1 + k tk − tk−1 Lemma 2.1. There is a constant C1 such that · ¸ √ 1 P1 5 2 . E sup |Yt − Yt | ≤ C1 |5|1−2/ − ln|5| ln |5| 0≤t≤T Proof. For any p ≥ 2, we have ¾ p/2 ½ P1 5 2 E sup |Yt − Yt | 0≤t≤T

≤ E P1 sup 0≤t≤T

©

|Yt5 − Yt | p

ª

An Approximation for the Zakai Equation

27

¯ ¯¾ p ¯ Yt − Ytk−1 ¯ (t − tk−1 )¯¯ |Yt − Ytk−1 | + ¯¯ k tk − tk−1 1≤k≤n tk−1 ≤t≤tk ½ ¾ ≤ 2 p E P1 sup sup |Yt − Ytk−1 | p ½

≤ E P1 sup

sup

1≤k≤n

≤2

p

n X

½

E

k=1

P1

tk−1 ≤t≤tk

¾

sup |Yt − Ytk−1 |

p

tk−1 ≤t≤tk

.

By the Burkholder–Davis–Gundy inequality (see Theorem 10.36 of [6]), there is a constant C2 independent of p such that E P1

p

sup |Yt − Ytk−1 | p ≤ C2 p p+1 |tk − tk−1 | p/2 .

tk−1 ≤t≤tk

Thus ¾ p/2 ½ n X p ≤ 2p C2 p p+1 |tk − tk−1 | p/2 E P1 sup |Yt5 − Yt |2 0≤t≤T

k=1 p

≤ 2 p C2 p p+1 |5| p/2−1

n X

|tk − tk−1 |

k=1 p

= 2 p C2 p p+1 |5| p/2−1 T. Therefore, we have E P1 sup |Yt5 − Yt |2 ≤ 4C22 p 2−2/ p T 2/ p |5|1−2/ p . 0≤t≤T

Taking p =

√

− ln|5|, the lemma follows from the above inequality.

For k = 1, 2, . . . , n, we consider the following SDEs for t ∈ [tk−1 , tk ): Z t f + σs5 (L(s) f ) ds σt5 f = σt5 k−1 Z +

tk−1

t tk−1

hσs5 (h(s, ·) f ), Y˙s5 i ds,

∀ f ∈ Cb2 (Rd ),

where Y˙t5 = (Ytk − Ytk−1 )/(tk − tk−1 ). To show that (2.1) has a unique solution, we make the following assumption: (B2) h is a bounded continuous function on [0, T ] × Rd . Let Ps,z be a probability measure on C([s, T ], Rd ) given by Ps,z (·) = P(X |[s,T ] ∈ ·|X s = z).

(2.1)

28

Y. Hu, G. Kallianpur, and J. Xiong

Lemma 2.2. Under Assumptions (B1) and (B2), the SDE (2.1) has a unique solution. Further, ∀k = 1, 2, . . . , n, t ∈ [tk−1 , tk ), we have σt5 f =

Z Rd

µZ

Z C([tk−1 ,tk ],Rd )

f (Z t ) exp

t

tk−1

¶ h(s, Z s ) ds Y˙t5 Ptk−1 ,z (d Z )σt5 (dz) k−1

for any f ∈ Cb (Rd ). Proof.

Let

1 νt f = σt5 f /σt5 k−1

and

λs (x) = hh(s, x), Y˙t5 i. k−1

Then νtk−1 is a probability measure on Rd and Z

t

νt f = νtk−1 f +

νs (L(s) f + λs (·) f ) ds.

tk−1

It is easy to verify the conditions of Theorem 5.3 (which is a generalization of the Feynman–Kac formula) in [2], and hence (2.1) has a unique solution. As Bhatt and Karandikar pointed out before Theorem 4.2 in their paper, the solution νt can be written as µZ t ¶ Z Z f (Z t ) exp λs (Z s ) ds Ptk−1 ,z (d Z )νtk−1 (dz). νt f = Rd

C([tk−1 ,tk ],Rd )

tk−1

1 on both sides, we see that the conclusion of the lemma holds. Multiplying by σt5 k−1 Now we patch the solutions of (2.1) over small intervals together. We consider the following SDE for t ∈ [0, T ]: µ5 t f = f (x 0 ) + ∀f ∈

Z 0

t

¢¢ ¡ 5 ˙5 ds, µ5 s (L(s) f ) + µs hh(s, ·), Ys i f

¡

Cb2 (Rd ).

(2.2)

Theorem 2.3. Under Assumptions (B1) and (B2), (2.2) has a unique solution. Further, ∀t ∈ [0, T ], ∀ f ∈ Cb2 (Rd ), µ5 f = σt5 f =

µZ

Z C([0,T ],Rd )

f (Z t ) exp 0

t

¶ hh(s, Z s ), Y˙s5 i ds P0,x0 (d Z ).

(2.3)

Proof. For t ∈ [0, t1 ), the SDE (2.2) coincides with the SDE (2.1) with k = 0. Therefore, the SDE (2.2) has a unique solution up to t < t1 . Suppose that (2.2) has a unique solution up to t < tk−1 . By the continuity, µtk−1 is uniquely determined. Now

An Approximation for the Zakai Equation

29

for t ∈ [tk−1 , tk ), we have 5 µ5 t f = µtk−1 f +

Z

t

¢¢ ¡ 5 ˙5 ds µ5 s (L(s) f ) + µs hh(s, ·), Ys i f

¡

tk−1

which coincides with the SDE (2.1) and hence (2.2) has a unique solution up to t < tk . By induction, we see that (2.2) has a unique solution and µ5 = σ 5 . It follows from Itˆo’s formula that the right-hand side of (2.3) is the unique solution of (2.2). For t ∈ [0, T ], f ∈ Cb (Rd ), let µZ

Z µt f =

C([0,T ],Rd )

t

f (Z t ) exp

¶ hh(s, Z s ), dYs i P0,x0 (d Z ).

(2.4)

0

To show that µ5 converges to µ, we make the following assumption: (B3) ∂h/∂t, ∂h/∂ xi , ∂ 2 h/(∂ xi ∂ x j ), 1 ≤ i, j ≤ d, are bounded functions on [0, T ] × Rd . Lemma 2.4. Under Assumptions (B1)–(B3), we have "

Z C([0,T ],Rd )

E

P1

¯2 # ¯Z t Z t ¯ ¯ 5 hh(s, Z s ), dYs i¯¯ P0,x0 (d Z ) sup ¯¯ hh(s, Z s ), Y˙s i ds −

0≤t≤T

≤ C3 |5|1−2/

0

0

√

− ln|5|

ln

1 . |5|

Let B 0 be an independent copy of the d-dimensional Brownian motion B. Let

Proof.

Z

t

Z t = x0 +

Z b(s, Z s ) ds +

0

0

t

σ (s, Z s ) d Bs0 .

By Itˆo’s formula, we have Z

t 0

hh(s, Z s ), Y˙s5 i ds − Z

t

=− 0

Z

t

hh(s, Z s ), dYs i

0

hYs5 − Ys , dh(s, Z s )i + hh(t, Z t ), Yt5 − Yt i µ

¶ ∂ + L(s) h(s, Z s )i ds + hh(t, Z t ), Yt5 − Yt i ∂s 0 À ¿ d Z t d X X ∂h 5 (s, Z s ) σik (s, Z s ) d Bs0k . Ys − Ys , − ∂ x i 0 i=1 k=1 Z

=−

t

hYs5 − Ys ,

30

Y. Hu, G. Kallianpur, and J. Xiong

It follows from Doob’s inequality (see, e.g., Theorem 10.5 on p. 55 of [9]) that " ¯2 # ¯Z t Z t Z ¯ ¯ E P1 sup ¯¯ hh(s, Z s ), Y˙s5 i ds − hh(s, Z s ), dYs i¯¯ P0,x0 (d Z ) d C([0,T ],R )

0≤t≤T

0

Z

Z ≤3 +

C([0,T ],Rd )

T

E P1

3khk2∞ E P1

0

0

µ° ° ¶2 ° ° 5 2 ° ∂h ° |Ys − Ys | ° ° + kL(s)hk∞ ds P0,x0 (d Z ) ∂s ∞

sup |Yt5 − Yt |2

0≤t≤T

+2 Z T* d X ∂h 5 + 12 E (s, Z s )σik (s, Z s ) ds P0,x0 (d Z ) Ys − Ys , d ∂ xi 0 k=1 C([0,T ],R ) i=1 ! Ã ° ° ° ∂h °2 2 2 2 2 ° ≤ 3 2T ° ° ∂s ° + 2T kL(·)hk∞ + khk∞ + 4T k∇hk∞ kσ k∞ ∞ · ¸ √ 1 P1 5 2 , sup |Yt − Yt | ≤ C3 |5|1−2/ − ln|5| ln ×E |5| 0≤t≤T d Z X

P1

where khk∞ = supt,x |h(t, x)|, and the boundedness of L(·)h follows from Assumptions (B1) and (B3). Lemma 2.5. Let ξ1 , ξ2 , . . . be i.i.d. N (0, 1). Then for any λ > 0, {ak } ⊂ R, n ≥ 1, we have µ ¶ λ|ξk | (2.5) ≤ C4 exp( 12 λ2 ) E max exp √ 1≤k≤n n and E max exp

à k X

1≤k≤n

Proof.

! ai ξi

i=1

Ã

! n 1X 2 ≤ 4 exp a . 2 k=1 k

Note that µ

E max exp 1≤k≤n

λ|ξk | √ n

¶ ≤

¶ µ ∞ X λ|ξk | ` 1 E max √ `! 1≤k≤n n `=0

¶ µ ∞ λ ` λ|ξk | X 1 = 1 + E max √ + E max |ξk |` √ 1≤k≤n 1≤k≤n `! n n `=2 µ ¶ X µ ¶ ∞ λ ` λ |ξk |2 1 1 + E max + ≤ 1+ nE|ξ1 |` √ 1≤k≤n n 2 `! n `=2 ≤

∞ X λ λ` (1 + E|ξ1 |2 ) + 1 + E|ξ1 |` 2 `! `=2

An Approximation for the Zakai Equation

≤

31

∞ X λ λ` (1 + 1) + E|ξ1 |` 2 `! `=0

= λ + Eeλ|ξ1 | ≤ λ + 2 exp( 12 λ2 ) ≤ C4 exp( 12 λ2 ). This proves (2.5). As Mk = 12 (a1 ξ1 + · · · + ak ξk ) is a martingale, hence exp(Mk ) is a submartingale. It follows from Doob’s inequality that n Y E exp(ak ξk ) E max exp(2Mk ) ≤ 4E exp(2Mn ) = 4 1≤k≤n

à = 4 exp

n 1X

2

!

k=1

ak2 .

k=1

We make the following assumption on the partition 5 (0 = t0 < t1 < t2 < · · · < tn = T ): (B4) {n|5|} is bounded. Lemma 2.6. Under Assumptions (B1)–(B4), we have Z C([0,T ],Rd )

E

P1

µ Z t ¶ 5 ˙ sup exp 2 hh(s, Z s ), Ys i ds P0,x0 (d Z ) ≤ C5

0≤t≤T

0

and Z C([0,T ],Rd )

µ Z t ¶ E P1 sup exp 2 hh(s, Z s ), dYs i P0,x0 (d Z ) ≤ C6 . 0≤t≤T

0

Proof. For simplicity of notation, we assume that m = 1. Let ξi = (Yti − Yti−1 )/ √ ti − ti−1 . It follows from Lemma 2.5 that µ Z t ¶ Z P1 5 ˙ E sup exp 2 hh(s, Z s ), Ys i ds P0,x0 (d Z ) C([0,T ],Rd )

Z

=

0≤t≤T

C([0,T ],Rd )

max

0

E P1 Ã

sup exp 2

1≤k≤n tk−1 ≤t
k−1 X i=1

1 ti − ti−1

Z

ti

ti−1

h(s, Z s ) ds(Yti − Yti−1 )

! Z t 1 h(s, Z s ) ds(Ytk − Ytk−1 ) P0,x0 (d Z ) +2 tk − tk−1 tk−1 ! Ã Z ti Z k−1 X 1 P1 E max exp 2√ h(s, Z s ) dsξi = 1≤k≤n ti − ti−1 ti−1 C([0,T ],Rd ) i=1

32

Y. Hu, G. Kallianpur, and J. Xiong

µ ¶ Z tk 1 × exp 2 √ |h(s, Z s )| ds|ξk | P0,x0 (d Z ) tk − tk−1 tk−1 Ã )1/2 ! (Z Z ti k−1 X 1 P1 E max exp 4√ h(s, Z s ) dsξi P0,x0 (d Z ) ≤ 1≤k≤n ti − ti−1 ti−1 C([0,T ],Rd ) i=1 ½Z × E P1 C([0,T ],Rd )

µ ¶ ¾1/2 Z tk 1 max exp 4 √ |h(s, Z s )| ds|ξk | P0,x0 (d Z ) 1≤k≤n tk − tk−1 tk−1 ( Z Ã )1/2 ¶2 ! Z ti n µ 1X 1 ≤ 4 exp h(s, Z s ) ds 4√ P0,x0 (d Z ) 2 i=1 ti − ti−1 ti−1 C([0,T ],Rd ) ½ ³ ´¾1/2 p × E P1 max exp 4khk∞ |5||ξk | 1≤k≤n

p ≤ 2 exp(4khk2∞ T ) C4 exp(8khk2∞ n|5|) ≤ C5 . By Doob’s inequality, we have µ Z t ¶ Z E P1 sup exp 2 hh(s, Z s ), dYs i P0,x0 (d Z ) 0≤t≤T C([0,T ],Rd ) 0 Z ¢ ¡ E P1 ≤ exp khk2∞ T C([0,T ],Rd ) µ ·Z t ¸¶ Z 1 t sup exp 2 hh(s, Z s ), dYs i − |h(s, Z s )|2 ds P0,x0 (d Z ) 2 0 0≤t≤T 0 Z ¢ ¡ E P1 ≤ 4 exp khk2∞ T µ ·Z exp 2

C([0,T ],Rd )

Z

T

1 hh(s, Z s ), dYs i − 2 0 Z ¢ ¡ E P1 ≤ 4 exp 2khk2∞ T µZ exp

C([0,T ],Rd )

T

h2h(s, Z s ), dYs i −

0

=

1 2

T

¸¶ |h(s, Z s )| ds 2

P0,x0 (d Z )

0

Z

t

¶ |2h(s, Z s )|2 ds P0,x0 (d Z )

0

4 exp(2khk2∞ T ).

This proves the conclusions of the lemma. Theorem 2.7. Suppose that Assumptions (B1)–(B4) hold. Let σ 5 and µ be defined by (2.3) and (2.4). Then there exists a constant C7 such that √ p (2.6) E P1 sup d(µt , σt5 ) ≤ C7 − ln|5| |5|1/2−1/ − ln|5| . 0≤t≤T

An Approximation for the Zakai Equation

33

As a consequence, ∀α < 12 , ∃C8 (α) such that E P1 sup d(µt , σt5 ) ≤ C8 (α)|5|α . 0≤t≤T

Proof.

Note that, using Lemmas 2.4 and 2.6, we have

E P1 sup |σt5 f − µt f | 0≤t≤T

¯ ¶ µZ t ¯ ¯exp ˙s5 i ds hh(s, Z ), Y ≤ k f k∞ E sup s ¯ 0≤t≤T C([0,T ],Rd ) 0 ¶¯ µZ t ¯ hh(s, Z s ), dYs i ¯¯ P0,x0 (d Z ) − exp 0 ¯ ·¯Z t Z t Z ¯ ¯ P1 ¯ ¯ hh(s, Z s ), Y˙ 5 i ds − hh(s, Z ), dY i ≤ k f k∞ E sup s s ¯ s ¯ Z

P1

C([0,T ],Rd )

0≤t≤T

0

¶ ½ µZ t 5 ˙ hh(s, Z s ), Ys i ds , × max exp 0 ¶¾¸ µZ t hh(s, Z s ),dYs i P0,x0 (d Z ) exp 0

"Z ≤ k f k∞

0

C([0,T ],Rd )

E P1

#1/2

¯2 ¯Z t Z t ¯ ¯ 5 ¯ ˙ sup ¯ hh(s, Z s ), Ys i ds − hh(s, Z s ), dYs i¯¯

0≤t≤T

0

0

P0,x0 (d Z ) ·Z ×

C([0,T ],Rd )

½ µ Z t ¶ E P1 sup max exp 2 hh(s, Z s ), Y˙s5 i ds , 0≤t≤T

0

¶¾ ¸1/2 µ Z t exp 2 hh(s, Z s ),dYs i P0,x0 (d Z ) 0

¸ · √ 1 1/2 ≤ k f k∞ C3 |5|1−2/ − ln|5| ln [C5 + C6 ]1/2 |5| √ p ≡ C7 k f k∞ − ln|5||5|1/2−1/ − ln|5| . Then E P1 sup d(µt , σt5 ) = 0≤t≤T

X j

≤

X j

2− j E P1 sup (|σt5 f j − µt f j | ∧ 1) 0≤t≤T

√ p 2− j C7 k f j k∞ − ln |5||5|1/2−1/ − ln |5|

√ p = C7 − ln|5||5|1/2−1/ − ln |5| , where { f j } is given in (1.5).

34

Y. Hu, G. Kallianpur, and J. Xiong

Theorem 2.8. µt f given by (2.4) is a solution of Z

t

µt f = f (x0 ) + 0

Proof. d

µ

Z µs (L(s) f + 12 |h(s, ·)|2 f ) ds +

t

hµs (h(s, ·) f ), dYs i.

0

It follows from Itˆo’s formula that µZ t ¶¶ hh(s, Z s ), dYs i f (Z t ) exp 0

=

µZ t ¶ ∂f (Z t ) exp hh(s, Z s ), dYs i d Z ti ∂ xi 0 i=1 µZ t ¶ + f (Z t ) exp hh(s, Z s ), dYs i hh(t, Z t ), dYt i

d X

0

µZ t ¶ ∂2 f 1 j (Z t ) exp hh(s, Z s ), dYs i d Z ti d Z t 2 i, j=1 ∂ xi ∂ x j 0 µZ t ¶ 1 + 2 f (Z t ) exp hh(s, Z s ), dYs i |h(t, Z t )|2 dt

+

d X

0

µZ = L(t) f (Z t ) exp

t

¶ hh(s, Z s ), dYs i dt

0

µZ t ¶ d d X X ∂f 0j (Z t ) exp hh(s, Z s ), dYs i σi j (t, Z t ) d Bt ∂ x i 0 i=1 j=1 µZ t ¶ + f (Z t ) exp hh(s, Z s ), dYs i (hh(t, Z t ), dYt i + 12 |h(t, Z t )|2 dt).

+

0

Taking integrals and expectations on both sides, we finish the proof. Note that the limit of σ 5 is not the solution of the Zakai equation. Further, by the Kallianpur–Striebel formula, we have µZ t ¶¶ µ Z Z 1 t 2 hh(s, Z s ), dYs i − |h(s, Z s )| ds f (Z t ) exp σt f = 2 0 C([0,T ],Rd ) 0 P0,x0 (d Z ). We modify σt5 f by Z σ˜ t5 f =

µZ

µ

C([0,T ],Rd )

f (Z t ) exp 0

t

hh(s, Z s ), Y˙s5 i ds −

1 2

Z

t

¶¶ |h(s, Z s )|2 ds

0

P0,x0 (d Z ). It is easy to see that ∀ f ∈ Cb2 (Rd ), Z t 5 (σ˜ s5 (L(s) f − 12 |h(s, ·)|2 f ) + σ˜ s5 (hh(s, ·), Y˙s5 i f )) ds. σ˜ t f = f (x0 ) + 0

An Approximation for the Zakai Equation

35

By the same arguments as those which led to the proof of Theorem 2.7, we have Theorem 2.9. C8 such that

Suppose that Assumptions (B1)–(B4) hold. Then there exists a constant

√ p E P1 sup d(σt , σ˜ t5 ) ≤ C9 − ln|5||5|1/2−1/ − ln |5| .

(2.7)

0≤t≤T

3.

Approximation of the Density

This section is concerned with the approximation of the density. We follow the notation introduced in previous sections. We make different assumptions. Let ai j (t, x), bi (t, x), h(t, x), i, j = 1, . . . , d, be continuous bounded functions. We assume that ai j (t, x) is uniformly elliptic, i.e., there are two constants 0 < µ < λ < ∞ such that µI ≤ (ai j (t, x))1≤i, j≤d ≤ λI,

(3.1)

where for two matrices A and B, A ≤ B means that B − A is positive definite. Under the above uniform ellipticity condition, σt has density, i.e., there is u t such that Z u t (x) f (x) d x σt f = Rd

and u t satisfies the following Zakai equation: du t = L(t)∗ u t dt + M(t)∗ u t dYt ,

(3.2)

where M∗ (t)u(x) = h(t, x)u(x). The Stratonovitch form of (3.2) is Z Z t Z t 1 t 2 ∗ L(s) u s ds − h u s ds + h s u s d ◦ Ys . ut = f + 2 0 s 0 0

(3.3)

We refer to [3] for a detailed study of this type of equation and also for the expansion of the solution according to both multiple Itˆo–Wiener integrals and multiple Stratonovitch integrals. The relation between these two types of expansion was also studied in their paper. Recall that 5: 0 = t0 < t1 < t2 < · · · < tn−1 < tn = T is a partition of the interval [0, T ]. Let Yt5 = Yti +

Yti+1 − Yti (t − ti ), ti+1 − ti

ti ≤ t < ti+1 .

Then the following parabolic partial differential equation has a unique solution u 5 t : ∗ 5 5 ˙5 1 2 5 u˙ 5 t = L(t) u t − 2 h t u t + h t u t Yt ,

5 u5 0 = f .

(3.4)

36

Y. Hu, G. Kallianpur, and J. Xiong

To prove that u 5 converges to the solution of the density-valued equation (3.2) and to estimate the convergence rate, we make the following assumptions: (L) ai j (t, x), bi (t, x), 1 ≤ i, j ≤ d, are bounded functions and have bounded derivative till third order. Moreover, ai j is uniformly elliptic, i.e., ai j (t, x) satisfies (3.1). (H) h t , 0 ≤ t ≤ T , is bounded in L q ∩ L ∞ for some constant q > p (q will be kept fixed) and for 0 ≤ s < t ≤ T there is a constant K such that |h t (x)| ≤ K < ∞,

sup

sup kh u − h v kq ≤ K |t − s|1/2 .

s≤u
0≤t≤T, x∈Rd

(F) For the initial condition we assume that f ∈ L p ∩ L pq/(q− p)

and

k f 5 − f k p ≤ K |5|1/2 ,

where µZ k f kp =

¶1/ p | f (x)| p d x

Rd

.

We shall prove Theorem 3.1.

Assume (L), (H), and (F). There is a constant C10 such that

p p/2 . Eku 5 t − u t k p ≤ C 10 |5|

Proof.

(3.5)

Set

bi∗ (t, x) =

d X ∂ai j j=1

∂ xj

(t, x) − bi (t, x)

and à ! d d X ∂ 2 ai j (t, x) ∂bi (t, x) 1 X − . c(t, x) = 2 j,k=1 ∂ x j ∂ xk ∂ xk i=1 Then L(t)∗ u(x) =

d X i, j=1

ai j (t, x)

d X ∂ 2u ∂u (x) + bi∗ (t, x) (x) + c(t, x)u(x). ∂ xi ∂ x j ∂ xi i=1

Let 0(x, t; ξ, τ ), 0 ≤ τ < t ≤ T , x, ξ ∈ Rd , be the fundamental solution of L(t)∗ u − ∂u/∂t = 0, i.e., for any fixed t > 0 and ξ ∈ Rd , ∂0(x, t; ξ, τ ) = L(t)∗ 0(x, t; ξ, τ ), ∂t

An Approximation for the Zakai Equation

37

and for any nice function g(x), we have Z lim

t→τ

Rd

0(x, t; ξ, τ )g(ξ ) dξ = g(x).

Under Assumption (L), we have that µ ¶ λ0 |x − ξ |2 |0(x, t; ξ, τ )| ≤ C11 (t − τ )−d/2 exp − 2(t − τ )

(3.6)

for some λ0 > 0, see, for example, p. 28 of [5]. Denote for f : Rd → R, Z Pt,τ f (x) =

Rd

0(x, t; ξ, τ ) f (ξ ) dξ.

Then by (3.6) we have for 1 ≤ p ≤ ∞ and 0 ≤ τ < t ≤ T , kPt,τ f k p ≤ C12 k f k p .

(3.7)

Let B 0 be a d-dimensional Wiener process, independent of Yt , on a probability space ˜ ˜ P). ˜ The expectation on this probability space is denoted by E. ˜ F, (Ä, In the backward equation (1.4) on p. 132 of [8], we take Yk = 0, namely, we consider the analogous backward equation ξ˜r,t = x −

Z

t

Z

b (s, ξ˜s,t ) ds − ∗

r

r

t

σ (s, ξ˜s,t ) d Bs0 .

Set ˜ s,t (x) = exp 8

½Z

t

h(r, ξ˜r,t ) dYr −

s

1 2

Z

t

h(r, ξ˜r,t )2 dr +

s

Z

t

¾ ˜ c(r, ξr,t ) dr .

s

We write ξ˜r,t (x) for ξ˜r,t to emphasize its dependency on x. Then by (1.6) of the above mentioned reference, the solution of (3.2) can be written as ˜ f (ξ˜0,t (x))8 ˜ 0,t (x)}. u(t, x) = E{ Let ˜ x) = c(t, x) − 1 h(t, x)2 . h(t, 2 Then ˜ u t (x) = E

½

f (ξ˜0,t (x)) exp

µZ s

t

h(r, ξ˜r,t (x)) dYr +

Z s

t

˜ ξ˜r,t (x)) dr h(r,

¶¾

38

Y. Hu, G. Kallianpur, and J. Xiong

is a solution of (3.2). Applying the result of the above mentioned reference in the case that Yk = 0, h = 0, and h 0 (t, x) = c(t, x) − 12 h(t, x) + h(t, x)Y˙t5 , we get u5 t (x)

˜ =E

½

f (ξ˜0,t (x)) exp 5

µZ

t

0

h s (ξ˜s,t (x))Y˙s5 ds +

Z

t

˜ ξ˜s,t (x)) ds h(s,

¶¾ .

0

Thus u5 t (x) − u t (x) µZ t ¶¾ ½ Z t 5 ˜ 5 ˜ ˙ ˜ ˜ ˜ h s (ξs,t (x))Ys ds + h(s, ξs,t (x)) ds = E f (ξ0,t (x)) exp ˜ −E

½

0

f (ξ˜0,t (x)) exp

µZ

t

h s (ξ˜s,t (x)) dYs +

0

=

I15 (x)

+

Z

0

t

˜ ξ˜s,t (x)) ds h(s,

¶¾

0

I25 (x),

where

½ ˜ [ f 5 (ξ˜0,t (x)) − f (ξ˜0,t (x))] I15 (x) = E ¶¾ µZ t Z t ˜ ξ˜s,t (x)) ds h(s, h s (ξ˜s,t (x))Y˙s5 ds + × exp 0

and I25 (x)

˜ =E

0

½

µZ t ¶ ˜ ˜ ˜ h(s, ξs,t (x)) ds f (ξ0,t (x)) exp 0 ¶ µZ t ¶¸¾ · µZ t 5 ˙ ˜ ˜ h s (ξs,t (x))Ys ds − exp h s (ξs,t (x))dYs . × exp 0

0

Introduce ϕt = max{n ≥ 0, tn ≤ t}, t5+1 = tϕt +1 . t5 = tϕt , For a function f : [0, T ] → R, we denote [ f ]s ≡ [ f ]5 s :=

1 tk − tk−1

Z

tk

f s ds

if

tk−1 ≤ s < tk .

tk−1

We omit the explicit dependence on 5 when there is no ambiguity. As h(t, x) ≤ c(t, x) ≤ kck∞ < ∞, we have |I15 (x)| p

µ Z t ¶¾ ½¯ ¯p ¯ ¯ 5 ˜ 5 ˜ ˙ ˜ ˜ ≤ C13 E ¯ f (ξ0,t (x)) − f (ξ0,t (x))¯ exp p h s (ξs,t (x))Ys ds , 0

where C13 = exp( pkck∞ T ). Thus E|I15 (x)| p

½ µ Z t ¶¾ 5 ˜ p 5 ˜ ˙ ˜ ˜ ≤ C13 E | f (ξ0,t (x)) − f (ξ0,t (x))| E exp p h s (ξs,t (x))Ys ds . 0

An Approximation for the Zakai Equation

39

The expectation E above can be computed and estimated explicitly as follows: ¶ µ Z t h s (ξ˜s,t (x))Y˙s5 ds E exp p 0

Ã

= E exp p

ϕt X k=1

1 tk − tk−1

Z

tk

h s (ξ˜s,t (x)) ds(Ytk − Ytk−1 )

tk−1

! Z t p ˜ h s (ξs,t (x)) ds(Ytk − Ytk−1 ) + t5+1 − t5 t5 Ã " ¶2 µZ tk ϕt 1 p2 X h s (ξ˜s,t (x)) ds = exp 2 k=1 tk − tk−1 tk−1 ¶2 #! µZ t 1 + h s (ξ˜s,t (x)) ds t5+1 − t5 t5 #! Ã " Z t ϕt Z tk p2 X |h s (ξ˜s,t (x))|2 ds + |h s (ξ˜s,t (x))|2 ds ≤ exp 2 k=1 tk−1 t5 ¶ µ 2 p khk∞ T ≡ C14 . ≤ exp 2

(3.8)

Thus ˜ f 5 (ξ˜0,t (x)) − f (ξ˜0,t (x))| p E|I15 (x)| p ≤ C15 E| = C15 Pt,0 (| f 5 − f | p )(x), where C15 = C13 C14 . Therefore we have Z E|I15 (x)| p d x EkI15 k pp = Rd

≤ C15 kPt,0 (| f 5 − f | p )k p ≤ C15 k| f 5 − f | p k1 = C15 k f 5 − f k pp . Now we estimate I25 . Using the fact that |e x − e y | ≤ (e x + e y )|x − y|, we have |I25 (x)|

· ˜ ≤ C16 E | f (ξ˜0,t (x))| ¯ ¶ µZ t ¶¯¸ µZ t ¯ ¯ h s (ξ˜s,t (x))Y˙s5 ds − exp h s (ξ˜s,t (x)) dYs ¯¯ × ¯¯exp 0

≤

C16 (I35 (x)

+

I45 (x)),

0

(3.9)

40

Y. Hu, G. Kallianpur, and J. Xiong

where C16 = (C13 )1/ p , ½ µZ t ¶ 5 5 ˜ ˙ ˜ ˜ h s (ξs,t (x))Ys ds I3 (x) = E | f (ξ0,t (x))| exp 0

¯¾ ¯Z t Z t ¯ ¯ 5 ¯ ˙ ˜ ˜ h s (ξs,t (x)) dYs ¯¯ × ¯ h s (ξs,t (x))Ys ds − 0

and

0

½ µZ t ¶ ˜ | f (ξ˜0,t (x))| exp h s (ξ˜s,t (x)) dYs I45 (x) = E 0 ¯¾ ¯Z t Z t ¯ ¯ 5 ¯ ˙ ˜ ˜ h s (ξs,t (x)) dYs ¯¯ . × ¯ h s (ξs,t (x))Ys ds − 0

0

Applying the H¨older inequality, we have ½ µ Z t ¶ ˜ | f (ξ˜0,t (x))| p exp p h s (ξ˜s,t (x))Y˙s5 ds E|I35 (x)| p ≤ C13 EE 0 ¯p¾ ¯Z t Z t ¯ ¯ h s (ξ˜s,t (x)) dYs ¯¯ × ¯¯ h s (ξ˜s,t (x))Y˙s5 ds − 0 0 ½ ¶ µ · µ Z t ˜ | f (ξ˜0,t (x))| p E exp p h s (ξ˜s,t (x))Y˙s5 ds = C13 E ¯Z t 0 ¯ × ¯¯ h s (ξ˜s,t (x))Y˙s5 ds 0 ¯ p ¸¶¾ Z t ¯ h s (ξ˜s,t (x)) dYs ¯¯ − 0 ( ¶¶1/r2 µ µ Z t ˜ | f (ξ˜0,t (x))| p E exp pr2 h s (ξ˜s,t (x))Y˙s5 ds ≤ C13 E 0

¯q ¶1/r1 ) µ ¯Z t Z t ¯ ¯ , × E ¯¯ h s (ξ˜s,t (x))Y˙s5 ds − h s (ξ˜s,t (x)) dYs ¯¯ 0

0

where r1 p = q and 1/r1 + 1/r2 = 1, i.e., r2 = q/(q − p). By (3.8), we have µ

Z

t

E exp pr2 0

h s (ξ˜s,t (x))Y˙s5 ds

¶

≤ C˜ 14 < ∞,

where C˜ 14 is C14 with p replaced by pr2 . Thus ˜ f (ξ˜0,t (x))| p (I55 (x))1/r1 }, E|I35 (x)| p ≤ C17 E{| where C17 = C13 (C˜ 14 )1/r2 and I55 (x) is defined and estimated as follows: ¯Z t ¯q Z t ¯ ¯ 5 5 ¯ ˙ ˜ ˜ h s (ξs,t (x)) dYs ¯¯ I5 (x) = E ¯ h s (ξs,t (x))Ys ds − 0

0

(3.10)

An Approximation for the Zakai Equation

41

¯ Z tk ϕt ¯X 1 ¯ = E¯ h s (ξ˜s,t (x)) ds(Ytk − Ytk−1 ) ¯ k=1 tk − tk−1 tk −1 Z t 1 h s (ξ˜s,t (x)) ds(Yt5+1 − Yt5 ) + t5+1 − t5 t5 ¯q Z t Z t5 ¯ h s (ξ˜s,t (x)) dYs − h s (ξ˜s,t (x)) dYs ¯¯ − ½ ¯Z ¯ q ≤ 3 E ¯¯

0

t5

t5

0

¯q ¯ ˜ ˜ {[h · (ξ·,t (x))]s − h s (ξs,t (x))} dYs ¯¯

¯ ¯q Z t ¯ ¯ 1 ¯ ˜ +¯ h s (ξs,t (x)) ds(Yt5+1 − Yt5 )¯¯ t5+1 − t5 t5 ¯Z t ¯q ¾ ¯ ¯ ¯ ˜ + ¯ h s (ξs,t (x)) dYs ¯¯ t5

½Z

t5

≤ C18 0

Z

× ( ≤ C18

|[h · (ξ˜·,t (x))]s − h s (ξ˜s,t (x))|q ds + |t5+1 − t5 |q/2−1 t

|h s (ξ˜s,t (x))|q ds + |t5+1 − t5 |q/2−1

t5 ϕt X k=1

Z

t

|h s (ξ˜s,t (x))|q ds

¾

t5

1 tk − tk−1

Z

Z

tk tk−1

Z

t

+ 2|5|

q/2−1

tk

|h r (ξ˜r,t (x)) − h s (ξ˜s,t (x))|q dr ds

tk−1

¾ q ˜ |h s (ξs,t (x))| ds .

t5

Thus we have Z Rd

EI55 (x) d x ≤ C18

( ϕt X k=1

1 tk − tk−1

+ 2|5|

Z

Z

tk tk −1

tk −1

Z tZ Rd

t5

≤ C18

k=1

1 tk − tk−1

Z

tk −1

Z + 2|5|

tk

t

q/2−1

Z Rd

Z

tk

tk −1

Z Rd

)

E|h r (ξ˜r,t (x)) − h s (ξ˜s,t (x))|q d x dr ds

kPs,t |h s | k1 ds q

t5

E|h r (ξ˜r,t (x)) − h s (ξ˜s,t (x))|q d x dr ds

E|h s (ξ˜s,t (x))|q d x ds

q/2−1

( ϕt X

tk

)

42

Y. Hu, G. Kallianpur, and J. Xiong

( ≤ C19

ϕt X k=1

+

Z

1 tk − tk−1

ϕt X k=1

Z

tk tk −1

1 tk − tk−1 )

Z

Z

tk tk −1

Z

tk tk −1

Rd tk

E|h r (ξ˜r,t (x)) − h s (ξ˜r,t (x))|q d x dr ds

Z

tk −1

Rd

E|h s (ξ˜r,t (x)) − h s (ξ˜s,t (x))|q d x dr ds

+|5|q/2 = C19 {I65 + I75 + |5|q/2 },

(3.11)

where the last inequality follows from assumption (H) and sup kPs,t |h s |q k1 ≤ sup k|h s |q k1 0≤s≤T

0≤s≤T

= sup kh s kqq < ∞. 0≤s≤T

By assumption (H) again, in a similar way one can prove that I65 ≤ C20 |5|q/2 .

(3.12)

To estimate I75 we need the following lemma. Lemma 3.2. ∇ f ∈ Lp Then Z Rd

Proof.

Let and

∇2 f ∈ L p.

E| f (ξ˜s,t (x)) − f (ξ˜r,t (x))| p d x ≤ C21 |s − r | p/2 .

(3.13)

Let r < s. By Itˆo’s formula we have

f (ξ˜s,t (x)) − f (ξ˜r,t (x)) Z s d Z s X ∂f ˜ L(t) f (ξv,t (x)) dv + σi j (ξ˜v,t (x)) (ξ˜v,t (x)) d Bi0 (v). = ∂ xj r i, j=1 r Thus E| f (ξ˜s,t (x)) − f (ξ˜r,t (x))| p ½ Z s ≤ C22 (s − r ) p−1 E|L(t) f (ξ˜v,t (x))| p dv r

+ (s − r )

p/2−1

(Z ≤ C23 (s − r ) p/2−1 r

d Z X j=1

s

r

s

¯ ¯p ) ¯ ∂f ¯ E ¯¯ (ξ˜v,t (x))¯¯ dv ∂ xj

E|L(t) f (ξ˜v,t (x))| p dv +

d Z X j=1

r

s

¯ ¯p ) ¯ ∂f ¯ E ¯¯ (ξ˜v,t (x))¯¯ dv ∂ xj

An Approximation for the Zakai Equation

( ≤ C24 (s − r )

p/2−1

43

¯ 2 ¯p ¯ ∂ f ¯ ¯ ˜ E¯ f (ξv,t (x))¯¯ dv ∂ xi ∂ x j i, j=1 r ¯ ¯p ) Z d s X ¯ ∂f ¯ E ¯¯ (ξ˜v,t (x))¯¯ dv . + ∂ xj j=1 r d Z X

s

Using the argument above, we prove (3.12). It follows easily from Lemma 3.2 that I75 ≤ C25 |5|q/2 .

(3.14)

Combining (3.11), (3.12), and (3.14) we see that Z Rd

EI55 (x) d x ≤ C26 |5| p/2 .

By (3.10) we have Z 5 p E|I35 (x)| p d x EkI3 k p = Rd

µZ

≤ C17 ≤ C17 k

Rd

˜ f (ξ˜0,t (x))| pr2 d x E|

¶1/r2 µZ Rd

EI55 (x) d x

¶1/r1

p f k pr (|5|q/2 )1/r1 2

≤ C17 |5| p/2 . In a similar way we can prove that EkI45 k pp ≤ C27 |5| p/2 . By (3.9), we have EkI25 k pp ≤ C28 |5| p/2 . By Assumption (F), we have EkI15 k pp ≤ C29 |5| p/2 . 5 5 Thus by the fact that u 5 t − u t = I1 + I2 , we complete the proof of the theorem.

Acknowledgment We thank an anonymous referee for the careful reading of the manuscript and for having brought reference [4] to our attention.

44

Y. Hu, G. Kallianpur, and J. Xiong

References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

A. Bhatt, G. Kallianpur and R. Karandikar, Uniqueness and robustness of solution of measure valued equations of nonlinear filtering, Ann. Probab., 23 (1995), 1895–1938. A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes characterized via martingale problems, Ann. Probab., 21 (1993), 2246–2268. A. Budhiraja and G. Kallianpur, Approximations to the solution of the Zakai equation using multiple Wiener and Stratonovitch integral expansions, Stochastics Stochastics Rep., 56 (1996), 271–315. M. Chaleyat-Maurel and D. Michel, A Stroock Varadhan support theorem in non-linear filtering theory. Probab. Theory Related Fields, 84 (1990), 119–139. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. S. W. He, J. G. Wang and J. A. Yan, Semimartingale and Stochastic Analysis (in Chinese), Science Press, Beijing, 1995. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusions, North-Holland, Amsterdam, 1989. H. Kunita, Stochastic partial differential equations connected with nonlinear filtering, in Nonlinear Filtering and Stochastic Control, ed. by S. K. Mitter and A. Moro, Lecture Notes in Mathematics 972, Springer-Verlag, Berlin, 1982, pp. 100–169. M. M´etivier, Semimartingales, de Gruyter, New York, 1982. J. M. Moulinier, Th´eor`eemes limites pour des processus a` un ou deux param`etres. Th`ese de 3`eme cycle, Universit´e de Paris VI, 1982. J. M. Moulinier, Th´eor`emes limites pour les e´ quations diff´erentielles stochastiques. Bull. Sci. Math. (2) 112 (1988), 118–210. E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213–229.

Accepted 23 April 2001. Online publication 14 August 2001.