NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH
PARTIAL INFORMATION
JIANHUI HUANG†, MICHAEL KOURITZIN‡
Abstract. Here, we investigate the nonlinear filtering problems of stochastic climate
models with the presence of mechanism noise in climate proxy datum. The underlying
state process is the intrinsic climate or temperature process which can be formulated by
a very general Markov process taking value in a complete, separable metric space. This
formulation includes the diffusion process driven by Brownian motion or α−stable process,
the jumping-diffusion process driven by (Poisson) random measure as its special cases.
All the processes arise naturally in the study of stochastic climatology when considering
different geophysical characters. The observation process is given by the climate proxy
(e.g., δ 18 O ratio from ice-core reproduction) which is additive by some observation noise
or ice-mass irregularities. The Duncan-Mortensen-Zakai (DMZ) equation and its robust
version for the climate filtering problem are derived based on the martingale problem
formulation, gauge transform and equivalent random measure transform. We also deal
with the model selection problem to different climate models. The Bayes factor and its
robust version are also derived to this end. For numerical computation purpose, the particle
filter and its particle MCMC method are also presented here.
1. Introduction
This paper focuses on the stochastic filtering and model selection (or calibration) problems for climate models. It is well known the climate change has been extensively studied
during the past few decades. Due to the intrinsic uncertainties inherited, considerable stochastic climate models are proposed to explain the climate mechanism. For example, the
atmosphere forcing as well as the glacial-interglacial climate cycles in the last 800kyr BP.
On the other hand, the paleoclimatic data is only available through some climate-proxy
such as the ice-core record. Nevertheless, it is well recognized that there exist considerable mechanism irregularities in the formulation of ice-core models including the ice-sheet
shifting, wind perturbation, air bubbles, etc. Therefore, it is more feasible to consider the
stochastic filtering to climate model by taking into account these mechanism irregularities or observation noises. As a response, this paper aims to develop the basic method of
stochastic filtering to climate models with observation noise.
Now, we first introduce the stochastic nonlinear filtering in a rather general framework.
Let (Ω, F, P) be a complete probability space on which all stochastic processes (e.g., the
underlying climate state and its proxy, e.g., ice-core) will be defined. Let N denote all
the null sets in (Ω, F, P) and for any stochastic process S, define its augmented natural
filtration as
FtS , σ{Su : 0 ≤ u ≤ t} ∨ N .
The filtering problem, can be formulated through a pair of processes (X, Y ), where the signal
or state X is a latent process (the climate state such as the temperature, or atmosphereocean interaction) which is always assumed to be Markov; the observation Y is some functional of X but corrupted with noise (for example, the climate proxy with additive ice-core
Key words and phrases. Nonlinear Filtering, Duncan-Mortensen-Zakai (DMZ) Equation, Robust Filter,
Stochastic Climate Models, α−Stable Temperature Process, Double-well Potential Climate Function, Icecore Climate Proxy, Glacial-interglacial Cycles.
1
2
JIANHUI HUANG†, MICHAEL KOURITZIN‡
mechanism irregularities, see McConnell et al. 2000). The available information to X is
just the observation filtration FtY and the primary goal of filtering is hence to recursively
characterize the conditional distribution
(1.1)
πt (·) = P[Xt ∈ ·|FtY ].
In this way, we get the stochastic filtering problem for climate models. Now we first briefly
introduce the mathematical expects of nonlinear filtering theory as follows. Stemming
from Fujisaki, Kallianpur, Kunita (1972) and Kushner (1967), it is known that under
suitable regularity conditions (see Kouritzin and Long (2006) for more general setup by
relaxing the finite energy condition), the optimal filter πt satisfies some stochastic differential equations driven by the observation process and these equations are often called the
Fujisaki-Kallianpur-Kunita (FKK) or Kushner-Stratonovich (KS) equation (for the conditional probability density process). An equivalent but much simpler equation is the DuncanMortensen-Zakai (DMZ) equation for the unnormalized conditional distribution σt which
can be linked to πt through the Kallianpur-Stribel (KS) formula. These filtering equations,
in principle, yield the optimal filter in the theoretical sense and the optimal filtering problem seems to be completely solved. However, as explained by Clark (1978), these filtering
equations are actually impractical to implement in real life as their sensitivity to the possible modeling errors is not readily apparent. The most common modeling inaccuracy is
caused by the popular use of Brownian motion as the idealized observation noise in filtering
framework. A robust representation of the conditional distribution was first introduced by
Clark (1978) to discern and potentially decrease the effects of modeling errors. Such robust
representation is called the “robust filter” and its rigorous mathematical deduction was
provided in Clark and Crisan (2005) in full detail. A substantial number of papers have
studied the robust filter since its introduction. The papers having a direct connection to
our work are the ones of Davis (1980, 1981) where a semi-group approach to robust filter
is proposed when the signal is Hunt process. Another important related work is Heunis
(1990) where a stochastic partial differential equation approach is given when the signal is
multi-dimensional diffusion process.
The robust filter has great importance to our climate model in partial information setup
due to the following reasons: (i) Due to the relatively few data presented in climate record,
it is more reliable to investigate the robust filter which is more immune to the possible errors
in interpolation in both spatial and temporal grid; (ii) It is always the case the underlying
climate state or its climate proxy mechanism are subject to some formulation uncertainties
thus it is more necessary to investigate the robust filter. To this end, our work aims to
derive the robust filter through two probability transforms: the first transform changes the
probability law of the observation and keep that of signal unchanged; the second pathdependent transform changes the law of the signal and the conditional expectation takes
the form of the robust filter is derived based on this new probability law. The advantage of
this new signal law is that, the conditional expectation takes some form of the FeynmanKac multiplicative functional under it and its operator can be easily derived. This is also
important to the potential computation of complex climate models.
The rest of this paper is organized as follows. Section 2 introduces the basic filtering
setup and some preliminary results when studying the stochastic climatology. Section 3 is
devoted to the robust filter via the path-dependent probability measure transform where
we impose the assumption A6. In section 4, we drop the assumption A6 and derive the
robust filter in a more general setup with the help of the random measures. Section 5
turns to the study of Bayes factor for climatology model selection. The robust Bayes factor
method is introduced in Section 6. The relation to mean-field linear-quadratic-Gaussian
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
3
(LQG) control, stochastic delayed control problem are also discussed. Section 7 discusses
the numerical computation to stochastic climatology through the particle filtering method.
2. Notations and Basic Setup
For any Polish space E, we let M (E) be the set of real-valued, Borel measurable functions
on E, B(E) denote the Banach space of bounded measurable functions with the sup-norms
∥ · ∥∞ and Cb (E) the subspace of real-valued bounded continuous functions on E. DE [0, ∞)
represents the space of all right-continuous left-limit (RCLL) functions from [0, ∞) into E
endowed with the Skorohod topology; CE [0, ∞) will be the subspace of continuous functions
from [0, ∞) into E with the topology of unform convergence. The underlying climatology
state Xt is of DE [0, ∞) sample paths, we let Xt− = lims↑t Xs and ∆Xt = Xt − Xt− for any
t > 0. For any semi-martingales M and N , [M, N ] denotes their cross variation process and
⟨M, N ⟩ their angle bracket process if it can be well defined. For generality, we use the local
martingale problem introduced by Either and Kurtz (1986, page 224).
Definition 2.1. For Lt ⊂ M (E) × M (E) with the common domain D(L), an E-valued process X is said to be a solution of the (Lt , µ)-local martingale problem if its initial distribution
is µ = P ◦ X0−1 and for all f ∈ D(L),
∫ t
(2.1)
Mtf = f (Xt ) − f (X0 ) −
Ls f (Xs )ds
0
{FtX }−local
is a
martingale. If its sample paths are in DE [0, ∞), we called it a DE [0, ∞)−
solution to local martingale problem.
The setup of local martingale problem allows us to work with more general domains than
those for the corresponding martingale problem. For ease of notation, we also define the
Lie bracket for operator Lt :
Definition 2.2. For f1 , f2 ∈ D(L), the Lie bracket for Lt is defined as
(2.2)
[f1 , f2 ]t , (Lt f1 f2 − f1 Lt f2 − f2 Lt f1 ).
2.1. Filtering Setup for Stochastic Climate Models. Consider the additive white
noise observation model:
∫ t
(2.3)
Yt =
g(s, Xs )ds + Bt ,
t ≥ 0,
0
where the climate state process X is a DE [0, ∞) solution of the (Lt , µ)-local martingale
problem; the time-dependent sensor function g = g(t, x) is a Rp -valued Borel function and
we write gt (·) = g(t, ·); the observation noise B is a Rp -valued Brownian motion independent
of X. One concrete example of X from stochastic climate model is the following stochastic resonance model (SRM) introduced by Benzi, Sutera, and Vulpiani (1981) through a
Langevin equation:
t
(2.4)
dXt = [−U ′ (Xt ) + A cos ]dt + σdWt
T
where the climate state Xt can represent the evolution of temperature; U (x) is some pseudoclimate potential, period T is related to the glaciation cycles, Bt is a standard Brownian
motion, A, σ are magnitude parameters. Based on SRM and motivated by Gihman and
Skorohod (1979), Heunis (1990), Itô (2004), the state process X is characterized by the
following stochastic differential equation
t
(2.5)
dXt = [−U ′ (Xt ) + A cos ]dt + σ1 dVt + σ2 dLt ,
T
4
JIANHUI HUANG†, MICHAEL KOURITZIN‡
where U (x) is the pseudo-climate potential, T is the magnitude of periodicity while A is
the scaling parameter, the first driving term Vt satisfies
√
2
(2.6)
dVt = −ρ Vt dt + ρ 1 + Vt2 dWt ,
where the scaling parameter ρ signifies the (inter-annual) correlation time, Bt is a standard
Brownian motion, the second driving term Lt is a alpha-stable process with stable index
α ≤ 2. It follows our formulation (2.5), (2.6) includes the SRM as its special case (σ1 = 0,
σ2 = σ, α = 2). However, our formulation is more flexible to accommodate the statistical
characters like the heavy-tail of era-transitions founded in Ditlevsen (1999). As to the
pseudo-potential function, we will apply the Fokker-Planck equation approach proposed in
Fujisaki, Kallianpur and Kunita (1972), Heunis (1990), Holley and Stroock (1981), and the
benchmark model will be the following Stommel double well-potential
( 4
)
x
x2
U (x) = 4∆
−
4
2
with ∆ as the height of the potential barrier. The observation process Y is modeled by the
white-noise additive equation
(2.7)
dYt = g(t, Xt )dt + dBt ,
where the sensor function g(t, x) depends on the observation mechanism. Some geophysical
factors of observation noise Y are specified as follows.
Time-scaling noise. This is due to the error embedded in the temperature reconstruction
from ice-core. In principle, for the ice-core timescale based on considering the seasonal
cycles (for example, in δ 18 O, sulfate, etc.), the uncertainty noise will increase with depth
(i.e., the historical time recorded) in an ice-core observation. Such uncertainty is also
affected by the possible eruption of volcanos. Steig et al. (2005) emphasized the need to
distinguish absolute accuracy from relative accuracy. For example, in the 200-year-long
U.S. ITASE ice cores from West Antarctica, they showed that while the absolute accuracy
of the dating was ±2 years, the relative accuracy among several cores was < ±0.5 year, due
to identification of several volcanic marker horizons in each of the cores. To remove the
effects of such uncertainty, we can make average of the observed ice-core data because the
systematic errors in timescale is consistent to all the measurement of our ice-core datum
collected.
Diffusion noise. This uncertainty noise is due to the migration of geochemical signals
occurring in polar ice cores primarily in the upper 60 − 80-m-thick firn layer; in glaciers experiencing summer melt, migration may be much faster and persist to much greater depths.
For cold firn, migration is essentially by molecular diffusion below the upper 10m. Diffusion
uncertainty is important to the extent that the characteristic length of diffusion exceeds the
characteristic depth-resolution at a proxy is being used. The diffusion uncertainty can be
formulated through the vapor diffusion models and the vapor transmission, snow accumulation rate functions can be constructed into our sensor function g(t, x). For more geophysical
mechanism of vapor diffusion and its quantitative analysis, see Cuffey and Steig (1998).
Sampling noise. This noise is due to the possible continuous snowfall in the ice-core sheet.
Consequently, there may raise some bias by snowfall occurring during a particular season,
or during a particular storm. Quantifying this uncertainty can probably be conducted by
evaluating the mean and variance of snowfall events in observational data sets and models.
Spatial noise. this type of noise can be attributed to the chemical composition amount
of the snowfall can varying significantly over short distances because of the local micrometerological effects (e.g., snow dunes). In general, the degree of spatial variance is reduced
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
5
when greater time averages are considered. Quantifying this uncertainty requires obtaining multiple ice cores from nearby locations. In mathematics, we can model such noise by
considering the spatial-temporal observation model (see Kurtz and Xiong 1999) by considering some space-time Gaussian noise, or considering the possible spatially-distributed
multi-agent data collection. The former generates the following more general index-valued
observation model
∫ t∫
(2.8)
Y (A, t) =
g(u, Xs )µ(du)dt + W (A, t).
0
A
Here, W is some spatial-time Gaussian white noise field, A denotes the possible spatial
distribution of ice-core sheets. The concrete example includes the spatially-distributed icecore datums from the Antarctic and Greenland. The latter will suggest some correlation
analysis on time-series, or the datum assimilation to mean-field stochastic models (e.g.,
the McKean-Vlasov equation). In both cases, the particle filtering representation will be
needed. This can be solved by using the particle representation of stochastic partial differential equations to the derived nonlinear filters. Furthermore, we list some mathematical
assumptions to our state and observation processes used here:
A 1. D(L) is closed under multiplication.
A 2. For f ∈ D(L) and T > 0,
sup E|f (Xt )|2 < ∞,
(2.9)
0≤t≤T
∫
T
E
(2.10)
|Lt f (Xt )|2 dt < ∞.
0
A 3. For all f ∈ D(L), t ≥ 0,
E[M f , M f ]t < ∞.
(2.11)
A 4. For any t ∈ [0, t], gt ∈ D(L).
A 5. For T > 0,
∫
T
(2.12)
|g(s, Xs )|2 ds < ∞
P a.s.,
0
(2.13)
1
E exp(
2
∫
t
[g, g]s (Xs )ds) < ∞
∀t ∈ [0, T ].
0
A 6. The sample paths of g(t, Xt ) is continuous.
Remark 2.1. Following Kouritzin and Long (2006), here we suppose the more general
condition (2.7) instead the following finite energy condition,
∫ T
E
|g(s, Xs )|2 ds < ∞.
0
Such condition relaxation is important when we deal with the stochastic climate models by
some α-stable process with double-well potential function where the empirical estimate of
stable index is around to be α = 1.75 therefore the second moment of state (climate) process
does not exist (see Ditlevsen 1999).
6
JIANHUI HUANG†, MICHAEL KOURITZIN‡
Remark 2.2. From A4,
(2.14)
∫
t
Ls g(s, Xs )ds + Mtg
g(t, Xt ) = g(0, X0 ) +
0
for some local martingale
(2.15)
Mtg .
If A 6 also holds true, then
∫ t
g
Mt = g(t, Xt ) − g(0, X0 ) −
Ls g(s, Xs )ds
0
is a continuous local martingale.
To facilitate the path-dependent probability transformation, we must determine a condition under which local martingale M f is actually a martingale for f ∈ D(L). Clearly, A3
is a sufficient condition for each M f to be a square-integrable martingale and A3 follows if
for all f ∈ D(L), ⟨M f c , M f c ⟩t is integrable and ∆f (Xt ) is bounded.
Proposition 2.1. Assume A2, then for f ∈ D(A), Mtf is a locally square integrable martingale.
Proof. Apply the Schwarz inequality directly.
From proposition 2.1, for any f1 , f2 ∈ D(L), the angle bracket process ⟨M f1 , M f2 ⟩t is
well defined whenever A2 holds true. Actually, we have the following important result.
Lemma 2.1. If A1, A2 hold true, then for f1 , f2 ∈ D(L),
∫ t
f1
f2
(2.16)
⟨M , M ⟩t =
[f1 , f2 ]s (Xs )ds,
0
Moreover, for f ∈ D(L),
Mtf
is quasi-left-continuous.
Proof. First, for f ∈ D(L),
∫
M (t) = f (Xt ) − f (X0 ) −
f
t
Ls f (Xs )ds
0
is a local martingale with M f (0) ≡ 0. Next, from A 1, f 2 ∈ D(L) and
∫ t
2
2
2
f (Xt ) = f (X0 ) +
Ls f 2 (Xs )ds + M f (t)
0
f2
for some RCLL local martingale M (t). Applying Itô’s formula,
∫ t
∫ t
2
2
f (Xt ) = f (X0 ) +
2f (Xs− )Ls f (Xs )ds +
2f (Xs− )dM f (s) + [M f , M f ]t .
0
0
Therefore,
∫ t
∫ t
∫ t
2
f
f
f2
2f (Xs− )Ls f (Xs )ds −
Ls f (Xs )ds + [M , M ]t = M (t) −
2f (Xs− )dM f (s).
0
0
0
However,
Kt = [M f , M f ]t − ⟨M f , M f ⟩t
(2.17)
is a local martingale so we have
∫
0
t
∫
2f (Xs− )Ls f (Xs )ds −
t
0
f2
∫
Ls f (Xs )ds + ⟨M , M ⟩t = M (t) −
2
f
f
0
t
2f (Xs− )dM f (s) − Kt .
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
7
The left hand side is a RCLL, adapted predictable finite variation process null at zero while
the right hand side is a local martingale starting from zero. Thus, from the decomposition
uniqueness to special semi-martingale, we have
∫ t
∫ t
2
f
f
(2.18)
(Ls f (Xs ) − 2f (Xs )Ls f (Xs ))ds =
[f, f ]s (Xs )ds.
⟨M , M ⟩t =
0
0
Following from the polarization, for f1 , f2 ∈ D(L),
∫ t
f1
f2
(2.19)
⟨M , M ⟩t =
[f1 , f2 ]s (Xs )ds.
0
⟨M f , M f ⟩t
Moreover, for f ∈ D(L),
is absolutely continuous with respect to the Lebesgue
f
measure, thus Mt is quasi-left-continuous.
2.2. Reference Probability Measure and Stochastic Filtering Equation. Now introduce the augmented filtration
Ft , σ{Xs , Ys : 0 ≤ s ≤ t} ∨ N .
Note that
(2.20)
∫
Λt , exp (−
t
g(s, Xs )dYs +
0
1
2
∫
t
|g(s, Xs )|2 ds)
0
is a {Ft }t≥0 -martingale by A5 and
dP
= ΛT
dP
(2.21)
defines a reference probability measure P equivalent to P on (Ω, FT ). As a standard result
in filtering theory, we have,
Proposition 2.2. Under P, Y is a Brownian motion independent of X and the law of X
under P is same as its law under P.
For f ∈ B(E), the Kallianpur-Striebel-Bayes formula (see Kallianpur 1980, Page 283)
links πt with the unnormalized filter σt by
(2.22)
πt (f ) =
σt (f )
σt (1)
where
Y
σt (f ) , E[f (Xt )Λ−1
t |Ft ].
(2.23)
Furthermore, πt (f ) and σt (f ) are characterized respectively by the following FKK and DMZ
equation. (We refer the reader to Fujisaki, Kallianpur, Kunita (1972) and Zakai (1969) for
the details)
Theorem 2.1. (FKK equation) Suppose A1-A4 hold true. If f ∈ D(L), then πt (f ) satisfies
the following stochastic differential equations.
∫ t
∫ t
(2.24)
πt (f ) = π0 (f ) +
πs (Ls f )ds +
πs (gs f ) − πs (f )πs (gs )(dYs − πs (g)ds).
0
0
Theorem 2.2. (DMZ equation) Under the conditions of Theorem (1.1), for f ∈ D(L), the
unnormalized filter σt (f ) satisfies
∫ t
∫ t
(2.25)
σt (f ) = σ0 (f ) +
σs (Ls f )ds +
σs (gs f )dYs ,
σ0 (f ) = E(f (X0 )).
0
0
8
JIANHUI HUANG†, MICHAEL KOURITZIN‡
Unfortunately, the representations of optimal filter through the FKK or DMZ equations
are not robust to observation equation modeling error and there are stochastic integrals
to be evaluated. However, in case of X taking values in locally compact space such as
the finite dimensional diffusion process, the above Zakai equation can be transformed to
some equivalent but non-stochastic partial differential equation parameterized by its sample
path which implies the robustness. As mentioned at beginning, Clark (1978) introduced
the notion of “robust filter” to create versions of the optimal filter that are continuously
dependent on the underlying observation path. Clark (1978) showed that such version, if
it exist, shall be unique under mild conditions. Pardoux (1979), Henuis (1990) obtained
the same result using the approach of Feynman-Kac formula and path-dependent measure
change. Davis (1980, 1981) generalized the above results to the case of the signal as some
Hunt process in locally compact Hausdorff spaces. Empirical results have demonstrated
that the robust filter does indeed perform favorably when applied to real data problem
where the Brownian observation noise assumptions are unrealistic. In particular, when we
study the stochastic climate datum such as the ice-core reconstruction.
3. Robust Filter via Path-dependent Probability Transform
In the remaining of this paper, denote y = {yt , t ≥ 0} an arbitrary but fixed observation
trajectory, in other words, yt = Y (t, ω) for all t ≥ 0 and some ω ∈ Ω.
Definition 3.1. For f ∈ B(E), define the gauge transform of X as
(3.1)
Y
νt (f ) , E[f (Xt )Λ−1
t exp(−Yt gt (Xt ))|Ft ],
and it is equivalent to σt in the following sense,
(3.2)
νt (f ) = σt (f exp(−Yt gt ))
and
σt (f ) = νt (f exp(Yt gt )).
The main result of this section characterizes the robust filter of νt (f ) recursively when
we assume A 6, that is,
Theorem 3.1. Assume A1-A6 hold true, then νt (f ) satisfies the evolution equation
2
d
b y f ) + νt (f {−yt Lt gt − 1 |gt |2 + yt [g, g]t })
νt (f ) = νt (L
t
dt
2
2
y
b y ) = D(L) where L
b f = Lt f − yt [g, f ]t .
for all f ∈ D(L
t
(3.3)
Its proof relies on some path-dependent probability transform and we first present some
preliminary results.
Proposition 3.1. Let M and N be two semi-martingales of which N is continuous such
that d[N ]t (ω) is absolutely continuous to Lebesgue measure a.s. , then we have
∫ t
∫ t
(3.4)
Ms dNs = Nt Mt −
Ns dMs − [M, N ]t .
0
0
Proof. By integration by parts with continuous N ,
∫ t
∫ t
(3.5)
Nt Mt =
Ms− dNs +
Ns dMs + [M, N ]t .
0
0
d[N ]t
Denote λN
t the Radon-Nikodym derivative dt , then on ([0, ∞) × Ω, B([0, ∞)) ⊗ F), we
introduce the Doléans measure νN as
∫ ∞
∫ ∞
νN (A) , E
1A (t, ω)d[N ]t = E
1A (t, ω)λN
t dt.
0
0
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
9
Moreover, because predictable process Mt− has at most a countable number of discontinuity
points, it is equivalent to Mt under νN and
∫ t
∫ t
Ms dNs
a.e.
Ms− dNs =
0
0
Hence the result.
Proposition 3.2. Suppose X and Y are respectively the signal and observation processes
introduced in (2.1), (2.3), then we have
∫ t
∫ t
(3.6)
g(s, Xs )dYs = Yt g(t, Xt ) −
Ys dg(s, Xs )
0
0
Proof. By the (Lt , µ)-local martingale problem and A 4,
∫ t
g(t, Xt ) = g(0, X0 ) +
g(s, Xs )ds + Mtg ,
0
∫t
and Yt = 0 g(t, Xt )dt + Bt are both special semi-martingales with Yt is continuous so
[Y, Y ]t = ⟨Y, Y ⟩t = t, then from Proposition 3.1, we have
∫
(3.7)
t
∫
g(s, Xs )dYs = Yt g(t, Xt ) −
0
t
Ys dg(t, Xt ) − [g(X), Y ]t .
0
On the other hand, we need show
[g(X), Y ]t = [M g , B]t = ⟨M g , B⟩t = 0.
(3.8)
For any two semi-martingales M, N , [M, N ] and ⟨M, N ⟩t are both preserved ( in the sense
of modifications) under equivalent measures so we focus ourself on P under which M g and
B are independent. The first identity is obvious and the second identity comes from the
fact that M g and B are continuous. Moreover, [M g , B]t = ⟨M g , B⟩t are also continuous
process so their versions under P or P are indistinguishable. To show [M g , B]t = 0, it is
sufficient to show (M g B)t is a local martingale under P. Due the localization, we need only
show in the martingale case, say, (M g B)t is a martingale under P provided M g and B are
both P-martingales. Following Fujisaki, Kallianpur and Kunita (1972),
E((M g B)t − (M g B)s |Fs ) = E((Mtg − Msg )(Bt − Bs )|Fs ) = 0,
where the second equality is due to the independence of Bt − Bs to Fs ∨ σ{Mtg − Msg } under
P, thus proving (3.8). Consequently,
∫ t
∫ t
g(s, Xs )dYs = Yt g(t, Xt ) −
Ys dg(s, Xs ).
0
0
From (2.16) and Proposition 3.2,
∫ t
∫ t
1
2
(3.9)
νt (f ) = E[f (Xt ) exp(− {Ys Lg(s, Xs ) + |g(s, Xs )| }ds) exp(−
Ys dMsg )|FtY ].
2
0
0
Moveover, the independence of X and Y under P implies
(3.10)
where
νt (f ) = E[f (Xt ) · Ξt · Ayt ],
10
(3.11)
JIANHUI HUANG†, MICHAEL KOURITZIN‡
∫ t
1
y2
Ξt , exp( {−ys Lg(s, Xs ) − |g(s, Xs )|2 + s [g, g]s (Xs )}ds),
2
2
0
and
∫
(3.12)
Ayt
, exp(−
∫
t
ys dMsg
−
0
Lemma 3.1. If A4, A5, A6 hold true, then
t
0
Ayt
ys2
[g, g]s (Xs )ds)
2
is a continuous martingale.
Proof. From A 6, we know Ayt is a continuous process and
(3.13)
dAyt = −Ayt yt dMtg ,
which implies Ayt is a local martingale. Meanwhile, from A 4,
∫
1 t
[g, g]s (Xs )ds) < ∞,
∀t ∈ [0, T ]
E exp(
2 0
then from the Novikov criteria, Ayt is a martingale.
After the first measure change to P, the law of the signal process X remains unchanged.
In contrast, with the second path-dependent measure change, the law of X will be changed
and characterized by some observation-dependent martingale problem. The gauge transform
ν(·) will then take the form of Feynman-Kac multiplicative functional. Now, we are ready
to introduce the path-dependent probability transform. For given y(·), introduce the pathby by
dependent probability measure P
by dP
y
= At ,
dP Ft
under which the gauge transform can be characterized in the form of Feynman-Kac multiplicative functional:
b y [f (Xt )Ξt ],
νt (f ) = E
where
(3.14)
∫ t
1
y2
Ξt = exp( {−ys Ls g(Xs ) − |g(Xs )|2 + s [g, g]s (Xs )}ds).
2
2
0
by , X is a RCLL solution to the L
b y martingale
Lemma 3.2. Assume A1-A6, then under P
problem
(3.15)
b y f (Xt )dt + dM
cf ,
df (Xt ) = L
t
t
b y ) = D(L) and for f ∈ D(L
b y ),
where D(L
(3.16)
b y f , Lt f − yt [g, f ]t ,
L
t
(3.17)
cf , dM f + yt [g, f ]t (Xt )dt.
dM
t
t
Moreover, the Kolmogorov forward equation of X is
∫ t
y
y
b
b
by (L
b ys f )ds.
(3.18)
Pt (f ) = P0 (f ) +
P
s
0
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
11
Proof. For f ∈ D(A), it is obvious that
b y f (Xt )dt + dM
cf
df (Xt ) = L
t
t
(3.19)
cf is a martingale. Due to the continuity of Ay , we have
and we only need to show M
t
∆[M f , Ay ]t = ∆(M f )t · ∆(Ay )t = 0,
therefore
∫
[M , A ]t = ⟨M , A ⟩t =
f
y
f
y
t
−Ays ys [f, g]s ds,
0
and
cf
dM
t
(3.20)
= dMtf −
1
d[M f , Ay ]t .
Ayt
by and
cf is a local martingale under P
Thus, from the Girsanov-Meyer theorem, M
t
b y [M
cf , M
cf ]t = E([M f , M f ]Ay )t < ∞.
E
by . cf is a martingale under P
Moreover, from A2, A3 and proposition 1.3, it follows that M
t
Proof of Theorem 3.1.
Proof. We have
b y (f (Xt )Ξt ).
νt (f ) = E
From (3.13) and integration by parts,
b y f (Xt )Ξt dt + f (Xt )Ξt {−yt Lt g(Xt ) − 1 |g(Xt )|2
d(f (Xt )Ξt ) = L
t
2
2
yt
cf .
+
[g, g]t (Xt )}dt + Ξt dM
t
2
∫t
by
by
cf
0 Ξs dMs is a P -martingale. Now, taking the expectation under P , we have
∫ t
∫ t
1 2 ys2
b y f )ds +
νt (f ) = ν0 (f ) +
νs (L
ν
(f
{−y
L
g
−
|g| + [g, g]s })ds.
s
s s
s
2
2
0
0
(3.21)
4. Robust Filter via Random Measure
In this section, we drop off the condition A6, that is, we no longer assume Mtg is a
continuous, instead, only a RCLL martingale and we derive the robust filter in this case
applying the random measure approach. Our result includes that of Davis (1980, 1981) as
its special case. An excellent account to the general random measure theory can be found
in Jacod and Shiryaev (1987). Here, we focus on the finite integer-valued random measure
which definition is
Definition 4.1. A finite integer-valued random measure ν defined on [0, t] × R is a random
measure satisfies (1) ν(ω, {t} × R) ≤ 1, (2) for each A ∈ B(R), ν(ω, [0, t] × A) ∈ N. (3) ν is
e − σ-finite. Here P
e = P × B(R) and P is the predictable algebra of Ω × [0, T ].
optional and P
A fundamental example of the integer-valued random measure of our interests will be
12
JIANHUI HUANG†, MICHAEL KOURITZIN‡
Definition 4.2. (Jump random measure of a RCLL process) Suppose Z is an arbitrary
R-valued RCLL process, then for any ω ∈ Ω,
∑
(4.1)
ν Z (ω, dt, dz) ,
δ{s,∆Zs (ω)} (dt, dz)1{∆Zs (ω)̸=0}
0≤s≤T
defines a finite random measure ν Z (ω, ·) on R+ × E where δ denotes the Dirac measure.
This formulation is introduced here to represent the possible volcano eruptions.
Definition 4.3. (Gihman, Skorohod, 1979, Page 88) An orthogonal local martingale measure is a random measure µ = µ(ω, ·) defined on R+ ×E if the stochastic process µ(ω, [0, t]×
A1 )·µ(ω, [0, t]×A1 ) is a locally square integrable martingale and for any A, the angle bracket
process ⟨µ(ω, [0, t] × A), µ(ω, [0, t] × A)⟩t = π(ω, [0, t] × A), where π(ω, [0, t] × A) is a random
measure which is a continuous monotonically nondecreasing integrable process for a fixed A.
π = π(ω, ·) is its compensator or characteristic.
A useful and general result will be the following
Proposition 4.1. (Gihman and Skorohod, 1979, page 85, 88) For an arbitrary integervalued random measure ν(t, A), if it satisfies (1) for ∀t ≥ 0; A ∈ B(R) such that A∩(−ε, ε) =
∅ for some ε > 0, then Eν(t, A) < ∞. (2) for a fixed A, the function ν(t, A) is monotonically
nondecreasing and RCLL. (3) for an arbitrary monotonically nondecreasing sequence of
stopping times τn such that lim τn = τ ≤ T ,
lim Eν(τn , A) = Eν(τ, A),
then it admits a unique decomposition with the form
(4.2)
ν(t, A) = µ(t, A) + π(t, A),
where µ is an orthogonal local martingale measure with the compensator π which is predictable.
Proposition 4.2. For the compensator π in Proposition 4.1, there exists a predictable, integrable and finite variation process A, a kernel K(ω, t, dz) from (Ω × [0, T ], P) into (R, B(R))
such that
(4.3)
π(ω, dt, dz) = dAt (ω)K(ω, t, dz).
A consequence of proposition 4.2 is the following lemma concerning our (Lt , µ)-local
martingale problem:
Lemma 4.1. If f ∈ D(L), then there exists a unique decomposition
Mtf = Mtf c + Mtf d
with
Mtf d
∫ t∫
zµf (ω, ds, dz).
=
0
R
Here, M f c is a continuous local martingale and M f d a purely discontinuous local martingale;
µf is an orthogonal local martingale measure with compensator π f and
(4.4)
ν f (t, A) = µf (t, A) + π f (t, A),
is the jumping random measure of Mtf .
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
13
For any predictable process H = H(t, ω) satisfying
∫ t∫
(4.5)
H 2 (s, ω)π f (ω, ds, dz) < ∞,
the stochastic integral
R
fd
0 Hs dMs is well defined and the following
∫ t
∫ t∫
fd
Hs dMs =
zH(s, ω)µf (ω, ds, dz).
0
0
R
0
∫t
identity holds true
A 7. For each f ∈ D(L), there exists a kernel K f (ω, t, dz) such that
π f (dt, dz) = K f (ω, t, dz)dt.
Lemma 4.2. Under A1, A2 and A9, for f1 , f2 ∈ D(L), we have
∫ t
∫
(4.6)
⟨M f1 c , M f2 c ⟩t =
{[f1 , f2 ]s (Xs ) −
z 2 K ⟨f1 ,f2 ⟩ (ω, s, dz)}ds,
R
0
where
and K f1 +f2 , K f1 −f2
1
K ⟨f1 ,f2 ⟩ = (K f1 +f2 − K f1 −f2 ),
4
are defined as in A 9.
Proof. First,
∫
∫ t∫
t
fc
f (Xt ) = f (X0 ) +
0
and
∫
2
(4.7)
zµf (ω, ds, dz)
Ls f (Xs )ds + M (t) +
0
2
R
t
2
Ls f 2 (Xs )ds + M f (t)
f (Xt ) = f (X0 ) +
0
2
M f (t)
where
is a RCLL local martingale. On the other hand, from Lemma 2.1, we know
Mtf is quiasi-left-continuous so we can apply the Itô’s formula from Gihman and Skorohod
(1979, page 105),
∫ t
∫ t
2
2
2f (Xs )dM f c (s) + ⟨M f c , M f c ⟩t
f (Xt ) = f (X0 ) +
2f (Xs )Ls f (Xs )ds +
0
0
∫ t∫
∫ t∫
2
f
+
[z + 2zf (X(s, ω)]µ (ω, ds, dz) +
z 2 K f (ω, s, dz)ds
0
R
0
R
∫ t
∫ t∫
(4.8)
= f 2 (X0 ) +
2f (Xs )Ls f (Xs )ds +
z 2 K f (ω, s, dz)ds + ⟨M f c , M f c ⟩t
0
0
R
∫ t∫
∫ t
2
f
+
[z + 2zf (X(s, ω)]µ (ω, ds, dz) +
2f (Xs )dM f c (s).
0
R
0
Note that
∫ t∫
[z 2 + 2zf (X(s, ω)]µf (ω, ds, dz)
0
R
is a local martingale and the decomposition uniqueness of special semi-martingale, then we
have
∫ t∫
fc
fc
f
f
⟨M , M ⟩t = ⟨M , M ⟩t −
z 2 K f (ω, s, dz)ds.
R
0
Thus we get
(4.9)
∫
⟨M , M ⟩t =
fc
fc
t
∫
{[f, f ] (Xs ) −
s
0
z 2 K f (ω, s, dz)}ds
R
14
JIANHUI HUANG†, MICHAEL KOURITZIN‡
From polarization, we have
1
⟨M f1 c , M f2 c ⟩t = {⟨M f1 c+f2 c , M f1 c+f2 c ⟩t − ⟨M f1 c−f2 c , M f1 c−f2 c ⟩t }.
4
As a result,
⟨M
f1 c
,M
f2 c
⟩t = ⟨M , M ⟩t −
f1
∫ t∫
f2
z 2 K f1 ,f2 (ω, s, dz)ds.
R
0
Thus
(4.10)
∫
⟨M
f1 c
,M
f2 c
t
⟩t =
∫
{[f1 , f2 ] (Xs ) −
s
z 2 K f1 ,f2 (ω, s, dz)}ds
R
0
Following Davis (1980), for 0 ≤ s ≤ t, we introduce
∫ t
∫ t
1
t
2
(4.11)
ks , exp{− (yu Lg(Xu ) + |g(Xu )| )du} · exp(−
yu dMug ),
2
s
s
which is a multiplicative functional. However, it is not Kac-type multiplicative functional
[see Itô (2003),
page 164. or Williams (1993), page 272 ] as there exists the stochastic
∫
integration yt dMtg . Next, we can define the two-parameter semi-group on the Banach
space B(E).
(4.12)
y
Ts,t
f (x) = Ex [f (Xt )kst ]
where Ex denotes expectation with Px , the probability measure starting from x ∈ E. Now,
note that the law of X remains unchanged and is still µ, thus we have
(4.13)
y
νt (f ) =< Ts,t
f, µ > .
Hence, to determine the dynamics of the robust filter νt (f ), we can determine the dynamics
y
of Ts,t
, namely, its extended generator which is defined in Davis (1980):
y
Definition 4.4. Suppose kst is a multiplicative functional and Ts,t
the corresponding twoy
parameter semi-group defined in (2.29), then (J, D(J)) is called the extended generator of Ts,t
f
if for each f ∈ D(J) ⊂ M (E), there exist Jf ∈ M (E) such that Ns,t
is a local martingale,
where
∫ t
f
(4.14)
Ns,t
, kst f (Xt ) − f (Xs ) −
ksu Jf (Xu )du.
s
Using the definition of extended generator, we turn to the path-dependent measure change
in the Polish space E. Then, we can employ the martingale problem technique to derive
the extended generator. To achieve this, first note that
∫ t
g
g
(4.15)
⟨M , M ⟩t =
[g, g]s (Xs )ds.
0
A 8. For any f ∈ D(L), there exists a kernel K f (x, dz) such that
K f (ω, t, dz) = K f (X(t, ω), dz)
In particular, such kernel exists for the Hunt process and in such case, it is just the Lévy
system (n(·, ·), ϕ).
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
15
Theorem 4.1. Suppose A1-A5, A8 hold true, then for any f ∈ D(L), νt (f ) satisfies the
evolution equation
∫ t
∫ t
1
f
νt (f ) = ν0 (f ) +
νs (Ls f )ds +
νs (f {−ys Ls gs − |gs |2
2
0
0
∫
2
ys
[g, g]s } + (e−zys − 1 + zys + z 2 )K g (s, dz))ds,
+
2
R
where
ft (f )(·) = Lt (f )(·) − yt [g, f ]t (·) +
L
(4.16)
∫
R
(e−yt z − 1)zK f,g (·, dz)
Proof. From lemma 4.2, we know
∫
⟨M gc , M gc ⟩t =
t
[g, g]s (Xs )ds −
∫ t∫
z 2 K g (ω, s, dz)ds
0
R
0
Thus, it follows that the pathwise evaluation representation takes the form,
νt (f ) = E[f (Xt ) · γt · δt ],
(4.17)
where
(4.18)
∫
δt = exp(
0
t
∫ t
∫ t∫
1 2
1
ys d⟨M gc , M gc ⟩t − {ys Ls gs (Xs )+ |g(s, Xs )|2 }ds+
(e−zys −1+zys )K g (ω, s, dz)}ds),
2
2
0
0
R
and
(4.19)
∫ t
∫ t
∫ t∫
∫ t∫
gc 1
2
gc
gc
g
γt = exp( −ys dMs −
y d⟨M , M ⟩t −
zys µ (ds, dz)−
(e−zys −1+zys )K g (ω, s, dz)}ds),
2 0 s
0
0
R
0
R
From the generalized Itô formula (Gihman and Skorohod, 1979, Page 104 ), it follows that
∫
γt = 1 −
(4.20)
∫ t∫
t
γs ys dMsgc
0
+
R
0
γs (e−zys − 1)µg (ds, dz).
So γt is a local martingale and
∫ t
∫ t
∫ t∫
γt f (Xt ) = γt f (Xs ) +
γu Lu f (Xu )du −
γu yu d⟨M f c , M gc ⟩u +
γu (e−yu z − 1)zK f,g (u, dz)du + It ,
s
s
s
R
where It is a local martingale. Therefore, the extended generator of νet (f ) satisfying
∫
t
f
(4.21)
Lt (f )(Xt ) = Lt (f )(Xt ) − yt [g, f ] (Xt ) + (e−yt z − 1)zK f,g (ω, t, dz)
R
From A8, there exist a K f,g (x, dz) such that
(4.22)
ft (f )(Xt ) = Lt (f )(Xt ) − yt [g, f ]t (Xt ) +
L
∫
R
(e−yt z − 1)zK f,g (X(t, ω), dz)
thus we have
(4.23)
ft (f )(·) = Lt (f )(·) − yt [g, f ]t (·) +
L
∫
R
(e−yt z − 1)zK f,g (·, dz).
16
JIANHUI HUANG†, MICHAEL KOURITZIN‡
On the other hand, δt is a Feynman-Kac multiplicative functional and its effect on νt (f ) is
just adding the following potential term
∫
1 2
1 2
gc
gc
(4.24)
y d⟨M , M ⟩t − yt Lt gt dt − gt dt − (e−zyt − 1 + zyt )K g (ω, t, dz)dt
2 s
2
R
on extended generator.
νet (f ) , E(f (Xt )γt ).
(4.25)
Finally, adding this potential term associated with Ξt , we can write the robust filter informally as
∫ t
∫ t
1
f
νt (f ) = ν0 (f ) +
νs (Ls f )ds +
νs (f {−ys Ls gs − |gs |2
2
0
0
∫
2
ys
+
[g, g]s } + (e−zys − 1 + zys + z 2 )K g (s, dz))ds.
2
R
Hence the result.
5. Bayes Factor for Climate Model Selection
Here, we aim to develop the basic model selection structure using the Bayes factor, and
discuss how to apply it to calibrate the more statistical reliable stochastic climate models.
For sake of presentation, we consider the underlying climate state (i.e., the temperature)
is in finite-dimensional space, that is, X ∈ Rnx . In this way, we do not consider the stochastic evolution equation of climate state with delayed operator (which thus lives in some
infinite-dimensional space). To this end, we can make average of our ice-core datum thus to
remove the possible time-scale inconsistency in historical climate datum). The possible climate evolution or measurement parameter θ ∈ Rnθ jointly satisfy the (D(A), A)-martingale
problem, that is
∫ t
(5.1)
Mtf = f (Xt , θ) − f (X0 , θ) −
Af (Xs , θ)ds
0
{FtX, θ }−martingale
is
for f ∈ D(A) ⊂
Note that the Bayes factor we plan to
introduced will be denoted by L thus here, we denote the generator of martingale problem by
A instead L as above. The martingale problem proposed by Stroock and Varadhan (1979)
provides a general formulation of the Markov processes, and many stochastic systems can
be nested into this setup (see Ethier and Kurtz (1986)). In particular, we can exam two
classes processes used in stochastic climate models:
B(Rnx +nθ ).
Example 1. (Diffusion Climate Process)
1
∂2f
∂f
Af (x, θ) = c2 (x, θ, t) 2 (x, θ, t) + b(x, θ, t) (x, θ, t),
2
∂x
∂x
where b, c are drift and diffusion functions. D(A) is the set of bounded second-order continuously differentiable functions on Rnx +nθ . This includes the SRM climate model as its special
case and in case the coefficients is non-Lipschtize, we can apply the truncation method to
the quadratic growth SDE.
Example 2. (Markov Jump Climate Process)
∫
Af (x, θ) = λ(x, θ)
(f (y, θ) − f (x, θ))µ(x, θ, dy),
Rnx +nθ
where µ(x, θ, dy) is a transition function on (Rnx +nθ × B(Rnx +nθ )) and λ ≥ 0. D(A) =
B(Rnx +nθ ). This type model includes the possible climate state transmission such as the
glacial-interglacial immigration due to the abrupt-change in climate mechanism.
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
17
Now consider the additive white noise observation model:
(5.2)
dYt = h(Xt , θ)dt + dWt ,
where the observation noise W is a standard Brownian motion independent of (X, θ); the
sensor function h = h(x, θ) satisfies the relaxed finite energy condition:
∫ T
(5.3)
|h(Xt , θ)|2 dt < ∞. a.s.
0
Note that the robust Bayes factor is denoted by g(t, x) thus here we denote the sensor
function by H = h(t, x) instead g(t, x) to avoid confusion. The partial information system
(1.1), (1.2) can be used to characterize a wide range of stochastic structures in applied
probability (see Kallianpur (1980), Kailath and Poor (1998) etc. in signal processing; Duffie
and Lando (2001), Back (2003), Frey and Runggaldier (2007) etc. in financial engineering).
The purpose of this section can be roughly expressed in the following question: suppose
there are two candidate models for partial information system,
∫ t
(1), f
M(1) : Mt
= f (Xt , θ) − f (X0 , θ) −
(5.4)
A(1) f (Xs , θ)ds,
0
dYt = h(1) (Xt , θ)dt + dWt ;
(5.5)
M
(2)
(2), f
Mt
:
∫
= f (Xt , θ) − f (X0 , θ) −
t
A(2) f (Xs , θ)ds,
0
dYt = h(2) (Xt , θ)dt + dWt .
We need evaluate which competing model best fits the observed data {Ys : 0 ≤ s ≤ t}.
We propose a Bayesian approach to solve this problem with the help of Bayes factor. Our
approach is efficient and reliable in that it can be updated recursively by incorporating the
new observations. We also illustrate how to define and compute the Bayes factor between
the competing models M(k) = (A(k) , h(k) ), k = 1, 2, and how it can be used to make model
calibration.
The key point of Bayes factor is to transform all observation models into the same canonical process via Girsanov measure change. Note that, for k = 1, 2,
(∫ t
)
∫
1 t (k)
(k)
(k)
2
(5.6)
Lt , exp
h (Xs , θ)dYs −
|h (Xs , θ)| ds
2 0
0
is a Ft -martingale where Ft , FtX,θ; Y . Thus
dP (k)
= Lt
(k)
Ft
dQ
defines a probability measure Q(k) which is mutually absolutely continuous to P. As a standard result of filtering theory (see Kallianpur (1980)), we have
Proposition 5.1. For M(k) , k = 1, 2, Y is a Brownian motion independent of (X, θ) and
the law of (X, θ) keeps unchanged under Q(k) .
For f ∈ B(Rnx +nθ ), introduce the unnormalized filter σ (k) of (X, θ) for M(k) , k = 1, 2:
(
)
(k)
(k)
(5.7)
σ (k) (f, t) , EQ
f (Xt , θ)Lt |FtY .
(k)
It is remarkable that Lt
is the joint likelihood of X, θ, Y in M(k) and
σ (k) (1, t) = EQ (Lt |FtY )
(k)
(k)
18
JIANHUI HUANG†, MICHAEL KOURITZIN‡
is the integrated or marginal likelihood of Y . It characterizes the possibility that the
historical observation {Ys , 0 ≤ s ≤ t} is generated by M(k) , k = 1, 2. The Bayes factors
between M(1) and M(2) are defined as the ratio of the integrated likelihoods:
Definition 5.1. (Bayes factor)
(5.8)
B12 (t) =
σ (1) (1, t)
σ (2) (1, t)
and
B
(t)
=
.
21
σ (2) (1, t)
σ (1) (1, t)
Now introduce the filter ratio process:
Definition 5.2. (Filter ratio)
q1 (f, t) =
(5.9)
σ (1) (f, t)
σ (2) (f, t)
and
q
(f,
t)
=
.
2
σ (2) (1, t)
σ (1) (1, t)
Then we have
(5.10)
B12 (t) = q1 (1, t); B21 (t) = q2 (1, t).
As discussed in Jeffreys (1961), the Bayes factor, say B12 , is the summary of the evidence
provided by the historical observation in favor of M(1) over M(2) . Following Kass and
Raftery (1995), Kouritzin and Zeng (2005), the Bayes factor can be explained by the following table:
B12
Evidence of M(1) over M(2)
1−3
Not worth more than a bare mention
3 − 12
Positive
12 − 150
Strong
> 150
Decisive
Now, the remaining issue is how to calculate the Bayes factor. As discussed in Kouritzin
and Zeng (2005), there exist two alternatives to calculate the Bayes factor. The first one is
calculating σ (k) (1, t), k = 1, 2 respectively and then take the ratio. However, this approach
is not always computationally efficient or numerically stable. It is quite possible that both
σ (1) (1, t) and σ (2) (1, t) get very large or very small as time evolves. Therefore, I focus on the
second approach, i.e., to characterize the dynamics of the Bayes factor (or filter ratio) and
then implement it numerically. The following evolution equation of the filter ratio process
is the main result of this paper.
Theorem 5.1. Suppose there are two models M(k) , k = 1, 2, then for f ∈ D(A(k) ),
(5.11)

(
dqk (f, t) = qk (f, t) ·
[
+ qk (h
(k)
(h(3−k) , t)
qk
q3−k (1, t)

)2
+ qk (A(k) f, t) − qk (h(k) f, t) ·
(h(3−k) , t)
qk
 dt
q3−k (1, t)
]
qk (h(3−k) , t)
dYt .
f, t) − qk (f, t) ·
q3−k (1, t)
Proof. Without loss of generality, we prove the result for k = 1. The unnormalized filter satisfies the following Duncan-Mortensen-Zakai (DMZ) equation (see Zakai (1969), Kallianpur
(1980)), for f ∈ D(A(1) ),
(5.12)
(5.13)
dσ (1) (f, t) = σ (1) (A(1) f, t)dt + σ (1) (h(1) f, t)dYt ,
∫ t
(2)
(2)
σ (1, t) = σ (1, 0) +
σ (2) (h(2) , s)dYs .
0
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
19
Note that σ (1) (f, t), σ (2) (1, t) are both semi-martingales, so we can apply the Itô formula to
σ (1) (f,t)
.
σ (2) (1,t)
(5.14)
For notation simplicity, denote U (t) = σ (1) (f, t), V (t) = σ (2) (1, t) and we have
(
)2
dV −1 (t) = −V −2 (t)σ (2) (h(2) , t)dYt + V −3 (t) σ (2) (h(2) , t) dt.
Moreover,
d(U V −1 )(t) = U (t)d(V −1 )(t) + V −1 (t)dU (t) + d[U −1 , V ]t
(
)
σ (1) (f, t)σ (1) (h(2) , t) σ (1) (h(1) f, t)
+
= −
dYt
V2
V
(
)
σ (1) (A(1) f ) σ (1) (f )σ (2) (h(2) )σ (2) (h(2) ) σ (1) (h(1) f )σ (2) (h(2) )
+
−
+
dt.
V
V3
V2
Note that
σ (2) (h(2) , t)
=
V
σ (2) (h(2) ,t)
σ (1) (1,t)
V
σ (1) (1,t)
=
q (2) (h(2) , t)
.
q (2) (1, t)
Therefore, we obtain the following evolution of the Bayes factor:


(
)2
(2)
(2)
q1 (h , t)
q1 (h , t) 
(5.15) dq1 (f, t) = q1 (f, t) ·
dt
+ q1 (A(1) f, t) − q1 (h(1) f, t) ·
q2 (1, t)
q2 (1, t)
[
]
(2) , t)
q
(h
1
+ q1 (h(1) f, t) − q1 (f, t) ·
dYt .
q2 (1, t)
Similarly, we have
(5.16)

(
dq2 (f, t) = q2 (f, t) ·
(h(1) , t)
q2
q1 (1, t)

)2
+ q2 (A(2) f, t) − q2 (h(2) f, t) ·
(h(1) , t)
q2
 dt
q1 (1, t)
[
]
(1) , t)
q
(h
2
+ q2 (h(2) f, t) − q2 (f, t) ·
dYt .
q1 (1, t)
Hence the result.
In particular,
Corollary 5.1. If M(k) , k = 1, 2 coincide, then we have the Fujisaki-Kunita-Kallianpur
(FKK) equation, for f ∈ D(A),
(5.17)
dπ(f, t) = π(Af, t)dt + [π(hf, t) − π(f, t) · π(h, t)] (dYt − π(h, t)dt) ,
where the normalized filter πt is defined as
(5.18)
(
)
π(f, t) , E f (Xt , θ)|FtY
for f ∈ B(Rnx +nθ ).
Proof. In case M(k) , k = 1, 2 coincide, we have
A(1) = A(2) = A, h(1) = h(2) = h.
In such case,
q1 (f, t) =
σ (1) (f, t)
σ (1) (f, t)
=
= π(f, t),
σ (2) (1, t)
σ (1) (1, t)
20
JIANHUI HUANG†, MICHAEL KOURITZIN‡
and the Bayes factor
q1 (1, t) = q2 (1, t) = 1.
Thus the evolution equation becomes
[
]
dπ(f, t) = π(f, t) · π 2 (h, t) + π(Af, t) − π(hf, t) · π(h, t) dt
+ [π(hf, t) − π(f, t) · π(h, t)] dYt ,
= π(Af, t)dt + [π(hf, t) − π(f, t) · π(h, t)] (dYt − π(h, t)dt) .
6. Robust Bayes Factor for Climate Model Selection
The DMZ equation of Bayes filter involves some stochastic integration so the unnormalized filter σt is not easy to implement in real time to catch the rapid data change. Moreover,
is is not robust to the possible modeling errors, as discussed in Clark (1978, 2005). Instead,
here we adopt the robust evolution equation of the Bayes filter to characterize the Bayes
factor. Empirical results show the robust filter does indeed performs favorably when applied
to real data problem. Clark (1978) introduces the robust filter and some other important
works on it include Davis (1980, 1981), Pardoux (1979), Heunis (1990). The robust Bayes
filter is closely related to the following gauge transform.
6.1. The evolution of robust Bayes factor.
Definition 6.1. Assume A1 and A2 hold true for M(k) , k = 1, 2, then the gauge transform
ν (k) of (X, θ) is
(6.1)
ν (k) (f, t) , EQ [f (Xt , θ)Lt exp(−Yt h(k) (Xt , θ))|FtY ]
(k)
(k)
for f ∈ B(Rnx +nθ ).
It is equivalent to σ (k) because
(6.2)
ν (k) (f, t) = σ (k) (f exp(−Yt h(k) )) and σ (k) (f, t) = ν (k) (f exp(Yt h(k) )).
The following result gives the robust evolution of the gauge transform.
Theorem 6.1. Assume A1 and A2 hold true for model M(k) , k = 1, 2, then ν (k) (f, t)
satisfies the evolution equation
(6.3)
dν (k) (f, t)
b y, (k) f, t)
= ν (k) (A
dt
Y2
1
+ ν (k) (f {−Yt A(k) h(k) − (h(k) )2 + t [h(k) , h(k) ]}, t)
2
2
b y, (k) ) = D(A(k) )and for f1 , f2 ∈ D(A(k) ),
for all f ∈ D(A
(6.4)
Proof. See the Appendix.
b y, (k) f = A(k) f − Yt [h(k) , f ].
A
Compared to DMZ equation, equation (6.3) has the property that its randomness only
appears in the coefficients of the equation and no stochastic integration involved. It can be
shown it is robust and continuously dependent to the modeling errors. This provides some
computational and practical advantages in real-time online computation. Now introduce
the robust filter ratio between M(1) and M(2) :
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
21
Definition 6.2. (Robust filter ratio)
(6.5)
qe1 (f, t) ,
ν (1) (f, t)
with g (2) = exp(Yt h(2) ),
ν (2) (g (2) , t)
qe2 (f, t) ,
ν (2) (f, t)
with g (1) = exp(Yt h(1) ).
(1)
(1)
ν (g , t)
Note that
(6.6)
q1 (f, t) =
σ (1) (f, t)
ν (1) (f exp(Yt h(1) ), t)
=
= qe1 (f g (1) , t).
σ (2) (1, t)
ν (2) (exp(Yt h(2) ), t)
Therefore,
B12 (t) = q1 (1, t) = qe1 (g (1) , t).
(6.7)
Theorem 6.2. Suppose A1 and A2 hold true for M(k) , k = 1, 2, then the robust filter
ratio qei , i = 1, 2 satisfies the following measure-valued evolution equation
(6.8)
(
)
y, (i)
b
de
qi (f, t) = qei At
f, t dt
(
)
1
Y2
+ qei f {−Yt A(i) h(i) − (h(i) )2 + t [h(i) , h(i) ]}, t dt
2
2
(
)
qei (f, t)
b y, (i) g (3−i) , t dt
−
·
q
e
A
3−i
t
qe3−i (g (3−i) , t)
(
)
qe3−i (f, t)
1 (3−i) 2 Yt2 (3−i) (3−i)
(3−i)
(3−i) (3−i)
−
· qe3−i g
{−Yt A
h
− (h
) +
[h
,h
]}, t dt.
2
2
qe3−i (g (3−i) , t)
Proof. We have
(6.9)
d (1)
b y, (1) f, t)
ν (f, t) = ν (1) (A
dt
1
Y2
+ ν (1) (f {−Yt A(1) h(1) − (h(1) )2 + t [h(1) , h(1) ]}, t)
2
2
and
(6.10)
d (2) (2)
b y, (2) g (2) , t)
ν (g , t) = ν (2) (A
dt
1
Y2
+ ν (2) (g (2) {−Yt A(2) h(2) − (h(2) )2 + t [h(2) , h(2) ]}, t)
2
2
Note that
ν (1) (f, t)
ν (2) (f, t)
and
q
e
(f,
t)
=
.
2
ν (2) (g (2) , t)
ν (1) (g (1) , t)
By the product rule of the common Lebesgue integral,
qe1 (f, t) =
de
q1 (f, t) =
ν (2) (g (2) , t)dν (1) (f, t) − ν (1) (f, t)dν (2) (g (2) , t)
(
)2
ν (2) (g (2) , t)
dν (1) (f, t)
−
= (2) (2)
ν (g , t)
Therefore
ν (1) (f, t)dν (2) (g (2) , t)
ν (2) (g (2) , t)ν (1) (g (1) , t)
ν (2) (g (2) , t)
ν (1) (g (1) , t)
.
22
JIANHUI HUANG†, MICHAEL KOURITZIN‡
dν (1) (f, t)
de
q1 (f, t) = (2) (2)
−
ν (g , t)
=
(2) (g (2) , t)
ν (1) (f, t)
· dν
ν (2) (g (2) , t)
ν (1) (g (1) , t)
ν (2) (g (2) , t)
ν (1) (g (1) , t)
b y, (1) f, t)
ν (1) (A
dt
ν (2) (g (2) , t)
ν (1) (f {−Yt A(1) h(1) − 12 (h(1) )2 +
+
ν (2) (g (2) , t)
−
ν (1) (f, t)
ν (2) (g (2) , t)
ν (2) (g (2) , t)
ν (1) (g (1) , t)
·
Yt2 (1) (1)
2 [h , h ]},
t)
dt
dν (2) (g (2) , t)
.
ν (1) (g (1) , t)
Thus
)
(
b y, (1) f dt
de
q1 (f, t) = qe1 A
t
(
)
1 (1) 2 Yt2 (1) (1)
(1) (1)
+ qe1 f {−Yt A h − (h ) +
[h , h ]}, t dt
2
2
(
)
(2) A
b y, (2) g (2) , t
ν
t
qe1 (f, t)
−
·
dt
qe2 (g (2) , t)
ν (1) (g (1) , t)
(
)
2
(2) g (2) {−Y A(2) h(2) − 1 (h(2) )2 + Yt [h(2) , h(2) ]}, t
ν
t
2
2
qe1 (f, t)
·
dt.
−
(2)
(1)
(1)
qe2 (g , t)
ν (g , t)
That is
(
)
b y, (1) f, t dt
de
q1 (f, t) = qe1 A
t
(
)
1 (1) 2 Yt2 (1) (1)
(1) (1)
+ qe1 f {−Yt A h − (h ) +
[h , h ]}, t dt
2
2
(
)
qe1 (f, t)
y, (2) (2)
b
−
·
q
e
A
g
,
t
dt
2
t
qe2 (g (2) , t)
)
(
qe2 (f, t)
1 (2) 2 Yt2 (2) (2)
(2)
(2) (2)
−
· qe2 g {−Yt A h − (h ) +
[h , h ]}, t dt.
2
2
qe2 (g (2) , t)
Similarly, we have
(
)
b y, (2) f, t dt
de
q2 (f, t) = qe2 A
t
(
)
1 (2) 2 Yt2 (2) (2)
(2) (2)
+ qe2 f {−Yt A h − (h ) +
[h , h ]}, t dt
2
2
(
)
qe2 (f, t)
b y, (2) g (1) , t dt
−
·
q
e
A
1
t
qe1 (g (1) , t)
(
)
qe1 (f, t)
1 (1) 2 Yt2 (1) (1)
(1)
(1) (1)
−
· qe1 g {−Yt A h − (h ) +
[h , h ]}, t dt.
2
2
qe1 (g (1) , t)
This completes the proof.
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
23
7. Particle filtering to model calibration
The evolution equation (6.3) does not admit the explicit solution excepts a few cases.
Instead, we need some efficient and recursive numerical algorithm to implement it. To
avoid the “curse of dimensionality,” we propose the particle filtering algorithm that can be
thought of a generalization of Del Moral, Noyer and Salut (1994). The particle filtering is
based on the following N -equalized time partitions
{τ0 = 0, τ1 =
(7.1)
T
iT
, . . . τi =
, · · · τN = T }.
N
N
The key point of particle filtering is to construct a sequence of particle pair such that their
empirical measures converge to the measure-valued process (e
q1 , qe2 ) as mN −→ ∞. Hereafter
(1), k
(2), k mN
we denote (Pt
, Pt
)k=1 the states of this particle pairs at time t.
7.1. Initialization.
• At τ0 = 0, we draw mN independent equally-weighted particle pairs with states
(1), k
(2), k mN
(P0
, P0
)k=1 satisfying the following conditions:
lim mN = ∞,
N −→∞
(1)
∀f ∈ B(Rnx +nθ ),
(2)
∀f ∈ B(Rnx +nθ ).
lim (φN (0), f ) = qe1 (f, 0)
N −→∞
lim (φN (0), f ) = qe2 (f, 0)
N −→∞
Here,
(1)
φN (0) ,
mN
1 ∑
δP (1), k
mN
0
(2)
φN (0) ,
k=1
mN
1 ∑
δP (2), k
mN
0
k=1
(k)
Remark 7.1. Here, δx (·) is the Dirac measure at x. Because Y0 = 0 and L0 = 1 for all
k = 1, 2, thus
σ (k) (f, 0) = ν (k) (f, 0) = qek (f, 0)
∀f ∈ B(Rnx +nθ ), k = 1, 2;
σ (k) (1, 0) = ν (k) (1, 0) = qek (1, 0) = 1, k = 1, 2.
7.2. Evolution.
N
• During [τi−1 , τi ), i = 1, 2, · · · , N, the particles {P (1), k , P (2), k }m
k=1 move independently to explore the state space. As θ is time-invariant, we only need consider the
dynamics of X during this interval which turns out to be:
b Y, (k) (defined in AppenLemma 7.1. Suppose model M(k) , k = 1, 2, then under Q
dix), X is a path-dependent diffusion process
dXt = bY, (k) (Xt , θ, t)dt + c(k) (Xt , θ, t)dBt ,
(7.2)
where
(7.3)
b
Y, (k)
[
(x, θ, t) = b
Proof. See Appendix.
(k)
(x, θ, t) − Yt · (c
(k)
(x, θ, t))
2 ∂h
(k) (x, θ)
∂x
]
∂f (x, θ)
.
∂x
24
JIANHUI HUANG†, MICHAEL KOURITZIN‡
7.3. Testing the weight.
(1), k
• At time τi , the particles are respectively given a weight (ωi
its trajectory realized on [τi−1 , τi ) :
(1), k
ωi
(2), k
ωi
(∫
(2), k mN
)k=1
, ωi
based on
) )
1 (1) 2 (1), k
Ys2 (1) (1) (1), k
= exp
−Ys A h
− (h ) (Ps
)+
[h , h ](Ps
) ds
2
2
τi−1
(∫
) )
τi (
1 (2) 2 (2), k
Ys2 (2) (2) (2), k
(2) (2)
(2), k
= exp
−Ys A h (Ps
) − (h ) (Ps
)+
[h , h ](Ps
) ds .
2
2
τi−1
τi
(
(1) (1)
(Ps(1), k )
7.4. Re-sampling.
• At time τi , i = 1, 2, · · · , N, we give each particle component (P (1), k , P (2), k ), k =
(1), k
(2), k
1, 2, · · · , mN a weight and these weights (ωi
, ωi
) are stored along with the
states of particles before re-sampling. Introduce the average weight at τi as
(1)
ωi
(7.4)
mN
mN
1 ∑
1 ∑
(1), k
(2)
(2), k
,
ωi
, ωi ,
ωi
mN
mN
k=1
k=1
(1), k
(1)
The re-sampling procedure is: if a particle P (1), k has a weight ωi
= rωi + z,
(1)
where r ∈ {0, 1, 2, · · · } and z ∈ (0, ωi ) before the re-sampling, then there will be
r or r + 1 particles at this state after the re-sampling with a probability selected
in order to leave the system unbiased. The same re-sampling procedure applies to
(2)
particle component P (2), k and average weight ωi .
It is important to consider the variance minimization of partial filter, see Crisan and Lyons
(1997), Crisan, Gaines and Lyons (1998).
The variance minimization is related to the linear-quadratic Gaussian (LQG) control
problem essentially, see Huang and Yu (2014), Chen and Huang (2014). Especially, Chen
and Huang (2014) is for delayed LQ problem.
8. Appendix
To simplify the notation, hereafter we suppress the superscripts and derive the results
for both M(1) , M(2) simultaneously. We use y = {ys , 0 ≤ s ≤ T } to denote a specific
observation realization, that is, yt = Yt (ω).
8.1. Path-dependent measure transform.
Proposition 8.1. For given observation path y,
ν(f, t) = EQ [f (Xt , θ) · Γyt · Ξyt ],
where
(8.1)
)
( ∫ t
∫ t 2
ys
h
[h, h](Xs , θ)ds ,
, exp −
ys dMs −
0
0 2
) )
(∫ t (
ys2
1 2
y
Ξt , exp
−ys Ah(Xs , θ) − h (Xs , θ) + [h, h](Xs , θ) ds .
2
2
0
Γyt
Proof. Due to the independence of B, (X, θ) and the integration by parts, we have
∫ t
∫ t
(8.2)
h(Xs , θ)dYs = Yt h(Xt , θ) −
Ys dh(Xs , θ).
0
0
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
25
It follows that
∫ t
∫
∫ t
∫
1 t 2
1 t 2
Lt = exp ( h(Xs , θ)dYs −
h (Xs , θ)ds) = exp (Yt h(Xt , θ) −
Ys dh(Xs , θ) −
h (Xs , θ)ds).
2 0
2 0
0
0
Consequently,
(8.3)
Q
[
ν(f, t) = E
( ∫ t
]
)
∫
1 t 2
Y
f (Xt , θ) exp −
Ys dh(Xs , θ) −
h (Xs , θ)ds |Ft .
2 0
0
Therefore, for given observation path y,
(8.4)
)]
[
( ∫ t
∫
∫ t
1 t 2
h
Q
h (Xs , θ)ds −
ys dMs
.
ν(f, t) = E f (Xt , θ) exp −
ys Ah(Xs , θ)ds −
2 0
0
0
Hence the result.
Lemma 8.1. Γyt is a continuous martingale.
Proof. It is easy to see that Γyt is a continuous process and
dΓyt = −Γyt yt dMth ,
(8.5)
which implies Γyt is a local martingale. Meanwhile, from A2,
∫
1 t
E exp(
[h, h](Xs , θ)ds) < ∞,
∀t ∈ [0, T ].
2 0
Then from the Novikov criteria, Γyt is a martingale.
Now, based on Proposition 8.1 and Lemma 8.1, we are ready to introduce the pathdependent probability transform.
Definition 8.1. For given yt , introduce the path-dependent probability measure Qy by
b y dQ
y
= Γt .
dQ Ft
After the first measure change to Q, the law of the signal process (X, θ) remains unchanged. In contrast, with the second path-dependent measure change, the law of X will
be changed and characterized by some observation-dependent martingale problem. The
gauge transform ν(·) then takes the form of Feynman-Kac multiplicative functional:
b y [f (Xt ) · Ξy ],
νt (f ) = E
t
where the expectation is under the measure
(∫ t (
y
Ξt = exp
−ys Ah(Xs , θ) −
0
b y and
Q
) )
1 2
ys2
h (Xs , θ) + [h, h](Xs , θ) ds .
2
2
b y , (X, θ) is the unique solution to the A
by
Lemma 8.2. Assume A1 and A2, then under Q
t
martingale problem
(8.6)
b y f (Xt , θ)dt + dM
cf ,
df (Xt , θ) = A
t
t
b y ) = D(A) and for f ∈ D(A
b y ),
where D(A
t
(8.7)
b y f , Af − yt [h, f ],
A
t
(8.8)
cf , dM f + yt [h, f ](Xt , θ)dt.
dM
t
t
26
JIANHUI HUANG†, MICHAEL KOURITZIN‡
Proof. For f ∈ D(A), it is obvious that
b y f (Xt , θ)dt + dM
cf
df (Xt , θ) = A
t
t
(8.9)
cf is a martingale. Due to the continuity of Γy , we have
and we only need to show M
t
∫ t
f
y
⟨M , Γ ⟩t =
−Γys ys [f, h]ds
0
and
cf
dM
t
(8.10)
= dMtf −
1
d[M f , Ay ]t .
Γyt
b y . Moreover,
cf is a local martingale under Q
Thus, from the Girsanov-Meyer theorem, M
t
from A2,
b y [M
cf , M
cf ]t = EQ ([M f , M f ]Γy )t < ∞.
E
b y . This complete the proof.
cf is a martingale under Q
Then it follows that M
t
8.2. Proof of Theorem 6.1.
Proof. We have
b y (f (Xt , θ) · Ξt ).
ν(f, t) = E
From Lemma 8.2 and integration by parts,
b y f (Xt , θ)Ξt dt + f (Xt , θ) · Ξt {−yt Ah(Xt , θ) − 1 h2 (Xt , θ)
(8.11) d(f (Xt , θ)Ξt ) = A
t
2
2
yt
cf .
+
[h, h](Xt , θ)}dt + Ξt dM
t
2
∫t
b y -martingale under A1 and A2, then taking the expectation
csf is a Q
Note that 0 Ξs dM
b y , we get
under Q
)
∫ t (
∫ t
2
1
y
s
y
2
b f, s)ds +
ν f {−ys Ah − h + [h, h]}, s ds.
ν(A
ν(f, t) = ν(f, 0) +
s
2
2
0
0
Replace the fixed observation path yt with the observation process Yt to get the result. 8.3. Proof of Lemma 7.1.
b Y , (X, θ) is a solution of the AY martingale
Proof. Therefore, it suffices to show that under Q
problem with generator
(
)
1 2
∂2f
∂h(x, θ) ∂f
Y
2
A f (x, θ) = c (x, θ, t) 2 (x, θ) + b(x, θ, t) − c (x, θ, t)
(x, θ).
2
∂x
∂x
∂x
b Y , (X, θ) is a solution of the A
b Y martingale problem with
Under Q
b Y f , At f − yt [h, f ].
A
t
Note that
(8.12)
then we have
∂ 2 f (x, θ)
∂f (x, θ)
1
+ b(x, θ, t)
,
Af (x, θ) = c2 (x, θ, t)
2
2
∂x
∂x
[ 2
]
1
∂ h(x, θ)
∂ 2 f (x, θ)
∂f ∂h
A(hf ) = c2 (x, θ, t) f
+
h
+
2
(x,
θ)
2
∂x2
∂x2
∂x ∂x
[
]
∂h(x, θ)
∂f (x, θ)
+ b(x, θ, t) f
+h
.
∂x
∂x
NONLINEAR FILTERING OF STOCHASTIC CLIMATE MODELS WITH PARTIAL INFORMATION
27
Therefore,
[h, f ] = c2 (x, θ, t)
and
∂f ∂h
,
∂x ∂x
[ 2
]
1 2
∂ f (x, θ)
∂f ∂h
∂f (x, θ)
Y
b
− 2yt
(x, θ) + b(x, θ, t)
At f = c (x, θ, t)
2
2
∂x
∂x ∂x
∂x
[
]
2
1 2
∂h(x, θ) ∂f (x, θ)
∂ f (x, θ)
2
= c (x, θ, t)
+ b(x, θ, t) − yt · c (x, θ, t)
.
2
∂x2
∂x
∂x
Thus
b Y = AY .
A
Hence the result.
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†Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong.
‡Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1 Canada.
The first author acknowledges the financial support from RGC Earmarked grant 500909.
E-mail address: [email protected];