Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs

0.58 MB·English

by  Harald Oberhauser

ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND
REFLECTED FBSDES
HARALDOBERHAUSERANDGONÇALODOSREIS
Abstract. Followingworkof Dupire(2005), Carr–Lee (2010) andCox–Wang (2011) on connections between
Root’s solutionoftheSkorokhod embeddingproblem, freeboundary PDEsandmodel-independent bounds on
optionsonvarianceweproposeanapproachwithviscositysolutions. Besidesextendingtheprevious results,it
gives a link with reflected FBSDEs as introduced by El Karoui et. al. (1997) and allows for easy convergence
proofsofnumericalschemesviatheBarles–Souganidis (1991)method.
3
1
0
1. Introduction
2
n Let B be standard, real-valued Brownian motion (started at 0) and µ a centered integrable probability
a measure on (R,B(R)). In 1961 Skorokhod[36] asked whether it is possible to find a stopping time τ such that
J B ∼µandBτ =(B ) isuniformlyintegrable;astoppingtimeτ withthesepropertiesisnowusuallyreferred
τ τ∧t t
6 to as a solution of the Skorokhod embedding problem or Skorokhod stopping problem. Further, he provided an
1
affirmativeanswerbyconstructingastoppingtimewhichdependsonanadditionalrandomvariableindependent

of B. Skorokhods’motivation came from invarianceproperties ofrandomwalks andsubsequently this question
]
R attractedtheinterestedofmanyresearchersandalargenumberofdifferentapproachesandgeneralizationswere
P developed, each approach giving rise to a different stopping time which solves the same Skorokhod embedding
. problem—wedonotevenattempttogiveabriefsummarybutrefertothesurveys[27,28,21]whichlistmany
h
different approaches and discusses several applications.
t
a
One of the earliest approaches after Skorokhod was initiated by Root in 1969 [30] who showed that if µ is
m
centeredandhasasecondmoment,thenitispossibletofindaasubsetRof[0,∞]×[−∞,∞],theso-calledRoot
[ barrier, such that its first hitting by the time–space process (t,B ), τ = inf{t≥0:(t,B )∈R}, solves the
t R t
1 Skorokhod embedding problem. In addition to this intuitive solution, Loynes [26] introduced extra conditions
v under which Root’s barrier R is uniquely defined (as a subset of [0,∞]×[−∞,∞]) and Rost [34] proved that
8 among all stopping times τ which solve the Skorokhod embedding problem, Root’s solution minimizes E τ2 ;
9
further he gave important generalizations and connections with potential theory, characterized the possible
7 (cid:2) (cid:3)
stopping distributions of Markov processes by introducing a filling scheme in continuous time and proved the
3
. existence of another barrier which leads to a hitting time which maximizes E τ2 . Despite all these nice
1
properties of Root’s and Rost’s approach they had until very recently the significant disadvantage that it was
0 (cid:2) (cid:3)
not known how to calculate the barriers — in fact, only for a handful of very simple target measures µ the
3
1 explicit form of the barriers was known.
: One of the more recent motivations to study the Skorokhod embedding problem comes from mathematical
v
financewhereSkorokhodembeddingsnaturallyappearifoneisinterestedinmodel-independentpricesofexotic
i
X
derivatives, i.e. “extremal” solutions of the Skorokhod problem lead to robust lower and upper bounds for
r arbitrage-free prices of exotics — we refer to the survey’s of Hobson [21] and Obłój [28]. Motivated by this,
a
Dupire[14]gavein2005apresentationwherehefirstlypointedoutthatinthecaseofoptionsonvariancethese
extremal solutions are the ones of Root [30] resp. Rost [34] and, secondly, that these barriers can be calculated
by solving nonlinear parabolic PDEs, namely solutions to free boundary problems. This was further developed
by Carr–Lee [8] and in 2011 Cox–Wang [11] made this connection precise using the variational approach as
developed in the 1970’s by A. Bensoussan–J.L. Lions et. al, cf. [6] (and do much more — we refer the reader
to [11]). This article is inspired by all these beautiful results and our goal is to study the connection of Root’s
barrier with the parabolic obstacle problem from the perspective of viscosity theory; more precisely, given two
probability measures µ,ν on (R,B(R)) and a function σ :[0,∞)×R →R we want to find (under appropriate
conditions on σ,µ,ν) the Root barrier R such that its first hitting time τ solves the following Skorokhod
R
embedding problem
dX = σ(t,X )dB , X ∼µ,
(1) t t t 0
(cid:26) XτR ∼ ν and XτR =(Xt∧τR)t is uniformly integrable.
Keywordsandphrases. Skorokhodembeddingproblem,RootandRostbarrier,viscositysolutionsofobstacleproblems,reflected
BSDEs,bounds onvarianceoptions.
1
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 2
by identifying R with a continuous function u∈C([0,∞)×R,R) which is a viscosity solution of
min u+ |.−x|ν(dx),∂u − σ2∆u = 0 on (0,∞)×R,
(2) R ∂t 2
( (cid:16) ´ u(0,.(cid:17)) = − |.−x|µ(dx) on R.
R
´
and with the quadruple of process (X,Y,Z,K), solution to the reflected Forward–BackwardSDE (for h(·) :=
− |·−y|ν(dy), T >0, (t,x)∈[0,T]×R and s∈[t,T])
R
´
Xt,x = x+ sσ(T −r,X )dW ,
s t r r
Yt,x = u 0,´Xt,x +Kt,x−Kt,x− T Zt,xdW ,
 s T T s s r r
 Yt,x ≥ h(Xt,x), t<s≤T and T´(Yt,x−h(Xt,x))dKt,x =0.
s (cid:0) s (cid:1) t s s s
´
What we consider among the benefits of the viscosity approach are that it
• coversthecasewhenσistimedependentandonlyelliptic,i.e.notime-homogeneityoruniformellipticity
assumptions as in [11] are needed (especially the important case for model-independent bounds when
σ(t,x)=x is immediately in our setting without an extra transformation),
• gives a direct link to reflected FBSDEs and their Feynman–Kac representation via classic work of El
Karoui, Kapoudjian, Pardoux, Peng and Quenez[15],
• uses continuous functions instead of functions with values in weighted Sobolev spaces and allows for
short proofs of convergence of numerical schemes via the Barles–Souganidis method.
Further, we introduce a generalization of Loynes’ notion of regular barriers, so-called (µ,ν)-regular barriers
which allows to establish a one–to–one correspondence between the problems (1) and (2).
In Section 2 we recall several results from the literature and discuss the general Root embedding problem,
in Section 3 we recall several results in viscosity theory and the connection with reflected FBSDES. Section
4 contains our main result: the link between the embedding problem, PDE in viscosity sense and reflected
FBSDE. In Section 5 apply the Barles–Souganidis [3] method in this context and we finish in Section 6 by
recalling the relevance of Root’s barrier for the derivation of bounds on variance options.
Acknowledgement 1. GdRisalsoaffiliatedwithCMA/FCT/UNL,2829-516Caparica,Portugal. GdRacknowl-
edges partial support by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and
Technology) through PEst-OE/MAT/UI0297/2011 (CMA). HO is grateful for partial support from the Euro-
pean Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC
grant agreement nr. 258237 and DFG Grant SPP-1324.
Acknowledgement 2. The authors would like to thank Alexander Cox, Paul Gassiat, Martin Keller–Resseland
Jan Obłój for helpful remarks.
2. Convex order, potential functions, root barriers and root embedding
Already for the case of Brownian motion one cannot hope to find Skorokhod embeddings for two arbitrary
probability measures µ and ν, i.e. B ∼ µ, B ∼ ν. Intuitively, a necessary condition is that the target
0 τ
distributionν shouldhaveahigherdispersionthanthe initialdistributionµ. This canbeexpressedbyaclassic
order relation on the set of probability measures, namely the so–called convex order. In the context of Root
solution this order relation appears also naturally when one thinks in terms of the potential theoretic picture
as developed in this context by Rost [31, 32, 33, 34], Chacon [10] and others.
Definition1. Letk ∈N,m∈RanddenotewithMk thesetofprobabilitymeasureson(R,B(R))withafinite
m
k-th moment and mean m (i.e. µ∈Mk iff xµ(dx)=m and |x|kµ(dx)<∞) and set Mk = Mk ,
m R R m∈R m
M= Mk. ´ ´
k∈N∪{0} S
DefinSition 2. Let m ∈ R. A pair of probability measures (µ,ν) ∈ M1×M1 is in increasing convex1 order,
µ≤ ν, if
cx
(3) f(x)µ(dx)≤ f(x)ν(dx) for every convex function f :R→R
ˆ ˆ
R R
(providedtheintegralsexist). Wealsosaythattwomeasuresµ,ν areinconvexorderifeitherµ≤ ν orν ≤ µ.
cx cx
Two real-valued random variables X,Y are in increasing convex order, denoted X ≤ Y, if L(X)≤ L(Y).
cx cx
Remark 1. Assume µ ∈ M1 , ν ∈ M1 and µ ≤ ν. Then m = n, this follows from choosing f(x) = x and
m n cx
f(x)=−x.
Theintuitionbeingthatµ≤ ν signifiesthataµdistributedrandomvariableislesslikelytotakeverylarge
cx
values than a ν distributed randomvariable since extremalvalues of convexfunctions are obtained onintervals
of the form (−∞,a)∪(b,∞). As it turns out, an especially important case is given when (3) is applied with
f(.)=|.−m| resp. f(.)=−|.−m| for fixed m.
1SometimesthisisalsocalledtheChoquetorder.
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 3
Definition 3 (Potential function). Associate with every µ ∈ M1 a continuous non-positive function u ∈
µ
C(R,(−∞,0]) given as
u (x):=− |x−y|µ(dy).
µ ˆ
R
We say that u is the potential function of the probability measure µ.
µ
Example 1. Denote with δ the Dirac measure at m ∈ R. Then u (x) = −|x−m| and it is easy to verify
m δm
that u ≤u for every µ∈M1 , resp. µ≥ δ .
µ δm m cx m
Lemma 1. Let m∈R and µ,ν ∈M1 . Then the following are equivalent
m
(1) µ≤ ν,
cx
(2) − (y−x)+µ(dy)≥− (y−x)+ν(dy),∀x∈R,
(3) u ´(x)≥u (x),∀x∈R. ´
µ ν
Moreover, µ≤ ν implies that xµ(dx)= xν(dx) and if µ,ν ∈M2 then x2µ(dx)≤ x2ν(dx).
cx R R m R R
´ ´ ´ ´
Proof. Follows from [35, Section 3.A]. (cid:3)
Before we justify the name “potential function” let us note that the classic connection between positive
[non-positive] Radon measures and second derivatives of concave [convex] functions reads in this setting as
Proposition1. Thereexistsaone–to–onecorrespondencebetweenpotentialfunctionsandprobabilitymeasures.
More precisely:
(1) Let µ∈M1 for some m∈R. Then u (x)=− |x−y|µ(dy) is a concave function satisfying
m µ R
´
u (x)≤u (x)=−|x−m| and lim u (x)−u (x)=0
µ δm x→±∞ µ δm
(2) Let u:R→R be a concave function satisfying
u(x)≤−|x−m| and lim u(x)+|x−m|=0
x→±∞
for some m∈R. Then there exists a probability measure µ∈M1 such that u is the potential function
m
of µ, i.e. u=u .
µ
Moreover, the connection is given by the relation µ= 1u′′ in the sense of distributions.
2
Proof. See [20, Proposition 2.1] and [5, Proposition 4.1]. (cid:3)
Remark 2. To explain the name “potential function” we briefly recall the classic potential theory for Markov
processes (cf. [7, 27, 28] for details): consider a real-valued Brownian motion B and denote its semigroup
operatorwith PB . ThepotentialkernelisdefinedasUB = ∞PBdt,i.e.UB canbeseenasalinearoperator
t 0 t
on the space of measures on (R,B(R)) by setting µUB = ∞´µPBdt which is of course nothing else than the
(cid:0) (cid:1) 0 t
occupation measure along Brownian trajectories started wi´th B ∼ µ. If µ is a signed measure with µ(R)= 0
0
and finite first moment, then the Radon–Nikodym density with respect to the Lebesgue measure is given as
dµUB
=− |x−y|µ(dy).
dx ˆ
Since (indimensionone)Brownianmotionis recurrent,µUB is infinite ifµ ispositive. However,the righthand
side − |x−y|µ(dy) is still well defined for every µ∈M1 and Chacon [10] demonstrated that this is indeed
R
a very´useful quantity in the study of Martingales and Markov processes with respect to hitting times.
We summarize some properties of the potential functions of probability measures which we use throughout.
The proofs are standard and can be found in [27, Proposition 2.3] and [28, Section 3.2].
Proposition 2. Let µ,ν ∈M1. Then
(1) µ≤ ν iff µ≥ ν iff u (.)≥u (.),
cx cv µ ν
(2) u is concave and Lipschitz continuous with Lipschitz constant equal to one,
µ
(3) if µ(R)=m then u (x)≤u (x)=−|x−m|,
µ δm
(4) if ν ∈M1 then ∀b∈R we have
u | =u | iff µ| ≡ν| ,
µ [b,∞) ν [b,∞) (b,∞) (b,∞)
(5) if µ ∈ M1, µ ⇒ µ weakly2 and u converges for some point x ∈ R then u converges for all
n n n µn 0 µn
x ∈ R and ∃c ≥ 0 s.th lim u = u +c. Further, if ∃ν ∈ M1 such that u ≥ u then µ ∈ M1
n→∞ µn µ µn ν
and c=0. Conversely, if u (x)→ u (x) ∀x∈R for some µ , µ∈M1 then µ ⇒µ weakly,
µn n µ n n
(6) if µ≤ ν then lim (u (x)−u (x))=0.
cx |x|→∞ µ ν
(7) u is almost everywhere differentiable on R and its right derivative at x equals 1−2µ((−∞,x]) and its
µ
left derivative equals 1−2µ((−∞,x)).
2Thesymbol“⇒” denotes weakconvergence ofprobabilitymeasures.
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 4
Since we wantto embed a measure via a Markovprocess,the potential function which is givenby the image
measure of this Markov process is a fundamental quantity.
Definition 4. If X is a real-valued, continuous Markov process carried on a probability space (Ω,F,(F ),P)
t
with P◦X−1 ∈M1 for every t≥0 then we call
t
uX(t,x) := uP◦X−1(x)
t
the potential function of X (at time t).
As shownby Chacon[10, Lemma 3.1a],potentialfunctions give an intuitive characterizationof Martingales.
Proposition 3. Let X be a real-valued, adapted process. Then X is a martingale iff for every pair τ ,τ of
1 2
bounded stopping times with τ ≤τ ,
1 2
u (τ ,.)≥u (τ ,.).
X 1 X 2
2.1. Root’s solution and classic results on barriers. Our aim is to provide explicit solutions of the Sko-
rokhod embedding problem not only for Brownian motion but for diffusion martingales.
Assumption 1. (Ω,F,(F ),P) is a filtered probability space that satisfies the usual conditions and carries a
t
standard Brownian motion B. Assume that σ ∈ C([0,∞)×R,R) is Lipschitz in space and of linear growth,
both uniformly in time, that is
|σ(t,x)−σ(t,y)| |σ(t,x)|
(4) σ := sup sup <∞ and σ := sup sup <∞.
Lip LG
|x−y| 1+|x|
t∈[0,∞)x6=y t∈[0,∞)x∈R
Further, let m ∈ R, (µ,ν) ∈ M2 , µ ≤ ν, and X ∼ µ, X independent of the σ-algebra generated by the
m cx 0 0
Brownian motion B and assume that with probability one σ2(t,X ) > 0 ∀t ≥ 0 where X = (X ) denotes the
t t
unique strong solution of the SDE (6), i.e. the real-valued, local martingale X such that
.
(5) X = X + σ(t,X )dB P−a.s
. 0 ˆ t t
0
X ∼ µ.
0
Further, assume that for every t>0, X has a continuous (but not necessarilty smooth) density with respect to
t
Lebesgue measure.
Definition 5 (Skorokhod embedding problem). Let (σ,µ,ν) and (Ω,F,(F ),(X ),P) be as in Assumption 1.
t t
If there exists a stopping time τ (wrt the filtration (F )) such that
t
X ∼ ν, Xτ =(X ) is uniformly integrable (u.i.)
τ t∧τ t≥0
then we say that τ solves the Skorokhod embedding problem given by (σ,µ,ν). We denote with S(σ,µ,ν) the
(possibly empty) set of stopping times τ which solve the Skorokhodembedding problem given by (σ,µ,ν).
Remark 3. Unless stated otherwise, all stopping times are stopping times in the filtration (F ).
t
As pointed out in the introduction there are many different solutions of the embedding problem, that is in
general the set S(σ,µ,ν) can contain much more than one stopping time (in the survey article of Obłój [28] at
least20differentsolutionsaregivenforembeddinglawsinBrownianmotion,i.e.(σ,µ,ν)=(1,δ ,ν)). Interms
0
of applications a natural requirement is to consider stopping times which are minimal.
Definition 6. A stopping time τ is called minimal for the process X if for every other stopping time ρ such
that ρ≤τ and X ∼X it follows that ρ=τ.
τ ρ
Thisarticleisfocusedonthe approachinitiatedbyRootin1968[30]andfurther developedbyRost,Loynes,
Kiefer et. al[34,32, 31,25, 26]ofsolvingSkorokhodembeddings by stopping times whichare the firstentrance
times of the time-space process (t,X ) into a closed subset R of [0,∞]×[−∞,∞].
t t≥0
Definition 7. A closed subset R of [0,∞]×[−∞,∞] is a Root barrier R if
(1) (t,x)∈R implies (t+r,x)∈R ∀r ≥0,
(2) (+∞,x)∈R ∀x∈[−∞,∞],
(3) (t,±∞)∈R ∀t∈[0,+∞].
We denote by R the set of all Root barriers R. Given R ∈ R, its barrier function f : [−∞,∞] → [0,∞] is
R
defined as
f (x):=inf{t≥0:(t,x)∈R}, x∈[−∞,∞].
R
Barrier functions have several nice properties such as being lower semi-continuous and that (f (x),x) ∈ R
R
for any x∈R (see [26, Proposition 3] for more properties).
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 5
Definition 8. We denote with R(σ,µ,ν) as the (possibly empty) subset of R of all R ∈ R(σ,µ,ν) such that
τ ∈S(σ,µ,ν) where τ is the first hitting time of R by (t,X ), i.e.
R R t
τ =inf{t>0:(t,X )∈R}.
R t
We also say that R∈R(σ,µ,ν) embeds the law µ into ν (via X) and call τ the Root stopping time.
R
Before we discuss conditions on σ,µ,ν which imply the existence of a Root barrier, i.e. that R(σ,µ,ν) 6=∅,
we note that simple examples show that the Root barrier is in general not unique, i.e. |R(σ,µ,ν)|>1 already
for generic situations.
Example 2. The set R = [0,∞]×[1,∞] [0,∞]×[−∞,−1] is an element of R 1,δ ,1(δ +δ ) , i.e. R
0 2 −1 1
embeds the sum of two Diracs into a standard Brownian motion but obviously one can define R differently,
S (cid:0) (cid:1)
in fact any other Root barrier with a barrier function which coincides with f on [−1,1] is also an element of
R
R 1,δ ,1(δ +δ ) .
0 2 −1 1
(cid:0)This non-uniquene(cid:1)ss problem was in the Brownian case resolved by Loynes [26] in 1970 who introduced the
notion of regular Root barriers.
Definition9 (Loynes[26]regularity). LetR∈R. WesaythatRresp.itsbarrierfunctionf isLoynes–regular
R
if f vanishes outside the interval xR,xR , where xR and xR are the first positive resp. first negative zeros of
R − + + −
fR. Given Q,R∈R we say that Q(cid:2),R are(cid:3)Loynes-equivalent if fQ =fR on xQ−,xQ+ and xR−,xR+ .
Remark 4. If Q,R are Loynes-equivalentthen xR =xQ and xR =xQ. h i (cid:2) (cid:3)
+ + − −
We advise the reader to be suspicious about the word “regular” since Loynes-regularRoot barriers can have
spikes(seeexamplesinSection2.2). Forthecase(σ,µ,ν)=(1,δ ,ν)existenceanduniquenessofLoynes-regular
0
Root barriers are standard.
Theorem1([30,26]). Foreveryν ∈M2 thesetR(1,δ ,ν)isnotemptyandthereexistsexactlyoneelementin
0 0
R(1,δ ,ν) which is Loynes-regular. The second moment of the Root stopping time, E τ2 for R∈R(1,δ ,ν),
0 R 0
is minimal among all elements of S(1,δ ,ν).
0
(cid:2) (cid:3)
An advantage of Root’s solution besides its intuitive appeal (the stopping time being a hitting time) the
propertythatτ minimizesE τ2 amongallsolutionsoftheSkorokhodembeddingproblem,i.e.inournotation
R
E τ2 = infE τ2 where the inf is taken over all τ ∈ S(σ,µ,ν). This was proven in Rost [34] (as well as in
R (cid:2) (cid:3)
[11]) and is the very reason why Root’s solution is important for financial applications, see Section 6.
(cid:2) (cid:3) (cid:2) (cid:3)
2.2. Regularity beyond Loynes. We nowextendthe discussiononRootbarriersto ourmoregeneralsetting
ofRootembeddings forItô-martingaleswithrandominitialdistribution, i.e.wediscussR(σ,µ,ν). Tothatend
weneedtoadaptseveralresultsofLoynes[26]. First,notethatalreadyinthe Browniancase(σ ≡1)weneeda
modification of the notion of Loynes-regularity,a notion that is tailor made for the Dirac as initial distribution
as the example below shows.
Example 3. Let µ = 1(δ +δ ) and ν = 1(δ +δ +δ +δ ). It is easy to see by symmetry properties
2 2 −2 4 3 1 −1 −3
of the Brownian motion that for a=b=0 the barrier
Q =[0,∞]×[3,∞] [a,∞]×{1} [b,∞]×{−1} [0,∞]×[−∞,−3] {+∞}×[−∞,+∞]
a,b
is an element of R(1,µ,ν), a[s is [ [ [
R=[0,∞]×[3,∞] [0,∞]×[1,−1] [0,∞]×[−∞,−3] {+∞}×[−∞,+∞].
However,neitherisLoynes-regular[andinfacttherec[annotexistaLoynes-[regularbarrierinR(1,µ,ν). (Assume
Q ∈ R(1,µ,ν), then a,b > 0 otherwise it would not be Loynes regular; now note that Q ∈ R(1,µ,ν),
a,b 0,0
hence every other Q puts under ν more mass on the point 3 than the required 1 since the geometry of Q
a,b 4 a,b
implies that only more trajectories can hit the line [0,∞]×{3} than in the case a = b = 0; further, every
element of R(1,µ,ν) must coincide with Q on [0,∞]×[1,3]∪[0,∞]×[−3,−1]).
0,0
Motivated by the above we introduce the notion of (µ,ν)-regular barriers.
Definition 10. For µ≤ ν define
cx
Nµ,ν :={x∈(−∞,+∞):u (x)=u (x)}∪{±∞} and Nµ,ν =[0,+∞]×Nµ,ν.
µ ν
A Root barrier R is (µ,ν)-regular if R = R∪Nµ,ν [or equivalently if f (x) = 0 ∀x ∈ Nµ,ν]. Denote with
R
R (σ,µ,ν)thesubsetofR(σ,µ,ν)of(µ,ν)-regularRootbarriers. Further,twoRootbarriersR,Qare(µ,ν)-
reg
equivalent if3 R\([0,∞]×(Nµ,ν)o)=Q\([0,∞]×(Nµ,ν)o) [or equivalently if f (x)=f (x) ∀x∈(Nµ,ν)c].
R Q
3WedenotewithAtheclosureandwithAo theinteriorofagivensetA.
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 6
If Fµ and Fν are the cumulative distribution functions of µ and ν respectively, then the set Nµ,ν coincides
with the set {x:Fµ(x)=Fν(x)}∪{±∞}. The next Lemma shows that in the case σ ≡ 1 and µ = δ the
0
(µ,ν)-regularity of a Root barrier is equivalent to Loynes regularity.
Lemma 2. Let ν ∈M2 and R∈R(1,δ ,ν). Then R is Loynes-regular iff R is (δ ,ν)-regular.
0 0 0
Proof. Assume R to be Loynes-regular, then f (x) = 0 for ∀x ∈/ xR,xR where xR,xR are as in Definition
R − + − +
9. Since the embedding is possible, i.e. X ∼ ν, then via the regularity of the barrier R one has that
τR (cid:0) (cid:1)
ν xR,xR = 1, i.e. ν puts no mass outside xR,xR (by definition so does δ ). In other words outside
− + − + 0
xR,xR bothmeasuresν andδ arethe samewhichby point(4) ofProposition2 implies that u (x)=u (x)
(cid:0)−(cid:2) + (cid:3)(cid:1) 0 (cid:2) (cid:3) δ0 ν
for x ∈/ xR,xR and by continuity of the potential functions for any x ∈/ xR,xR . We conclude that R is
(cid:2) (cid:3) − + − +
(δ ,ν)-regular.
0
(cid:2) (cid:3) (cid:0) (cid:1)
For the inverse direction, just remark that being R a (δ ,ν)-regular barrier we have due to δ ≤ ν that
0 0 cx
Nδ0,ν ∩R has the form Nδ0,ν ∩R=R\(a,b) for some a<0<b, in other words f (x)=0 for any x∈/ (a,b).
R
Notice that the convex order relation (with δ ) implies that if u (c)=0 for some c>0 then u (x)=0 for any
0 ν ν
x ≥ c (and corresponding condition for the case c < 0), this implies that a and b are the first negative resp.
positive zero of f . Hence R is Loynes-regular. (cid:3)
R
Definition10allowstoextendLoynes’[26,Theorem1]tothepresentsetting,i.e.ifthereexistsanembedding
with a Root barrier then there also exists an embedding with a (µ,ν)-regular Root barrier.
Theorem 2. Let µ,ν ∈M2, µ ≤ ν and let X be a real-valued continuous square integrable local Martingale
cx
X with a quadratic variation that is strictly increasing and X ∼ µ. Assume additionally that there exists
0
Q ∈ R s.th. XτQ ∼ ν and XτQ is u.i.. Then Q is (µ,ν)-equivalent to a unique (µ,ν)-regular barrier R ∈ R
s.th. XτR ∼ν and XτR is u.i.; moreover E[[X]τR]<∞.
We prepare the proof with a lemma concerning the union and intersection of Root barriers.
Lemma 3. Take ν ∈ M1 and let Y be a process with continuous paths. Assume there exists Q,R ∈ R with
first hitting times τ and τ respectively and assume that Y ∼ ν and Y ∼ ν. Then Q∪R,Q∩R ∈R and
Q R τQ τR
both generate the same law ν with respective stopping times min{τ ,τ } and max{τ ,τ }.
Q R Q R
Proof. This proof is a straightforward modification of [26, Proposition 4 ], we fill in some details and sketch
the proof for the case of Q ∪ R: let f and f be the barrier functions of R and Q and define the set
R Q
K :={x:f (x)<f (x)}. Since both barriers (with their respective stopping times) generate ν we have that
Q R
P Y ∈K =P[Y ∈K] iff
τQ τR
(cid:2) P YτQ(cid:3) ∈K, YτR ∈K +P YτQ ∈K, YτR ∈Kc =P YτQ ∈K, YτR ∈K +P YτQ ∈Kc, YτR ∈K
iffP YτQ(cid:2) ∈K, YτR ∈Kc =(cid:3)P Yτ(cid:2)Q ∈Kc, YτR ∈K .(cid:3)Aquic(cid:2)kanalysisshowstha(cid:3)tfor(cid:2)atrajectoryY (ω)s.t(cid:3).YτR(ω)
ends in K it is impossible that Y (ω)∈Kc, inother words,since the paths ofY are continuousandin the set
(cid:2) (cid:3) (cid:2) τQ (cid:3)
[0,∞)×K the barrier Q is to the left of R then any path that hits Q must hit R before (hitting Q) and hence
P Y ∈Kc, Y ∈K =0=P Y ∈K, Y ∈Kc .
τQ τR τQ τR
Finally, by decomposing the probability space in disjoint events (based on K and Kc) and using the just
(cid:2) (cid:3) (cid:2) (cid:3)
obtained result one concludes P Y ∈K, Y ∈K ∪ Y ∈Kc, Y ∈Kc = 1. The rest of the proof
τQ τR τQ τR
follows as in [26]. (cid:3)
(cid:2)(cid:8) (cid:9) (cid:8) (cid:9)(cid:3)
We can now give the proof of Theorem 2.
Proof of Theorem 2. To see that Q is (µ,ν)-equivalent to a (µ,ν)-regular barrier note that since u ≤u (and
ν µ
weembedbyassumption)thecontinuoustime-spaceprocess t∧τ ,X doesnotenter[0,∞]×Nµ,ν,hence
Q t∧τQ
R:=Q∪Nµ,ν is a also an element of R(σ,µ,ν), and moreoveran element of R (σ,µ,ν).
reg
(cid:0) (cid:1)
To see the uniqueness, i.e. |R (σ,µ,ν)| = 1 suppose that there are two (µ,ν)-regular barriers B,C, each
reg
embedding ν (via X) with u.i. stopping times τ and τ respectively. Then Γ = B ∪C also embeds ν with
B C
stopping time γ =min{τ ,τ } (see Lemma 3); γ is also u.i. Furthermore, since X is a local martingale,
B C
E[[X] ]=E X2 =E X2 =E[[X] ].
τB τB γ γ
Since the quadratic variation t 7→ [X]t is strict(cid:2)ly inc(cid:3)reasin(cid:2)g an(cid:3)d γ ≤ τB this already implies [X]γ = [X]τB.
Thestrictmonotonicityoft7→[X] thenimpliesγ =τ . Thesameargumentsappliestoτ ,henceweconclude
t B C
that τ =γ =τ .
B C
To see that B,C and B ∪C are the same (outside Nµ,ν since in Nµ,ν this must hold) we argue as in the
proof of Lemma 2 in [26] by showing that if B 6= Γ then also τ 6= τ . Note that B 6= Γ implies that the
B γ
existence of x ∈R such that x ∈/ Nµ,ν and (wlog) f (x )<f (x ) where f (x )6=0. Since the barriers are
0 0 B 0 Γ 0 Γ 0
regular and lower semi-continuous functions attain their minimum on a compact set, ∃ǫ>0 s.th f (x) >ǫ on
Γ
some neighborhoodof x . Moreoverǫ can be chosen in such a way that 2ǫ<f (x )−f (x ), implying that a
0 Γ 0 B 0
δ >0 exists s.th f (x)>f (x )−ǫ>f (x )+ǫ for any x in the set {x:|x−x |<δ}. Since the embedding
Γ Γ 0 B 0 0
via X ispossible,there existsa positiveprobabilitythatthe time-spaceprocess(t,X ) hits the line segment
t t≥0
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 7
[f (x ),f (x )+ǫ]×{x } while hitting Γ only after time f (x )+ǫ. This implies that P[τ 6=τ ]>0 and
B 0 B 0 0 B 0 B Γ
the result follows. Since Q∈R(σ,µ,ν), E [X] <∞ and hence by construction also E[[X] ]<∞. (cid:3)
τQ τR
2.3. The general Root embedding and(cid:2) optim(cid:3)ality. We give a generalization of Theorem 1 showing that
R(σ,µ,ν) is not empty, i.e. embedding ν ∈M2 via the solution process of the SDE
(6) dX =σ(t,X )dB , X ∼µ.
t t t 0
Root’s original construction (for Brownian motion, i.e. σ ≡ 1 and µ = δ ) relies only on the continuity of the
0
Brownian trajectories, the strong Markov property, that trajectories exit compact intervals in finite time and
the fact that the process has no interval of constancy – hence (as remarked several times in the literature) the
same construction generalizes. Here we give only the results, complete proofs are postponed to Appendix 7. It
turns out to be natural for the proof to distinguish between the cases [X] = ∞ and when X is a geometric
∞
Brownian motion (which implies only [lnX] =∞).
∞
2.3.1. Existence and uniqueness of a Root barrier if [X] =∞. We nowstateageneralresultonthe existence
∞
of the Root embedding via the solution of (6) (the proof is given in the appendix).
Theorem 3 (Root Embedding). Let (σ,µ,ν) be such that Assumption 1 is fulfilled and that [X] = ∞ a.s.
∞
Then |R (σ,µ,ν)|=1. Moreover,
reg
E[[X] ]= x²ν(dx)<∞ and E[τ ]<∞ ∀R∈R(σ,µ,ν).
τR ˆ R
R
Remark 5. One can also apply the results of Rost and Cox–Wang [11, 34] to show that τ minimizes E[f(τ)]
R
among allτ ∈S(σ,µ,ν) where f canbe a convex,increasingfunction with f(0)=0 whichfurther fulfills some
properties which depend in general on σ.
2.3.2. ExistenceandUniquenessforgeometricBrownian motion(σ(t,x)=x). ThecasewhenX isaGeometric
Brownianmotion(GBMhenceforth),i.e.σ(t,x)=x,isnotcoveredbyTheorem3sincealthoughE[[X] ]=∞
∞
wehave[X] <∞a.s.(otherwiseonecouldwriteX asatime-changedBrownianmotionviaDambis/Dubins–
∞
Schwarz). However,this case is especially important for the applications in finance (see Section 6).
Assumption 2. (µ,ν)∈M2, µ≤ ν and there exists an ǫ>0 such that
cx
supp(µ)⊂(0,+∞) and supp(ν)⊂[ǫ,+∞).
Theassumptionthatǫisstrictlygreaterthan0isneededforourarguments(seeRemark17intheappendix)
and does not cover the case ν([0,∞)) = 1,ν({0}) = 0. (However, note that one cannot hope for Root
embeddings where the targetmeasure has an atom at 0). We now state the embedding result for this situation
(again, the proof is given in the appendix).
Theorem 4 (Root Embedding via Geometric Brownianmotion). Let σ(x)=x and Assumption 2 hold. Then
|R (σ,µ,ν)|=1. Moreover
reg
E[[X] ]= x2µ(dx)<∞, [lnX] =τ ∀R∈R(σ,µ,ν)
τR ˆ τR R
R
Further, for every other stopping time τ ∈S(σ,µ,ν),
E[f(τ )]≤E[f(τ)]
R
for every function f :[0,∞)→[0,∞) which is convex, f(0)=0 and f has a bounded right-derivative.
3. Background on viscosity solutions and the obstacle problem
We briefly recall the definition and some basic stability properties of viscosity solutions in the parabolic
setting (for a detailed exposition we refer to the User’s guide [12]) with some additional lemmas which will
become useful in the subsequent sections. We then discuss the connection to reflected FBSDEs.
3.1. Sub- and supersolutions, properness and semirelaxed limits.
Definition 11. Let O be a locally compact subset of R and denote O =(0,T)×O for T ∈(0,∞]. Consider
T
a function u : O → R and define for (s,z) ∈ O the parabolic superjet P2,+u(s,z) as the set of triples
T T O
(a,p,m)∈R×R×R which fulfill
u(t,x) ≤ u(s,z)+a(t−s)+hp,x−zi
1
(7) + hm(x−z),x−zi+o |t−s|+|x−z|2 as O ∋(t,x)→(s,z)
T
2
(cid:16) (cid:17)
Similarly we define the parabolic subjet P2,−u(s,z) such that P2,−u=−P2,+(−u).
O O
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 8
Example 4. If a function u∈C1,2(O ,R) then
T
∂u ∂u ∂u
P2,−u(t,x) = | , | ,m :m≤ |
∂t (t,x) ∂x (t,x) ∂2x (t,x)
(cid:26)(cid:18) (cid:19) (cid:27)
∂u ∂u ∂u
P2,+u(t,x) = | , | ,m | :m≥ |
∂t (t,x) ∂x (t,x) (t,x) ∂2x (t,x)
(cid:26)(cid:18) (cid:19) (cid:27)
hence
∂u ∂u ∂u
P2,−u(t,x)∩P2,+u(t,x)= , , | .
∂t ∂x ∂2x (t,x)
(cid:26)(cid:18) (cid:19) (cid:27)
(If u is only differentiable from the left or right, a similar statement holds and is important for the proofof our
main result, see Lemma 5).
Remark 6. An equivalent definition is to consider the extrema of u−ϕ when ϕ is taken from an appropriate
set of smooth test functions, see [12].
Definition 12. A function F :O ×R×R×R×R→R is proper if ∀(t,x,a,p)∈O ×R×R
T T
F(t,x,r,a,p,m)≤F (t,x,s,a,p,m′) ∀m≥m′, s≥r.
Denote the real-valued, upper semicontinuous functions on O with USC(O ) and the lower semicontinuous
T T
functions with LSC(O ). A subsolution of the (forward problem)
T
F t,x,u,∂ u,Du,D2u = 0
(8) t
u(0,.) = u (.)
0
(cid:26) (cid:0) (cid:1)
is a function u∈USC(O ) such that
T
F (t,x,u,a,p,m) ≤ 0 for (t,x)∈O and (a,p,m)∈P2,+u(t,x)
T O
u(0,.) ≤ u (.) on O
0
The definitionofa supersolution followsby replacingUSC(O ) by LSC(O ), P2,+ byP2,− and≤ by ≥. If u
T T O O
is a supersolution of (8) then we also say that F t,x,u,∂ u,Du,D2u ≥ 0, u(0,x) ≥ u (x) holds in viscosity
t 0
sense (similar for subsolutions). Similarly we call a function v a viscosity (sub-,super-)solution of the backward
(cid:0) (cid:1)
problem
G t,x,v,∂ v,Dv,D2v = 0
(9) t
v(T,.) = v (.)
T
(cid:26) (cid:0) (cid:1)
if
G(t,x,v,a,p,m) ≤ 0 for (t,x)∈O and (a,p,m)∈P2,+v(t,x)
T O
v(0,.) ≤ v (.) on O.
T
The next lemma shows that finding a viscosity solution to (9) is equivalent to finding one to (8).
Lemma 4. Let T < ∞. Then u is a viscosity solution of the forward problem (8) iff v(t,x) := u(T −t,x) is
a viscosity solution of the backward problem (9) with v (.)=u (.) and
T 0
G(t,x,v,a,p,m)=F (T −t,x,v,−a,p,m).
Proof. Assume u is a viscosity solution. Let ϕ ∈ C∞(O ,R) and assume (t,x) 7→ v(t,x)−ϕ(t,x) attains
T
a local maximum at tˆ,xˆ ∈ O . Then T −tˆ,xˆ is a local maximum of (t,x) 7→ u(t,x) − ϕ˜(t,x) with
T
ϕ˜(t,x)=ϕ(T −t,x), hence
(cid:0) (cid:1) (cid:0) (cid:1)
F ·,·,u,∂ ϕ˜,Dϕ˜,D2ϕ˜ | =G ·,·,v,−∂ ϕ,Dϕ,D2ϕ |
t (T−tˆ,xˆ) t (tˆ,xˆ)
which is by Remark 6 su(cid:0)fficient to see that(cid:1)v is a viscos(cid:0)ity solution of (9). T(cid:1)he other implication follows
similarly. (cid:3)
A strong feature of viscosity solutions, which is also the key for the proof of our main theorem, is that
they are very robust under perturbations. The type of reasoning presented below is classic and called the
Barles–Perthame method of semi-relaxed limits.
Definition 13. LetA⊂Rm and(g ) asequenceoffunctionsg :A→R whichis locallyuniformly bounded.
n n n
We define g,g¯:A→R as
g(p) = limsup g (q).
n
q→p,n→∞
g(p) = liminf g (q).
n
q→p,n→∞
ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 9
Proposition 4. Let (un) ⊂USC(O ), O a locally compact subset of R, (F ) a sequence of functions
n T n n
F :O ×R×R×R→R
n T
such that each un is a subsolution of
F t,x,v,∂ v,D2v =0.
n t
Further, assume (un) and (Fn) are locally uniformly bounded. Then u is a subsolution of
n n (cid:0) (cid:1)
F t,x,v,∂ v,D2v =0 on O.
t
The analogous statement holds for a sequ(cid:0)ence of LSC(O(cid:1)T) functions which are supersolutions. Further, if
u=u then the convergence of (u ) to u=u is locally uniform.
n
Proof. ThisissimplyarestatementforparabolicPDEsofProposition4.3,Lemma6.1foundintheUser’sguide
[12] combined with their subsequent remarks: the definition of u as limsup un(s,y) guarantees
n→∞,(s,y)→(t,x)
the existence of sequences
n →∞, (t ,x )∈O s.t. (t ,x )→(t,x) and u (t ,x )→u
j j j T j j nj j j
and it is clear that for every sequence O ∋(t ,x ) converging to (t,x) one has limsup u (t ,x ) ≤ u. This
T j j j nj j j
allows to apply [12, proposition 4.3 ] which implies that ∀(a,p,X)∈P2,+u(t,x) there exists a sequence
(t ,x )⊂O ,(p ,m )∈P2,+u (t ,x ) s.t. t ,x ,u (x ),a ,m →(t,x,u(x),a,m)∈P2,+u(t,x).
j j T j j O nj j j j j nj j j j O
However, each unj is a subsolution so that for eve(cid:0)ry j it holds that (cid:1)
F (t ,x ,a ,p ,m )≤0
nj j j j j j
and hence from the definition of F it follows that
F (t,x,a,p,m)≤0.
The proof for subsolutions follows the same argument. Finally, if u = u =: u then u must be continuous and
the convergence u →u must be uniform (as seen by [12, Remark 6.4]). (cid:3)
n
3.2. Reflected FBSDEs and viscosity solutionsof obstacle problems. Ourgoalistoshowthatknowing
theRootbarrierisequivalenttoknowingtheviscositysolutionofacertainobstaclePDE,thatisweareinterested
inthecaseF (t,x,r,a,p,m)=min r−h(x),a− σ2(t,x)m wheretheinitialconditionu andthebarrierhare
2 0
givenasthepotentialfunctionofth(cid:16)einitialandtargetmea(cid:17)suresandσ iscontinuous(wediscussthe motivation
for looking for such a PDE solution at the beginning of Section 4).
Definition 14. Let T ∈(0,∞], u :R→R and h,σ ∈[0,T)×R→R. Denote with V (σ,u ,h) the (possibly
0 T 0
empty) set of viscosity solutions
min u(t,x)−h(x), ∂ − σ2∆ u(t,x) = 0,(t,x)∈(0,T)×R
(10) t 2
( (cid:16) (cid:16) (cid:17) u(0,.(cid:17)) = u0(x),x∈R.
whichareoflineargrowthuniformly int,i.e. ∀(t,x)∈[0,T)×R,sup |u(t,x)| <∞. We referto this
(t,x)∈[0,T)×R 1+|x|
PDE as the obstacle problem on (0,T)×R with barrier h.
Remark 7. F (t,x,r,a,p,m) = min r−h(x),a− σ2(t,x)m is proper and moreover the barrier h introduces
2
anasymmetryinthedefinitionofsub(cid:16)-andsupersolutions: in(cid:17)thiscaseDefinition12saysthatuisasubsolution
of (10) if u(0,.)≤u (.) and for every (t,x)∈(0,T)×R such that u(t,x)>h(x) we have
0
σ2(t,x)
(11) a− m≤0 ∀(a,p,m)∈P2,+u(t,x).
2 R
Whereas u is a supersolution if u(0,.)≥u (.) and ∀(t,x)∈(0,T)×R
0
σ2(t,x)
(12) a− m≥0 ∀(a,p,m)∈P2,−u(t,x) and u(t,x)≥h(x).
2 R
Wenowdiscusstheabstractproblemofexistenceanduniquenessofaviscositysolutionto(2). Firstwerecall
thedefinitionofasubclassofreflectedforward-backwardstochasticequation(RFBSDE)([15,Section8]applied
with f =0). Therefore fix T ∈(0,∞) and for each t∈[0,T] denote with {Ft,t≤s≤T} the natural filtration
s
of the Brownianmotion {W −W , t≤s≤T}, augmentedby the P-nullsets ofF. Take(t,x)∈[0,T]×R and
s t
define for s∈[t,T] the forward-backwarddynamics whose solution is the F-adapted quadruple (X,Y,Z,K)
Xt,x = x+ sσ(T −r,Xt,x)dW ,
s t r r
(13) Yt,x = u X´t,x +Kt,x−Kt,x− T Zt,xdW ,
 s 0 T T s s r r
 Yt,x ≥ h(Xt,x), t<s≤T and T´(Yt,x−h(Xt,x))dKt,x =0,
s (cid:0) s (cid:1) t s s s
´

ROOT’S BARRIER, VISCOSITY SOLUTIONS OF OBSTACLE PROBLEMS AND REFLECTED FBSDES 10
where (Kt,x) is an increasing and continuous process verifying Kt,x = 0. In the above equation the
s s∈[t,T] t
meaning of the processes X and Y is clear and a roughinterpretation of the processes Z and K is a follows: Z
guides the evolution of Y via the Itô integral so that Y can hit the random variable u (X ) at horizon time
0 T
T. Note that both processes Y,Z are (F )-adapted and so they do not “see the future”, nonetheless, u (X ) is
t 0 T
attained at time t = T. The process K ensures that Y does not go below the barrier h: K pushes Y upwards
whenever Y touches and tries to go below the barrier h, else it remains inactive (that is constant).
Proposition 5 (Stochastic representation for the obstacle PDE). Let u ,h ∈ C(R,R),σ ∈ C([0,∞)×R,R),
0
h(.)≤u (.) and assume that for every t∈[0,∞), σ(t,.),h(.),u (.) are Lipschitz continuous (σ uniformly in
0 0
t). Moreover, assume that
|u (x)|+|σ(t,x)|+|h(x)|
0
sup sup <∞.
1+|x|
t∈[0,∞)x∈R(cid:18) (cid:19)
Then there exists a v ∈C([0,∞)×R,R) which is a viscosity solution of
∞
min v(t,x)−h(x), ∂ − σ2∆ v(t,x) = 0, (t,x)∈(0,∞)×R
(14) t 2
( (cid:16) (cid:16) (cid:17) v(0,x(cid:17)) = u0(x), x∈R.
Moreover, for every T ∈(0,∞)
(1) there exists ∀(t,x) ∈ [0,T]×R a unique quadruple (Xt,x,Yt,x,Zt,x,Kt,x) of {Ft}-progressively
s s s s s∈[t,T] s
measurable processes which satisfies for some c >0
T
T
sup E sup Xt,x 2+ Yt,x 2 + Zt,x 2dr+Kt,x ≤c (1+|x|)
s s ˆ r T T
t∈[0,T] "s∈[t,T] t #
(cid:8)(cid:12) (cid:12) (cid:12) (cid:12) (cid:9) (cid:12) (cid:12)
and this quadruple fulfills (cid:12)(13).(cid:12)Mor(cid:12)eover(cid:12), ∀(s,x) ∈(cid:12) [t,(cid:12)T]×R the mapping (s,x) 7→ (Xt,x,Yt,x) is
s s
P-a.s. jointly continuous and in particular Xt,x,Yt,x,Zt,x,Kt,x ∈R4 (i.e. deterministic),
t t t t
(2) (t,x)7→YT−t,x is the unique element of V (σ,u ,h),
T−t T(cid:0) 0 (cid:1)
(3) v | (t,x)≡YT−t,x.
∞ [0,T]×R T−t
Proof. Existence & uniqueness in [0,T]×R, T <∞: Time reversion of (2) (see (9) and Lemma 4) shows that
it is enough to deal with
min v(t,x)−h(x), −∂ − σ2∆ v(t,x) = 0, (t,x)∈O =(0,T)×R
t 2 T
( (cid:16) (cid:16) (cid:17) v(T,x(cid:17)) = u0(x), x∈R.
where σ(t,x) = σ(T −t,x). Continuity, existence and uniqueness of the viscosity solution v follows from
Lemma8.4,Theorems8.5and8.6in[15]respectively. Thelineargrowthofuinitsspatialvariablefollowsfrom
standardmanipulationsforRBSDEs. [15,Proposition3.5]appliedtothe RFBSDEsettingabove(i.e.(13)with
σ replaced by σ) yields the existence of a constant k >0 such that ∀(t,x)∈[0,T]×R
T
Yt,x 2 ≤E sup Yt,x 2 ≤k E u Xt,x 2 +E sup h+ s,Xt,x 2 ≤c 1+|x|2
t s T 0 T s T
"s∈[t,T] # "s∈[t,T] #!
(cid:12) (cid:12) (cid:12) (cid:12) (cid:2)(cid:12) (cid:0) (cid:1)(cid:12) (cid:3) (cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
with h(cid:12)+ :=(cid:12)max{0,h} and(cid:12) whe(cid:12)re the last ine(cid:12)quality fo(cid:12)llows from the(cid:12)linear growt(cid:12)h assumptions on u and h
0
along with standard SDE estimates: sup E Xt,x 2 ≤cˆ 1+|x|2 (see e.g. equation (4.6) of [16]). The
t∈[0,T] T T
solution to (2) follows via our Lemma 4.
(cid:2)(cid:12) (cid:12) (cid:3) (cid:0) (cid:1)
Existence & uniqueness in [0,∞)×R: Above(cid:12)give(cid:12)s a unique solution for every finite T > 0. First note
that for T,T′ >0, YT−t,x and YT′−t,x coincide on [0,T ∧T′)×R. By the comparison result in the appendix,
T−t T′−t
Theorem 9, we can define v (t,x) as YT−t,x where T can be arbitrary chosen subject to T > t. Since the
∞ T−t
definition of a viscosity solution just relies on local properties, v is a viscosity solution of (14). (cid:3)
∞
Remark 8. In the above link between the solutions of (13) and (the time reversed version) of (10), (14) one
can in general not expect that the solutions of (14) exist in a classical sense. The following formal argument
gives at leastan intuition why RFBSDE and obstacle PDEs are in a similar relationas SDEs and linear PDEs:
suppose a sufficiently regular solution v of (14) exists. Via Itô’s formula it follows that:
s σ2
Yt,x =v s,Xt,x , Zt,x =(σ¯∇ v) s,Xt,x and Kt,x = −∂ v− ∆v r,Xt,x dr.
s s s x s s ˆ t 2 r
t (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
The last condition in (13) then reads as
T T σ2
Yt,x−h Xt,x dKt,x =0 iff (v−h) −∂ v− ∆v r,Xt,x dr =0
ˆ r r r ˆ t 2 r
t t (cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)
and the rhs explains the form of the PDE fulfilled by v.