Robust Filtering · 2015-05-18 · Dan Crisan (Imperial College London) Robust Filtering 15 May 2015 13 / 48 Preliminary Bounds y · – arbitrary element of the set C R m [0 ,t ],

. . . . . .

.

......Robust Filtering

Dan Crisan

Imperial College London

45th Annual John H. Barrett Memorial LecturesStochastic Filtering, Computations and Their Applications

University of Tennessee, KnoxvilleMay 13-16, 2015

Dan Crisan (Imperial College London) Robust Filtering 15 May 2015 1 / 48

. . . . . .

Talk Synopsis

Part 1: Independent signal/observation noiseThe frameworkThe Importance of a robust representationPreliminary boundsThe robustness resultBibliographical notes

Part II: Correlated noiseThe frameworkNon-existence of a robust representation formula a la ClarkObservation lift to a rough pathStochastic differential equations with rough driftThe robust representation formula on the rough path space

Joint work with Martin Clark, Joscha Diehl, Harald Oberhauser, Peter Friz.

J. M. C. Clark; D. C, On a robust version of the integral representation formulaof nonlinear filtering, Probab. Theory Related Fields 133 (2005).D. C.; J. Diehl; P. K. Friz; H. Oberhauser, Robust filtering: correlated noise andmultidimensional observation. Ann. Appl. Probab. 23 (2013).A. Bain, D. Crisan, Fundamentals of stochastic filtering. Stochastic Modellingand Applied Probability, 60. Springer, New York, (2009).


. . . . . .

What is stochastic filtering

PART I


. . . . . .


(Ω,F ,P) – complete probability space.(Ft)t≥0 – satisfies the usual conditions.(S,S) – the state space for the signal: S complete separable metric space andS is the associated Borel σ-field B(S).B(S) – the space of bounded B(S)-measurable functions.A – a (possibly unbounded) operator from B(S) to B(S). 1 ∈ D(A) and A1 = 0.D(A) domain of A: If f ∈ D(A) then Af is bounded.P(S) – the space of probability measures over (S,S)

The signalX = Xt , t ≥ 0 an Ft -adapted process X = Xt , t ≥ 0 with values in (S,S)(the state space) and cadlag paths. Assume that X is a solution of themartingale problem for (A, π0). That is, the distribution of X0 is π0 ∈ P(S) andthe process Mφ = Mφ

t , t ≥ 0 defined as

Mφt = φ(Xt)− φ(X0)−

∫ t

0Aφ(Xs) ds, t ≥ 0, (1)

is an Ft -adapted martingale for any φ ∈ D(A). The operator A is called thegenerator of the process X .


. . . . . .


The observation process

Y an Ft -adapted process satisfying the following evolution equation

Yt = Y0 +

∫ t

0h(Xs) ds + Wt , (2)

where h = (hi)mi=1 : S → Rm is a measurable function such that

P

(∫ t

0∥h(Xs)∥2 ds < ∞

)= 1 (3)

for all t ≥ 0 and W is a standard Ft -adapted m-dimensional Brownian motionon (Ω,F ,P) independent of X . The condition (3) ensures that the Riemannintegral in the definition of Yt exists a.s. Define

Yt = σ(Ys, s ∈ [0, t ]) ∨N ⊂ Ft (4)


. . . . . .


.Definition..

......

The filtering problem consists in determining the conditional distribution πt ofthe signal X at time t given the information accumulated from observing Y inthe interval [0, t ]. That is for φ ∈ B(S), computing

πt(φ) = E[φ(Xt) | Yt ]. (5)

y· – arbitrary element of the set CRm [0, t ], where t ≥ 0 is arbitrary but fixed.s 7→ ys is a continuous function y· : [0, t ] → Rm.Y· – the path-valued random variable

Y· : Ω → CRm [0, t ], Y·(ω) = (Ys(ω), 0 ≤ s ≤ t).

If φ is a bounded Borel-measurable function, then πt(φ) can be written as afunction of the observation path. That is, there exists a bounded measurablefunction fφ : CRm [0, t ] → R such that

πt(φ) = E[φ(Xt) | Yt ] = fφ(Y·) P-a.s. (6)


. . . . . .


Remarksfφ is not unique. Any other function fφ such that

P Y−1·(fφ = fφ

)= 0,

where P Y−1· is the distribution of Y· on the path space CRm [0, t ] can

replace fφ in (6).In the following, we obtain a robust representation of the conditionalexpectation πt(φ). That is, we show that there exists a continuousfunction fφ : CRm [0, t ] → R (with respect to the supremum norm onCRm [0, t ]) such that

πt(φ) = fφ(Y·) P-a.s. (7)

A continuous fφ is unique: Since P Y−1· positively charges all

non-empty open sets in CRm [0, t ], there exists a unique continuousfunction fφ : CRm [0, t ] → R for which (7) holds true.


. . . . . .


The need for this type of representation arises when the filteringframework is used to model and solve ‘real-life’ problems. The model Ychosen for the “real-life” observation process Y may not be a perfect one.However, as long as the distribution of Y· is close in a weak sense to thatof Y· (and some integrability assumptions hold), the estimate f (Y·)computed on the actual observation will still be reasonable, as

E[(φ(Xt)− fφ(Y·))2]

is well approximated by the idealized error E[(φ(Xt)− fφ(Y·))2].

Even when Y and Y coincide, one is never able to obtain and exploit acontinuous stream of data as modelled by the continuous path Y·(ω).Instead the observation arrives and is processed at discrete moments intime

0 = t0 < t1 < t2 < · · · < tn = t .

However the continuous path Y·(ω) obtained from the discreteobservations (Yti (ω))

ni=1 by linear interpolation is close to Y·(ω) (with

respect to the supremum norm on CRm [0, t ]); hence, by the sameargument, fφ(Y·) will be a sensible approximation to πt(φ).


. . . . . .


Since Y0 = 0, π0, the initial distribution of X , is identical with the conditionaldistribution of X0 given Y0 and we will use the same notation for both.

A particular case: X a diffusion process

X = (X i)di=1 – the solution of a d-dimensional stochastic differential equation

driven by a p-dimensional Wiener process V = (V j)pj=1:

X it = X i

0 +

∫ t

0f i(Xs) ds +

p∑j=1

∫ t

0σij(Xs) dV j

s. (8)

where f = (f i)di=1 : Rd → Rd and σ = (σij)i=1,...,d,j=1,...,p : Rd → Rd×p are

globally Lipschitz. The SDE (8) has a unique solution. The generator Aassociated to the process X is the second order differential operator,

A =d∑

i=1

f i ∂

∂xi+

d∑i,j=1

aij ∂2

∂xi∂xj, (9)

where a = (aij)i,j=1,...,d : Rd → Rd×d is the matrix valued function defined as

aij =12

p∑k=1

σikσjk =12(σσ⊤)ij

. i , j = 1, . . . , d . (10)


. . . . . .


The domain:

D(A) = C2k (Rd )

D(A) = SL2(Rd ) be the subset of all twice continuously differentiablereal-valued functions on Rd for which there exists a constant C such that forall i , j = 1, . . . , d and x ∈ Rd we have for x ∈ Rd

|∂iφ(x)| ≤C

1 + ∥x∥,∣∣∂i∂jφ(x)

∣∣ ≤ C1 + ∥x∥2 .

D(A) to be the maximal domain of A. That is, D(A) is the set of allφ ∈ B(Rd ) for which Aφ ∈ B(Rd ) and Mφ is a martingale. In the following,unless otherwise stated, D(A) is the maximal domain of A.


. . . . . .


Let Z = Zt , t > 0 be the process defined by

Zt = exp

(−

m∑i=1

∫ t

0hi(Xs) dW i

s −12

m∑i=1

∫ t

0hi(Xs)

2 ds

), t ≥ 0. (11)

We assume (in addition to (3)) that

E

[∫ t

0Zs ∥h(Xs)∥2 ds

]< ∞, ∀t > 0. (12)

Under (12), the process Z is a martingale. Let X be the solution of (8). If hhas linear growth i.e. there exists C such that

∥h(x)∥2 ≤ C(1 + ∥x∥2) ∀x ∈ Rd , (13)

and E[X 20 ] < ∞, then (12) is satisfied. For further details, see

T. Cass, J.M.C. Clark, D.C., The filtering equations revisited, in StochasticAnalysis and Applications, 2014.


. . . . . .


Define a probability measure P on on∪

0≤t<∞ Ft such that

dPdP

∣∣∣∣∣Ft

= Zt .

.Theorem..

......

If condition (12) is satisfied then under P, the observation process Y is aBrownian motion independent of X ; additionally the law of the signal processX under P is the same as its law under P.

Let Z = Zt , t ≥ 0 be the process defined as Zt = Z−1t for t ≥ 0,

Zt = exp

(m∑

i=1

∫ t

0hi(Xs) dY i

s −12

m∑i=1

∫ t

0hi(Xs)

2 ds

), (14)

Then Z is an Ft -adapted martingale under P anddPdP

∣∣∣∣Ft

= Zt for t ≥ 0.


. . . . . .


.Definition..

......

Define the unnormalized conditional distribution of the signal to be themeasure-valued process ρ = ρt , t ≥ 0 determined by

ρt(φ) = E[Ztφ(Xt) | Yt

], φ ∈ B(S), t ≥ 0.

.Theorem (Kallianpur-Striebel)..

......

Assume that condition (12) holds. Then for every φ ∈ B(S),

πt(φ) =E[Ztφ(Xt) | Y]

E[Zt | Y]=

ρt(φ)

ρt(1)P(P)-a.s. (15)

The Kallianpur-Striebel formula explains the usage of the term unnormalisedin the definition of ρt as the denominator ρt(1) can be viewed as thenormalizing factor. The formula (15) holds true for any Borel-measurable φ,not necessarily bounded, such that E [|φ(Xt)|] < ∞.


. . . . . .

Preliminary Bounds

y· – arbitrary element of the set CRm [0, t ], y· : [0, t ] → Rm, continuous.Y· – the path-valued r.v. Y·(ω) = (Ys(ω), 0 ≤ s ≤ t), Y· ∈ CRm [0, t ], P-a.s. CS

Let Θ(y·) be the following random variable

Θ(y·) , exp

(h(Xt)

⊤yt − I(y·)−12

∫ t

0∥h(Xs)∥2 ds

), (16)

where I(y·), is a version of the stochastic integral∫ t

0 y⊤s dh(Xs). The argument

of the exponent in the definition of Θ(y·) will be recognized as a formalintegration by parts of the argument of the exponential

Zt = exp

(∫ t

0h⊤(Xs) dY i

s −12

∫ t

0∥h(Xs)∥2 ds

). (17)

Let gφ, g1, fφ : CRm [0, t ] → R be the following functions:

gφ(y·) = E [φ(Xt)Θ(y·)] , g1(y·) = E [Θ(y·)] , f (y·) =gφ(y·)g1(y·)

. (18)

Then gφ(Y·) is the robust representation for ρt(φ) and that f (Y·) is the robustrepresentation for πt(φ).


. . . . . .

Preliminary Bounds

In the following, we denote by ∥ξ∥Ω,p the Lp norm of ξ, ∥ξ∥Ω,p = E [|ξ|p]1/p.

Additional Conditions

Assume φ Borel-measurable function such that ∥φ(Xt)∥Ω,p < ∞ for p > 1. Note that ∥φ(Xt)∥Ω,p is the same whether we integrate w.r.t. P or P. hi , (hi)2 ∈ D(A), i = 1, . . . ,m. Then t 7→ h(Xt) is a semimartingale

h(Xt) = h(X0) +

∫ t

0Ah(Xs) ds + Mh

t , t ≥ 0,

The stochastic integral I(y·) is well defined:

gφ(y·) = E[φ(Xt)exp

(h(Xt)

⊤yt −

I(y·)︷︸︸︷∫ t

0y⊤

s dh(Xs)−12

∫ t

0∥h(Xs)∥2 ds

)︸︷︷︸

Θ(y·)

],


. . . . . .

Preliminary Bounds

.Lemma..

......

For any R > 0 and p ≥ 1 there exists a positive constant MΘR,p such that

sup∥y·∥≤R

∥Θ(y·)∥Ω,p ≤ MΘR,p. (19)

.Lemma..

......

For any R > 0 and q ≥ 1 there exists a positive constant MΘR,q such that∥∥Θ(y1

· )−Θ(y2· )∥∥Ω,q ≤ MΘ

R,q

∥∥y1· − y2

·∥∥ (20)

for any two paths y1· , y2

· such that |y1· |, |y2

· | ≤ R. In particular, (20) implies thatg1 is locally Lipschitz; more precisely∣∣g1 (y1

·)− g1 (y2

·)∣∣ ≤ MΘ

R

∥∥y1· − y2

·∥∥

for any two paths y1· , y2

· such that∥∥y1

·∥∥ , ∥∥y2

·∥∥ ≤ R and MΘ

R = infq≥1 MΘR,q .


. . . . . .

Preliminary Bounds

.Proof...

......

For the two paths y1· , y2

· denote by y12· the difference path defined as

y12· , y1

· − y2· . Then

∣∣Θ (y1·)−Θ

(y2·)∣∣ ≤ (Θ (y1

·)+Θ

(y2·)) ∣∣∣∣∣∫ t

0

(y12

s)⊤

dh(Xs)

∣∣∣∣∣ ,Using the Cauchy–Schwartz inequality

∥∥Θ(y1· )−Θ(y2

· )∥∥Ω,q ≤ 2MΘ

R,2q

∥∥∥∥∥∫ t

0

(y12

s)⊤

dh(Xs)

∥∥∥∥∥Ω,2q

. (21)

Finally, since∥∥y12

·∥∥ ≤ 2

∥∥y1· − y2

·∥∥, a standard argument based on

Burkholder–Davis–Gundy’s inequality shows that the expectation on theright-hand side of (21) is bounded by a constant

CR∥∥y1

· − y2·∥∥

where CR is a constant depending on R and independent of the paths. Hence(20) holds true.


. . . . . .

Preliminary Bounds

.Lemma..

......The function gφ is locally Lipschitz and locally bounded.

.Proof...

......

Fix R > 0 and let y1· , y2

· be two paths such that ∥y1· ∥, ∥y2

· ∥ ≤ R. By Holder’sinequality and (20), we see that

E[∣∣φ(Xt)

∣∣ ∣∣Θ (y1·)−Θ

(y2·)∣∣] ≤ ∥φ(Xt)∥Ω,pMΘ

R,q

∥∥y1· − y2

·∥∥ . (22)

where q is such that p−1 + q−1 = 1. Hence gφ is locally Lipschitz, since

gφ(y1· )− gφ(y2

· ) = E[φ(Xt)

(Θ(y1

· )−Θ(y2· ))]

and R > 0 was arbitrarily chosen. Next let y· be a path such that ∥y·∥ ≤ R.Again, by Holder’s inequality and (19), we get that

sup∥y·∥≤R

|gφ(y·)| = sup∥y·∥≤R

∣∣∣E [φ(Xt)Θ(y1·)]∣∣∣ ≤ ∥φ(Xt)∥pMΘ

R,q < ∞.

Hence gφ is locally bounded.Dan Crisan (Imperial College London) Robust Filtering 15 May 2015 18 / 48

. . . . . .

Preliminary Bounds

.Theorem........The function fφ is locally Lipschitz.

.Proof...

......

The ratio gφ/g1 of the two locally Lipschitz functions gφ and g1 (Lemma 6 andLemma 7) is locally Lipschitz provided both gφ and 1/g1

t are locally bounded.The local boundedness property of gφ is shown in Lemma 7 and that of 1/g1

tfollows from the following simple argument. If ∥y·∥ ≤ R Jensen’s inequalityimplies that

E [Θ(y·)] ≥ exp

(E

[∫ t

0(yt − ys)

⊤ dh(Xs)−12

∫ t

0∥h(Xs)∥2 ds

])

≥ exp

(−2R

m∑i=1

E

[∫ t

0|Ah(Xs)| ds

]− 1

2E

[∫ t

0∥h(Xs)∥2 ds

]). (23)

Note that both expectations in (23) are finite.


. . . . . .

The Robustness Result

We proceed next to show that fφ(Y·) is a version of πt(φ). This fact is muchmore delicate than showing that fφ is locally Lipschitz. The main difficulty isthe fact that the mapping

(y·, ω) ∈ CRm [0, t ]× Ω → I(y·) ∈ R

is not B (CRm [0, t ])×F -measurable since the integral I(y·) is constructed pathby path (where B(CRm [0, t ]) is the Borel σ-field on CRm [0, t ]).For α < 1

2 , let Hα be the following subset of CRm [0, t ],

Hα =

y· ∈ CRm [0, t ] : K (y·) , sup

s1,s2∈[0,t]

∥ys1 − ys2∥∞|s1 − s2|α

< ∞

.

Remark. Y ∈ Hα, P(P)-a.s..Lemma..

......

There exists a version of the stochastic integral I(y·) which has the propertythat the mapping (y·, ω) ∈ CRm [0, t ]× Ω → I(y·) ∈ R, whilst still (possibly)non-measurable, is equal on Hα × Ω to a B (CRm [0, t ])×F-measurablemapping.


. . . . . .


.Proof...

......

The mapping (y·, ω) ∈ CRm [0, t ]× Ω → J(y·) ∈ R, defined as

J(y.)(ω) = lim supk→∞

2k−1∑i=0

y⊤it/2k (h(X(i+1)t/2k (ω))− h(Xit/2k (ω))). (24)

is B (CRm [0, t ])×F -measurable. Define I(y·) to be

I(y·)(ω) ,

J(y.)(ω) if (y·, ω) ∈ Hα × Ω

any version of∫ t

0 y⊤s dh(Xs) if (y·, ω) /∈ Hα × Ω

For y. /∈ Hα it is quite possible that J(y.) differs from I(y·).


. . . . . .


We decouple the two processes X and Y .

Let (Ω, F , P) be copy of (Ω,F , P) and let X be the copy of X within the newspace (Ω, F , P). Then the function gφ has the following representation,

gφ(y·) = E[φ(Xt)Θ(y·)

](25)

Θ(y·) = exp

(h(Xt)

⊤yt − I(y·)−12

∫ t

0∥h(Xt)∥2 ds

), (26)

where E denotes integration on (Ω, F , P), and I(y·) is the measurable versionof the stochastic integral

∫ t0 y⊤

s dh(Xs) corresponding to I(y·) as constructedabove. Then, for y· ∈ Hα, Θ(y·) can be written as

Θ(y·) = exp

(h⊤(Xt)yt − I(y·)−

12

∫ t

0∥h(Xs)∥2 ds

). (27)

Finally, let (Ω, F , P) be the product space (Ω, F , P) = (Ω× Ω,F ⊗ F , P⊗ P) onwhich we ‘lift’ the processes H and Y from the component spaces. In otherwords, Y (ω, ω) = Y (ω) and H(ω, ω) = H(ω) for all (ω, ω) ∈ Ω× Ω.


. . . . . .


.Lemma..

......There exists a null set N ∈ F such that the mapping (ω, ω) ∈ Ω 7→ I(Y (ω))(ω)coincides on (Ω\N )× Ω with an F-measurable mapping.

.Proof...

......

Consider the F-measurable random variable (ω, ω) 7→ Jm(Y (ω))(ω) definedas

J(Y (ω))(ω) = lim supk→∞

2k−1∑i=0

Y⊤it/2k (ω)h(X(i+1)t/2k (ω))− h(Xit/2k (ω))). (28)

Then (ω, ω) 7→ J(Y (ω))(ω) coincides with the mapping

(ω, ω) ∈ Ω 7→ I(Y (ω))(ω)

on (Ω\N )× Ω, where N ∈ F is the null set N , ω ∈ Ω : Y·(ω) ∈ Hα.


. . . . . .


.Lemma..

......

P-almost surely ∫ t

0Y⊤

s dh(Xs) = I(Y (ω)). (29)

Proof.We only need to prove that, for arbitrary K > 0, P-almost surely,∫ t

0Y K

s dh(Xs) = I(Y K (ω)). (30)

where

Y Ks =

Ys if |Ys| ≤ KK otherwise.


. . . . . .


In turn, (30) follows once we prove that

limn→∞

E

(n−1∑i=0

(Y K

it/n

)⊤ (h(X(i+1)t/n)− h(Xit/n)

)− J

(Y K·

))2 = 0.

By Fubini’s theorem, using the F-measurability of J(Y K· ) and the fact that

I(Y K· ) coincides with J(Y K

· ) on (Ω\N )× Ω we have

E

(n−1∑i=0

(Y K

it/n

)⊤ (h(X(i+1)t/n)− h(Xit/n)

)− J

(Y K·

))2=

∫Ω\N

E[(

In(

Y K· (ω)

)− J

(Y K·

))2]

dP(ω)

=

∫Ω\N

E[(

In(Y K· (ω))− I(Y K

· ))2]

dP(ω).


. . . . . .


Now since s 7→ Y Ks (ω) is a continuous function and I(Y K

· (ω)) is a version ofthe stochastic integral

∫ t0

(Y K

s)⊤

(ω) dh(Xs), it follows that

limn→∞

E[(

In(Y K· (ω))− I(Y K

· (ω)))2]= 0

for all ω ∈ Ω\N . Since

supω∈Ω

E[(

Imn (Y

K· (ω))− Im(Y K

· (ω)))2]< ∞.

the result follows by the dominated convergence theorem.


. . . . . .


.Theorem..

......

The random variable fφ(Y·) is a version of πt(φ); that is, πt(φ) = fφ(Y·),P-almost surely. Hence fφ(Y·) is the unique robust representation of πt(φ).

It suffices to prove that, P-almost surely (or, equivalently, P-almost surely),

ρt(φ) = gφ(Y·) and ρt(1) = g1(Y·).

We need only prove the first identity as the second is just a special caseobtained by setting φ = 1 in the first. From the definition of abstractconditional expectation therefore it suffices to show

E [ρt(φ)b(Y·)] = E [gφ(Y·)b(Y·)] , (31)

where b is an arbitrary continuous bounded function b : CRm [0, t ] → R. SinceX and Y are independent under P, it follows that the pair processes (X ,Y )

under P, and (X ,Y ) under P have the same distribution. Hence, the left-handside of (31) has the following representation,


. . . . . .


E [ρt(φ)b(Y·)]

= E

[φ(Xt)exp

(∫ t

0h(Xs)

⊤ dYs −12

∫ t

0∥h(Xs)∥2 ds

)b(Y·)

]

= E

[φ(Xt)exp

(∫ t

0h(Xs)

⊤ dYs −12

∫ t

0∥h(Xs)∥2 ds

)b(Y·)

]

= E

[φ(Xt)exp

(h(Xt)

⊤Yt −∫ t

0Y⊤

s dh(Xs)−12

∫ t

0∥h(Xs)∥2 ds

)b(Y·)

].

On the other hand, using (27), the right-hand side of (31) has therepresentation


. . . . . .


E [gφ(Y·)b(Y·)]

= E[b(Y·)E

[φ(Xt)exp

(h(Xt)

⊤Yt − I(Y·)−12

∫ t

0∥h(Xs)∥2 ds

)]]= E

[b(Y·)E

[φ(Xt)exp

(h(Xt)

⊤Yt − J(Y·)−12

∫ t

0∥h(Xs)∥2 ds

)]].

Hence by Fubini’s theorem (using, again the F-measurability of J(Y·))

E [gφ(Y·)b(Y·)] = E[φ(Xt)exp

(h(Xt)

⊤Yt − J(Y·)−12

∫ t

0∥h(Xs)∥2 ds

)b(Y·)

].

Finally, from Lemma 11, the two representations coincide.


. . . . . .


Remarks.

Lemma 11 appears to suggest a pathwise construction for the stochasticintegral ∫ t

0h(Xs)

⊤ dYs,

but we know that for cases such as∫ t

0 Bs dBs a stochastic integral cannot bedefined pathwise. However, this apparent paradox is resolved by noting thatthe terms appearing in the lemma are only constructed on the space Ω.

In the second part, this will no longer be possible. We will need to employthe methods of rough paths theory to circumvent this problem.


. . . . . .


PART II


. . . . . .

Bibliographic Notes

The robust representation was introduced by Clark in

J. M. C. Clark. The design of robust approximations to the stochasticdifferential equations of nonlinear filtering. In J. K. Skwirzynski, editor,Communication Systems and Random Process Theory, volume 25 of Proc.2nd NATO Advanced Study Inst. Ser. E, Appl. Sci., pages 721734. SijthoffNoordhoff, Alphen aan den Rijn, 1978.

Kushner also showed that the associated robust expression for the conditionaldistribution fφ given by (18) is locally Lipschitz continuous in the observationpath y .H. J. Kushner. A robust discrete state approximation to the optimal nonlinearfilter for a diffusion. Stochastics, 3(2):7583, 1979.

Very general robustness results have been obtained by Gyongy

I. Gyongy. On stochastic partial differential equations. Results onapproximations. In Topics in Stochastic Systems: Modelling, Estimation andAdaptive Control, volume 161 of Lecture Notes in Control and Inform. Sci.,pages 116 136. Springer, Berlin, 1991.


. . . . . .

Bibliographic Notes

and Gyongy and Krylov

I. Gyongy and N. V. Krylov. On stochastic partial differential equations withunbounded coefficients. In Stochastic partial differential equations andapplications (Trento, 1990), volume 268 of Pitman Res. Notes Math. Ser.,pages 191203. Longman Sci. Tech., Harlow, 1992.

As shown by Davis, this type of robust representation is also possible whenthe signal and the observation noise are correlated, provided the observationprocess is scalar.

M.H.A. Davis. Pathwise nonlinear filtering. Stochastic systems: themathematics of filtering and identification and applications, Proc. NATO Adv.Study Inst., Les Arcs/Savoie/ France 1980, 505-528 (1981)., 1981.

M. H. A. Davis and M. P. Spathopoulos. Pathwise nonlinear filtering fornondegenerate diffusions with noise correlation. SIAM J. Control Optim. 1987.


. . . . . .

Bibliographic Notes

Additional results for the correlated noise case with scalar observation havebeen obtained by Florchinger and Zakai.

P. Florchinger. Zakai equation of nonlinear filtering with unboundedcoefficients. the case of dependent noises. Systems & control letters, 21(5)413–422, 1993.

The extension of the robustness result to special cases of the correlated noiseand multidimensional observation has been tackled in several works: Elliott and Kohlmann deduce robustness in under a commutativity conditionon the signal vector fields and a constraint on the sensor function.

R.J. Elliott and M. Kohlmann. Robust filtering for correlated multidimensionalobservations. Mathematische Zeitschrift, 178(4):559–578, 1981.

Florchinger and Nappo do not have correlated noise, but allow the coefficientsto depend on the signal and the observation.

P. Florchinger and G. Nappo, Continuity of the filter with unboundedobservation coefficients. Stochastic analysis and applications, (29) :612 - 630,2011.


. . . . . .

Bibliographic Notes

In parallel, Bagchi and Karandikar treat a different model with “finitely additive”state white noise and “finitely additive” observation noise. Robustness there isvirtually built into the problem.

A. Bagchi and R. Karandikar. White noise theory of robust nonlinear filteringwith correlated state and observation noises. Systems & control letters,23(2):137–148, 1994.

In the following, I discuss the correlated multidimensional noise case.

D. C.; J. Diehl; P. K. Friz; H. Oberhauser, Robust filtering: correlated noise andmultidimensional observation. Ann. Appl. Probab. 23 (2013).


. . . . . .

Part II Corellated Noise Framework

In the following, we will assume that the pair of processes (X ,Y ) satisfy theequation

dXt = f0(Xt ,Yt)dt +∑

k

σk (Xt ,Yt)dW kt +

∑j

σj(Xt ,Yt)dBjt , (32)

dYt = h(Xt ,Yt)dt + dWt , (33)

with X0 being a bounded random variable and Y0 = 0. In (32) and (33), theprocess X is the dX -dimensional signal, Y is the dY -dimensional observation,B and W are independent dB-dimensional, respectively, dY -dimensionalBrownian motions independent of X0. Suitable assumptions on thecoefficients f0, σ1, . . . , σdB : RdX+dY → RdX , σ1, . . . , σdY : RdX+dY → RdX andh = (h1, . . . , hdY ) : RdX+dY → RdY will be introduced later on.Let Z = Zt , t > 0 be the process defined by

Zt = exp

[−

dY∑i=1

(∫ t

0hi(Xs,Ys)dW i

s −12

∫ t

0(hi(Xs,Ys))

2ds

)]. (34)


. . . . . .


Then, under suitable assumptions (Novikov’s condition), Z is a martingalewhich is used to construct the probability measure P0 equivalent to P on∪

0≤t<∞ Ft whose Radon–Nikodym derivative with respect to P is given by u,viz

dP0

dP

∣∣∣∣Ft

= Zt .

.Theorem..

......

Under P0, Y is a Brownian motion independent of B. Moreover the equationfor the signal process X becomes


k

σk (Xt ,Yt)dY kt +

∑j

σj(Xt ,Yt)dBjt . (35)

Observe that equation (44) is now written in terms of the pair of Brownianmotions (Y ,B) and the coefficient f0 is given by f0 = f0 +

∑k σk hk .


. . . . . .


.Theorem (Kallianpur-Striebel)..

......

For any measurable, bounded function φ : RdX+dY → R, we have the followingformula called the Kallianpur-Striebel’s formula,

πt(φ) =pt(φ)

pt(1), pt(f ) := E0[φ(Xt ,Yt)Zt |Yt ] (36)

where Z = Zt , t > 0 is the process defined as Zt := exp(It), t ≥ 0 and

It :=dY∑i=1

(∫ t

0hi(Xr ,Yr )dY i

r −12

∫ t

0(hi(Xr ,Yr ))

2dr

), t ≥ 0. (37)

The representation (36) suggests the following three-step methodology toconstruct a robust representation formula for πt(φ):

Step 1. We construct the triplet of processes (X y ,Y y , Iy ) corresponding tothe pair (y ,B) where y is now a fixed observation path y. = ys, s ∈ [0, t ]belonging to a suitable class of continuous functions and prove that therandom variable f (X y ,Y y )exp(Iy ) is P0-integrable.


. . . . . .


Step 2. We prove that the function y . → gφt (y .) defined as

gφt (y.) = E0

[φ(X y

t ,Yyt ) exp(Iy

t )]

(38)

is continuous. Step 3. We prove that gφ

t (Y .) is a version of pt(f ). Then, following (36), thefunction, y . → θφt (y .) defined as

θt =gφ

t

g1t

(39)

provides the robust version of πt(φ).

Step 1 is immediate in the uncorrelated case. In this the process (X y ,Y y ) canbe taken to be the pair (X , y). Moreover, we can define Iy by the formula

Iyt :=

dY∑i=1

(hi(Xt)y i

t −∫ t

0y i

r dhi(Xr )−12

∫ t

0(hi(Xr ,Yr ))

2dr

), t ≥ 0. (40)

provided the processes hi(X ) are semi-martingales. The construction of theprocess (X y ,Y y , Iy ) is no longer immediate in the correlated noise case.


. . . . . .

Part II A counterexample

Consider the filtering problem where the signal and the observation processsolve the following pair of equations

Xt = X0 +

∫ t

0Xr d

[Y 1

r + Y 2r]+

∫ t

0Xr dr

Yt =

∫ t

0h(Xr )dr + Wt ,

where Y is 2-dimensional and P(X0 = 0) = P(X0 = 1) = 12 . Then with f , h

such that f (0) = h1(0) = h2(0) = 0 one can explicitly compute

E[f (Xt)|Yt ]

=f (eY 1

t +Y 2t )

1 + exp(−∑

k=1,2

∫ t

0hk (eY 1

s +Y 2s ))dY k

s +12

∫ t

0||h(eY 1

t +Y 2t )||2ds

) .(41)

Following the findings of rough path theory the expression on the right handside of (41) is not continuous in supremum norm (nor in any other metric onpath space) because of the stochastic integral.


. . . . . .


Nevertheless, a variation of the robustness representation formula still existsin this case. For this we ”enhance” the original process Y by adding a secondcomponent to it which consists of its iterated integrals (that, knowing the path,is in a one-to-one correspondance with the Levy area process).

Explicitly we consider the process Y = Y t , t ≥ 0 defined as

Y t =

Yt ,

∫ t0 Y 1

r dY 1r · · ·

∫ t0 Y 1

r dY dY

r· · · · · · · · ·∫ t

0 Y dY

r dY 1r · · ·

∫ t0 Y dY

r dY dY

r

, t ≥ 0. (42)

The stochastic integrals in (42) are Stratonovich integrals. The state space ofY is G2(RdY ) ∼= RdY ⊕ so(dY ), where so(dY ) is the set of anti-symmetricmatrices of dimension dY . Over this state space we consider not the space ofcontinuous function, but a subspace C0,α that contains paths

η : [0, t ] → G2(RdY )

that are α-Holder in the RdY -component and “2α-Holder” in theso(dY )-component, where α is a suitably chosen constant α < 1/2.


. . . . . .


Remark. There exists a modification of Y such that Y (ω) ∈ C0,α for all ω.

The space C0,α is endowed with the α-Holder rough path metric under whichC0,α becomes a complete metric space..Theorem..

......Under additional assumptions, θt : C0,α → R is locally Lipschitz.

Denote by Y ·, as before, the canonical rough path lift of Y to C0,α. We thenhave.Theorem..

......Under additional assumptions, θ(Y ·) = πt(f ), P− a.s.


. . . . . .

Part II Notation

Notation:

Lipγ - the set of γ-Lipschitz functions a : Rm → Rn where m and n arechosen according to the context (in the sense of E. Stein, i.e. bounded k-thderivative for k = 0, . . . , ⌊γ⌋ and γ − ⌊γ⌋-Holder continuous ⌊γ⌋-th derivative)

G2(RdY ) ∼= Rd ⊕ so(dY ) –the state space for a dY -dimensional Brownianmotion (or, in general for an arbitrary semi-martingale) and its correspondingLevy area.

C0,α := C0,α−Hol0 ([0, t ],G2(RdY )) – the set of geometric α-Holder rough paths

η : [0, t ] → G2(RdY ) starting at 0. We use the non-homogenous metric ρα−Holon this space.

(Ω, F , (Ft)t≥0, P) – an auxiliary filtered probability space carrying adB-dimensional Brownian motion B.

Let S0 = S0(Ω) – the space of adapted, continuous processes in RdS , withthe topology of uniform convergence in probability. For q ≥ 1 we denote bySq = Sq(Ω) the space of processes X ∈ S0 such that

||X ||Sq :=

(E[sup

s≤t|Xt |q]

)1/q

< ∞.


. . . . . .

Part II SDE with rough drift

Stochastic differential equations with rough drift.

On (Ω, F , (Ft)t≥0, P) choose a dB-dimensional Brownian motion B and abounded dS-dimensional random vector S0 independent of B. In the following,we fix ϵ ∈ (0, 1) and α ∈ ( 1

2+ϵ ,12 ). Let ηn : [0, t ] → RdY be smooth paths, such

that ηn → η in α-Holder norm, for some η ∈ C0,α and let Sn be adS-dimensional process which is the unique solution to the classical SDE

Snt = S0 +

∫ t

0a(Sn

r )dr +∫ t

0b(Sn

r )dBr +

∫ t

0c(Sn

r )dηnr ,

where we assume that (a1.) a ∈ Lip1(RdS ), b1, . . . , bdB ∈ Lip1(RdS ) and c1, . . . , cdY ∈ Lip4+ϵ(RdS )

(a1’) a ∈ Lip1(RdS ), b1, . . . ,bdB ∈ Lip1(RdS ) and c1, . . . , cdY ∈ Lip5+ϵ(RdS ))..Theorem..

......

Under assumption (a1), there exists a dS-dimensional process S∞ ∈ S0 suchthat Sn → S∞, in S0. In addition, the limit Ξ(η) := S∞ only depends on η andnot on the approximating sequence.Moreover, for all q ≥ 1, η ∈ C0,α it holds that Ξ(η) ∈ Sq and the correspondingmapping Ξ : C0,α → Sq is locally uniformly continuous (and locally Lipschitzunder assumption (a1’)).


. . . . . .


Following Theorem 17, we say that Ξ(η) is a solution of the SDE with roughdrift

Ξ(η)t = S0 +

∫ t

0a(Ξ(η)r )dr +

∫ t

0b(Ξ(η)r )dBr +

∫ t

0c(Ξ(η)r )dηr . (43)

Recall that (Ω,F ,P0) carries, as above, the dY -dimensional Brownian motionY and let Ω = Ω× Ω be the product space, with product measure P := P0 ⊗ P.Let S be the unique solution on this probability space to the SDE

St = S0 +

∫ t

0a(Sr )dr +

∫ t

0b(Sr )dBr +

∫ t

0c(Sr ) dYr .

Denote by Y the rough path lift of Y (i.e. the enhanced Brownian Motion overY ).

The following result establishes some of the salient properties of solutions ofSDEs with rough drift:


. . . . . .


.Theorem..

......

Under assumption (a1) we have thatFor every R > 0, q ≥ 1

sup||η||α−Hol<R

E[exp(q|Ξ(η)|∞;[0,t])] < ∞.

For P0 − a.e. ω

P[Ss(ω, ·) = Ξ(Y (ω))s(·), s ≤ t ] = 1.


. . . . . .


In the following we will make use of the Stratonovich version of equation


k

σk (Xt ,Yt)dY kt +

∑j

σj(Xt ,Yt)dBjt , (44)

where f j0(x , y) = f j

0(x , y)−12

∑k∑

i ∂xi σjk (x , y)σ

ik (x , y)−

12

∑k ∂yk σ

jk (x , y).

Recall that, under P0 the observation Y is a Brownian motion independent ofB. We fix ϵ ∈ (0, 1) α ∈ ( 1

2+ϵ ,12 ), t > 0 and X0 is a bounded random vector

independent of B and Y . We will use one of the following assumptions:

Z1, . . . ,ZdY ∈ Lip4+ϵ, h1, . . . ,hdY ∈ Lip4+ϵ and L0, L1, . . . ,LdB ∈ Lip1

Z1, . . . ,ZdY ∈ Lip5+ϵ, h1, . . . ,hdY ∈ Lip5+ϵ and L0, L1, . . . ,LdB ∈ Lip1

Remark. Assumption (A1) and (A1’) lead to the existence of a solution of anSDEs with rough driver . Under (A1) the solution mapping is locally uniformlycontinuous, under (A1’) it is locally Lipschitz .


. . . . . .


Assume either (A1) or (A1’). For η ∈ C0,α there exists a solution (Xη, Iη) tothe following SDE with rough drift

Xηt = X0 +

∫ t

0f (Xη

r ,Yηr )dr +

∫ t

0σ(Xη

r ,Yηr )dηr +

∑j

∫ t

0σj(X

ηr ,Yη

r )dBjr ,

Yηt =

∫ t

0dηr ,

Iηt =

∫ t

0h(Xη

r ,Yηr )dηr −

12

∑k

∫ t

0Dk hk (Xη

r ,Yηr )dr .

(45)

Note that formally (!) when replacing the rough path η with the process Y ,Xη,Yη yields the solution to the SDE (??) and exp(Iηt ) yields the (Girsanov)multiplicator in (36). This observation is made precise in the statement ofTheorem 17.


. . . . . .


We introduce the functions gf ,g1, θ : C0,α → R defined as

gf (η) := E[f (Xηt ,Yη

t ) exp(Iηt )], g1(η) := E[exp(Iηt )], θ(η) :=gf (η)

g1(η), η ∈ C0,α.

.Theorem..

......Assume that (A1) holds, then θ is locally uniformly continuous. Moreover if(A1’) holds, then θ is locally Lipschitz.

Denote by Y ·, as before, the canonical rough path lift of Y to C0,α. We thenhave.Theorem..

......Assume either (A1) or (A1’). Then θ(Y ·) = πt(f ), P− a.s.


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