Convergence in Nonlinear Filtering for Stochastic Delay Systems

TLDR: An approximation scheme for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of X for t-tau, where $\tau$ is a constant.

Abstract: We study an approximation scheme for a nonlinear filtering problem when the state process $X$ is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of $X(s)$ for $s\in [t-\tau,t]$, where $\tau$ is a constant. The approximating state is the piecewise linear Euler-Maruyama scheme, and the observation process is a noisy function of the approximating state. The rate of convergence of this scheme is computed.

Full text

Convergence in nonlinear filtering
for stochastic delay systems
∗
Antonella Calzolari
†
Patrick Florchinger
‡
Giovanna Nappo
§
Abstract
We study some approximation schemes for a nonlinear filtering problem when the
state process X is the solution of a stochastic delay diffusion equation, and the
observation process is a noisy function of X
s
for s ∈ [t − τ, t], where τ is a constant.
The approximating state is given by means of an Euler discretization scheme, and
the observation process is a noisy function of the approximating state.
Contents
1 Introduction 2
2 The model and its approximation 3
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Approximation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Some related results 9
4 Approximation results 13
4.1 General considerations on approximation for filters . . . . . . . . . . . . . 13
4.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
∗
Partially supported by MURST project ”Modelli probabilistici e applicazioni”
†
Dipartimento di Matematica - Universit`a di Roma ”Tor Vergata”, via della Ricerca Scientifica 1, I
00133 Roma, Italy
‡
D´epartement de Math´ematiques, Universit´e de Metz, 23 All´ee des Oeillets F 57160 Moulins les Metz,
France
§
Dipartimento di Matematica - Universit`a di Roma ”La Sapienza”, piazzale A. Moro 2, I 00185 Roma,
Italy
1

1 Introduction
Let (X, Y ) = (X
t
, Y
t
)
t≥0
be a stochastic system. Assume that the state process X =
(X
t
)
t≥0
of the system cannot be directly observed, while the other component Y = (Y
t
)
t≥0
is completely observable, and therefore is referred to as the observation process. The aim
of stochastic nonlinear filtering is to compute the conditional law π
t
of the state process
at time t, given the observation pro cess up to time t, i.e. the computation of
π
t
(ϕ) = E
£
ϕ(X
t
)/F
Y
t
¤
,
for all functions ϕ belonging to a determining class, i.e. the best estimate of ϕ(X
t
) given
the σ−algebra of the observations up to time t, F
Y
t
= σ{Y
s
, s ≤ t}.
A classical model of partially observed system extensively studied in the last past years
arises when both the state and the observation pro cesses are diffusion processes.
For this model, under suitable hypotheses on the coefficients, the filtering equations
are well known (see for example Pardoux [15] or Kallianpur [6] and the references therein)
and different approximation schemes have been studied (see for example Kushner [10] or
Le Gland [12] and the references therein).
In this paper we are interested in nonlinear filtering of partially observed delay systems
and in computable approximations of the filter.
To our knowledge there are only three papers dealing with nonlinear filtering for delay
systems: Kwong and Willsky (1978) [11], Chang (1987) [4], and Kallianpur and Mandal
(2002) [7].
In [11] Kwong and Willsky give a characterization of the optimal filter when dealing
with nonlinear delay systems with gaussian noises. A Fujisaki-Kallianpur-Kunita equa-
tion for the filter is deduced from a representation result which characterizes conditional
moment functionals of nonlinear delay systems. However the uniqueness of the solution
of this equation is not guaranteed.
In [4] Chang gives a computable approximation for the optimal filter when dealing with
one dimensional nonlinear delay filtering systems with gaussian noises. The original model
is approximated by a discrete-time model obtained by applying an Euler discretization
scheme. An optimal filter for the approximate system is obtained by an explicit procedure
and the weak convergence of the approximating process and the approximating filter to
the original ones are verified.
In [7] Kallianpur and Mandal study a nonlinear filtering problem where the state pro-
cess is given as the solution of a stochastic delay differential equation and the observation
depends not only on a function of the instantaneous value of the signal but also on the
values of the signal from the past. By using some extensions of results obtained by Mo-
hammed [14] for stochastic delay differential equations they prove that the signal process
is the unique solution to an appropriate martingale problem. By taking this fact into
account the authors prove that the optimal filter corresponding to the nonlinear filtering
problem is the unique solution of a Zakai-type equation. A Fujisaki-Kallianpur-Kunita
equation for the filter is also deduced from the Zakai-type equation.
2

The main results of these papers will be presented in a more extensive way in Section 3.
The main result of this paper concerning the convergence of the approximation scheme
introduced in Section 2.2 to the system introduced in Section 2.1 is stated and proved
in Section 4.2. The proof is based on the convergence results of Bhatt, Kallianpur and
Karandikar (1999) [2] and Bhatt and Karandikar (2002) [3], which are recalled in Section
4.1.
2 The model and its approximation
2.1 The model
The state satisfies the following kind of delay differential equation on the probability space
(Ω, F, (F
t
)
t∈[0,T ]
, P )





X(t) = η(t), −τ ≤ t ≤ 0,
X(t) = η(0) +
R
t
0
a(u, Π
u
X)du +
R
t
0
b(u, Π
u
X)d
˜
W
u
, 0 ≤ t ≤ T,
(1)
where
(H1) τ > 0, and Π
t
X is a C([−τ, 0], R) random valued process defined by
Π
t
X(s) = X(t + s) − τ ≤ s ≤ 0,
(H2) {
˜
W (t); t ≥ 0} is a standard Brownian motion,
(H3) η is a F
0
square integrable C([−τ, 0], R) valued random variable, that is
E
¡
kΠ
0
Xk
2
¢
= E
³
sup
s∈[−τ,0]
|η(s)|
2
´
< ∞,
(H4) a and b are two Borel measurable functionals on [0, T ]× C([−τ, 0], R) satisfying the
Lipschitz condition
|a(t, θ) − a(t,
¯
θ)|
2
+ |b(t, θ) − b(t,
¯
θ)|
2
≤ Kkθ −
¯
θk
2
, (2)
and the growth condition
|a(t, θ)|
2
+ |b(t, θ)|
2
≤ K
¡
1 + kθk
2
¢
, (3)
for some constant K > 0.
3

The observation process is given by
Y (t) =
Z
t
0
h(u, Π
u
X)du + W (t), 0 ≤ t ≤ T, (4)
where
(H5) {W (t); t ≥ 0} is a standard Brownian motion, independent of {
˜
W (t); t ≥ 0},
(H6) h : [0, T ] × C([−τ, 0], R) → R is a Borel measurable function such that
Z
T
0
E[|h(u, Π
u
X)|
2
]du < ∞.
In this paper we consider the filtering problem of the delay system (Π
t
X, Y (t))
t≥0
, i.e.
we want to compute
π
t
(φ) = E[φ(Π
t
X)|F
Y
t
],
for measurable and bounded functions φ mapping C([−τ, 0], R) into R.
Conditions (H1) to (H4) imply the existence and uniqueness of the solution of the
stochastic delay differential equation (1) (see Mohammed [14]), though the square in-
tegrability condition (H3) is not necessary (see Kallianpur and Mandal [7]). However
condition (H3), together with the growth condition (3) in (H4), implies that
E
h
sup
u∈[0,T ]
kΠ
u
Xk
2
i
< ∞, (5)
(see [14], Theorem II.2.1 and Lemma I II.1.2).
Note that in the homogeneous case, i.e. a(t, θ) = a(θ) and b(t, θ) = b(θ), the Lipschitz
assumption (2) in (H4) implies the growth condition (3). Moreover in the homogeneous
case Kallianpur and Mandal prove in [7] that the process (Π
t
X)
t≥0
is Markov. This result
extends a previous result by Mohammed [14] for the case a and b bounded.
Conditions (H5) and (H6) are classical assumptions in nonlinear filtering theory which
guarantee that the filter π
t
can be represented via a Kallianpur–Striebel formula
π
t
(φ) =
σ
t
(φ)
σ
t
(1)
,
with
σ
t
(φ) = E
0
·
φ(Π
t
X) exp
½
Z
t
0
h(s, Π
s
X)dY
s
−
1
2
Z
t
0
|h(s, Π
s
X)|
2
ds
¾
¯
¯
¯
F
Y
t
¸
,
where E
0
denotes the expectation w.r.t. the reference measure P
0
, defined by the Radon-
Nikodym derivative
dP
0
dP
= exp
½
−
Z
T
0
h(s, Π
s
X)dY
s
+
1
2
Z
T
0
|h(s, Π
s
X)|
2
ds
¾
. (6)
4

This fact will play a fundamental role in the proof of our approximation results.
Furthermore, the following conditions on the observation function h will also be used
in our analysis.
(H7) h is sublinear, i.e. |h(t, θ )|
2
≤ K(1 + kθk
2
),
(H8) h is jointly continuous.
Note that condition (H7) together with the integrability condition (5) on the state
process implies that (H6) is also satisfied.
Condition (H8) will be used in the proof of our approximation results, together with
the following condition.
(H9) The functions a(t, θ) and b(t, θ) are jointly globally Lipschitz, i.e.
|a(t, θ) − a(t
0
,
¯
θ)|
2
+ |b(t, θ) − b(t
0
,
¯
θ)|
2
≤ K
¡
kt − t
0
k
2
+ kθ −
¯
θk
2
¢
.
Note that the above joint global Lipschitz condition is stronger than condition (H4),
but in the homogeneous case, i.e. when a(t, θ) = a(θ) and b(t, θ) = b(θ), both conditions
are clearly equivalent to (2).
In addition, to prove our approximation results, condition (H3) is not strong enough
and we will assume that the initial condition of the stochastic delay differential equation
(1) is such that
(H10) E
¡
kΠ
0
Xk
4
¢
= E
³
sup
s∈[−τ,0]
|η(s)|
4
´
< ∞.
Note that under this condition it is easy to prove that E
£
sup
t∈[0,T ]
kΠ
t
Xk
4
¤
is finite.
To conclude this section, note that conditions (H7) to (H9) are clearly satisfied if, for
example, the functions a, b and h are given by
a(t, θ) = ϕ
a
¡
t,
Z
0
−τ
ψ
a
(u, θ(u))du
¢
, b(t, θ) = ϕ
b
¡
t,
Z
0
−τ
ψ
b
(u, θ(u))du
¢
,
h(t, θ) = ϕ
¡
t,
Z
0
−τ
ψ(u, θ(u))du
¢
,
where ϕ
a
, ϕ
b
, ϕ, ψ
a
, ψ
b
and ψ are joint Lipschitz functions.
5

Frequently asked questions

Q1. What are the contributions in "Convergence in nonlinear filtering for stochastic delay systems" ?

The authors study some approximation schemes for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation, and the observation process is a noisy function of Xs for s ∈ [ t− τ, t ], where τ is a constant. 

Q2. What is the last statement of the Theorem?

Since the filter π is continuous in time, the last statement of the Theorem is an immediate consequence of the convergence in probability of π̃n to π. 

Q3. what is the approximation xn of the state process?

The approximation Xn of the state process is the linear interpolation of the Euler discretization scheme with step δ = δn = T/n, with τ = mδ (as in Chang [4], for the sake of simplicity, the authors assume that T/τ is rational)1: Xn(`δ) = η(`δ), −m ≤ ` ≤ 0, Xn((` + 1)δ) = Xn(`δ) + a(`δ, Π`δX n)δ+b(`δ, Π`δX n) [ W̃ ((` + 1)δ)− 

Q4. what is the optimal filtering problem for a partially observable model?

In [4], Chang considers the optimal filtering problem for a partially observable model, when the state process and the observation process are real valued and satisfy the stochastic nonlinear differential equationsdX(t) = a(t, ΠtX)dt + dW̃t, Π0X = ηdY (t) = h(t, ΠtX)dt + dWt, Y (0) = 0where W̃ and W are two independent Brownian motions. 

Q5. what is the law of the approximation of the state process?

under P 0,n, the processes Xn and Y n are independent and the law of the approximated state process is invariant under P and P 0,n, hence, for t ∈ [`δ, (` + 1)δ), 0 ≤ ` ≤ n,σnt (φ) = E [φ(ΠtX n)Lnt (X n(·), y0, yδ, · · · , y`δ, y)] ∣∣∣∣ y0=Y n0 ,y1=Y n δ ,··· ,y`=Y n`δ,y=Y nt . 

Q6. What is the typical situation when dealing with approximation problems?

When dealing with approximation problems the following typical situation can arise: the model is (X,Y ), and therefore the authors observe Y , while the models (Xn,Y n) are merely more manageable approximations, despite the fact that it may be impossible to observe the approximate process Y n. 

Q7. what is the probability density of a dimensional random vector?

τ0 < · · · < τn = 0 the (n + 1) − dimensional random vector (η(τi); 0 ≤ i ≤ n) has a probability density w.r.t. the Lebesgue measure in Rn+1. 

Q8. What is the simplest way to prove the weak convergence of the observation process?

With the approximations (14) or (15) for the observation process it is natural to takeX nt = (t, ΠtXn), (22)orX nt = (t, Πδ·bt/δcXn), (23)as an approximation for the signal process. 

Q9. What is the condition for (B4) to hold?

To conclude this section observe that a sufficient condition for (B4’) to hold is that(B4”) lim n→∞ E (∫ T 0 |hn(X ns )− h(Xs)|2 ds ) =0. 

Q10. What is the way to describe the optimal filter?

Y n(`δ) = h(`δ, Π`δXn)δ + [ W ((` + 1)δ)−W (`δ)], 0 ≤ ` ≤ n. (17)Then, the author shows that the optimal filter for the discrete time approximation can be designed by an explicit procedure and he verifies the weak convergence of the approximation process and the approximation filter to the original ones. 

Q11. What is the main result of the theorem 4.2?

The authors will prove the convergence of Xn to X , in a sense stronger than weak convergence, in Proposition 4.5 below (see also Remark 4.6). 

Q12. what is the bound for n,(T )?

taking into account (29) and (31), and invoking Gronwall’s inequality the authors get a bound for φnσ,`(T ) = φ n σnN ,`(T ), uniform in n and N .