Table Of ContentRough path stability of SPDEs arising in non-linear filtering
Peter Friz Harald Oberhauser
TU Berlin and WIAS TU Berlin
0
1 July 21, 2010
0
2
l Abstract
u
J We consider linear and semi-linear stochastic partial differential equations and establish
0 stability in rough path sense. The main example we have in mind is the Zakai equation from
2 the theory of non-linear filtering; in this case our results can be viewed as ”robustification”,
applicable even when thenoise in theobservation process dependsof thesignal.
]
R
P 1 Introduction
.
h
t 1.1 Motivation from filtering
a
m
Stochastic filtering is concerned with the estimation of the conditional law of a Markov process,
[ given observations of some function of it. The classical formulation looks at the case where the
process is of diffusion type and splits into a first part (known as the signal) and a second part,
2
v known as the observation process with values in a vector space, and whose martingale part has
1 stationary increments independent of the signal. (This amounts to σ = 0 in what follows and we
8 willspeakoftheuncorrelated case.) However,veryoftenfilteringproblemspresentthemselveswith
7
the noiseinthe observationprocessbeing dependentofthe signal. Inthis caseitis muchharderto
1
achieve robust and stable filtering as the transformation; we will speak of the correlated case. To
.
5 be specific, let B,B˜ be independent multidimensional Brownian motions and consider1
0
0 d
1 dX = V (X)dt+ V (X)dB˜i+ σ (X)dBj (signal)
0 i j
:
v i j=1
X X
Xi dZ = η(X)dt+dB ∈Rd (observation).
r Under generic conditions, the (unnormalized) conditional density u of X given σ(Z :s≤t)
a t t s
is then known to satisfy Zakai’s equation
d
du = G∗u dt+ N∗u dZj
t t j t t
j=1
X
d d
1
= G∗− N∗N∗ u dt+ N∗u ◦Zj
2 j j t j t t
j=1 j=1
X X
1Thepresentdiscussionispurelymotivationalandweshallnotgivepreciseconditionsonthecoefficients.
1
whereG=hV ,D·i+(1/2)Tr VVT +σσT D2· isthegeneratorofX andN u=hσ (x),Dui+
0 j j
η (x)u. In other words, u=u(t,x;ω) satisfies a stochastic partialdifferential equation(SPDE) of
j (cid:2)(cid:0) (cid:1) (cid:3)
parabolic type, namely
d
du=L x,u,Du,D2u dt+ Λ (x,u,Du) ◦dZj.
j
j=1
(cid:0) (cid:1) X
NotethatwehavewrittentheobservationdifferentialsinStratonovichform;thiswillbeconvenient
for the rough path approach taken later in this paper.
Classical robustification: Intheuncorrelated case (σ ≡0)onehasM =M (x,u)=η (x)u
j j j
anditispartofthefolkloreofthesubjectthatZakai’sequationadmitsapathwisesolution. Indeed,
take a smooth path z and solve the auxilary differential equation φ˙ = φ η (x)dzj ≡ φη·dz.
j j
The solution is given by
P
t
φ =φ exp η(x)·dz =φ exp(η(x)·z )
t 0 0 t
(cid:18)Z0 (cid:19)
and induces the flow map φ(t,φ ) := φ exp(η(x)·z ); observe that these expressions can be
0 0 t
extended by continuity to any continuous path z such as a typical realization of Z (ω). The
·
point is that this transform allows to transform the Zakai SPDE into a random PDE, sometimes
called the Zakai equation in its robust form: it suffices to introduce v via the ”outer transform”
u(t,x)=φ(t,v(t,x)) which leads immediately to
v(t,x):=exp(−η(x)·z )u(t,x).
t
An elementary computation then shows that v solves a linear PDE, in the sense that φL is affine
linear in v,Dv,D2v with coefficients that will depend on z resp. Z (ω),
t
dv = φL x,v,Dv,D2v dt.
Moreover,onecanconcludefromthisrepresen(cid:0)tationthatu(cid:1)=u(z)iscontinuouswithrespecttothe
uniformmetric|z−z˜| . Thisprovidesafullypathwise”robust”approachtoZakai’sequation.
∞;[0,T]
Rough path robustification: The classical robustification does not work in presence of
correlation (σ 6=0). In fact, we cannot expect PDE solutions to
d
du=L x,u,Du,D2u dt+ Λ (x,u,Du) dz,
j
j=1
(cid:0) (cid:1) X
surely well-defined for smooth z : [0,T] → Rd, to depend continuously on z in uniform topology.
Our main result is that u = u(z) is continuous with respect to rough path metric2. That is, if
(zn) ⊂ C1 [0,T],Rd is Cauchy in rough path metric then (un) will converge to a limit which ill
be seen to depend only on the (rough path) limit of zn (and not on the approximating sequence).
(cid:0) (cid:1)
As a consequence,it is meaningful to replace z aboveby an abstract(geometric)roughpath z and
the analogue of Lyons’ universallimit theorem [11] holds. Let us briefly mention that this leads to
numerous (essentially immediate) corollaries for our class of SPDEs, including Stroock-Varadhan
2Two(smooth)pathsz,z˜arecloseinroughpathmetriciffzisclosetoz˜ANDsufficienlymanyiteratedintegrals
ofz areclosetothoseofz˜. Moredetailsaregivenlaterinthisarticleasneeded.
2
type support theorems, Freidlin-Wentzell type large deviations, maximum principle and splitting
results. Since this has been rather well understood in the (rough path) literature, we shall not go
into explicit detail here.
At last, we should mention that the relevance of rough path theory for filtering in presence of
correlation appears as motivating conjecture in Lyons’ seminal 1998 paper [10] on rough paths,
a theory that proposes to understand paths together with their iterated integrals (”areas”) in
order to guarantee robustness of differential equations driven by such paths.This conjecture is
supported in particular by the works of I. Gy¨ongy [8, 7] who obtained explicit results of how
“correct” approximations to the observation process, but wrong approximationof its area process,
leadto convergencetothe “wrong”solution. Aquantitativeformofthis conjectureis the assertion
that solutions to the filtering problem (i.e. the afore-mentioned SPDEs) depend continuously on
the observation process seen as a rough path, the proof of which is the content of this paper.
1.2 Structure and outline
Consider the SPDE
d
du+L t,x,u,Du,D2u dt= Λ (t,x,u,Du)◦dZk, (1)
k
k=1
(cid:0) (cid:1) X
withscalarinitialdatau(0,·)=u (·),assumedtobeBUC (boundedanduniformlycontinuous)on
0
Rn,andgivemeaningandroughpathstabilityforthesolutiontotheaboveequationsasafunction
of Z, the (canonical semi-martingale) enhancement of the d-dimensional observation process Z.
Here L is a linear second order operator of the form
L(t,x,r,p,X)=−Tr[A(t,x)·X]+b(t,x)·p+c(t,x,r).
Actually, our approach allows to deal with L semi-linear in the sense that f above may depend
non-linearly on r with no extra effort. The first order operators Λ =Λ (t,x,r,p) are affine linear
k k
in r,p. That is,
Λ (t,x,r,p)=(p·σ (x))+rν (t,x)+g (t,x).
k k k k
We shall prefer to write the rhs of (1) in the equivalent form
d1 d2 d3
(Du·σ (x))◦dY1;i+u ν (t,x)◦dY2;j + g (t,x)◦dY3;k
i t j k
i=1 j=1 k=1
X X X
where Y ≡(Y1,Y2,Y3) is a (d +d +d )-dimensionalsemimartingale. Our approachis basedon
1 2 3
a pointwise (viscosity) interpretation of (1): we successively transform away the noise terms such
as to transform the SPDE, ultimately, into a random PDE. The big scheme of the paper is
u Transfo7→rmation1 u1 where u1 has the (gradient) noise driven by Y1 removed;
u1 Transfo7→rmation2u12 where u12 has the remaining noise driven by Y2 removed;
u12 Transfo7→rmation3u˜ where u˜ has the remaining noise driven by Y3 removed.
None of these transformations is new on its own. The first is an example of Kunita’s stochastic
characteristics method; the second is known as robustification (also know as Doss-Sussman trans-
form); the third amounts to change u12 additively by a random amount and has been used in
3
virtually every SPDE context with additive noise.3 What is new is that the combined transforma-
tion can be managed and is compatible with rough path convergence; for this we have to remove
all probability from the problem: In fact, we will transform an RPDE (rough PDE) solution u
into a classical PDE solution u˜ in which the coefficients depend on various rough flows (i.e. the
solution flows to rough differential equations) and their derivatives. Stability results of rough path
theory and viscosity theory, first introduced for a class non-linear problems [2], then play together
to yield the desired result. Upon using the canonical rough path lift of the observation process
in this RPDE, a truly (rough)pathwise point of view, one has constructed a robust version of the
SPDE solution in equation. We note that the viscosity/Stratonovich approach allows us to avoid
any ellipticity assumption on L; we can even handle the fully degenerate first order case. In turn,
we only obtain BUC solutions. Stronger assumptions would allow to discuss all this in a classical
context (i.e. u˜ would be a C1,2 solution); SPDE solution can then be seen to have certain spatial
regularity.
Inthe presenceofgradient(rough)noise onlythis approachwasimplemented in[1]. We should
remark that the usual way to deal with (1), which goes back to Krylov, Rozovskii, Pardoux and
others, is to find solutions in a suitable functional analytic setting; e.g. such that solutions evolve
insuitable Sobolev spaces. Interestingly,there hasbeen no successuntilnow (despite the advances
by Gubinelli–Tindel [6] and Teichmann [13]) to include (1) in a setting of abstractrough evolution
equations on infinite-dimensional spaces.
Acknowledgement 1 The first author would like to thank the organizers and participants of the
Filtering Workshop in June 2010, part of the Newton Institute’s SPDE program, where this work
was first presented. The second author would like to express his gratitude for a Newton Institute
Junior membership grant.
2 Recalls on viscosity theory and rough paths
Let us recall some basic ideas of (second order) viscosity theory [3, 4] and rough path theory
[11, 12]. As for viscosity theory, consider a real-valued function u=u(t,x) with t∈[0,T],x∈Rn
and assume u∈C2 is a classical subsolution,
∂ u+F t,x,u,Du,D2u ≤0,
t
(cid:0) (cid:1)
where F is a (continuous) function, degenerate elliptic in the sense that F (t,x,r,p,A+B) ≤
F (t,x,r,p,A) whenever B ≥ 0 in the sense of symmetric matrices. The idea is to consider a
(smooth) test function ϕ and look at a local maxima tˆ,xˆ of u−ϕ. Basic calculus implies that
Du tˆ,xˆ =Dϕ tˆ,xˆ , D2u tˆ,xˆ ≤Dϕ tˆ,xˆ and, from degenerate ellipticity,
(cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) ∂ ϕ+(cid:0)F tˆ(cid:1),xˆ,u,Dϕ,D2ϕ ≤0. (2)
t
This suggests to define a viscosity supersol(cid:0)ution (at the poi(cid:1)nt xˆ,tˆ ) to ∂ +F =0 as a continuous
t
function u with the property that (2) holds for any test function. Similarly, viscosity subsolutions
(cid:0) (cid:1)
are defined by reversing inequality in (2); viscosity solutions are both super- and subsolutions. A
3Transformation2and3couldactuallybeperformedin1step;however,theseparationleedstoasimpleranalytic
tractabilityofthetransformedequations.
4
different point of view is to note that u(t,x)≤ u tˆ,xˆ −ϕ tˆ,xˆ +ϕ(t,x) for (t,x) near tˆ,xˆ . A
simple Taylor expansion then implies
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1
u(t,x)≤u tˆ,xˆ +a t−tˆ +p·(x−xˆ)+ (x−xˆ)T ·X ·(x−xˆ)+o |xˆ−x|2+ tˆ−t (3)
2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:12) (cid:12)(cid:17)
as |xˆ−x|2+ tˆ−t →0 with a=∂ ϕ tˆ,xˆ , p=Dϕ tˆ,xˆ , X =D2ϕ tˆ,xˆ . Moreover(cid:12),if (3(cid:12)) holds
t
for some (a,p,X) and u is differentiable, then a=∂ u tˆ,xˆ , p=Du tˆ,xˆ , X ≤D2u tˆ,xˆ , hence
(cid:12) (cid:12) (cid:0) (cid:1) t(cid:0) (cid:1) (cid:0) (cid:1)
by degenerate(cid:12) ellip(cid:12)ticity
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
∂ ϕ+F tˆ,xˆ,u,p,X ≤0.
t
Pushing this idea further leads to a definitio(cid:0)n of viscosit(cid:1)y solutions based on a generalized notion
of “ ∂ u,Du,D2u ” for nondifferentiable u, the so-called parabolic semijets, and it is a simple
t
exercise to show that both definitions are equivalent. The resulting theory (existence, uniqueness,
(cid:0) (cid:1)
stability,...) iswithout doubtoneofthe mostimportantrecentdevelopmentsinthe fieldofpartial
differential equations. As a typical result4, the initial value problem (∂ +F)u = 0, u(0,·) =
t
u ∈ BUC(Rn) has a unique solution in BUC([0,T]×Rn) provided F = F(t,x,u,Du,D2u) is
0
continuous,degenerateelliptic,proper(i.e. increasingintheuvariable)andsatisfiesa(well-known)
technical condition5. In fact, uniqueness follows from a stronger property known as comparison:
assume u (resp. v) is a supersolution (resp. subsolution) and u ≥ v ; then u ≥ v on [0,T]×Rn.
0 0
A key feature of viscosity theory is what workers in the field simply call stability properties. For
instance, it is relatively straight-forward to study (∂ +F)u = 0 via a sequence of approximate
t
problems, say (∂ +Fn)un =0, provided Fn →F locally uniformly and some apriori information
t
on the un (e.g. locally uniform convergence, or locally uniform boundedness6. Note the stark
contrast to the classical theory where one has to control the actual derivatives of un.
The idea of stability is also central to rough path theory. Given a collection (V ,...,V ) of
1 d
(sufficiently nice) vector fields on Rn and z ∈C1 [0,T],Rd one considers the (unique) solution y
to the ordinary differential equation
(cid:0) (cid:1)
d
y˙(t)= V (y)z˙i(t), y(0)=y ∈Rn. (4)
i 0
i=1
X
The questionis,if the outputsignaly depends inastable wayonthe drivingsignalz. Theanswer,
of course, depends strongly on how to measure distance between input signals. If one uses the ∞
norm, so that the distance between driving signals z,z˜is given by |z−z˜| , then the solution
∞;[0,T]
will in general not depend continuously on the input.
Example 2 Take n=1,d=2, V =(V ,V )=(sin(·),cos(·)) and y =0. Obviously,
1 2 0
1 1
zn(t)= cos 2πn2t , sin 2πn2t
n n
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
converges to 0 in ∞-norm whereas the solutions to y˙n =V (yn)z˙n,yn =0, do not converge to zero
0
(the solution to the limiting equation y˙ =0).
4BUC(...)denotes thespaceofbounded, uniformlycontinuous functions.
5(3.14)oftheUser’sGuide[3].
6WhatwehaveinmindhereistheBarles–Perthame method of semi-relaxed limits [4].
5
If |z−z˜| is replaced by the (much) stronger distance
∞;[0,T]
|z−z˜| = sup z −z˜ ,
1-var;[0,T] ti,ti+1 ti,ti+1
(ti)⊂[0,T]
X(cid:12) (cid:12)
(cid:12) (cid:12)
itiselementarytoseethatnowthesolutionmapiscontinuous(infact,locallyLipschitz);however,
this continuitydoes not lenditself to pushthe meaning of(4): the closureofC1 (or smooth)paths
in variation is precisely W1,1, the set of absolutely continuous paths (and thus still far from a
typical Brownian path). Lyons’ theory of rough paths exhibits an entire cascade of (p-variation or
1/p-H¨older type rough path) metrics, for each p ∈ [1,∞), on path-space under which such ODE
solutionsarecontinuous(andevenlocallyLipschitz)functions oftheir drivingsignal. Forinstance,
the ”rough path” p-variation distance between two smooth Rd-valued paths z,z˜is given by
1/p
p
(j) (j)
max sup z −z˜
j=1,...,[p] (ti)⊂[0,T]X(cid:12) ti,ti+1 ti,ti+1(cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
wherez(j) = dz ⊗···⊗dz withintegrationoverthej-dimensionalsimplex{s<r <···<r <t}.
s,t r1 rj 1 j
This allows to extend the very meaning of (4), in a unique and continuous fashion, to driving sig-
nals which livRe in the abstract completion of smooth Rd-valued paths (with respect to rough path
p-variation or a similarly defined 1/p-H¨older metric). The space of so-called p-rough paths7 is
precisely this abstract completion. In fact, this space can be realized as genuine path space,
C0,p-var [0,T],G[p] Rd resp. C0,1/p-Ho¨l [0,T],G[p] Rd
(cid:16) (cid:0) (cid:1)(cid:17) (cid:16) (cid:0) (cid:1)(cid:17)
where G[p] Rd is the free step-[p] nilpotent group over Rd, equipped with Carnot–Caratheodory
metric; realized as a subset of 1+t[p] Rd where
(cid:0) (cid:1)
t[p] Rd =(cid:0) R(cid:1)d⊕ Rd ⊗2⊕···⊕ Rd ⊗[p]
isthenaturalstatespacefor(up(cid:0)to[p(cid:1)])iterated(cid:0) int(cid:1)egralsofas(cid:0)moo(cid:1)thRd-valuedpath. Forinstance,
almost every realization of d-dimensional Brownian motion B enhanced with its iterated stochastic
integrals in the sense of Stratonovich, i.e. the matrix-valued process given by
·
B(2) := Bi◦dBj (5)
(cid:18)Z0 (cid:19)i,j∈{1,...,d}
yields a path B(ω) in G2 Rd with finite 1/p-H¨older (and hence finite p-variation) regularity, for
any p > 2. (B is known as Brownian rough path.) We remark that B(2) = 1B ⊗B +A where
(cid:0) (cid:1) 2
A:=Anti B(2) is knownas L´evy’s stochastic area; in other words B(ω) is determined by (B,A),
i.e. Brownian motion enhanced with L´evy’s area. A similar construction work when B is replaced
(cid:0) (cid:1)
by a generic multi-dimensional continuous semimartingales; see [5, Chapter 14] and the references
therein.
7Inthestrictterminologyofroughpaththeory: geometricp-roughpaths.
6
3 Transformations
3.1 Inner and outer transforms
Throughout,F =F (t,x,r,p,X)isacontinuousscalar-valuedfunctionon[0,T]×Rn×R×Rn×S(n),
S(n) denotes the space of symmetric n × n-matrices, and F is assumed non-increasing in X
(degenerate elliptic). Time derivatives of functions are denoted by upper dots, spatial deriva-
tives (with respect to x) by D,D2, etc. Further, we use h.,.i to denote tensor contraction8, i.e.
hp,qi ≡ p qi1,...,im, p∈ Rl ⊗m,q ∈ Rl ⊗n⊗ Rl ′ ⊗m.
j1,...,jn i1,...,im i1,...,im j1,...,jn
P (cid:0) (cid:1) (cid:0) (cid:1) (cid:16)(cid:0) (cid:1)(cid:17)
Lemma 3 (Inner Transform) Let z ∈C1 [0,T],Rd , σ =(σ ,...,σ )⊂C2(Rn,Rn) and ψ =
1 d b
ψ(t,x) the ODE flow of dy =V (y)dz, i.e.
(cid:0) (cid:1)
d
ψ˙ (t,x)= σ (ψ(t,x))z˙i, ψ˙ (0,x)=x∈Rn.
i t
i=1
X
Then u is a viscosity subsolution (always assumed BUC) of
d
∂ u+F t,x,r,Du,D2u − (Du·σ (x))z˙i =0; u(0,.)=u (.) (6)
t i t 0
i=1
(cid:0) (cid:1) X
iff w(t,x):=u(t,ψ(t,x)) is a viscosity subsolution of
∂ w+Fψ t,x,w,Dw,D2w =0; w(0,.)=u (.) (7)
t 0
(cid:0) (cid:1)
where
Fψ(t,x,r,p,X)=F t,ψ (x),r, p,Dψ−1| , X,Dψ−1| ⊗Dψ−1| + p,D2ψ−1|
t t φt(x) t ψt(x) t ψt(x) t ψt(x)
(cid:0) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)(cid:1)
and
∂ ψ−1(t,x) k k=1,...,n ∂ ψ−1(t,x) k k=1,...,n
Dψ−1| = t and D2ψ−1| = .
t x ∂xi t x ∂xixj
(cid:0) (cid:1) ! (cid:0) (cid:1) !
i=1,...,n i,j=1,...,n
The same statement holds if one replaces the word subsolution by supersolution throughout.
Corollary 4 (Transformation 1) Let ψ = ψ(t,x) be the ODE flow of dy = V (y)dz, as above.
Define L=L(t,x,r,p,X) by
L=−Tr[A(t,x)·X]+b(t,x)·p+c(t,x,r);
define also the transform
Lψ =−Tr Aψ(t,x)·X +bψ(t,x)·p+cψ(t,x,r)
8Wealsouse·todenote contractiono(cid:2)veronlyindex(cid:3)ortodenotematrixmultiplication.
7
where
Aψ(t,x) = A(t,ψ (x)),Dψ−1| ⊗Dψ−1| ,
t t ψt(x) t ψt(x)
bψ(t,x)·p = b(cid:10)(t,ψt(x))· p,Dψ−t 1|ψt(x) −Tr A(t,(cid:11)ψt)· p,D2ψ−t 1|ψt(x) ,
cψ(t,x,r) = c(t,ψt(x),r)(cid:10). (cid:11) (cid:0) (cid:10) (cid:11)(cid:1)
Then u is a solution (always assumed BUC) of
d
∂ u+L t,x,u,Du,D2u = (Du·σ (x))z˙i; u(0,.)=u (.)
t i t 0
i=1
(cid:0) (cid:1) X
if and only if u1(t,x):=u(t,ψ(t,x)) is a solution of
∂ +Lψ =0; u1(0,.)=u (.) (8)
t 0
Proof. Set y = ψ (x). When u is a classical sub-solution, it suffices to use the chain-rule and
t
definition of Fψ to see that
w˙ (t,x) = u˙(t,y)+Du(t,y)·ψ˙ (x)=u˙(t,y)+Du(t,y)·V (y)z˙
t t
≤ F t,y,u(t,y),Du(t,y),D2u(t,y) =Fψ t,x,w(t,x),Dw(t,x),D2w(t,x) .
The case when u i(cid:0)s a viscosity sub-solution of (6)(cid:1)is not(cid:0)much harder: suppose that (t¯,x¯)(cid:1)is a
maximum of w − ξ, where ξ ∈ C2([0,T]×Rn) and define ϕ ∈ C2([0,T]×Rn) by ϕ(t,y) =
ξ t,ψ−1(y) . Set y¯=ψ (x¯) so that
t t¯
(cid:0) F(cid:1) t¯,y¯,u(t¯,y¯),Dϕ(t¯,y¯),D2ϕ(t¯,y¯) =Fψ t¯,x¯,w(t¯,x¯),Dξ(t¯,x¯),D2ξ(t¯,x¯) .
Obviously, (t¯(cid:0),y¯) is a maximum of u−ϕ, and(cid:1)since u(cid:0)is a viscosity sub-solution of (6)(cid:1)we have
ϕ˙ (t¯,y¯)+Dϕ(t¯,y¯)V (y¯)z˙(t¯)≤F t¯,y¯,u(t¯,y¯),Dϕ(t¯,y¯),D2ϕ(t¯,y¯) .
Ontheotherhand,ξ(t,x)=ϕ(t,ψ (x))implies(cid:0)ξ˙(t¯,x¯)=ϕ˙ (t¯,y¯)+Dϕ(t¯,y¯)V (y¯)(cid:1)z˙(t¯)andputting
t
things together we see that
ξ˙(t¯,x¯)≤Fψ t¯,x¯,w(t¯,x¯),Dξ(t¯,x¯),D2ξ(t¯,x¯)
which says precisely that w is a viscos(cid:0)ity sub-solution of (7). Replacin(cid:1)g maximum by minimum
and ≤ by ≥ in the preceding argument, we see that if u is a super-solution of (6), then w is a
super-solution of (7).
Conversely, the same arguments show that if v is a viscosity sub- (resp. super-) solution for (7),
then u(t,y)=w t,ψ−1(y) is a sub- (resp. super-) solution for (6).
We prepare the next lemma by agreeing that for a sufficiently smooth function φ = φ(t,r,x) :
[0,T]×R×Rn →(cid:0) R we sha(cid:1)ll write
∂φ(t,r,x) ∂φ(t,r,x)
φ˙ = ,φ′ = ,
∂t ∂r
∂φ(t,r,x) ∂2φ(t,r,x)
Dφ = and D2φ= .
∂xi ∂xi∂xj
(cid:18) (cid:19)i=1,...,n (cid:18) (cid:19)i,j=1,...,n
8
Lemma 5 (Outer transform) Let φ=φ(t,r,x)∈C1,2,2 and strictly increasing in r. (It follows
that φ(t,.,x) is an increasing diffeomorphism on the real line). Then u is a subsolution of ∂ u+
t
F t,x,r,Du,D2u =0, u(0,.)=u (.) if and only if
0
(cid:0) (cid:1) v(t,x)=φ−1(t,u(t,x),x)
is a subsolution of ∂ v+φF t,x,r,Dv,D2v =0, v(0,.)=φ−1(0,u (x),x) with
t 0
(cid:0) φ˙ (cid:1)1
φF (t,x,r,p,X) = + F t,x,φ,Dφ+φ′p, (9)
φ′ φ′
φ′′p⊗p+(cid:0)Dφ′⊗p+p⊗Dφ′+D2φ+φ′X
where φ and all derivatives are evaluated at (t,r,x). The same statement holds if(cid:1)one replaces the
word subsolution by supersolution throughout.
Proof. (=⇒) We show the first implication, i.e. assume u is a subsolution of ∂ u+F =0 and set
t
v(t,x)=φ−1(t,u(t,x),x). By definition, (a,p,X)∈P2;+v(s,z) iff
1
v(t,x)≤v(s,z)+a(t−s)+p·(x−z)+ (x−z)T·X·(x−z)+o |t−s|+|x−z|2 as (t,x)→(s,z).
2
(cid:16) (cid:17)
Since φ(t,.,x) is increasing,
1
φ(t,v(t,x),x)≤φ t,v(s,z)+a(t−s)+p·(x−z)+ (x−z)T ·X ·(x−z)+o |t−s|+|x−z|2 ,x
2
(cid:18) (cid:16) (cid:17) (cid:19)
and using a Taylor expansion on φ in all three arguments we see that the rhs equals
1
φ(s,v(s,z),z)+φ˙ (t−s)+φ′ a(t−s)+φ′ p·(x−z)+ φ′ (x−z)T ·X ·(x−z)
s,v(s,z),z s,v(s,z),z s,v(s,z),z 2 s,v(s,z),z
1
+Dφ ·(x−z)+ (x−z)T ·D2φ ·(x−z)
s,v(s,z),z 2 s,v(s,z),z
+(x−z)T · D φ′ ⊗p·(x−z)
s,v(s,z),z
+(x−z)T ·p(cid:0)⊗(cid:0)(D(cid:1)φ(cid:1))′ ·(x−z)
s,v(s,z),z
+(x−z)T ·φ′′ p⊗p·(x−z)+o |t−s|+|x−z|2 as (s,z)→(t,x)
s,v(s,z),z
(cid:16) (cid:17)
Hence,
φ˙ +φ′ a,Dφ +φ′ p,
s,v(s,z),z s,v(s,z),z s,v(s,z),z s,v(s,z),z
(cid:16)
φ′′ p⊗p+D φ′ ⊗p+p⊗(Dφ)′ +D2φ +φ′ X
s,v(s,z),z s,v(s,z),z s,v(s,z),z s,v(s,z),z s,v(s,z),z
(cid:0) (cid:1) (cid:17)
belongs to P2;+u(s,z) and since u is a subsolution this immediately shows
φ˙ +φ′ a+F s,z,φ ,Dφ +φ′ p,
s,v(s,z),z (s,v(s,z),z) (s,v(s,z),z) s,v(s,z),z s,v(s,z),z
(cid:16)
φ′′ p⊗p+D φ′ ⊗p+p⊗(Dφ)′ +D2φ +φ′ X ≤ 0.
s,v(s,z),z s,v(s,z),z s,v(s,z),z s,v(s,z),z s,v(s,z),z
(cid:17)
(cid:0) (cid:1)
9
Dividing by φ′ (>0) shows that v is a subsolution of ∂ v+Fφ =0.
t
(⇐=)Assume v isasubsolutionof∂ v+φF =0, φF definedasin(9)forsomeF. Setu(t,x):=
t
φ(t,v(t,x),x). By above argument we know that v is a subsolution of φ−1 φF (t,x,r,p,X). For
brevity write ψ(t,.,x)=φ−1(t,.,x). Then
(cid:0) (cid:1)
φ−1 φF (t,x,r,p,X)
ψ 1
= t,r,(cid:0)x +(cid:1) φF t,x,ψ ,Dψ +ψ′ p,
ψ′ ψ′ (t,r,x) t,r,x t,r,x
t,r,x t,r,x
(cid:16)
ψ′′ p⊗p+D ψ′ ⊗p+p⊗(Dψ)′ +D2ψ +ψ′ X
t,r,x t,r,x t,r,x t,r,x t,r,x
= ψt,r,x + 1 (cid:0)φ˙t,(cid:1)ψt,r,x,x+ (cid:17)
ψ′t,r,x ψ′t,r,x "φ′t,ψ ,x
t,r,x
1
F t,x,φ t,ψ ,x ,Dφ +φ′ Dψ +ψ′ p
φ′ t,r,x t,ψt,r,x,x t,ψt,r,x,x t,r,x t,r,x
t,ψ ,x
t,r,x (cid:16) (cid:0) (cid:1) (cid:8) (cid:9)
,φ′′ p⊗p+D φ′ ⊗p+p⊗(Dφ)′ +D2φ
t,ψt,r,x,x t,ψt,r,x,x t,ψt,r,x,x t,ψt,r,x,x
+φ′ ψ′′ p⊗(cid:0) p(cid:1)+D ψ′ ⊗p+p⊗(Dψ)′ +D2ψ +ψ′ X .
t,ψt,r,x,x t,r,x t,r,x t,r,x t,r,x t,r,x
n (cid:0) (cid:1) o(cid:17)i
Usingseveraltimes equalitiesofthe type f ◦f−1 ′ =f′ f−1 ′ =idcancelsthe terms involving
f−1
φ,ψ and their derivatives and we are left with F, i.e.
(cid:0) (cid:1) (cid:0) (cid:1)
φ−1 φF =F.
This finishes the proof. (cid:0) (cid:1)
Corollary 6 (Transformation 2) Assume ν = (ν ,...,ν ) ⊂ C0,2([0,T]×Rn). Assume φ =
1 d b
φ(t,x,r) is determined by the ODE
d
φ˙ =φ ν (t,x)z˙j ≡φν(t,x)·z˙ , φ(0,x,r)=r.
j t t
j=1
X
Define L=L(t,x,r,p,X) by
L=−Tr[A(t,x)·X]+b(t,x)·p+c(t,x,r);
define also
φL(t,x,r,X) = −Tr[A(t,x)·X]+ φb(t,x)·p+ φc(t,x,r) where (10)
2
φb(t,x)·p ≡ b(t,x)·p− Tr A(t,x)·Dφ′⊗p
φ′
1 (cid:2) 1 (cid:3) 1
φc(t,x,r) ≡ − Tr A(t,x)·(D2φ) + b(t,x)·(Dφ)+ c(t,x,φ)
φ′ φ′ φ′
(cid:2) (cid:3)
where φ and all its derivatives are evaluated at (t,r,x). Then
∂ w+L t,x,r,Dw,D2w −wν(t,x)·z˙(t)=0
t
(cid:0) (cid:1)
10
