•
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PARAMETER ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS
DRIVEN BY WIENER AND POISSON NOISE
Charles E. Smith and Loren Cobb
Biomathematics Series No. 26
Institute of Statistics Mimeo Series No. 1679
Raleigh, North Carolina
June 1986
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NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
PARAMETER ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS
DRIVEN BY WIENER AND POISSON NOISE
Charles E. Smith
Department of Statistics, Biomathematics Program
North Carolina State University
Raleigh, North Carolina 27695-8203, USA
Loren Cobb
Department of Biometry
Medical University of South Carolina
Charleston, South Carolina 29425, USA
Abstract
Ensemble and temporal parameter estimators are developed for linear
and nonlinear stochastic differential equations driven by both Wiener
and Poisson processes.
Linear moment recursion relations are obtained for
the stationary moments of the process.
Consistency and asymptotic
normality of the resulting ensemble estimators are demonstrated.
The
temporal estimators, i.e., using a single temporal record, are shown to
coincide with
maximum likelihood estimators in the special case of linear
systems driven by Wiener noise.
Keywords:
stochastic differential equations, parameter estimators, moment
recursion relation, nonlinear stochastic systems.
Acknowledgement:
Support for this work was furnished by National Science
Foundation Grant ISP 80-11451 and" Office Naval Research Contract N00014-85-K-0105.
* Author to whom correspondence should be addressed.
2
1.
Introduction
This report is concerned with estimating the parameters of a class of
linear and nonlinear stochastic differential equations driven by Wiener
and Poisson processes.
drift term.
Specifically, we consider two models with a polynomial
The first model (1) has state-dependent noise, while the
second (2) has an additive noise term.
The first model is:
dx
(1)
t
= -Y
(x ) dt + x dn ,
t
t·
t
= cr dW t + a (dP t - A dt),
t
d
where y(x) = 6 + 6 x +
+ 6 x , 6
1 0, 6 > 0, W is a standard
t
0
d
0
d
l
Wiener process (zero mean and variance t), P is a homogeneous Poisson process
dn
t
with intensity A and unit jump size, and cr, a are constants.
We assume
the two noise processes, W and P , are independent of each other and of
t
t
the initial condition x .
o
The second model is similar to the first:
(2)
dn
t
= cr dW
t
+
a (dP
t
- A dt),
with all conditions identical to (1) except that eO is no longer constrained
to be nonzero.
Systems of this form, (1) and (2) have been used as models for a number
of physical and biological processes (see, for example, Arnold and Lefever,
1981).
The polynomial structure of y permits the modelling of systems with
multi-model stationary distributions.
For example bistable systems can be
modelled with y(x) as a cubic polynomial, see Kipnis and Newman (1985),
Cobb and Zacks (1985).
State-dependent noise, as in (1), has been applied
to a variety of nonlinear systems (population growth: Hanson and Tuckwell
(1981); optimal harvesting: Ryan and Hanson (1985); economics: Malliaris and
3
Brock (1982); neural models: Tuckwell (1979). Wilbur and Rinzel (1983).
Hanson and Tuckwell (1983). Smith and Smith (1984); overview of Poisson
models: Tuckwell (1981)).
Let f(x.t.x ) denote the transition probability density function (pdf)
O
for the Markov process x ' given the initial value xo.
t
With 6
d
> 0 the
transition pdf approaches a nondegenerate pdf. denoted by f*. as t
+
00.
is independent of x . In the next section we
O
show that the moments of f* satisfy a linear recursion relation which
The stationary density f*
depends only on the parameters 8 ,
0
. .. ,
6
d'
0 •
A, a.
This recursion relation
permits the construction of an estimator of the parameters (6 ,
0
... ,
6 )
d
i f A. a and o are known. This estimator is shown to be consistent and
asymptotically normal.
Since this estimator is constructed from i.i.d.
samples of the ensemble, we refer to it as an "ensemble" estimator.
The
problem of estimation from a single temporal record is then addressed.
This report extends earlier results on versions of (1) and (2) with
Wiener input alone (Cobb et al., 1983; also see Lanska, 1979), and with
Poisson input alone (Smith and Cobb. 1982).
2.
Estimation from observations of an ensemble
In this section. we derive linear moment recursion relations for
models (1) and (2) and construct ensemble estimators for 8 based on these
relations.
Theorem 1 (Moment Recursion Relations).
Let ~ denote the nth noncentral moment of f*. the stationary pdf.
For models (1) and (2) these moments satisfy a recursion relation for
n = 0. 1. 2•... that is linear in terms
of~.
For model (1).
4
d
e·1 m.l+n = 1/1 n+l mn+l
i=O
r
(3a)
with
1/Ik = A((l + a)
k
- l)/k
2
(k-l) a 12.
+
For model (2),
d
r e.1
i=O
(3b)
m = n a 2 m 12
i+n
n-l
n
+
r (n)
k 1\ mn _k
k=O
with
ak
=
Aa
k+l
I(k+l).
Proof.
Model (1):
Let a denote the operator for partial differentiation with respect to x.
x
It can be shown (Gihman and Skorohod, 1972, p299) that f(x,t,x
o)
satisfies
the forward Feller-Kolmogorov equation
(4)
-A f + A f(x/(l+a)) 1
Il+al.
The stationary pdf f* is the solution of dtf = 0 in (4).
conditions are dk f * (+ ~) = 0 for k = 0, 1, 2,
x
the Fourier transform of a function f, i.e.,
F {f}
v
d
= J
~
The boundary
Now let Fv{f} denote
f(x) exp(-iv x) dx,
-~
where v is the transform variable, and i is the unit imaginary number.
Taking the Fourier transform of our equation for f*, we· obtain
(5)
=A (Fv {f*} - Fv (1 +a ) {f*}),
5
where for the last term we have used the scaling property of the Fourier
transform, i. e.,
F {f(bx)}
v
for some constant b.
= l/lbi
Ib {f(x)}
Fv
The LHS is evaluated using
F)a?Cf(x)}
= (i v )
m
Fv {x f(x)}
=i
F if}
v
and
m
m
av F,v if}
to obtain
(6)
( i v) y ( i a ) F {f*} +
V
V
r/ /2
( i v ) 2 (i a ) 2 F" {f *}
'J
v
A (Fv{f*} - FV(l+a){f*}).
=
we can now derive the moments m by using the well-known fact that
n
m
n
= (idv)n F{f*} when
v
= o.
Using the (iav)n operator on both sides of (6), the RHS becomes
which simplifies to
n
for v
A[l - (1 + a) ] m
n
= o.
For the LHS of (6), we use the product rule of differentiation, namely
n
a (u·v)
x
=
~
n
(k) (a
l:
k=O
n-k
x
u) (a
k
x
v)
to obtain (Sl + S2) Fv{f * }, where
S
=
(-n) (ia )n-l y(ia ) + O(v),
S
=
02/2 n(n-l) (ia )n + O(v).
v
1
v
y
and
2
6
When
v
=0
the LHS becomes
-n
d
r
8
k=O
k
I1l
+ _ +
k n l
2
0
/2 n(n-l) mn
so, combining these results and shifting the index by 1, we obtain
A /(n+l) [(l+a)n+l - 1] m
+ n
n+l
-
m
n+l'
,I.
- 'f'n+l
0
2 /2 m
n+l
for n = 0, 1, 2, ...
the desired result (3a).
For model (2), the corresponding Feller-Kolmogorov equation is
at f
=
ax [y(x)f]
Except for the term
0
2/2
+
ax2f ,
Smith and Cobb (1982, p703).
0
2
/2
ax2f
+ A (f(x - a) -f).
this equation is identical to the one in
The result follows from a minor adjustment
of their development.
Remark.
These results carryover easily to finite sums of independent
Wiener and ·Poisson noise.
A concrete example gives some feeling for the mechanics of the use of
2
3
the relationship. Let y(x) = 80 + 8 X + 8 x + 8 X , a cubic polynomial.
l
2
3
We now obtain four equations for the four unknowns 8 , 8 , 8 and 83 in
1
0
2
terms of the parameters of the noise and the zeroth through sixth moments
of f*.
Ensemble estimators of 8 can be derived from the moment recursion
relations as follows.
Suppose that {X }, k = 1, t are independent random
k
variables, each with density f*. The moment recursion relations for
both (1) and (2) reduce to
7
Me
= WB,
where
e
= [eO'
e- l , ... , ed ] I ~
with (') indicating transpose;
t
= mi + j _ 2 = lit
M'1J.
"
i+j-2
~ Xk
'
k=l
(Modell)
Wij = mi °ij'
with O.. being the Kronecker delta,
1J
for j
>i
for j
< i.
(Model 2)
=
m. .
l-J
and
(Modell)
B. = ljJ •
J
J
{::-:
=
for j
r
for j
= 2.
2,
(Model 2)
Theorem 2.
A_lA
Under models (1) and (2) the ensemble estimator e = M WB is consistent
A
and Ir (e - e) is asymptotically normal N(O,V), where V is a dxd matrix
that satisfies
A
[MVM] ..
lJ
Proof:
=
A
A
E{([WB]. - [Me]l') ([WE]. - [Me].)}
1
J
J
(following Cobb et al., 1983, Theorem 2)
d
Let p (x) = a + a x + ... + a x
Because x is a random
Old
a
variable with a continuous nondegenerate density, we have E[p~(X)] = a'M a
Consistency:
for any vector a
invertible.
~
O.
From this it follows that M is positive definite and
Furthermore, M is invertible w.p. 1
by a similar argument, as
>0
8
A
(M)
-1
-1
A
W~ M
M~
A
> d.
long as n
Since
A
All
M and W ~ W, we also have (M)-
~ M-
and
0
A
Consistency (8 ~ 8) follows immediately because B is
W.
nonstochastic.
We have I~ (M - M) = 0 (1) and 8 - 8 =
Normality:
p
---A.c...-<.-
0
P
(1).
Rewrite
It M (8 - 8) as follows, collecting terms of similar order in probability:
It
M
(~
It
8) =
-
(WB - M8) - Ir (M -
M)
(6 - 8)
Each entry of the second term on the right-hand side is Op'(l) opel) = o (1),
P
where ('), indicates transpose.
A
Thus
A
ItM (8 - 8) -It
(WB -
M8) ~
0,
and
-1
(8 - 8) -/~ M
(WBA
It
-1
since M
_
is invertible.
~fl)~
The vector It
0,
-1
M
.(WB
- M8) can be written as
R.
k:1 h(xk)/If , where hex) is a vector of polynomials in x.
that E[h(X)]
=
0, due to the linear moment recursion relations.
Note
Let
.-
V..
1
J
=
E[h i (X) h.(X)].
J
Then If (8 - 8)
multivariate Central Limit Theorem.
is asymptotically N(O,V) by the
9
3.
Estimation from continuous observations
In this section we present a method for estimating the coefficients
of (1) and (2) given continuous observations of the stochastic process
x
t.
on [0, T].
Explicit maximum likelihood estimation is not possible
because the noise process is a mixture of Wiener and Poisson noise.
We shall
instead construct an estimator which reduces to the MLE when there is no
Poisson noise, and which always satisfies a minimum mean squared error
criterion.
context.
8
0
This estimator is best developed in a slightly more general
Let 8 = [8 , 8 , ... , 8 d]
~ 0, 8
d
0
<0,
1
and d odd.
,
d
I
and g(x) = [1, x, ... , X]
Suppose that x
with
is a stochastic process which
t
satisfies the stochastic differential equation
dx
= 8
t
,
g(x) dt
+
t
cr(x) dn ,
t
where cr(x) is a smooth non-anticipatory function, cr(x)
~ixture
is, as before, a
~
0 if x
~
0, and n
t
of an arbitrary number of independent Wiener and
compensated Poisson processes, such that the process x
t
is ergodic.
We
shall make use of a weighting function introduced by Lipster and
Shiryayev (1977, p. 274):
vex)
=
{o cr2
1/
i f cr(x) = 0
Cx)
otherwise.
~Qt(x,c)
Consider the function
defined by
2
~Q
2
(x,c) = [(~x - c'g(x ) ~t)/crCx )] - [~x /cr (x )] ,
t
t
t
t
t
t
where the right hand side is evaluated at x = x, and where ~x ~
t
t
x
t
t+~t
E:
- x
t
[0,
The function
T-~t],
~Q
t
(x,c) has a graph that for
is a paraboloid with a single minimum.
converges in probability as
=-
~t +
n' and
The ratio
~Q Cx,c)/~t
t
0 to the stochastic differential
2 c' g(x ) V(x ) dX
t
t
t
with the RHS evaluated at x = x.
t
dQt(x,c)
W E:
+
c
[g(x )g(x )']c vex ) dt
t
t
t
10
The mean squared error of c for the stochastic process x
t
on [O,t]
is given by
MSE(c,T) =
=-
T
f o dQt(Xt,C)
f~
g(x ) V(X ) dX +
t
t
t
Let S (w) be the solution of V MSE(c,T)
T
c
2 c'
A
c'[f~
g(xt)g(x )' V(X ) dt] c.
t
t
= 0 for fixed w € ~.
Thus
A
T
= [fo
g(Xt)g(X ), V(X ) dt]
t
t
In the special case in which cr(x)
STew)
-1
T
[fo
g(x ) v(x ) dx ]·
t
t
t
= x, d = 1, and A = 0, we have an
estimator ST which coincides with the maximum likelihood estimator (see
Basawa and RaQ (1980, Theorem 5.1) for example).
In the general case
the consistency of the estimator depends crucially on the ergodicity of
the process.
11
4. Discussion
In the preceding sections we have considered both ensemble and temporal
estimation.
There are interesting connections between the two methods.
If the systems described by (1) and (2) are ergodic and satisfy the
appropriate mixing conditions, then the moment recursion relations of
Section 2 can be used with temporal moments (instead of ensemble moments)
to generate estimates of the parameters.
It is interesting to observe that,
in the case of Wiener noise only, this method coincides with the "minimum
contrast" procedure of Lanska (1979).
Lanska showed that the minimum
contrast estimates are strongly consistent and asymptotically normal.
With respect to the question of
existence and uniqueness of solutions
for (1) and (2), note that the usual Lipschitz condition is not met by the
function y(x), except when d = 1.
yeA)
y(x) =
+
y(x)
y(B)
+
However if we modify y so that it reads
y (A) (x - A) for x
x
for B
> A,
y (B) (x - B) for x
< B,
x
< x < A,
where yx denotes differentiation with respect to x.
The solutions of
the modified process coincide exactly with the solutions of (1) and (2)
up until a random hitting time
T,
when x t first reaches A or B.
Using
this construction and a theorem of Kallianpur and Wolpert (1984), a
heuristic argument for the existence and uniqueness of solutions for (1)
and (2) in the presence of both Wiener and Poisson noise is obtained.
In the foregoing we have assumed that the noise process has already been
characterized, and that we need only to estimate the parameters of y(x).
Jointly estimating all parameters is a problem that has been addressed in
some special cases by Lansky (1983) and Habib (1985).
12
ACKNOWLEDGEMENTS
We would like to thank several of our colleagues, particularly Shelly
Zacks, Ian McKeague and David Dickey, for their comments on a draft of
this manuscript.
REFERENCES
ARNOLD, L. AND LEFEVER, R., Eds. (1981) Stochastic Nonlinear Systems in
Physics, Chemistry, and Biology.
Proceedings of the Workshop Bielefeld,Fed.
Rep. of Germany, Oct. 5-11, 1980.
Springer-Verlag, Berlin.
BASAWA, I. V. AND PRAKASA RAO, B.L.S. (1980) Statistical Inference for
Stochastic Processes.
Academic Press, New York.
COBB, L., KOPPSTEIN, P., CHEN, N. H. (1983) Estimation and moment
recursion relations for multimodal distributions of the exponential family.
J. Amer. Stat. Assoc. 78, 124-130.
COBB, L. AND ZACKS, S. (1985) Applications of catastrophe theory for
statistical modeling in the biosciences. J. Amer. Stat. Assoc. 80, 793-802.
GIHMAN, I. I. AND SKOROHOD, A. V. (1972) Stochastic Differential
Equations.
Springer-Verlag, New York.
HABIB, M. K. (1985) Parameter estimation for randomly stopped diffusion
processes and neuronal modeling.
Institute of Statistics Mimeograph
Series #1492, Department of Biostatistics, University of North Carolina at
Chapel Hill.
HANSON, F. B. AND TUCKWELL, H. C.
density independent disasters.
(1981) Logistic growth with random
Theoret. Popn. BioI. 19, 1-18.
HANSON, F. B. AND TUCKWELL, H. C. (1983) Diffusion approximations for
neuronal activity including synaptic reversal potentials. J. Theoret.
Neurobiol. 2, 127-153.
13
KALLIANPUR, G. AND WOLPERT, R. (1984) Weak convergence of solutions of
stochastic differential equations with applications to nonlinear neuronal
models.
Center for Stochastic Processes Technical Report 60, Department of
Statistics, University of North Carolina at Chapel Hill.
KIPNIS, C_AND NEWMAN, C. M.
(1985) The metastable behaviour of
infrequently observed, weakly random, one-dimensional diffusion processes.
SIAM J. Appl. Math. 45, 972-982.
LANSKA, V. (1979) Minimum contrast estimation in diffusion processes.
J. Appl. Probe 16, 65-75.
LANSKY, P. (1983) Inference for the diffusion models of neuronal
activity.
Math. Biosci. 67, 247-260.
LIPSTER, R. S. AND SHIRYAYEV, A. N (1977) Statistics of Random Processes I:
General Theory.
Springer-Verlag, New York.
MALLIARIS, A, G. AND BROCK, W. A. (1982) Stochastic Methods in Economics
and Finance. North-Holland, Amsterdam.
RYAN, D. AND HANSON, F. B. (1985) Optimal harvesting with exponential
growth in an environment with random disasters and bonanzas. Math. Biosci. 74,
37-57.
SMITH, C. E. AND COBB. L. (1982) The stationary moments of Poissondriven nonlinear dynamical systems. J. Appl. Probe 19, 702-706.
SMITH, C. E. AND SMITH, M. V. (1984) Moments of voltage trajectories
for Stein's
model with synaptic reversal potentials.
J. Theoret. Neurobiol. 3,
67-77.
TUCKWELL, H. C. (1979) Synaptic transmission in a model for stochastic
neural activity. J. Theor. BioI. 77, 65-81.
14
TUCKWELL, H. C. (1981) Poisson processes in Biology.
In Stochastic
Nonlinear Systems, eds. L. Arnold and R. Lefever. Distributed by SpringerVerlag, Berlin, 162-173.
WILBUR, J. A. AND RINZEL, J. A. (1983) A theoretical basis for large
coefficient of variation and bimodality in neuronal interspike interval
distributions. J. Theor. BioI. 105, 345-368.
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Parameter Estimation for Stochastic Differential Equations Driven by Wiener and
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12. PERSONAL AUTHOR(S)
Charles E. Smith and Loren Cobb
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COSATI CODES
FIELD
GROUP
19 ABSTRACT (Continu.
'8 SUBJECT TERMS (Continue on ,everse if necessary and identIfy by blo<lc
num~'J
SUB·GROUP
on
"verse if neceuoJry and idMti(y by blo<k num"','
Ensemble and temporal parameter estimators are developed for linear and nonlinear
stochastic differential equations driven by both Wiener and Poisson processes.
Linear moment recursion relations are obtained for the stationary moments of the process.
Consistency and asymptotic normality of the resulting ensemble estimators are demonstrate~
The temporal estimators, i. e. , using a single temporal record, are shown to coincide with
maximum likelihood estimators in the special case of linear systems driven by Wiener
D
I
noise.
!
!
I
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UNClASSIF IE OIUNlIMlT E0
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