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DECOMPOSITION OF THE SNELL ENVELOPE 2

SAUL D. JACKA† AND DOMINYKAS NORGILAS†

3

Abstract. LetGbe a semimartingale, andS its Snell envelope. Under the assumption that

4

G∈ H1_{, we show that the finite-variation part of} _{S}_{is absolutely continuous with respect to the}

5

decreasing part of the finite-variation part ofG. In the Markovian setting, this enables us to identify

6

sufficient conditions for the value function of the optimal stopping problem to belong to the domain

7

of the extended (martingale) generator of the underlying Markov process. We then show that the

8

dualof the optimal stopping problem is a stochastic control problem for a controlled Markov process,

9

and the optimal control is characterised by a function belonging to the domain of the martingale

10

generator. Finally, we give an application to the smooth pasting condition.

11

Key words. Doob-Meyer decomposition, optimal stopping, Snell envelope, stochastic control,

12

martingale duality, smooth pasting

13

AMS subject classifications. 60G40, 60G44, 60J25, 60G07, 93E20

14

1. Introduction. Given a (gains) process G = (Gt)t≥0, living on the usual

15

filtered probability space (Ω,F,_{F}= (Ft)t≥0,P), the classical optimal stopping
prob-16

lem is to find a maximal reward v0 = sup_{τ≥0}_{E}[Gτ], where the supremum is taken

17

over all_{F} - stopping times. In order to compute v0, we consider, the value process

18

vt = ess sup_{τ≥t}E[Gτ|Fσ], t ≥ 0. It is, or should be, well-known (see, for example,

19

El Karoui [16], Karatzas and Shreve [31]) that under suitable integrability and

regu-20

larity conditions on the processG, the Snell envelope ofG, denoted byS = (St)t≥0,

21

is the minimal supermartingale which dominates G and aggregates the value

pro-22

cess at each F-stopping time σ ≥ 0, so that Sσ = vσ almost surely. Moreover, 23

τσ := inf{r ≥σ :Sr =Gr} is the minimal optimal stopping time, so, in particular,

24

Sσ=vσ =E[Gτσ|Fσ] almost surely. A successful construction of the processS leads, 25

therefore, to the solution of the initial optimal stopping problem.

26

In the Markovian setting the gains process takes the formG=g(X), whereg(·) is

27

some payoff function applied to an underlying Markov processX. Under very general

28

conditions, the Snell envelope is then characterised as the least super-mean-valued

29

function V(·) that majorizesg(·). A standard technique to find the value function

30

V(·) is to solve the corresponding obstacle (free-boundary) problem. For an exposition

31

of the general theory of optimal stopping in both settings we also refer to Peskir and

32

Shiryaev [39].

33

The main aim of this paper is to answer the following canonical question of

in-34

terest:

35

Question. When does the value function V(·) belong to the domain of the

ex-36

tended (martingale) generator of the underlying Markov processX?

37

Very surprisingly, given how long general optimal stopping problems have been

38

studied (see Snell [49]), we have been unable to find any general results about this.

39

As the title suggests, we tackle the question by considering the optimal stopping

40

∗_{Submitted to the editors 14/04/2018.}

Funding:This work was funded by the EPSRC under the grant EP/P00377X/1, the Alan Tur-ing Institute under the EPSRC grant EP/N510129/1 and the EPSRC Doctoral TrainTur-ing Partnerships grant EP/M508184/1.

†_{Department of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom}
(s.d.jacka@warwick.ac.uk,d.norgilas@warwick.ac.uk).

problem in a more general (semimartingale) setting first. If a gains process G is

41

sufficiently integrable, then S is of class (D) and thus uniquely decomposes into the

42

difference of a uniformly integrable martingale, sayM, and a predictable, increasing

43

process, sayA, of integrable variation. From the general theory of optimal stopping

44

it can be shown that ¯τσ := inf{r≥σ:Ar>0}is the maximal optimal stopping time,

45

while the stopped processSτ¯σ _{= (}_{St∧¯}

τσ)t≥0 is a martingale. Now suppose thatGis a 46

semimartingale itself. Then its finite variation part can be further decomposed into

47

the sum of increasing and decreasing processes that are, as random measures, mutually

48

singular. Off the support of the decreasing one, Gis (locally) a submartingale, and

49

thus in this case it is suboptimal to stop, and we again expect S to be (locally) a

50

martingale. This also suggests that A increases only if the decreasing component

51

of the finite variation part of G decreases. In particular, we prove the following

52

fundamental result (seeTheorem 3.6):

53

the finite-variation process in the Doob-Meyer decomposition of S

is absolutely continuous with respect to the decreasing part of the corresponding finite-variation process in the decomposition of G.

54

This being a very natural conjecture, it is not surprising that some variants of it

55

have already been considered. As a helpful referee pointed out to us, several versions

56

of Theorem 3.6were established in the literature on reflected BSDEs under various

57

assumptions on the gains process, see El Karoui et. al. [17] (Gis a continuous

semi-58

martingale), Crep´ey and Matoussi [9] (G is a c`adl`ag quasi-martingale), Hamad´ene

59

and Ouknine [23] (G is a limiting process of a sequence of sufficiently regular

semi-60

martingales). We note that these results (except Hamad´ene and Ouknine [23], where

61

the assumed regularity of Gis exploited) are proved essentially by using (or

appro-62

priately extending) the related (but different) result established in Jacka [27]. There,

63

under the assumption that S and Gare both continuous and sufficiently integrable

64

semimartingales, the author shows that a local time of S−Gat zero is absolutely

65

continuous with respect to the decreasing part of the finite-variation process in the

66

decomposition ofG. Our proof ofTheorem 3.6relies on the classical methods

estab-67

lishing the Doob-Meyer decomposition of a supermartingale.

68

The first part of section 3is devoted to the groundwork necessary to establish

69

Theorem 3.6. It turns out that an answer to the motivating question of this paper

70

then follows naturally. In particular, in the second part ofsection 3, inTheorem 3.18,

71

we show that, under very general assumptions on the underlying Markov processX,

72

if the payoff functiong(·) belongs to the domain of the martingale generator ofX, so

73

does the value functionV(·) of the optimal stopping problem.

74

Insection 4we discuss some applications. First, we consider a dual approach to

75

optimal stopping problems due to Davis and Karatzas [10] (see also Rogers [43], and

76

Haugh and Kogan [24]). In particular, from the absolute continuity result announced

77

above, it follows that the dual is a stochastic control problemfor a controlled Markov

78

process, which opens the doors to the application of all the available theory related

79

to such problems (see Fleming and Soner [19]). Secondly, if the value function of the

80

optimal stoping problem belongs to the domain of the martingale generator, under a

81

few additional (but general) assumptions, we also show that the celebratedsmooth fit

82

principle holds for (killed) one-dimensional diffusions.

2. Preliminaries. 84

2.1. General framework. Fix a time horizonT ∈(0,∞]. LetGbe an adapted,

85

c`adl`ag gains process on (Ω,F,F= (Ft)0≤t≤T,P), where Fis a right-continuous and 86

complete filtration. We suppose thatF0 is trivial. In the case T =∞, we interpret

87

F∞ = σ

∪0≤t<∞Ft

and G∞ = lim inft→∞Gt. For two F-stopping timesσ1, σ1

88

with σ1 ≤σ2 P-a.s., by Tσ1,σ2 we denote the set of allF-stopping timesτ such that

89

P(σ1≤τ≤σ2) = 1. We will assume that the following condition is satisfied: 90

(2.1) _{E}h sup

0≤t≤T

|Gt|

i

<∞,

91

and let

¯

Gbe the space of all adapted, c`adl`ag processes such that(2.1)holds.

Theoptimal stopping problemis to compute the maximal expected reward

92

v0:= sup τ∈T0,T

E[Gτ].

93

94

Remark 2.1. First note that by (2.1), _{E}[Gτ]< ∞ for allτ ∈ T0,T, and thus v0

95

is finite. Moreover, most of the general results regarding optimal stopping problems

96

are proved under the assumption thatGis a non-negative (hence thegains) process.

97

However, under (2.1), N = (Nt)0≤t≤T given by Nt = E[sup0≤s≤T|Gs||Ft] is a

uni-98

formly integrable martingale, while ˆG:=N+Gdefines a non-negative process (even

99

ifGis allowed to take negative values). Then

100

ˆ

v0:= sup τ∈T0,T

E[Nτ+Gτ] =E h

sup 0≤t≤T

|Gt|

i

+ sup τ∈T0,T

E[Gτ],

101

and finding ˆv0 is the same as finding v0. Hence we may, and shall, assume without

102

loss of generality thatG≥0.

103

The key to our study is provided by the family{vσ}σ∈T0,T of random variables 104

(2.2) vσ:= ess sup

τ∈Tσ,T

E[Gτ|Fσ], σ∈ T0,T.

105

Note that, since each deterministic timet∈[0, T] is also a stopping time, (2.2) defines

106

an adaptedvalue process (vt)0≤t≤T. Forσ∈ T0,T, it is tempting to regard vσ as the

107

process (vt)0≤t≤T evaluated at the stopping timeσ. It turns out that there is indeed a

108

modification (St)0≤t≤T of the process (vt)0≤t≤T that aggregates the family{vσ}σ∈T0,T 109

at each stopping timeσ(see Theorem D.7 in Karatzas and Shreve [31]). This process

110

S is the Snell envelope ofG.

111

Theorem 2.2 (Characterisation ofS). Let G∈G¯. The Snell envelope process 112

S of Gsatisfies

113

(2.3) Sσ = ess sup

τ∈Tσ,T

E[Gτ|Fσ], P−a.s., σ∈ T0,T.

114

Moreover,S is the minimal c`adl`ag supermartingale that dominates G.

For the proof of Theorem 2.2 under slightly more general assumptions on the

116

gains processGconsult Appendix I in Dellacherie and Meyer [12] or Proposition 2.26

117

in El Karoui [16].

118

IfG∈G¯, it is clear thatGis a uniformly integrable process. In particular, it is also 119

of class (D), i.e. the family of random variables {Gτ1{τ <∞} : τ is a stopping time}

120

is uniformly integrable. On the other hand, a right-continuous adapted process Z

121

belongs to the class (D) if there exists a uniformly integrable martingale ˆN, such

122

that, for allt∈[0, T],|Zt|≤Ntˆ P-a.s. (see e.g. Dellacherie and Meyer [12], Appendix 123

I and references therein). In our case, by (2.3) and using the conditional version of

124

Jensen’s inequality, fort∈[0, T], we have

125

|St|≤E h

sup 0≤s≤T

|Gs|

Ft

i

:=Nt P-a.s. 126

But, sinceG∈G¯,N is a uniformly integrable martingale, which proves the following 127

Lemma 2.3. SupposeG∈G¯. ThenS is of class (D). 128

LetM0denote the set of right-continuous martingales started at zero. LetM0,loc

129

andM0,U Idenote the spaces of local and uniformly integrable martingales (started at

130

zero), respectively. Similarly, the adapted processes of finite and integrable variation

131

will be denoted byF V and IV, respectively.

132

It is well-known that a right-continuous (local) supermartingale P has a unique

133

decompositionP =B−IwhereB ∈ M0,locandIis an increasing (F V) process which

134

is predictable. This can be regarded as the general Doob-Meyer decomposition of a

135

supermartingale. Specialising to class (D) supermartingales we have a stronger result

136

(this is a consequence of, for example, Protter [40] Theorem 16, p.116 and Theorem

137

11, p.112):

138

Theorem 2.4 (Doob-Meyer decomposition). Let G∈G¯. Then the Snell enve-139

lope processS admits a unique decomposition

140

(2.4) S =M∗−A,

141

whereM∗∈ M0,U I, andA is a predictable, increasingIV process.

142

Remark 2.5. It is normal to assume that the process A in the Doob-Meyer

de-143

composition ofS is started at zero. The duality result alluded to in the introduction

144

is one reason why we do not do so here.

145

An immediate consequence of Theorem 2.4 is that S is a semimartingale. In

146

addition, we also assume thatGis a semimartingale with the following decomposition:

147

(2.5) G=N+D,

148

whereN ∈ M0,locandD is aF V process. Unfortunately, the decomposition(2.5)is not, in general, unique. On the other hand, uniqueness isobtained by requiring the F V term to also be predictable, at the cost of restricting only to locally integrable processes. If there exists a decomposition of a semimartingale X with a predictable F V process, then we say that X is special. For a special semimartingale we always choose to work with itscanonicaldecomposition (so that aF V process is predictable). Let

Gbe the space of semimartingales in ¯G.

Lemma 2.6. SupposeG∈G. ThenGis a special semimartingale. 150

See Theorems 36 and 37 (p.132) in Protter [40] for the proof.

151

The following lemma provides a further decomposition of a semimartingale (see

152

Proposition 3.3 (p.27) in Jacod and Shiryaev [28]). In particular, theF V term of a

153

special semimartingale can be uniquely (up to initial values) decomposed in a

pre-154

dictable way, into the difference of two increasing, mutually singularF V processes.

155

Lemma 2.7. Suppose that K is a c`adl`ag, adapted process such that K ∈ F V.

156

Then there exists a unique pair (K+_{, K}−_{)} _{of adapted increasing processes such that}

157

K−K0 = K+−K− and

R

|dKs|=K++K−. Moreover, if K is predictable, then

158

K+_{,}_{K}− _{and}R

|dKs|are also predictable.

159

2.2. Markovian setting. 160

The Markov process. Let (E,E) be a metrizable Lusin space endowed with the

161

σ-field of Borel subsets of E. Let X = (Ω,G,Gt, Xt, θt,Px : x ∈ E, t ∈ R+) be a

162

Markov process taking values in (E,E). We assume that a sample space Ω is such that

163

the usual semi-group of shift operators (θt)t≥0 is well-defined (which is the case, for

164

example, if Ω =E[0,∞)is the canonical path space). If the corresponding semigroup

165

of X, (Pt), is the primary object of study, then we say that X is a realisation of a

166

Markov semigroup (Pt). In the case of (Pt) being sub-Markovian, i.e. Pt1E ≤ 1E,

167

we extend it to a Markovian semigroup over E∆ = E∪ {∆}, where ∆ is a

coffin-168

state. We also denote by C(X) = (Ω,F,Ft, Xt, θt,Px:x∈E, t∈R+) thecanonical

169

realisation associated with X, defined on Ω with the filtration (Ft) deduced from

170

F0

t =σ(Xs: s≤t) by standard regularisation procedures (completeness and

right-171

continuity).

172

In this paper our standing assumption is that the underlying Markov processXis

173

a right process (consult Getoor [20], Sharpe [46] for the general theory). Essentially,

174

right processes are the processes satisfying Meyer’s regularity hypotheses (hypoth`eses

175

droites) HD1 and HD2. If a given Markov semigroup (Pt) satisfies HD1 andµis an

176

arbitrary probability measure on (E,E), then there exists a homogeneous E-valued

177

Markov processX with transition semigroup (Pt) and initial lawµ. Moreover, a

real-178

isation of such (Pt) is right-continuous (Sharpe [46], Theorem 2.7). Under the second

179

fundamental hypothesis, HD2, t → f(Xt) is right-continuous for every α-excessive

180

function f. Recall, for α > 0, a universally measurable function f : E → R is α -181

super-median ife−αtPtf ≤f for allt≥0, andα-excessive if it isα-super-median and

182

e−αtPtf →f ast→0. If (Pt) satisfies HD1 and HD2 then the corresponding

realisa-183

tionX is strong Markov (Getoor [20], Theorem 9.4 and Blumenthal and Getoor [7],

184

Theorem 8.11).

185

Remark 2.8. One has the following inclusions among classes of Markov processes:

186

(Feller)⊂(Hunt)⊂(right)

187

LetLbe a given extended infinitesimal (martingale) generator ofXwith a domain

188

D(L), i.e. we say a Borel functionf :E →Rbelongs toD(L) if there exists a Borel 189

functionh:E →R, such thatR0t|h(Xs)|ds <∞, ∀t≥0,Px-a.s. for each xand the

190

processMf = (M_{t}f)t≥0, given by

191

(2.6) M_{t}f :=f(Xt)−f(x)−

Z t

0

h(Xs)ds, t≥0, x∈E,

192

is a local martingale under eachPx(see Revuz and Yor [42] p.285), and then we write

193

h=Lf.

Remark 2.9. Note that if A ∈ E and _{P}x(λ({t : Xt ∈ A} = 0) = 1 for each

195

x∈E, where λis Lebesgue measure, thenhmay be altered onA without affecting

196

the validity of (2.6), so that, in general, the mapf →his not unique. This is why

197

we refer toamartingale generator.

198

Optimal stopping problem. Let X = (Ω,G,Gt, Xt, θt,Px : x ∈ E, t ∈ R+) be
a right process. Given a function g : E → _{R} (with g(∆) = 0), α ≥ 0 and T ∈

R+∪ {∞} define a corresponding gains process Gα (we simply write G if α = 0) by Gα

t = e−αtg(Xt) for t ∈ [0, T]. In the case of T = ∞, we make a convention thatGα

∞= lim inft→∞Gαt. LetEe,Eu be theσ-algebras onE generated by excessive
functions and universally measurable sets, respectively (recall thatE ⊂ Ee_{⊂ E}u_{). We}
write

g∈ Y, given thatg(·) isEe_{-measurable and}_{G}α _{is of class (D).}

For a filtration ( ˆGt), and ( ˆGt) - stopping timesσ1andσ2, withPx[0≤σ1≤σ2≤T] =

199

1,x∈E, letTσ1,σ2( ˆG) be the set of ( ˆGt) - stopping timesτ withPx[σ1≤τ ≤σ2] = 1.

200

Consider the following optimal stopping problem:

201

V(x) = sup τ∈T0,T(G)

Ex[e−ατg(Xτ)], x∈E.

202

By convention we set V(∆) = g(∆). The following result is due to El Karoui et

203

al. [18].

204

Theorem 2.10. Let X = (Ω,G,Gt, Xt, θt,Px:x∈E, t∈R+) be a right process

205

with canonical filtration(Ft). Ifg∈ Y, then

206

V(x) = sup τ∈T0,T(F)

Ex[e−ατg(Xτ)], x∈E,

207

and(e−αtV(Xt))is a Snell envelope ofGα_{, i.e. for all} _{x}_{∈}_{E} _{and}_{τ} _{∈ T}
0,T(F)

208

e−ατV(Xτ) = ess sup σ∈Tτ,T(F)

Ex[Gασ|Fτ] Px-a.s.

209

The first important consequence of the theorem is that we can (and will) work with

210

the canonical realisation C(X). The second one provides a crucial link between the

211

Snell envelope process in the general setting and the value function in the Markovian

212

framework.

213

Remark 2.11. The restriction to gains processes of the formG=g(X) (orGα if

214

α >0) ismuch less restrictive than might appear. Given that we work on the canonical

215

path space withθbeing the usual shift operator, we can expand the state-space ofX

216

by appending an adapted functional F, taking values in the space (E0_{,}_{E}0_{), with the}

217

property that

218

(2.7) {Ft+s∈A} ∈σ(Fs)∪σ(θs◦Xu: 0≤u≤t), for allA∈ E0.

219

This allows us to deal with time-dependent problems, running rewards and other

220

path-functionals of the underlying Markov process.

221

Lemma 2.12. SupposeX is a canonical Markov process taking values in the space

222

(E,E)whereEis a locally compact, countably based Hausdorff space andEis its Borel

223

σ-algebra. Suppose also that F is a path functional of X satisfying (2.7) and taking

224

values in the space (E0,E0_{)} _{where}_{E}0 _{is a locally compact, countably based Hausdorff}

space with Borel σ-algebraE0_{, then, defining} _{Y} _{= (}_{X, F}_{)}_{,}_{Y} _{is still Markovian. If} _{X}

226

is a strong Markov process andF is right-continuous, thenY is strong Markov. IfX

227

is a Feller process andF is right-continuous , then Y is strong Markov, has a c`adl`ag

228

modification and the completion of the natural filtration ofX,F, is right-continuous 229

and quasi-left continuous, and thus Y is a right process.

230

Example 2.13. IfX is a one-dimensional Brownian motion, thenY, defined by

231

Yt= Xt, L0t, sup 0≤s≤t

Xs,

Z t

0 exp(−

Z s

0

α(Xu)du)f(Xs)ds

!

, t≥0,

232

whereL0 is the local time ofX at 0, is a Feller process on the filtration ofX.

233

3. Main results. In this section we retain the notation of subsection 2.1and

234

subsection 2.2.

235

3.1. General framework. The assumption thatG∈G(i.e. Gis a semimartin-236

gale with integrable supremum andG=N+Dis its canonical decomposition), neither

237

ensures that N ∈ M0, nor that D is anIV process, the latter, it turns out, being

238

sufficient for the main result of this section to hold. In order to prove Theorem 3.6

239

we will need a stronger integrability condition onG.

240

For any adapted c`adl`ag processH, define

241

(3.1) H∗= sup

0≤t≤T

|Ht|

242

and

243

(3.2) ||H||Sp=||H∗||_{L}p:=_{E}|H∗|p

1/p

, 1≤p≤ ∞.

244

245

Remark 3.1. Note that ¯_{G} = S1_{, so that under the current conditions we have}

246

thatG∈ S1_{.}

247

For a special semimartingale X with canonical decomposition X = ¯B + ¯I, where

248

¯

B ∈ M0,loc and ¯I is a predictableF V process (with I0=X0), define theHp norm,

249

for 1≤p≤ ∞, by

250

(3.3) ||X||Hp=||B¯||_{S}p+

Z T

0

|dI¯s|

_{L}p+||I0||L
p,
251

and, as usual, writeX ∈ Hp _{if}_{||}_{X}_{||}

Hp<∞. 252

Remark 3.2. A more standard definition of theHp_{norm is with}_{||}_{B}_{¯}_{||}

Spreplaced 253

by ||[ ¯B,B¯]1/2_{T} ||Lp. However, the Burkholder-Davis-Gundy inequalities (see Protter
254

[40], Theorem 48 and references therein) imply the equivalence of these norms.

255

The following lemma follows from the fact that ¯I∗≤RT

0 |dIs¯|+|I0|,P−a.s:

256

Lemma 3.3. On the space of special semimartingales, the Hp norm is stronger

257

thanSp _{for}_{1}_{≤}_{p <}_{∞, i.e. convergence in} _{H}p _{implies convergence in}_{S}p_{.}

258

In general, it is challenging to check whether a given process belongs toH1_{, and thus}

259

the assumption that G∈ H1 _{might be too stringent. On the other hand, under the}

260

assumptions in the Markov setting (seesubsection 3.2), we will have thatGislocally

261

inH1_{. Recall that a semimartingale}_{X} _{belongs to}_{H}p

loc, for 1≤p≤ ∞, if there exists

a sequence of stopping times{σn}n∈N, increasing to infinity almost surely, such that 263

for eachn≥1, the stopped processXσn _{belongs to}_{H}p_{. Hence, the main assumption}
264

in this section is the following:

265

Assumption3.4. Gis a semimartingale in bothS1 andH1loc.

266

Remark 3.5. Given that G ∈ H1_{,} _{Lemma 3.3} _{implies that} _{Assumption 3.4} _{is}

267

satisfied, and thus all the results ofsubsection 2.1 hold. Moreover, we then have a

268

canonical decomposition ofG

269

(3.4) G=N+D,

270

with N ∈ M0,U I and a predictable IV process D. On the other hand, under

As-271

sumption 3.4, the decomposition(3.4)still holds, however,N andDare only locally

272

uniformly integrable martingale (started at zero) and the process of integrable

varia-273

tion, respectively, i.e. Gσn _{∈ M}

0,U I andIσn is a process of IV, where {σn}n≥1 is a

274

localising sequence.

275

We finally arrive to the main result of this section:

276

Theorem 3.6. SupposeAssumption 3.4holds. LetAbe a predictable, increasing

277

IV process in the decomposition of the Snell envelope S, as inTheorem 2.4. Let D−

278

(D+_{)}_{denote the decreasing (increasing) components of} _{D}_{, as in} _{Lemma} _{2.7}_{. Then}

279

A is, as a measure, absolutely continuous with respect toD− almost surely on [0, T],

280

andµ, defined by

281

µt:= dAt

dD_{t}−, 0≤t≤T,

282

has a version that satisfies 0≤µt≤1 almost surely.

283

Remark 3.7. As is usual in semimartingale calculus, we treat a process of bounded

284

variation and its corresponding Lebesgue-Stiltjes signed measure as synonymous.

285

The proof of Theorem 3.6 is based on the discrete-time approximation of the

pre-286

dictableF V processes in the decompositions ofS (2.4)andG(2.5). In particular, let

287

Pn ={0 =tn0 < tn1 < tn2 < ... < tnkn =T}, n= 1,2, ..., be an increasing sequence of 288

partitions of [0, T] with max1≤k≤knt

n k −t

n

k−1→0 asn→ ∞. Note that here T <∞

289

is fixexd, but arbitrary. Let Sn t = Stn

k if t

n

k ≤ t < t n

k+1 and S n

T = ST define the

290

discretizations ofS, and set

291

An_{t} = 0 if 0≤t < tn_{1},

292

An_{t} =
k

X

j=1

E[Stn

j−1−Stnj|Ftnj−1] ift n

k ≤t < t n

k+1,k= 1,2, ..., kn−1,

293

An_{T} =
kn
X

j=1

E[Stn

j−1−Stnj|Ftnj−1].

294

295

If S is regular in the sense that for every stopping time τ and nondecreasing

296

sequence (τn)n∈N of stopping times with τ = limn→∞τn, we have limn→∞E[Sτn] = 297

E[Sτ], or equivalently, ifAis continuous, Dol´eans [14] showed thatAnt →Atuniformly

298

in L1 _{as} _{n} _{→ ∞} _{(see also Rogers and Williams [44], VI.31, Theorem 31.2). Hence,}

299

given thatSis regular, we can extract a subsequence{Anl

t }, such that liml→∞Antl=

300

Ata.s. On the other hand, it is enough forGto be regular:

Lemma 3.8. Suppose G ∈ G¯ is a regular gains process. Then so is its Snell 302

envelope processS.

303

SeeAppendix Afor the proof.

304

Remark 3.9. If it is not known that Gis regular, Kobylanski and Quenez [32],

305

in a slightly more general setting, showed thatS is still regular, provided that Gis

306

upper semicontinuous in expectation along stopping times, i.e. for all τ ∈ T0,T _{and}

307

for all sequences of stopping times (τn)n≥1 such thatτn ↑τ, we have

308

E[Gτ]≥lim sup n→∞ E

[Gτn]. 309

The case where S is not regular is more subtle. In his classical paper Rao [41]

310

utilised the Dunford-Pettis compactness criterion and showed that, in general,An t →

311

At only weakly in L1 _{as} _{n} _{→ ∞} _{(a sequence (}_{Xn}_{)}

n∈N of random variables in L1 312

converges weakly in L1 _{to} _{X} _{if for every bounded random variable} _{Y} _{we have that}

313

E[XnY]→E[XY] asn→ ∞). 314

Recall that weak convergence in L1 does not imply convergence in probability,

315

and therefore, we cannot immediately deduce an almost sure convergence along a

316

subsequence. However, it turns out that by modifying the sequence of approximating

317

random variables, the required convergence can be achieved. This has been done

318

in recent improvements of the Doob-Meyer decomposition (see Jakubowski [29] and

319

Beiglb¨ock et al. [4]. Also, Siorpaes [48] showed that there is a subsequence that

320

works for all (t, ω)∈[0, T]×Ω simultaneously). In particular, Jakubowski proceeds

321

as Rao, but then uses Koml´os’s theorem [34] and proves the following (Jakubowski

322

[29], Theorem 3 and Remark 1):

323

Theorem 3.10. There exists a subsequence {nl} such that fort ∈ ∪∞

n=1Pn and

324

asL→ ∞

325

(3.5) 1

L

_{X}L

l=1 Anl

t

→At, a.s. and in L1.

326

In particular, in any subsequence we can find a further subsequence such that (3.5)

327

holds.

328

Proof of Theorem 3.6. Let (σn)n≥1 be a localising sequence for Gsuch that, for

329

eachn≥1,Gσn _{= (}_{G}

t∧σn)0≤t≤T is in H

1_{. Similarly, set} _{S}σn _{= (}_{S}

t∧σn)0≤t≤T for a 330

fixedn≥1. We need to prove that

331

(3.6) 0≤Aσn

t −A σn

s ≤(D
−_{)}σn

t −(D
−_{)}σn

s a.s.,

332

since then, as σn ↑ ∞ almost surely, as n → ∞, and by uniqueness of A and D−,

333

the result follows. In particular, since A is increasing, the first inequality in (3.6) is

334

immediate, and thus we only need to prove the second one.

335

After localisation we assume thatG∈ H. For any 0≤t≤T and 0≤≤T−t

336

we have that

337

E[St+|Ft] =E h

ess sup τ∈Tt+,T

E[Gτ|Ft+]

Ft

i 338

≥_{E}h_{E}[Gτ|Ft+]

Ft

i 339

=_{E}[Gτ|Ft] a.s.,

whereτ ∈ Tt+,T is arbitrary. Therefore

342

(3.7) E[St+|Ft]≥ ess sup τ∈Tt+,T

E[Gτ|Ft] a.s.

343

Then by (2.3) and using(3.7)together with the properties of theessential supremum

344

(see also LemmaA.1in theAppendix A) we obtain

345

E[St−St+|Ft]≤ess sup τ∈Tt,T

E[Gτ|Ft]− ess sup τ∈Tt+,T

E[Gτ|Ft]

346

≤ess sup τ∈Tt,T

E[Gτ−Gτ∨(t+)|Ft]

347

= ess sup τ∈Tt,t+

E[Gτ−Gτ∨(t+)|Ft] (3.8)

348

= ess sup τ∈Tt,t+

E[Gτ−Gt+|Ft] a.s.

349 350

The first equality in (3.8) follows by noting that Tt+,T ⊂ Tt,T, and that for any

351

τ ∈ Tt+,T the term inside the expectation vanishes. Using the decomposition of G

352

and by observing that, for all τ ∈ Tt,t+, (Dτ+−D +

t+)≤0, whileN is a uniformly

353

integrable martingale, we obtain

354

E[St−St+|Ft]≤ess sup τ∈Tt,t+

E[D−t+−D−τ|Ft]

355

=_{E}[D−_{t+}−D−_{t}|Ft] a.s.
(3.9)

356 357

Finally, for 0≤s < t≤T, applying Theorem3.10toAtogether with(3.9)gives

358

At−As= lim L→∞

1 L

XL

l=1 k

X

j=k0

E[Stnl_{j}_{−}_{1}−Stnl_{j} |Ftnl_{j}_{−}_{1}]

359

≤ lim L→∞

1 L

XL

l=1 k

X

j=k0

E[D−_{t}nl
j

−D_{t}−nl
j−1

|F_{t}nl
j−1]

a.s., (3.10)

360

361

wherek0 ≤kare such thattnl

k0 ≤s < t

nl

k0_{+1}andt

nl

k ≤t < t nl

k+1. Note that D− is also

362

the predictable, increasingIV process in the Doob-Meyer decomposition of the class

363

(D) supermartingale (G−D+_{). Therefore we can approximate it in the same way as}

364

A, so that D−_{t} −D−_{s} is the almost sure limit along, possibly, a further subsequence

365

{nlk}of{nl}, of the right hand side of (3.10). 366

We finish this section with a lemma that gives an easy test as to whether the given

367

process belongs toH1

loc(consult Appendix Afor the proof).

368

Lemma 3.11. Let X ∈ G with a canonical decomposition X = L+K, where 369

L ∈ M0,loc and K is a predictable F V process. If the jumps of K are uniformly

370

bounded by some finite constant c >0, thenX∈ H1 loc.

371

3.2. Markovian setting. In the rest of the section (and the paper) we consider

372

the following optimal stopping problem:

373

(3.11) V(x) = sup

τ∈T0,TE

x[g(Xτ)], x∈E,

374

for a measurable functiong:E→Rand a Markov processX satisfying the following 375

set of assumptions:

Assumption3.12. X is a right process.

377

Assumption3.13. sup_{0≤t≤T}|g(Xt)|∈L1(Px),x∈E.

378

Assumption3.14. g ∈ D(L), i.e. g(·) belongs to the domain of a martingale 379

generator ofX.

380

Remark 3.15. Lemma 2.12tells us that if X is Feller and F is an adapted

path-381

functional of the form given in(2.7)then (a modification of) (X, F) satisfies

Assump-382

tion 3.12.

383

Example 3.16. LetX= (Xt)t≥0be a Markov process and letD( ˆL) be the domain 384

of a classical infinitesimal generator ofX, i.e. the set of measurable functionsf :E→

385

R, such that limt→0(Ex[f(Xt)]−f(x))/texists. ThenD( ˆL)⊂D(L). In particular, 386

1. ifX = (Xt)t≥0 is a solution of an SDE driven by a Brownian motion in Rd, 387

thenC2 b(R

d_{,}

R)⊂D( ˆL); 388

2. if the state spaceE is finite (so that X is a continuous time Markov chain),

389

then any measurable and boundedf :E→_{R}belongs to_{D}( ˆL)

390

3. ifXis a L´evy process onRdwith finite variance increments thenCb2(Rd,R)⊂

391

D( ˆL) 392

Note that the gains process is of the form G = g(X), while by Theorem 2.10, the

393

corresponding Snell envelope is given by

394

S_{t}T :=

(

V(Xt) :t < T, g(XT) :t≥T.

395

In a similar fashion to that in the general setting,Assumption 3.13ensures the class

396

(D) property for the gains and Snell envelope processes. Moreover, under

Assump-397

tion 3.14,

398

(3.12) g(Xt) =g(x) +Mtg+

Z t

0

Lg(Xs)ds, 0≤t≤T, x∈E,

399

and theF V process in the semimartingale decomposition of G=g(X) is absolutely

400

continuous with respect to Lebesgue measure, and therefore predictable, so that(3.12)

401

is a canonical semimartingale decomposition ofG=g(X). Then, byAssumption 3.13,

402

and usingLemma 3.11, we also deduce thatg(X)∈ H1 loc.

403

Remark 3.17. WhenT < ∞, the optimal stopping problem, in general, is

time-404

inhomogeneous, and we need to replace the processXt by the processZt= (t, Xt),

405

t∈[0, T], so that (3.11) reads

406

(3.13) V˜(t, x) = sup τ∈T0,T−t

Et,x[˜g(t+τ, Xt+τ)], x∈E,

407

where ˜g : [0, T]×E →R is a new payoff function (consult Peskir and Shiryaev [39] 408

for examples). In this case, Assumption 3.14 should be replaced by a requirement

409

that there exists a measurable function ˜h: [0, T]×E→Rsuch thatMt˜g:= ˜g(Zt)−

410

˜

g(0, x)−Rt

0˜h(Zs)dsdefines a local martingale.

411

The crucial result of this section is the following:

412

Proof. In order to be consistent with the notation in the general framework, let

414

Dt:=g(X0) +

Z t

0

Lg(Xs)ds, 0≤t≤T.

415

RecallLemma 2.7. ThenD+ andD− are explicitly given (up to initial values) by

416

D+_{t} : =

Z t

0

Lg(Xs)+ds,

417

D−t : =

Z t

0

Lg(Xs)−ds.

418 419

In particular, D− is, as a measure, absolutely continuous with respect to Lebesgue

420

measure. By applyingTheorem 3.6, we deduce that

421

(3.14) V(Xt) =V(x) +M_{t}∗−

Z t

0

µsLg(Xs)−ds, 0≤t≤T, x∈_{R},

422

whereµis a non-negative Radon-Nikodym derivative with 0≤µs≤1. Then we also

423

have thatR_{0}t|µsLg(Xs)−|ds <∞, for every 0≤t≤T.

424

In order to finish the proof we are left to show that there exists a suitable

mea-425

surable functionλ:E→Rsuch thatAt=

Rt

0µsLg(Xs)

−_{ds}_{=}Rt

0λ(Xs)dsa.s., for all

426

t∈[0, T]. For this, recall that a processZ (on (Ω,G,Gt, Xt, θt,Px:x∈E, t∈R+) or

427

just onC(X)) isadditiveifZ0= 0 a.s. andZt+s=Zt+Zs◦θta.s., for alls, t∈[0, T].

428

Then, for any measurable functionf :E→R,Ztf =f(Xt)−f(x) defines an additive

429

process. (C¸ inlar et al. [8] gives necessary and sufficient conditions for Zf _{to be a}

430

semimartingale.) More importantly, if Zf is a semimartingale, then the martingale

431

andF V processes in the decomposition ofZf are also additive, see Theorem 3.18 in

432

C¸ inlar et al. [8].

433

Finally, we have that At=R_{0}tµsLg(Xs)−ds, t∈[0, T], is an increasing additive

434

process such thatdAtdt. SetKt= lim infs↓0,s∈Q(At+s−At)/sandβ(x) =Ex[K0],

435

x ∈E. Then by Proposition 3.56 in C¸ inlar et al. [8], we have that, for t ∈ [0, T],

436

At=R_{0}tβ(Xs)dsPx-a.s. for eachx∈E.

437

Remark 3.19. In some specific examples it is possible to relax Assumption 3.14.
Let S := {x ∈ E : V(x) = g(x)} be the stopping region. It is well-known that
S=V(X) is a martingale on the go region Sc_{, i.e.} _{M}c _{given by}

M_{t}cdef=

Z t

0

1(Xs−∈Sc)dSs

is a martingale (see Lemma A.2). This implies that R_{0}t1(Xs−∈Sc)dAs = 0, and

438

therefore we note that in order for V ∈ D(L), we need D to be absolutely con-439

tinuous with respect to Lebesgue measure λ only on the stopping region i.e. that

440 R·

01(Xs−∈S)dDs λ. For example, let E =R, fix K ∈ R+ and consider g(·) given

441

byg(x) = (K−x)+_{,} _{x}_{∈}_{E}_{. We can easily show, under very weak conditions, that}

442

S ⊂[0, K] and so we need only have thatR·

01(Xs−<K)dDs is absolutely continuous.

443

4. Applications: duality, smooth fit. In this section we retain the setting of

444

subsection 3.2.

4.1. Duality. Let x∈ E be fixed. As before, let Mx

0,U I denote all the

right-446

continuous uniformly integrable c`adl`ag martingales (started at zero) on the filtered

447

space (Ω,F,F,Px), x∈E. The main result of Rogers [43] in the Markovian setting

448

reads:

449

Theorem 4.1. SupposeAssumption 3.12and3.13hold. Then

450

(4.1) V(x) = sup τ∈T0,TE

x[Gτ] = inf M∈Mx

0,U I Ex

h

sup 0≤t≤T

Gt−Mt

i

, x∈E.

451

We call the right hand side of (4.1)thedualof the optimal stopping problem. In

452

particular, the right hand side of (4.1) is a ”generalised stochastic control problem

453

of Girsanov type”, where a controller is allowed to choose a martingale fromMx 0,U I,

454

x∈E. Note that an optimal martingale for the dual isM∗, the martingale appearing

455

in the Doob-Meyer decomposition of S, while any other martingale inMx

0,U I gives

456

an upper bound of V(x). We already showed that M∗ = MV_{, which means that,}

457

when solving the dual problem, one can search only over martingales of the formMf,

458

for f ∈ D(L), or equivalently over the functions f ∈ D(L). We can further define 459

DM0,U I ⊂D(L) by 460

DM0,U I :={f ∈D(L) :f ≥g, f is superharmonic,M

f

∈ M0,U I}.

461

To conclude that V ∈ DM0,U I we need to show that V is superharmonic, i.e. for 462

all stopping times σ ∈ T0,T _{and all} _{x} _{∈} _{E}_{,}

Ex[V(Xσ)] ≤ V(x). But this follows

463

immediately from the Optional Sampling theorem, since S = V(X) is a uniformly

464

integrable supermartingale. Hence, as expected, we can restrict our search for the

465

best minimising martingale to the setDM0,U I. 466

Theorem 4.2. Suppose that G = g(X) and the assumptions of Theorem 3.18

467

hold. Let DM0,U I be the set of admissible controls. Then the dual problem, i.e. the 468

right hand side of (4.1), is a stochastic control problem for a controlled Markov process

469

(X, Yf, Zf),f ∈ DM0,U I (defined by (4.2) and (4.3)), with a value functionVˆ given 470

by (4.4)

471

Proof. For anyf ∈ DMx

0,U I,x∈E andy, z∈R, define processesY

f _{and} _{Z}f _{via}

472

Y_{t}f :=y+

Z t

0

Lf(Xs)ds, 0≤t≤T, (4.2)

473

Z_{s,t}f := sup
s≤r≤t

f(x) +g(Xr)−f(Xr) +Y_{r}f, 0≤s≤t≤T,
(4.3)

474 475

and to allow arbitrary starting positions, setZ_{t}f =Z_{0,t}f ∨z, forz ≥g(x) +y. Note

476

that, for anyf ∈D(L),Yf is an additive functional of X. Lemma 2.12implies that 477

iff ∈ DM0,U I then (X, Y

f_{, Z}f_{) is a Markov process.}

478

Define ˆV :E×R2→Rby 479

(4.4) Vˆ(x, y, z) = inf f∈DMx

0,U I

Ex,y,z[Z f

T], (x, y, z)∈E×R×R.

480

It is clear that this is a stochastic control problem for the controlled Markov process

481

(X, Yf_{, Z}f_{), where the admissible controls are functions in}_{D}

M0,U I. Moreover, since 482

V ∈ DM0,U I, by virtue ofTheorem 4.1, and adjusting initial conditions as necessary, 483

we have

484

V(x) = ˆV(x,0, g(x)) =Ex,0,g(x)[ZTV], x∈E.

a

486

4.2. Some remarks on thesmooth pasting condition. We will now discuss

487

the implications ofTheorem 3.18for the smoothness of the value functionV(·) of the

488

optimal stopping problem given in(3.11).

489

Remark 4.3. While inTheorem 4.4(resp. Theorem 4.9) we essentially recover (a

490

small improvement of) Theorem 2.3 in Peskir [37] (resp. Theorem 2.3 in Samee [45]),

491

the novelty is that we prove the results by means of stochastic calculus, as opposed

492

to the analytic approach in [37] (resp. [45]).

493

In addition toAssumption 3.13andAssumption 3.14, we now assume thatX is a

494

one-dimensional diffusion in the Itˆo-McKean [26] sense, so thatX is a strong Markov

495

process with continuous sample paths. We also assume that the state spaceE⊂Ris 496

an interval with endpoints−∞ ≤a≤b≤+∞. Nnote that the diffusion assumption

497

impliesAssumption 3.12. Finally, we assume thatX isregular: for anyx, y∈int(E),

498

Px[τy <∞]>0, whereτy= min{t≥0 :Xt=y}. Letα≥0 be fixed; αcorresponds

499

to a killing rate of the sample paths ofX.

500

The case without killing: α = 0. Let s(·) denote a scale function of X, i.e. a

501

continuous, strictly increasing function onE such that forl, r,x∈E, witha≤l <

502

x < r≤b, we have

503

(4.5) Px(τr< τl) =

s(x)−s(l) s(r)−s(l),

504

see Revuz and Yor [42], Proposition 3.2 (p.301) for the proof of existence and

prop-505

erties of such a function.

506

From (4.5), using regularity of X and thatV(X) is a supermartingale of class

507

(D) we have thatV(·) iss-concave:

508

V(x)≥V(l)s(r)−s(x)

s(r)−s(l) +V(r)

s(x)−s(l)

s(r)−s(l), x∈[l, r]. (4.6)

509 510

Theorem 4.4. Suppose the assumptions of Theorem 3.18 are satisfied, so that

511

V ∈D(L). Further assume thatX is a regular, strong Markov process with continuous 512

sample paths. LetY =s(X), wheres(·)is a scale function of X.

513

1. Assume that for eachy∈[s(a), s(b)], the local time ofY aty,Ly, is singular

514

with respect to Lebesgue measure. Then, if s ∈ C1, V(·), given by (3.11),

515

belongs toC1_{.}

516

2. Assume that([Y, Y]t)t≥0 is, as a measure, absolutely continuous with respect

517

to Lebesgue measure. Ifs0(·)is absolutely continuous, thenV ∈C1 _{and}_{V}0_{(}_{·}_{)}

518

is also absolutely continuous.

519

Remark 4.5. IfG is the filtration of a Brownian motion,B, then Y =s(X) is a

520

stochastic integral with respect toB (a consequence of martingale representation):

521

(4.7) Yt=Y0+

Z t

0

σsdBs.

522

Moreover, Proposition 3.56 in C¸ inlar et al. [8] ensures thatσt=σ(Yt) for a suitably measurable functionσand

[Y, Y]t=

Z t

0

In this case, both, the singularity of the local time of Y and absolute continuity

523

of [Y, Y] (with respect to Lebesgue measure), are inherited from those of Brownian

524

motion. On the other hand, if X is a regular diffusion (not necessarily a solution to

525

an SDE driven by a Brownian motion), absolute continuity of [Y, Y] still holds, if the

526

speed measure ofX is absolutely continuous (with respect to Lebesgue measure).

527

Proof. Note thatY =s(X) is a Markov process, and letKdenote its martingale

528

generator. Moreover, V(x) = W(s(x)) (see Lemma 4.7 and the following remark),

529

where, on the interval [s(a), s(b)],W(·) is the smallest nonnegative concave majorant

530

of the function ˆg(y) =g◦s−1(y). Then, sinceV ∈D(L), 531

V(Xt) =V(x) +MtV +

Z t

0

LV(Xu)du, 0≤t≤T,

532

and thus

533

W(Yt) =W(y) +MtV +

Z t

0

(LV)◦s−1(Yu)du, 0≤t≤T.

534 535

Therefore,W ∈D(K), since 536

(4.8) W(Yt) =W(y) +MtV +

Z t

0

KW(Yu)du,

537

fory∈[s(a), s(b)], 0≤t≤T, withKW =LV ◦s−1≤0.

538

On the other hand, using the generalised Itˆo formula for concave/convex functions

539

(see e.g. Revuz and Yor [42], Theorem 1.5 p.223) we have

540

W(Yt) =W(y) +

Z t

0

W_{+}0(Yu)dYu−

Z s(b)

s(a)

Lz_{t}ν(dz),

541

for y ∈ [s(a), s(b)], 0 ≤ t ≤ T, where Lz_{t} is the local time of Yt at z, and ν is a

542

non-negative σ-finite measure corresponding to the second derivative of −W in the

543

sense of distributions. Then, by the uniqueness of the decomposition of a special

544

semimartingale, we have that, fort∈[0, T],

545

(4.9) −

Z t

0

KW(Yu)du=

Z s(b)

s(a)

Lz_{t}ν(dz) a.s.

546

We prove the first claim by contradiction. Suppose that ν({z0}) >0 for some

547

z0∈(s(a), s(b)). Then, using (4.9) we have that

548

(4.10) −

Z t

0

KW(Yu)du=Lz0

t ν({z0}) +

Z s(b)

s(a)

1{z6=z0}Lztν(dz) a.s.

549

SinceLz0_{t} is positive with positive probability and, by assumption,Ly_{,}_{y}_{∈}_{[}_{s}_{(}_{a}_{)}_{, s}_{(}_{b}_{)],}

550

is singular with respect to Lebesgue measure, the process on the right hand side of

551

(4.10) is not absolutely continuous with respect to Lebesgue measure, which

contra-552

dicts absolute continuity of the left hand side. Therefore,ν({z0}) = 0, and since z0

553

was arbitrary, we have thatν does not charge points. It follows thatW ∈C1_{. Since}

554

s∈C1 _{by assumption, we conclude that}_{V} _{∈}_{C}1_{.}

We now prove the second claim. By assumption, [Y, Y] is absolutely continuous with respect to Lebesgue measure (on the time axis). Invoking Proposition 3.56 in C¸ inlar et al. [8] again, we have that

[Y, Y]t=

Z t

0

σ2(Yu)du

(as in Remark4.5). A time-change argument allows us to conclude thatY is a
time-change of a BM and that we may neglect the set{t:σ2_{(}_{Y}

t) = 0}in the representation (4.8). Thus

W(Yt) =W(Y0) +

Z t

0

1Nc(Yu)dM_{u}V +
Z t

0

1Nc(Yu)KW(Yu)du

where N is the zero set of σ. Then, using the occupation time formula (see, for

556

example, Revuz and Yor [42], Theorem 1.5 p.223) we have that

557

−

Z t

0

KW(Yu)du=

Z t

0

f(Yu)d[Y, Y]u=

Z s(b)

s(b)

f(z)Lz_{t}dz a.s.,

558

where f : [s(a), s(b)] →R is given by f : y 7→ −KWσ2 1Nc(y). Now observe that, for 559

0≤r≤t≤T,η([r, t]) :=Rs(b)

s(a)f(z)

Lz t−Lzr

dzandπ([r, t]) :=Rs(b)

s(a)

Lz t−Lzr

ν(dz)

560

define measures on the time axis, which, by virtue of (4.9), are equal (and thus both

561

are absolutely continuous with respect to Lebesgue measure). Now define Tl,¯l :=

562

{t : Yt ∈ [l,¯l]}, s(a) ≤ l ≤ ¯l ≤ s(b). Then the restrictions of η and π to Tl,¯l,

563

η|_{T}l,¯l and π|_{T}l,¯l, are also equal. Moreover, since Y is a local martingale, it is also a
564

semimartingale. Therefore, for every 0≤t≤T, Lz

t is carried by the set{t:Yt=z}

565

(see Protter [40], Theorem 69 p.217). Hence, for eacht∈[0, T],

566

(4.11) η|_{T}l,¯l([0, t]) =
Z ¯l

l

Lz_{t}f(z)dz=

Z ¯l

l

Lz_{t}ν(dz) =π|_{T}l,¯l([0, t]),
567

and, since l and ¯l are arbitrary, the left and right hand sides of (4.11) define

mea-568

sures on [s(a), s(b)]⊆R, which are equal. It follows that ν is absolutely continuous 569

with respect to Lebesgue measure on [s(a), s(b)] and f(z)dz = ν(dz). This proves

570

thatW ∈C1 andW0(·) is absolutely continuous on [s(a), s(b)] with Radon-Nikodym

571

derivativef. Since the product and composition of absolutely continuous functions

572

are absolutely continuous, we conclude thatV0(·) is absolutely continuous (sinces0(·)

573

is, by assumption).

574

Remark 4.6. We note that for a smooth fit principle to hold, it is not necessary

575

that s∈C1_{. Given that all the other conditions of}_{Theorem 4.4} _{hold, it is sufficient}

576

that s(·) is differentiable at the boundary of the continuation region. On the other

577

hand, ifg∈_{D}(L),V ∈C1_{, even if}_{g /}_{∈}_{C}1_{.}

578

Moreover, sinceV =g on the stopping region,Theorem 4.4tells us thatg ∈C1

579

on the interior of the stopping region. However, the question whether this stems

580

already from the assumption thatg∈D(L) is more subtle. For example, ifg∈D(L) 581

andgis a difference of two convex functions, then by the generalised Itˆo formula and

582

the local time argument (similarly to the proof of Theorem 4.4) we could conclude

583

thatg∈C1 _{on the whole state space}_{E}_{.}

Case with killing: α >0. We now generalise the results of theTheorem 4.4in the

585

presence of a non-trivial killing rate. Consider the following optimal stopping problem

586

(4.12) V(x) = sup

τ∈T0,TE

x[e−ατg(Xτ)], x∈E.

587

Note that, sinceα >0, using the regularity of X together with the supermartingale

588

property ofV(X) we have that

589

V(x)≥V(l)_{E}x[e−ατl1τl<τr] +V(r)Ex[e

−ατr_{1}

τr<τl], x∈[l, r]⊆E.

(4.13)

590 591

Define increasing and decreasing functionsψ, φ:E→R, respectively, by 592

ψ(x) =

(

Ex[e−ατc], ifx≤c 1/Ec[e−ατx], ifx > c

φ(x) =

(

1/_{E}c[e−ατx], ifx≤c

Ex[e−ατc], ifx > c (4.14)

593 594

wherec∈E is arbitrary. Then, (Ψt)0≤t≤T and (Φt)0≤t≤T, given by

595

Ψt=e−αtψ(Xt), Φt=e−αtφ(Xt), 0≤t≤T,

596

respectively, are local martingales (and also supermartingales, since ψ, φ are

non-597

negative); see Dynkin [15] and Itˆo and McKean [26].

598

Letp1, p2: [l, r]→[0,1] (where [l, r]⊆E) be given by

599

p1(x) =Ex[e−ατl1τl<τr], p2(x) =Ex[e

−ατr_{1}

τr<τl]. 600

Continuity of paths ofX implies thatpi(·), i= 1,2, are both continuous (the proof

601

of continuity of the scale function in (4.5) can be adapted for a killed process). In

602

terms of the functionsψ(·),φ(·) of(4.14), using appropriate boundary conditions, one

603

calculates

604

(4.15) p1(x) =ψ(x)φ(r)−ψ(r)φ(x)

ψ(l)φ(r)−ψ(r)φ(l) , p2(x) =

ψ(l)φ(x)−ψ(x)φ(l)

ψ(l)φ(r)−ψ(r)φ(l), x∈[l, r].

605

Let ˜s:E→_{R}+ be the continuous increasing function defined by ˜s(x) =ψ(x)/φ(x).

606

Substituting(4.15)into(4.13)and then dividing both sides byφ(x) we get

607

V(x) φ(x) ≥

V(l) φ(l) ·

˜

s(r)−s˜(x) ˜

s(r)−˜s(l) + V(r) φ(r) ·

˜

s(x)−˜s(l) ˜

s(r)−˜s(l), x∈[l, r]⊆E,

608

so thatV(·)/φ(·) is ˜s-concave.

609

Recall that(4.13)essentially follows fromV(·) beingα-superharmonic, so that it

610

satisfiesEx[e−ατV(Xτ)]≤V(x) forx∈E and any stopping time τ. Since Φ and Ψ

611

are local martingales, it follows that the converse is also true, i.e. given a measurable

612

function f : E → R, f(·)/φ(·) is ˜s-concave if and only if f(·) is α-superharmonic 613

(Dayanik and Karatzas [11], Proposition 4.1). This shows that a value function V(·)

614

is the minimal majorant ofg(·) such thatV(·)/φ(·) is ˜s-concave.

615

Lemma 4.7. Suppose[l, r]⊆E and letW(·)be the smallest nonnegative concave

616

majorant of g˜ := (g/φ)◦ ˜s−1 on [˜s(l),s˜(r)], where s˜−1 is the inverse of s˜. Then

617

V(x) =φ(x)W(˜s(x))on[l, r].

618

Proof. Define ˆV(x) =φ(x)W(˜s(x)) on [l, r]. Then, trivially, ˆV(·) majorizes g(·)

619

and ˆV(·)/φ(·) is ˜s-concave. ThereforeV(x)≤Vˆ(x) on [l, r].

On the other hand, let ˆW(y) = (V /φ)(˜s−1(y)) on [˜s(l),s˜(r)]. SinceV(x)≥g(x)

621

and (V /φ)(·) is ˜s-concave on [l, r], ˆW(·) is concave and majorizes (g/φ)◦s˜−1(·) on

622

[˜s(l),s˜(r)]. Hence,W(y)≤Wˆ(y) on [˜s(l),˜s(r)].

623

Finally, (V /φ)(x)≤( ˆV /φ)(x) =W(˜s(x))≤Wˆ(˜s(x)) = (V /φ)(x) on [l, r].

624

Remark 4.8. Whenα= 0, let (ψ, φ) = (s,1). ThenLemma 4.7is just Proposition

625

4.3. in Dayanik and Karatzas [11].

626

With the help of Lemma 4.7and using parallel arguments to those in the proof

627

of Theorem 4.4 we can formulate sufficient conditions for V to be in C1 _{and have}

628

absolutely continuous derivative.

629

Theorem 4.9. Suppose the assumptions of Theorem 3.18 are satisfied, so that

630

V ∈ D(L). Further assume that X is a regular Markov process with continuous 631

sample paths. Letψ(·), φ(·)be as in (4.14)and consider the process Y = ˜s(X).

632

1. Assume that, for each y∈[˜s(a),s˜(b)], the local time ofY aty ∈[˜s(a),˜s(b)],

633

ˆ

Ly_{, is singular with respect to Lebesgue measure. Then if} _{ψ, φ} _{∈} _{C}1_{,} _{V}_{(}_{·}_{)}_{,}

634

given by (4.12), belongs toC1_{.}

635

2. Assume that [Y, Y] is, as a measure, absolutely continuous with respect to

636

Lebesgue measure. If ψ0(·), φ0(·) are both absolutely continuous, thenV0(·)is

637

aslo absolutely continuous.

638

Proof. First note thatY is not necessarily a local martingale, while ΦY is. Indeed,

639

ΦY = Ψ. Hence

640

(Nt)0≤t≤T :=

Z t

0

ΦtdYt+ [Φ, Y]t

0≤t≤T

641

is the difference of two local martingales, and thus is a local martingale itself. Using

642

the generalised Itˆo formula for concave/convex functions, we have

643

(4.16) ΦtW(Yt) = Φ0W(y) +

Z t

0

W(Ys)dΦs+

Z t

0

W_{+}0(Ys)dNs−

Z ˜s(r)

˜ s(l)

ΦtLˆztν(dz),

644

for y ∈ [˜s(l),˜s(r)], 0 ≤ t ≤ T, where ˆLzt is the local time of Yt at z, and ν is a

645

non-negative σ-finite measure corresponding to the derivative W00 in the sense of

646

distributions.

647

On the other hand, ifg∈D(L), thenV ∈D(L). Therefore, 648

(4.17) e−αtV(Xt) =V(x) +

Z t

0

e−αsdM_{s}V +

Z t

0

e−αs{L −α}V(Xs)ds, 0≤t≤T.

649

Then, similarly to before, from the uniqueness of the decomposition of the Snell

650

envelope, we have that the martingale and F V terms in (4.16) and (4.17) coincide.

651

Hence, fort∈[0, T],

652

Z s(r)˜

˜ s(l)

e−αtφ(Xt) ˆLz_{t}ν(dz) =−

Z t

0

e−αs{L −α}V(Xs)ds a.s.

653

Using the same arguments as in the proof of Theorem 4.4 we can show that both

654

statements of this theorem hold. The details are left to the reader.

655

Acknowledgments. We are grateful to two anonymous referees and Prof. Goran

656

Peskir for useful comments and suggestions.

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Appendix A. . 753

Lemma A.1. For each0≤t≤T, the family of random variables {E[Gτ|Ft] :τ ∈

754

Tt,T}is directed upwards, i.e. for any σ1,σ2∈ Tt,T, there existsσ3∈ Tt,T, such that

755

E[Gσ1|Ft]∨E[Gσ1|Ft]≤E[Gσ3|Ft], a.s.

756

Proof. Fix t ∈ [0, T]. Suppose σ1, σ2 ∈ Tt,T and define A := {E[Gσ1|Ft] ≥

757

E[Gσ2|Ft]}. Letσ3:=σ11A+σ21Ac. Note thatσ3∈ T_{t,T}. UsingF_{t}-measurability of
758

A, we have

759

E[Gσ3|Ft] =1AE[Gσ1|Ft] +1AcE[Gσ2|Ft]

760

=_{E}[Gσ1|Ft]∨E[Gσ2|Ft] a.s.,

761 762

which proves the claim.

763

Lemma A.2. Let G ∈ G¯ and S be its Snell envelope with decomposition S = 764

M∗−A. For 0≤t≤T and >0, define

765

(A.1) K_{t}= inf{s≥t:Gs≥Ss−}.

766

ThenAK

t =At a.s. and the processes(AKt)andA are indistinguishable. 767

Proof. From the directed upwards property (Lemma A.1) we know thatE[St] = 768

sup_{τ∈T}_{t,T}E[Gτ]. Then for a sequence (τn)n∈N of stopping times in Tt,T, such that