Stopping of functionals with discontinuity at the boundary of an open set

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Palczewski, J and Stettner, L (2011) Stopping of functionals with discontinuity at the

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STOPPING OF FUNCTIONALS WITH DISCONTINUITY AT THE BOUNDARY OF AN OPEN SET

JAN PALCZEWSKI AND ŁUKASZ STETTNER

A. We explore properties of the value function and existence of optimal stopping times for functionals

with discontinuities related to the boundary of an open (possibly unbounded) setO. The stopping horizon is either random, equal to the first exit from the setO, or fixed: finite or infinite. The payofffunction is continuous with a possible jump at the boundary ofO. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal orε-optimal stopping times.

Keywords:optimal stopping, Feller Markov process, discontinuous functional, penalty method

1. I

The problem of optimal stopping of Markov processes has received continuous attention for last fourty years and produced diverse approaches for its solution. Foundations and general existence results can be found, e.g., in Bismut and Skalli [4], El Karoui [8], El Karoui et al. [9], Fakeev [10], and Mertens [16]. From 1980s functional analytic methods gave way to a more explicit approach initiated by Bensoussan and Lions [3]: value function was characterized as a solution to a variational inequality, which could be solved analytically or numerically. The main limitation of this method is the requirement of a particular differential form of the generator of the underlying Markov process. This paper belongs to another strand of literature which initially aimed at studying smoothness of the value function but also provides a different approach for the numerical approximation to the value function for a more general class of Markov pro-cesses (see Zabczyk [22] for a survey). These methods are not constrained by the form of generators and the development of the theory of PDEs. Specifically, we build on the penalty method introduced by Robin [19] and generalized by Stettner and Zabczyk [20] (see also [21]), which originates in ideas developed for partial differential equations but follows a purely probabilistic route. Of interest to numerical methods discussed in this paper is also a time-discretization technique explored by Mackevicius [14] and further applied by Kushner and Dupuis [12] for numerical algorithms; see also Palczewski and Stettner [18] for its application to stopping of time-discontinuous functionals.

We assume that the state of the world is described by a standard Markov process X(t)_{defined on a}

locally compact separable space E endowed with a metricρwith respect to which every closed ball is compact (see the Appendix for the definition and properties of standard Markov processes). The Borel

σ-algebra onEis denoted by_E. The process X(t)

satisfies the weak Feller property:

PtC0 ⊆ C0,

whereC0is the space of continuous bounded functionsE→Rvanishing in infinity, andPtis the transition

semigroup of the process X(t)

, i.e.,Pth(x)=Exh X(t) for any bounded measurableh:E→R.

Let_{O ⊂}Ebe an open set andτ_O=inf_{t:X(t)<_O}– the first exit time from_O. We study maximization of several classes of functionals:

(1) Stopping is allowed up to timeτ_O. The payoffis described by a functionGbeforeτ_O and by a functionHatτ_O:

(1)

J(s,x, τ)=Exn

Z τ_∧τ_O

0

e−αuf s+u,X(u)_du

+1_{τ<τ_O}e−ατG s+τ,X(τ)+1_{τ_≥τ_O}e−ατOH s+τ_O,X(τ_O)

o

,

Date: Date: 2011-04-28 13:26:31 .

Research of both authors supported in part by MNiSzW Grant no. NN 201 371836.

where (s,x)∈ [0,_∞)×E,α > 0,τ _≥0 and f,G,H : [0,_∞)×E _→ R_{are continuous bounded}

functions.

(2) Stopping is allowed up to timeτ_Oand the payoffis given by a functionF: [0,_∞)_×E_→R_which

is continuous on [0,_∞)_{× O}and possibly discontinuous in the space variable on [0,_∞)_{× O}c_:

(2) J(s,x, τ)=Exn

Z τ_∧τO

0

e−αuf s+u,X(u)_du+e−α(τ_∧τO)_{F s+(τ∧τ}

O),X(τ∧τO) o

.

This, in particular, covers a complementary problem to (1): withFcontinuous on [0,_∞)_×O¯and on [0,_∞)_×O¯c_{with a possible jump at the boundary [0,}

∞)_×∂_O.

(3) Stopping is unconstrained (infinite horizon,T =_∞) or constrained by a constantT (finite horizon) with the following functional:

(3) J(s,x, τ)=Exn

Z τ_∧(T₋s)

0

e−αuf s+u,X(u)_du+e−α(τ_∧T)_F

(s+τ)_∧T,X(τ_∧(T₋s))o_,

where the payofffunctionFis continuous apart from a possible discontinuity on [0,_∞)_×∂_O. Optimal stopping problems of the first type were studied by Bensoussan and Lions [3] for non-degenerate diffusion processes under assumptions thatG _≤Hand_Ois bounded with a smooth boundary∂_O. They used penalization techniques similar to ours but applied them on the level of variational inequalities. Gen-eralizations were attempted by many authors in two directions: to extend the class of processes for which this approach applies and to relax assumptions on the functional; see, e.g., Menaldi [15] for the removal of restrictions on degeneracy of the diffusion, and Fleming, Soner [7] for relaxation of many assumptions regarding the functional and the coefficients of the diffusion via viscosity solutions approach. Functionals of the third type recently gained a lot of attention. Lamberton [13] obtained continuity and variational characterization of the value function for stopping of one-dimensional diffusions with bounded and Borel-measurable payofffunction F. His result, however, cannot be extended to multidimensional diffusions. Bassan and Ceci studied stopping of semi-continuous payofffunctionsFfor diffusions and certain jump-diffusions in one dimension ([1, 2]). They proved that value function for a functional with lower/upper semi-continuous functionFis lower/upper semi-continuous. The existence of optimal stopping times was also shown but without an explicit construction.

This paper complements existing theory in two aspects. Firstly, it provides results for a far larger family of Markov processes (in particular, in dimensions higher than 1) and enables numerical treatment of the value function. Secondly, it relaxes constraints on the region_O, which can be unbounded and with non-smooth boundary. Our main assumption is that the mapping x_7→ Ex_{1

{τ_O<t_}h(Xt)}is continuous for any

t >0 and a continuous bounded functionh. This assumption is non-restrictive as we show in Section 5. It is usually satisfied by solutions to nondegenerate stochastic differential equations driven by Brownian or Levy noise. Consequently our results, based on probabilistic arguments, provide regularity of solutions to differential or integrodifferential variational inequalities, related to appropriate stopping problems, with various types of discontinuity.

In our approach, the value function is approximated by a sequence of penalized value functions which are unique fixed points of contraction operators. These operators do not involve stopping or any other type of control, which makes them easier to compute numerically. Moreover, a discrete approximation of the state space can be used because we prove that the penalized functions are continuous.

2. P 

We solve the stopping problem (1) using the penalty method introduced by Robin [19] and generalized by Stettner and Zabczyk [20]. Forβ >0 consider a penalized equation

(4)

wβ(s,x)=Exn

Z τO

0

e−αuhf s+u,X(u)_{+βG s+u,X(u)}

−wβ s+u,X(u)+i_du

+e−ατO_{H s+τ}

O,X(τO) o

.

LEMMA 2.1. Assume that g and h are bounded functions andα > 0. For any bounded progressively measurable process(b(t)), the following formulae

z(s,x)=Exn

Z τO

0

e−αug(s+u,x(u))du+e−ατO_h(s+τ

O,x(τO)) o

,

(5)

z(s,x)=Exn

Z τO

0 e−αu−

Ru

0b(t)dtg(s+u,x(u))+b(u)z(s+u,x(u))du

+e−ατO−

Rτ_O

0 b(t)dth(s+τ

O,x(τO)) o

(6)

are equivalent in the following sense: z defined in(5)is a solution to(6); and any solution to(6)is of the form(5).

Proof. We use similar arguments as in Lemma 1 of [21]. The only difference is that now we haveτ_O

instead of the deterministic timeT ₋s.

Using this lemma, in a similar way as in Proposition 1 of [21], we show

LEMMA 2.2. There is exactly one bounded measurable function wβthat satisfies(4). Proof. By Lemma 2.1 the penalized functionwβ_{can be equivalently written as}

(7)

wβ(s,x)=Exn

Z τO

0

e−(α+β)uhf s+u,X(u)

+βG s+u,X(u)

−wβ s+u,X(u)+_+βwβ _s+u,X

(u)i_du

+e−(α+β)τO_{H s+τ}

O,X(τO) o

.

Hence,wβis a fixed point of the operatorT defined for measurable bounded functionsφas follows: Tφ(s,x)=Exn

Z τO

0

e−(α+β)uhf s+u,X(u)

+βG s+u,X(u)

−φ s+u,X(u)+_{+βφ s+u,X(u)}i_du

+e−(α+β)τO_{H s+τ}

O,X(τO) o

.

This operator is a contraction on the space of measurable bounded functions for anyβ >0. Indeed,_Tφ

is identically equal toHon [0,_∞)× Oc_{, whereas the contraction property on [0,∞)× Ofollows from the}

estimate

Tφ1− Tφ2≤

β

α+βkφ1−φ2k∞.

This implies thatwβ_{is a unique fixed point of}

T.

We make the following assumption

(A1) The stopped semigroupPτO

t h(x)=E x

{1_{t<τ_O}h(X(t))}maps the space of continuous bounded

func-tions into itself.

The following three lemmas prove continuity results which, in particular, will be used to show thatwβ _is

LEMMA 2.3. Under (A1), for a continuous bounded function h: [0,_∞)×E_→(−∞,_∞)the mapping

(s,x)_7→PτO

t h(s,x) :=E x

{1_{t<τO}h(s+t,X(t))}

is continuous.

Proof. Let (sn,xn)→(s,x). By Proposition A.1 for a givenε >0 there is a compact setK⊂Esuch that

Pxn

∃u_∈[0,s+t+1]X(u)<K ≤ε.

Fornlarge enough, i.e., such that|s₋sn| ≤1, we have

P

τO

t h(sn,xn)−PτtOh(s,x)

≤

Exn_{1

{t<τ_O}h(sn+t,X(t))} −Exn{1{t<τ_O}h(s+t,X(t))}

+

Exn_{1

{t<τ_O}h(s+t,X(t))} −Ex{1_{t<τ_O}h(s+t,X(t))}

≤ε_kh_k+

Exn_{1

{t<τ_O}1_{X(t)_∈K_}h(sn+t,X(t))−h(s+t,X(t))}

+

Exn_{1

{t<τ_O}h(s+t,X(t))} −Ex{1_{t<τ_O}h(s+t,X(t))}

=ε_kh_k+an+bn.

The sequenceanconverges to 0 by uniform continuity ofhis on [0,s+t+1]×K. Assumption (A1) implies

bn →0, which completes the proof.

LEMMA 2.4. Under assumption (A1) the mapping

(s,x)_7→Exn

Z τO

0

e−γuh(s+u,X(u))duo is continuous for any function h_{∈ C}([0,_∞)_×E)andγ >0.

Proof. Fubini’s theorem implies

Exn

Z τO

0

e−γuh(s+u,X(u))duo=

Z ∞

0

e−γuϕ(s,u,x)du,

whereϕ(s,u,x) =Ex_{1

{u<τ_O}h(s+u,X(u)}. This function is continuous in (s,x) for any fixedu ≥0 by

Lemma 2.3. Dominated convergence theorem concludes. LEMMA 2.5. Under (A1) forα > 0and a continuous bounded function h: [0,_∞)_×E _7→ (_−∞,_∞)the mapping

(s,x)7→Exn_e−ατO_h(s+τ

O,X(τO)) o

is continuous.

Proof. Assume first that

(8) h(s,x)=Exn

Z ∞

0

e−αuh s˜ +u,X(u)_duo

for a continuous bounded function ˜h. Using this decomposition we write

H(s,x)=Exn_e−ατO_h(s+τ

O,X(τO)) o

=Exn

Z ∞

τO

e−αuh s˜ +u,X(u)_duo

=h(s,x)₋Exn

Z τO

0

e−αuh s˜ +u,X(u)_duo_.

Hence,His continuous by Lemma 2.4.

By the weak Feller property of (X(t)) functions of the form (8) are dense inC0([0,∞)×E) (see Lemma

3.1.6 in [6]). Hence, H is continuous forh inC0. The extension of this result to continuous bounded

functionshuses Proposition A.1 in the appendix. Fix a compact setK _⊆EandS _≥0. For anyT, ε >0 there is a compact setL_⊆Esuch that

Px _X(t)<Lfor somet_∈[0,T]_{< ε,}

Definer(s,x)=e−ρ(x,L)−(s−(S+T))+h(s,x), whereρ(x,L) denotes the distance ofxfrom the setL. Suchris in

C0([0,∞)×E) and by preceding resultsR(s,x) =Ex

n

e−ατO_r(s+τ

O,X(τO)) o

is continuous. By definition

r_≡hon [0,S +T]×L. LetA=_{X(t)<Lfor somet_∈[0,T]}. The distance ofRandHis bounded in the following way:

kH(s,x)−R(s,x)k=Ex_e−ατO_(h−r)(s+τ

O,X(τO))

=Ex_e−ατ_O∧T

1_{Ac_}(h−r)(s+τ_O∧T,X(τ_O∧T))

+Ex

1_{A_}e−ατO(h−r)(s+τ_O,X(τ_O))

+Exn₁

{Ac_}

e−ατO_(h−r)(s+τ

O,X(τO))

−e−ατO∧T(h₋r)(s+τ_O∧T,X(τ_O∧T))o ≤0+_kh₋r_kε+e−αT2_kh₋r_k.

This implies_kH(s,x)₋R(s,x)_{k ≤}2_kh_k(ε+2e−αT_{) for (s,x)}

∈[0,S]_×K. SinceT andεare arbitrary this implies continuity ofHon [0,S]_×K. Hence,His continuous on its whole domain by the arbitrariness of

S,K.

COROLLARY 2.6. Under (A1), the unique bounded solution wβ_{of (4)is continuous.}

Proof. Lemmas 2.4 and 2.5 imply that the operator_T introduced in the proof of Lemma 2.2 maps the space of continuous bounded functions into itself. Since this operator is a contraction on the space of bounded measurable functions it is a contraction on the space of continous bounded functions. This implies thatwβ

as a unique fixed point is continuous.

To establish convergence ofwβ_towasβ

→ ∞we introduce two additional representations ofwβ_.

LEMMA 2.7. Under assumption (A1), the function wβ_{has the following equivalent representation:}

(9) wβ(s,x)=sup

τ

n

J(s,x, τ)₋Ex

1_{τ<τ_O}e−ατ G−wβ+ s+τ,X(τ)

o

.

Proof. Markov property implies that for any stopping timeσthe following equality is satisfied:

wβ(s,x)=Exn

Z τO∧σ

0

e−αuhf s+u,X(u)_{+βG s+u,X}

(u)

−wβ s+u,X(u)+i_du

+e−α(τO∧σ)wβ(s+τ_O∧σ,X(τ_O∧σ))o.

This gives the lower bound: (10) wβ(s,x) ≥ Exn

Z τO∧σ

0

e−αuf s + u,X(u)_{du + e}−α(τO∧σ)_wβ

(s + τ_{O ∧} σ,X(τ_{O ∧} σ))o.

Further,

(11)

wβ(s,x)_≥Exn

Z τ_O∧σ

0

e−αuf s+u,X(u)_du

+1_{σ<τ_O}e−ασG−(G−wβ)+ s+σ,X(σ)+1_{σ_≥τ_O}e−ατOH(s+τ_O,X(τ_O))

o

,

becausewβ_=Hon

Oc_andG

−(G₋wβ₎+

≤wβ_{. Define a stopping time}

σ∗=inf_{u_≥0 :wβ(s+u,X(u))_≤G(s+u,X(u))_}.

Due to the continuity ofGandwβ(see Corollary 2.6) we have

1_{σ∗<τ_O}wβ(s+σ∗,X(σ∗))≤1_{σ∗<τ_O}G(s+σ∗,X(σ∗)).

This implies that forσ∗_{the inequalities in (10) and (11) become equalities and (9) follows easily.} LEMMA 2.8. The function wβ_{has the following equivalent representation:}

(12)

wβ(s,x)=sup

b_∈Mβ

Exn

Z τO

0 e−αu−

Ru

0b(t)dthf s+u,X(u)+b(u)G s+u,X(u)idu

+e−ατO−R0τOb(t)dtH s+τ_O,X(τ_O)

o

where Mβis the class of progressively measurable processes with values in[0, β]. Proof. By Lemma 2.1 the functionwβ_{has the following equivalent formulation:}

wβ(s,x)=Exn

Z τO

0 e−αu−

Ru

0b(t)dt

h

f s+u,X(u)

+βG s+u,X(u)

−wβ s+u,X(u)+_+b

(u)wβ(s+u,x(u))idu

+e−ατO−

Rτ_O

0 b(t)dth(s+τ

O,x(τO)) o

for any progressively measurable processb(t) with values in [0, β]. Sinceb(t)≤βwe have

βG s+u,X(u)

−wβ s+u,X(u)+_+b

(u)wβ(s+u,X(u))≥b(u)G s+u,X(u)_,

which implies

wβ(s,x)_≥Exn

Z τO

0 e−αu−

Ru

0b(t)dthf s+u,X(u)+b(u)G s+u,X(u)idu

+e−ατO−R0τOb(t)dth(s+τ_O,X(τ_O))

o

.

This is an equality forb(t) given by

b(t)=

      

β, G s+t,X(t)

≥wβ _s+t,X(t),

0, otherwise.

Hence, formula (12) is proved.

PROPOSITION 2.9. Under (A1), the functions wβ(s,x)increase pointwise to w(s,x)asβ_{→ ∞}.

Proof. Equation (12) implies that the functionswβ_{(s,x) are increasing inβ. Hence the limitw∞(s,x) =}

limβ→∞wβ(s,x) exists. By (9) we havewβ ≤wand, therefore,w∞ ≤w. To prove thatw∞ = wwe first

show thatw∞_{≥G. Letx∈ Oand, forη >0, putb}η_(u)=1

{u≤η}β. Then by (12) we have

wβ(s,x)≥Exn

Z τO

0 e−αu−

Ru

0b

η_(t)dt

f s+u,X(u)_du

+

Z τ_O∧η

0

e−(α+β)uβG s+u,X(u)_du+e−ατ_O−RτO

0 b

η_(t)dt

H τ_O,X(τ_O)o

=Ex

(I)+(II)+(III).

Lettingβ_{→ ∞}we can make (I) and (III) arbitrarily small and for sufficiently smallηand largeβthe term (II) is arbitrarily close toG(s,x). Dominated Convergence Theorem impliesw∞(s,x)≥G(s,x).

From (9) for any stopping timeτwe have

wβ(s,x)_≥J(s,x, τ)₋Ex

1_{τ<τ_O}e−ατ G−wβ+ s+τ,X(τ) .

By lettingβ_{→ ∞}we obtain

(13) w∞(s,x)_≥J(s,x, τ),

because limβ_→∞(G−wβ)+(s,x)=0 forx∈ O. Sinceτis arbitrary we conclude thatw∞(s,x)=w(s,x) for x_{∈ O}. Forx_∈E_{\ O}we havewβ_{(s,x)=H(s,x)=w(s,x).}

COROLLARY 2.10. Under (A1), the value function w is lowersemicontinuous. Moreover, if w is contin-uous then wβ_{approaches w uniformly on compact sets.}

3. P    w

In this section we explore the properties of the value function, in particular, its behaviour on the boundary of_O.

THEOREM 3.1. Under (A1), for x_∈∂_Owe have

(14) lim

y_→x,y_∈Ow(s,y)=G∨H(s,x).

The proof of this theorem consists of several steps which are of interest on their own. They are formu-lated and proved as separate results below.

It is clear thatw _≥ G on_Oandw = H on the complement of_O. It is therefore natural to expect a discontinuity at the boundary of_OifG > H. The following proposition shows that this discontinuity is constrained to the minimum: the absolute value of the difference betweenGandH.

PROPOSITION 3.2. Assume (A1) and G_≥H. For any x_∈∂_Owe have

(15) lim

y_→x,y_∈Ow(s,y)=G(s,x)

and the convergence is uniform in s and x from compact sets.

Proof. Since w(s,x) ≥ G(s,x) for x _{∈ O} and G is continuous we obtain that lim infy_→x,y_∈Ow(s,y)

≥ G(s,x). In the remaining part of the proof we show that lim supy_→x,y_∈Ow(s,y) ≤ G(s,x), which

im-plies that the limit in (15) exists and equalsG(s,x).

Fix a compact setK _⊆E,T >0 andε >0. First we make preparatory steps. By Proposition A.1 in the Appendix, there is a compact setL_⊆Esuch that

(16) sup

x_∈K

Px _∃t∈[0,T+1]X(t)<L

≤ε.

The extension of the time interval by one unit to [0,T +1] is required to allow the initial timesto be in [0,T] and leave time for the process (X(t)) to evolve. Notice that belowδandηare both bounded by 1.

Letδ_∈(0,1) be such that for (s,x)∈[0,T]×B(L, δ),y_∈L,kx₋y_{k ≤}δandt_∈[0, δ] (17) |G(s,x)−G(s+t,y)| ≤ε,

Proposition A.3 implies that there isη >0, which, for convenience, is bounded byδ_∧ε, such that

(18) sup

x_∈L

sup

t≤η

Px X(t)<B(x, δ)

≤ε.

Fixx_∈∂_{O ∩}Kands_∈[0,T]. For anyy_{∈ O ∩}Kwe have

w(s,y)=sup

τ

J(s,y, τ) ≤sup

τ

Eyn

Z τ∧τO

0

e−αuf(s+u,X(u))su+e−α(τ∧τO)_G(s+τ∧τ

O,X(τ∧τO)) o

≤Py_(τ

O > η) k

f_k

α +kGk

_+Py

(τ_O≤η)η_kf_k+G(s,y)

+sup

τ

Ey₁ {τO≤η}

e−

α(τ_∧τ_O)_G

(s+τ_∧τ_O,X(τ_∧τ_O))₋G(s,y) .

Consider the last term. For any stopping timeτwe have

Ey₁ {τ_O≤η_}

e−

α(τ_∧τ_O)

G(s+τ_∧τ_O,X(τ_∧τ_O))₋G(s,x)

≤ kG_k(1−e−αη)+Ey

G(s+τ∧τO∧η,X(τ∧τO∧η))−G(s,x)

≤αη_kG_k+Ey

G(s+τ∧τO∧η,X(τ∧τO∧η))−G(s+η,X(η))

+Ey

G(s+η,X(η))−G(s,y)

The first term is bounded byαε_kG_k. The estimate of the second term requires conditioning onX(τ_∧τ_O∧η), the use of the strong Markov property and inequalities (16), (18):

Ey

G(s+τ∧τO∧η,X(τ∧τO∧η))−G(s+η,X(η))

=EynEX(τ∧τO∧η)n

G(s+τ∧τO∧η,X(0))−G(s+η,X(η−τ∧τO∧η)) oo

≤2_kG_kPy_{{∃s∈[0, η]X(s)}<L_}

+2kG_kPy

∀s_∈[0, η]X(s)∈L and X(η)<B X(τ_∧τ_O∧η), δ +ε

≤2_kG_kε+2_kG_kε+ε=ε(1+4_kG_k).

Term (III) is estimated similarly knowing thatyis inLby assumption: (III)_≤ε(1+2_kG_k). Combining these estimates we obtain

w(s,y)_≤hη(y) kfk

α +kGk

₊

(1₋hη(y))η_kf_k+G(s,y)_+ε

2+(6+α)_kG_k

≤G(s,y)+hη(y) kfk

α +2kGk

_+η

kf_k+ε2+(6+α)kG_k,

wherehη(y)=Py

{τ_O> η). Assumption (A1) implies thathηis continuous onE. Clearly,hη(x)=0. Hence,

lim sup

y_→x,y_∈O

w(s,y)_≤G(s,x)+η_kf_k+ε 2+(6+α)_kG_k

and the limit in the right-hand side is uniform in (s,x)∈[0,T]×(∂_{O ∩}K). Sinceε > 0 is arbitrary and

η < εthis implies (15).

COROLLARY 3.3. Under (A1), for any x_{∈ O}we have

(19) lim sup

y_→x,y_∈O

w(s,y)_≤G_∨H(s,x)

Proof. Notice that

w(s,y) ≤ sup

τ

Eyn

Z τ∧τO

0

e−αuf(s + u,X(u))du + e−α(τ∧τO)_{G ∨ H(s + τ ∧ τ}

O,X(τ ∧ τO)) o

and then continue as in the proof of Proposition 3.2 replacingGwithG_∨H.

The following proposition explores the impact of the value of the functional on the complement ofOon the value function close to the boundary ofO.

PROPOSITION 3.4. Assume (A1). For eachε >0, T >0and a compact set K _⊆E there is a compact set Kε_{⊂ O}such that for x_∈K_\Kε, s_∈[0,T]andβ >0we have

(20) wβ(s,x)≥H(s,x)−ε.

Proof. Fixε′>0 and chooseη >0 such that (16)-(18) hold for the functionH. By the definition ofwβwe have

wβ(s,x)_{≥ −k}f_kη₋kfk

α P

x

{τ_O> η_}+Ex_e−ατO_H(s+τ

O,X(τO)) .

Splitting the last term depending on whetherτ_Ois greater or smaller thanηand doing analogous estimates as in the proof of Proposition 3.2 we obtain the following lower bound

wβ(s,x)≥(1−hη(x))H(s,x)−hη(x) kfk

α +kHk

−ε′(2+6kH_k+_kf_k).

By arbitrariness ofε′>0 and continuity ofhηwe can chooseKεsuch that (20) is satisfied.

According to Proposition 3.4, Assumption (A1) guarantees the ”migration” ofHinto_O, i.e., the function

Hprovides a lower bound forwβ_{whenxapproaches∂}

O. Aswβ_{is the lower bound forw(see Proposition}

Proof of Theorem 3.1. From (20), letting firstβ_{→ ∞}and thenε_→0 we obtain lim inf

y_→x,y_∈Ow(s,x)≥H(s,x).

SinceGis continuous andw(s,y)_≥G(s,y) on_Othis extends to lim inf

y→x,y∈Ow(s,x)≥G∨H(s,x).

Corollary 3.3 and the above inequality imply that the limit in (14) exists and equalsG_∨H.

4. C w     

LetDdenote the set of functionsϕ(s,x) admitting the following decomposition: (21) ϕ(s,x)=Exn

Z τO

0

e−αuϕ1(s+u,X(u))du+e−ατOϕ2(s+τ_O,X(τ_O))

o

forϕ1, ϕ2∈C0([0,∞)×E).

LEMMA 4.1. The set_Dis a dense subset of_C0([0,∞)×E).

Proof. It follows immediately from the proof of Lemma 4 in [21].

LEMMA 4.2. If G has decomposition(21)withϕ1=g1, ϕ2=g2∈ C0([0,∞)×E)then wβ(s,x)−G(s,x)≥ −kf_α−₊g1_βk−Ex_e−(α+β)τO_kH−g2k.

If, moreover, G_≤H then

wβ(s,x)−G(s,x)≥ −kf_α−₊g1_βk.

Proof. Define ¯wβ(s,x)=wβ(s,x)−G(s,x). Decomposition (21) ofGand representation (4) ofwβimply ¯

wβ(s,x)=Exn

Z τO

0

e−αu_f

−g1+β( ¯wβ)− _s+u,X

(u)_du+e−ατO_{(H−g2) s+τ}

O,X(τO) o

.

By Lemma 2.1 we have the following equivalent form of the above equation ¯

wβ(s,x)=Exn

Z τO

0

e−(α+β)u_f

−g1+β( ¯wβ)−+βw¯β s+u,X(u)du

+e−(α+β)τO_{(H−g2) s+τ}

O,X(τO) o

.

Since ( ¯wβ)−+w¯β _≥0, we obtain ¯

wβ(s,x)_≥Exn

Z τO

0

e−(α+β)u(f₋g1) s+u,X(u)_du+e−(α+β)τO_{(H−g2) s+τ}

O,X(τO) o

.

This implies the first statement of the lemma.

Due to decomposition (21) we haveg2(s,x) =G(s,x) forx< _O. Together with the conditionG _≤H

this yields (H₋g2) s+τ_O,X(τ_O)

≥0.

THEOREM 4.3. Assume (A1) and G _≤ H. The value function w is continuous on E and an optimal

stopping moment is given by

(22) τ∗(s)=inf{t_≥0 :w(s+t,X(t))≤G(s+t,X(t)) or X(t)<_O}.

Proof. Functions wβ are continuous (by Lemma 2.2), increasing in βand dominated byw. Therefore, it suffices to estimate the differencew₋wβ. For functionsGwith decomposition (21), Lemma 4.2 and equation (9) give

wβ(s,x)_≥w(s,x)₋kf−g1k

α+β .

SinceDis dense inC0([0,∞)×E) (Lemma 4.1) we also obtain the continuity ofwforG∈ C0([0,∞)×E).

The extension of this result to continuous boundedG uses Proposition A.1 in the appendix. Fix a compact setK_⊆EandS _≥0. For anyT, ε >0 there is a compact setL_⊆Esuch that

Px _X(t)<Lfor somet_∈[0,T]_{< ε, x}

Define ˜G(s,x) = e−ρ(x,L)−(s−(S+T))+G(s,x), where ρ(x,L) denotes the distance ofxfrom the setL. Let ˜w

be the value function corresponding to ˜G. Since ˜G _{∈ C}0([0,∞)×E), preceding results imply that ˜wis

continuous. We also have_kw(s,x)₋w˜(s,x)_{k ≤}(e−αT _+ε)

kf_k/α+_kG_k+_kH_k) forx_∈ Kands_∈[0,S]. SinceTandεare arbitrary this implies continuity ofwon [0,S]_×K. By the arbitrariness ofS,Kthe value functionwis continuous on its whole domain.

Define forε >0

(23) τε(s)=inf{t_≥0 :w(s+t,X(t))≤G(s+t,X(t))+ε or X(t)<_O}

and

(24) τβ(s)=inf{t≥0 :wβ(s+t,X(t))≤G(s+t,X(t)) or X(t)<O}.

Fixδ >0 andT >0. By Proposition A.1 for a givenx_∈Ethere is a compact setKδsuch thatPx

{Aδ_{} ≥}1₋δ, whereAδ=_{X(t)_∈Kδ_∀t_∈[0,T]_}. From (4), due to the Markov property of (X(t)), we obtain

wβ(s,x)≤hPx_{Ac

δ}+e− αT_Px

{Aδandτβ(s)∨τε(s)>T}

i

kG_k+_kH_k+kfk α

+Exn₁

{Aδ}

Z σ∗_ε,β,T(s)

0

e−αu_{f +β}

(G₋wβ)+ _s+u,X

(u)_du

+1_{Aδ}e−

ασ∗

ε,β,T(s)_wβ _s+σ∗

ε,β,T(s),X(σ∗ε,β,T(s)) o_,

whereσ∗

ε,β,T(s)=τβ(s)∧τε(s)∧τO∧T. Notice that by the uniform convergence on compact subsets of

wβ_{towwe haveP}x_τ

β(s)∧T < τε(s)∧T andAδ →0 asβ→ ∞. Therefore lettingβ→ ∞we obtain by

Dominated Convergence Theorem

w(s,x)≤ δ+e−αT

kG_k+_kH_k+kfk α

+Exn_1{Aδ}

Z τε_(s) ∧τ_O∧T

0

e−αuf s+u,X(u)_du

+1_{Aδ}e−

ατε_(s)

∧τ_O∧T_{w s+τ}ε

(s)∧τ_O∧T,X(τε(s)∧τ_O∧T)o_.

Proposition A.1 implies that (Aδ) form an increasing sequence of subsets whenδ_→0 and limδ_→0Px{Aδ}=

1. Therefore, lettingδ_→0 we get

w(s,x)_≤e−αT _kG_k+_kH_k+kfk α

+Exn

Z τε_(s) ∧τ_O∧T

0

e−αuf s+u,X(u)_du

+e−ατε(s)∧τO∧Tw s+τε(s)∧τ_O∧T,X(τε(s)∧τ_O∧T)o_.

Now taking the limitT _{→ ∞}yields (25) w(s,x)_≤Exn

Z τε_(s) ∧τO

0

e−αuf s+u,X(u)_du+e−ατε_(s)

∧τO_{w s+τ}ε_(s)∧τ

O,X(τε(s)∧τO) o

.

Note thatτε(s)→τ∗(s), asε_→0, and, further, by quasi-leftcontinuity of the process (X(t)) (see, e.g., [6]) we also haveX(τε

(s))→X(τ∗_(s)),Px_{-a.s.. Consequently letting}ε_→0 in (25) and using the continuity of

wgive

w(s,x)_≤Exn

Z τ∗_(s)∧τ O

0

e−αuf s+u,X(u)_du+e−ατ∗_(s)∧τ

O_{w s+τ}∗_(s)∧τ

O,X(τ∗(s)∧τO) o

.

This implies that there is an optimal stopping time dominating τ∗(s). The optimality of τ∗(s) is now

obvious.

To prove the continuity of the value functionwwithout the requirement of an upward jump we introduce the following assumptions:

(A2) limη_→0Px{τ_O < η}=0 uniformly inxfrom compact subsets ofO.

(A3) (X(t)) is strongly Feller, i.e., the mappingx_7→Ex_{{h(X(t))}is continuous for any measurable bounded}

Before we formulate Theorem 4.8 we prove three auxiliary results. Lemma 4.4 shows that under (A3) the time-state process semigroup maps time-continuous bounded functions into functions continuous in both parameters. Lemma 4.5 states that the weak Feller continuity of the processX(t) is sufficient for the continuity of the value functionwin the time parameters. Lemma 4.6 shows that (A2) follows from (A1).

LEMMA 4.4. Under assumption (A3), the mapping

(s,x)_7→Ex_{F(s+h,X(h))_}

is continuous for h >0and a bounded measurable function F, provided that the mapping s_7→ F(s,x)is continuous uniformly in x in compact subsets of E.

Proof. Fix a compact setK _⊆EandT, ε >0. By Proposition A.1 there is a compact setL_⊆Esuch that

supx∈KPx{X(h)<L}< ε. Hence for (s,x)∈[0,T]×Kwe have

E

x_{F s+h,X}

(h) −Φ(s,x)

<kFkε,

whereΦ(s,x)=Ex

1_{X(h)_∈L_}F s+h,X(h) . Let (sn,xn)→ (s,x) such that (sn,xn)∈[0,T]×Kfor alln.

By the continuity ofFinsand by assumption (A3), for sufficiently largek, we have lim

n_→∞

Φ(sn,xn) − Φ(s,x)

≤ _nlim →∞

Φ(sk,xn) − Φ(s,xn) + _nlim

→∞

Φ(s,xn) − Φ(s,x)

= ε + 0.

By the arbitrariness ofεthis completes the proof. LEMMA 4.5. The mapping s_7→w(s,x)is continuous uniformly in x in compact subsets of E.

Proof. Assume thatsn→ sand fix a compact setK⊆E. Since functions f,GandHare bounded and the

discount rateα >0, for anyε >0 there isT >0 such that_|J(sn,x, τ)−J(sn,x, τ∧T)| ≤εfor allx∈Kand

n=1,2, . . .. By Proposition A.1 there is a compact setL_⊆Esuch that for allx_∈Kandτ_≤Twe have

Ex₁

{∃t∈[0,T]X(t)<L}

nZ τ∧τO

0

e−αuf s+u,X(u)_du

+1_{τ<τ_O}e−ατG s+τ,X(τ)+1_{τ_≥τ_O}e−ατOH s+τ_O,X(τ_O)

o

≤ε.

Uniform continuity of the functions f,G, andHon [0,T]×Lyields

Ex₁

{∀t∈[0,T]X(t)∈L}

nZ τ∧τO

0

e−αu_{f s+u,X}

(u)

−f sn+u,X(u)du

+1_{τ<τ_O}e−ατ G s+τ,X(τ)−G sn+τ,X(τ)

+1_{τ_≥τ_O}e−ατO H s+τ_O,X(τ_O)−H sn+τ_O,X(τ_O) o

≤ε

for τ _≤ T, a sufficiently largen andx _∈ K. Consequentlyw(sn,x) → w(s,x) asn → ∞uniformly in

x_∈K.

LEMMA 4.6. Assumption (A1) implies (A2).

Proof. By Lemma 2.4 the functiongγ(x) =Ex_{e−γτO_{}is continuous onOfor anyγ > 0. By dominated}

convergence theoremgγ(x) converges to 0 whenγ_{→ ∞}andx_{∈ O}. This convergence is monotone and, due to Dini’s theorem, uniform on compact subsets of_O. Chebyshev’s theorem yields

Px_{τ_O< η_}=Px_{e−τO/η_>e−1_{} ≤e g}

1/η(x).

The right-hand side converges to 0, whenη_→0, uniformly on compact subsets ofO, which completes the

proof.

REMARK4.7. Assumption (A2) holds if the process (X(t)) satisfies the following continuity condition:

(A2’) for anyε >0

lim

t_→0

Px

sup

s_∈[0,t]

ρ(x,X(s))_≥ε =0

Such assumption is satisfied for a wide variety of Markov processes which are solutions to the stochas-tic differential equations with Levy noise with bounded coefficients. To prove this we simply use the Doob’s maximal inequality to the martingale terms in the stochastic differential equation (see e.g. Theo-rem 1.3.8(iv) in [11]).

Let forh>0

(26) wh(s,x)=Ex nZ h

0

e−αuf s+u,X(u)_du+e−αh_w

(s+h,X(h))o.

THEOREM 4.8. Under (A2) and (A3), the function w is continuous on[0,_∞)× O. Assume additionally (A1). The penalized functions wβ are continuous and converge to w uniformly on compact subsets of

[0,_∞)× O. Anε-optimal stopping time is given by

τε(s)=inf{t_≥0 :w(s+t,X(t))≤G(s+t,X(t))+ε or X(t)<_O}.

Proof. By Lemmas 4.4 and 4.5 the functionwhis continuous in (s,x). Letτh_O=inf{t≥h:X(t)<O}and

Jh(s,x, τ)=Exn

Z τ∧τh O

0

e−αuf s+u,X(u)_du

+1_{τ<τh O}e

−ατ_{G s+τ,X}

(τ)₊

1_{τ_≥τh O}e

−ατh

OH s+τ_O,X(τh O)

o_.

By Theorem 3b of [16] applied to the Markov process consisting of a pair (s+t,X(t)) we have

wh(s,x)=sup

τ≥h

Jh(s,x, τ).

Consider an auxiliary value function ˜

wh(s,x)=sup

τ_≥h

J(s,x, τ).

We have the following inequalities

(27) _|wh(s,x)−w˜h(s,x)| ≤CPxτO<h

and

(28) 0_≤w(s,x)₋w˜h(s,x)≤sup

τ

_J

(s,x, τ)₋J(s,x, τh) =:Ih(s,x),

whereτh =τ∨handC >0. Assumption (A2) implies the difference|wh−w˜h|converges to 0 ash→ 0

uniformly on compact subsets of [0,_∞)× O. The proof of uniform convergence ofIhis more involved.

First notice

Ih(s,x)=sup

τ≤h _J

(s,x, τ)−J(s,x,h)

≤ kf_kh+2(_kG_k+_kH_k)Px_{τ_O <h_}+sup

τ_≤h

Ex_{G(s+τ,X(τ))_{} −}Ex_{G(s+h,X(h))_}.

It suffices to prove that ash_→0

(29) sup

τ_≤h

Ex_{G(s+τ,X(τ))_{} −}Ex_{G(s+h,X(h))_{} →}0 uniformly ins,xin compact subsets of [0,_∞)_{× O}. By Proposition A.2 we have

lim

h_→0

Ex_{G(s+h,X(h))_}=G(s,x)

uniformly ins,xin compact subsets. Letvh(s,x)=supτ_≤hEx{G(s+τ,X(τ))}. By weak Feller property this

function is continuous (see, e.g., [18, Corollary 3.6] or [22]). By dominated convergence theorem and the right-continuity of trajectories ofXwe have limh→0vh(s,x)=G(s,x). Since this convergence is monotone

and functionsvh andGare continuous Dini’s theorem implies thatvh tends toGuniformly on compact

sets. This completes the proof of (29). Consequently,wh(s,x) converges tow(s,x) ash→ 0 uniformly in

compact subsets of (s,x)_∈[0,_∞)_{× O}andwis continuous on [0,_∞)_{× O}.

Assume (A1). By Corollary 2.6 functionswβ_{are continuous. Dini’s Theorem and Proposition 2.9 imply}

their uniform convergence towon compact sets. In an identical way as in Theorem 4.3 we prove thatτε_(s)

Theorem 4.8 states the continuity ofwin [0,_∞)× O. It is also clear thatwis continuous on [0,_∞)× Oc

because on this setwcoincides with H. However, if there is a downward jump on the boundary of_O (G(s,x)>H(s,x) for somex_∈∂_O) the functionwhas a discontinuity in this point. This follows from the observation thatw _≥Gon the set [0,_∞)_{× O}andw=Hon [0,_∞)_{× O}c_{. Therefore, the statement of the}

above theorem cannot be strengthened. This also implies that an optimal stopping time might not exist as the following example shows.

EXAMPLE4.9. LetE=R_{andX(t) be a Brownian motion. TakeO=(−∞,1) andα < 1/2. It is easy to}

see that assumptions (A1)-(A3) are satisfied. PutG(s,x) =min(ex,e) andH(s,x)= f(s,x) =0. Notice that these functions do not depend ons, which implies that the value function is also time-independent. We shall, therefore, skipsin the notation.

LEMMA 4.10. In the setting of the example,

(1) for x<1and t_≥0we have

(30) l(t,x) :=Ex_e−αt₁

{X(t)<1}eX(t) =e(

1

2−α)t+xΦ1−_√x−t

t

,

whereΦis the standard normal cumulative distribution function,

(2) w(x)_≥l(t,x), for x<1and t_≥0,

(3) w(x)>G(x), for x<1.

Sketch of the proof. The formula (30) can be calculated directly using the normality ofX(t). To prove (2), define a sequence of stopping timesτn=inf{t:X(t)≥1−1/n}. Clearly,

w(x)≥Ex_e−α(τn∧t)_eX(τn∧t)

and

lim

n_→∞

Ex_e−α(τ_n∧t)_eX(τ_n∧t)

≥l(t,x).

The proof of the last assertion rests on the observation thatG(x)=l(0,x) and _∂∂_tl(0,x)>0 forx<1.

Assume that there exists an optimal stopping momentτ∗for somex∗<1, i.e.,w(x∗)=Ex∗_{e−ατ∗

1_{τ∗<τ_O}G(X(τ∗))}.

From the strong Markov property of the process X(t) we infer that G(X(τ∗)) = w(X(τ∗)), Px∗

-a.s. on

{τ∗< τ_O}. Sincew(x∗)_≥ex∗

we havePx∗

(τ∗< τ_O)>0. This is a contradiction with assertion (3) of Lemma

4.10.

REMARK4.11. Penalty method offers a numerical procedure for solution of optimal stopping problems. Lemma 2.7 provides an estimate of the error:kw₋wβ_{k ≤ k}(G₋wβ)+k. This error decreases asβincreases: by Proposition 2.9wβforms a non-decreasing sequence of functions converging tow. Under (A1) functions

wβ_{are continuous (c.f. Corollary 2.6). Theorems 4.3 and 4.8 state assumptions under whichwis continuous}

and is approximated bywβ_{uniformly on compact sets. The continuity ofwandw}

βimplies that state space

discretization methods can be safely applied. Following Lemma 2.2 functionwβ can be computed as a fixed point of a contraction operator_T given by

Tφ(s,x)=Exn

Z τO

0

e−(α+β)uhf s+u,X(u)_{+βG s+u,X}

(u)

−φ s+u,X(u)+

+βφ s+u,X(u)i_du+e−(α+β)τO_{H s+τ}

O,X(τO) o

for a bounded measurable functionφ. This operator can be implemented via PDE or Kushner-Dupuis space-time discretization approach (see [12]). The fixed point is approximated by an iterative procedure with an exponential decrease of the error (due to the contraction property ofT).

5. S  (A1)

Define forη >0

hη(x)=Px_{τ

O> η}.

Consider the following assumption:

(A4)

lim

This assumption ensures that when approaching the boundary ofOthe probability of crossing it in a short time converges to 1. It is clearly satisfied (by Chebyshev inequality) whenever the mappingx_→Ex_{τ_O}is continuous. It can be viewed as a complementary assumption to (A2). We will show that (A2)-(A4) imply (A1) and (A1) is sufficient for (A4).

LEMMA 5.1. The function hηis continuous on E under assumptions (A2)-(A4).

Proof. Forδ _∈(0, η) definerδ(x)=Ex_{hη

−δ(X(δ))}. This function is continuous by (A3). The difference

betweenrδandhηcan be bounded in the following way:

0_≤rδ(x)₋hη(x)_≤Px_{τ_O< δ_}.

Assumption (A2) states that the right-hand side of the above inequality converges to 0 asδ_→0 uniformly inxfrom compact subsets of_O. Hence,hηis continuous in_O. It is identically zero onE_{\ O}. These two pieces fit continuously at the boundary of_Obecause, due to (A4),hη(x) converges to 0 asxapproaches the

boundary of_O.

The continuity of hη implies uniformity of the limit in assumption (A4), which is formalized in the following corollary.

COROLLARY 5.2. If hηis continuous then for any compact set L_⊆E and constantsη, ε >0there is an open set Lη,εsuch that Lη,ε_{⊂ O}and for each x_∈L_\Lη,εwe havePx_(τ

O> η)≤ε.

PROPOSITION 5.3. Under (A2)-(A3) the mapping x_7→PτO

t h(x)is continuous on E\∂Ofor any bounded

measurable function h and t>0. If additionally (A4) holds then PτO

t maps the space of bounded measurable

functions into the space of continuous bounded functions and as a result condition (A1) is satisfied.

Proof. Lethbe a bounded measurable function. By the strong Feller property (A3), fors<t, the mapping

x_7→ExnEX(s)₁

{t₋s<τ_O}h(X(t−s))

o

is continuous. Furthermore,

(31)

Ex_{1

{t<τ_O}h(X(t))}=Ex

n

1_{s<τ_O}EX(s)1_{t₋s<τ_O}h(X(t−s))

o

=ExnEX(s)

1_{t−s<τO}h(X(t−s)) o

−Exn_1{τO≤s}EX(s)

1_{t−s<τO}h(X(t−s)) o

.

Therefore

Ex_{1

{t<τ_O}h(X(t))} −Ex

n

EX(s)

1_{t₋s<τ_O}h(X(t−s))

o ≤ khk

Px_{τ

O≤s} ≤ khkPx{τO<2s} →0

uniformly on compact subsets of_Oass_→0 by (A2). This shows the continuity ofx_7→PτO

t h(x) forx∈ O.

ForxinE_{\ O}we clearly haveEx_{1{t<τO}h(X(t))_} = 0. Now we prove continuous fit at the boundary of O. Assumption (A4) implies that_|Ex_{1

{t<τ_O}h(X(t))}| ≤Px{t< τ_O} khkdecreases to 0 as xapproaches the

boundary. Hence,PτO

t his continuous onE.

Proposition 5.3 states that (A2)-(A4) are sufficient for Assumption (A1). The following lemma shows that (A1) implies (A4). Recall that (A1) also implies (A2), see Lemma 4.6.

LEMMA 5.4. Under (A1) the function hηis continuous on E, which, in particular, implies (A4).

Proof. Follows from the identityhη=PτO

η 1, where1denotes a function identically equal 1.

6. S   Oc

In this section we explore a stopping problem with a more general payofffunctionF:

J(s,x, τ)=Exn

Z τ∧τO

0

e−αuf s+u,X(u)_du+e−α(τ∧τO)_{F s+(τ∧τ}

O),X(τ∧τO) o

,

where f,Fare measurable bounded functions that are continuous in suniformly in xfrom compact sets andFis continuous on [0,_∞)× O. In particular,Fcan be of the form

F(s,x)=1_{x_∈O}¯ G(s,x)+1_{x<_O}¯ H(s,x),

whereG,Hare continuous bounded functions. This is a complementary problem to the one described in preceding sections: a discontinuity of the payoffmanifests itself only when the process (X(t)) jumps to ¯_Oc