Baye , Ch is ian; Pelizza i, Luca; Schoenmake s, John
A icle — Published Ve sion
P imal and dual op imal s opping wi h signa u es
Finance and S ochas ics
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Sugges ed Ci a ion: Baye , Ch is ian; Pelizza i, Luca; Schoenmake s, John (2025) : P imal and dual
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Finance and S ochas ics (2025) 29:981–1014
h ps://doi.o g/10.1007/s00780-025-00570-8
P imal and dual op imal s opping wi h signa u es
Ch is ian Baye 1·Luca Pelizza i1,2 ·John Schoenmake s1
Recei ed: 7 Decembe 2023 / Accep ed: 4 No embe 2024 / Published online: 16 June 2025
© The Au ho (s) 2025
Abs ac
We p opose wo signa u e-based me hods o sol e an op imal s opping p oblem – ha
is, o p ice Ame ican op ions – in non-Ma ko ian amewo ks. Bo h me hods ely on
a global app oxima ion esul o Lp- unc ionals on ough-pa h spaces, using linea
unc ionals o obus , ough-pa h signa u es. In he p imal o mula ion, we p esen
a non-Ma ko ian gene alisa ion o he amous Longs a –Schwa z algo i hm, us-
ing linea unc ionals o he signa u e as eg ession basis. Fo he dual o mula ion,
we pa ame ise he space o squa e-in eg able ma ingales using linea unc ionals
o he signa u e and apply a sample a e age app oxima ion. We p o e con e gence
o bo h me hods and p esen i s nume ical examples in non-Ma ko ian and non-
semima ingale egimes.
Keywo ds Signa u e ·Op imal s opping ·Rough pa hs ·Mon e Ca lo ·Rough
ola ili y
Ma hema ics Subjec Classifica ion 60L10 ·60L20 ·91G20 ·91G60
JEL Classifica ion C63 ·G12
All au ho s g a e ully acknowledge unding by he Deu sche Fo schungsgemeinscha (DFG,
Ge man Resea ch Founda ion) unde Ge many’s Excellence S a egy – The Be lin Ma hema ics
Resea ch Cen e MATH+(EXC-2046/1, p ojec ID: 390685689).
✉L. Pelizza i
[email p o ec ed]
C. Baye
[email p o ec ed]
J. Schoenmake s
[email p o ec ed]
1Weie s ass Ins i u , Moh ens asse 39, 10117 Be lin, Ge many
2Ins i u ü Ma hema ik, Technische Uni e si ä Be lin, S . des 17. Juni 136, 10587 Be lin,
Ge many
982 C. Baye e al.
1In oduc ion
S ochas ic p ocesses wi h memo y play a mo e and mo e impo an ole in he mod-
elling o inancial ma ke s. In he modelling o equi y ma ke s, ough s ochas ic
ola ili y models a e now pa o he s anda d oolbox; see e.g. Ga he al e al. [27]
and Baye e al. [4]. In he same a ea, pa h-dependen s ochas ic ola ili y models
(e.g. Guyon e al. [30]) a e a e y powe ul al e na i e o cap u ing memo y e ec s.
P ocesses wi h memo y a e also an essen ial ool o modelling he mic os uc u e o
inancial ma ke s, d i en by he ma ke p ac ice o spli ing la ge o de s in o many
medium-size ones, as well as by he eac ion o algo i hmic ade s o such o de s.
Seen om ou side, his ma e ialises as sel -exci a ion o he o de low and, conse-
quen ly, Hawkes p ocesses a e a undamen al ool o modelling o de lows; see e.g.
Bouchaud e al. [15, Chap. 9.4]. Beyond inance, p ocesses wi h memo y play an
impo an ole in he modelling o many na u al phenomena (e.g. ea hquakes, see
Oga a e al. [35]) o social phenomena.
In his pape , we s udy op imal s opping p oblems in non-Ma ko ian amewo ks,
ha is, he unde lying p ice is possibly a s ochas ic p ocess wi h memo y. Fo con-
c e eness’ sake, le us in oduce wo p ocesses de e mining he op imal s opping
p oblem: an unde lying s a e p ocess X, oge he wi h i s na u al il a ion 𝔽X, and a
ewa d p ocess Z, which is 𝔽X-adap ed – hink abou X=(S, ) o a s ock p ice
p ocess Sd i en by a s ochas ic a iance p ocess and Z =ϕ( ,S ). The op imal
s opping p oblem hen consis s o sol ing he op imisa ion p oblem
y0=sup
τ∈𝒮0
E[Zτ],(1.1)
whe e 𝒮0deno es he se o 𝔽X-s opping imes alued in [0,T] o some T>0. We
me ely assume α-Hölde -con inui y o Xin ou amewo k, see Sec . 3.1 below, in
pa icula allowing non-Ma ko ian and non-semima ingale s a e p ocesses X.
The lack o a Ma ko p ope y leads o se e e heo e ical and compu a ional chal-
lenges in he con ex o op imal con ol p oblems, and hus in pa icula in he op-
imal s opping p oblem (1.1). Indeed, he p ima y analy ical and nume ical ame-
wo k o s ochas ic op imal con ol p oblems a guably is he associa ed Hamil on–
Jacobi–Bellman (HJB) PDE, which in he con ex o op imal s opping leads o so-
called ee-bounda y p oblems; see Peski and Shi yae [36, Chap. 4]. When he
s a e p ocess is no a Ma ko p ocess, such PDEs do no exis a p io i. As no ed
abo e, in ini e-dimensional (BS)PDE o mula ions can be gi en; see o ins ance
Baye e al. [6] o a BSPDE desc ip ion o he Ame ican op ion p ice in ough
ola ili y models. When he unde lying dynamics is o s ochas ic Vol e a ype, pa h-
dependen HJB PDEs could p obably be de i ed ollowing he app oach o Bonesini
and Jacquie [14]. Howe e , mos nume ical app oxima ion me hods c ucially ely
on he Ma ko p ope y as well.
I should be no ed ha a leas in ui i ely, all p ocesses wi h memo y can be
u ned in o Ma ko p ocesses by adding he his o y o he cu en s a e – bu see
e.g. Ca mona and Cou in [16] o a mo e sophis ica ed app oach in he case o ac-
ional B ownian mo ion. Hence heo e ical and nume ical me hods om he Ma ko-
ian wo ld a e in p inciple a ailable, bu a he cos o ha ing o wo k in in ini e-
dimensional (o en e y ca e ully d a ed, see e.g. Cuchie o and Teichmann [21])
P imal and dual op imal s opping wi h signa u es 983
s a e spaces. On he o he hand, Ma ko ian app oxima ions, i.e., ini e-dimensional
Ma ko p ocesses closely mimicking he p ocess wi h memo y, can some imes be
a e y e icien su oga e model, especially when high accu acy is achie able wi h
low-dimensional Ma ko ian app oxima ions; see e.g. Baye and B eneis [3].
Inspi ed by many success ul uses in machine lea ning ( o ime-se ies da a), Kalsi
e al. [31] in oduced a model- ee me hod o nume ically sol ing a s ochas ic op i-
mal execu ion p oblem. The me hod is based on he pa h signa u e, see e.g. F iz and
Vic oi [26, Chap. 7], and is applicable in non-Ma ko ian se ings. This app oach
was ex ended o op imal s opping p oblems in Baye e al. [5], whe e s opping imes
we e pa ame ised as i s hi ing imes o a ine hype planes in he signa u e space.
A igo ous ma hema ical analysis o ha me hod was pe o med and nume ical ex-
amples e i ying i s e iciency we e p o ided.
The signa u e X<∞o a pa h X:[0,T]→ℝdis gi en (a leas o mally) as he
in ini e collec ion o i e a ed in eg als, ha is, o 0 ≤ ≤s≤T,
X<∞
s, ={︃∫︂
s∫︂ k
s···∫︂ 2
s
dXi1
1···dXik
k:i1,...,i
k∈{1,...,d},k ≥0}︃.
The signa u e cha ac e ises he his o y o he co esponding ajec o y and hence p o-
ides a sys ema ic way o “li ing” a p ocess wi h memo y o a Ma ko p ocess by
adding he pas o he s a e. Relying only on minimal egula i y assump ions, he
co esponding encoding is e icien and has nice algeb aic p ope ies. In many ways,
(linea unc ionals o ) he pa h signa u e beha es like an analogue o polynomials on
pa h space and can be seen as a canonical choice o basis unc ions on pa h space.
Fo example, a S one–Weie s ass ype esul shows ha when es ic ed o compac s,
con inuous unc ionals on pa h spaces can be app oxima ed by linea unc ionals o
he signa u e, ha is, by linea combina ions o i e a ed in eg als; see o ins ance
Kalsie al.[31, Lemma 3.4].
As a i s con ibu ion, we p o ide in Sec . 2an abs ac app oxima ion esul on
α-Hölde ough-pa h spaces by linea unc ionals o he obus signa u e, wi h espec
o he Lp-no m; see Theo em 2.8 below. As a di ec consequence and unde e y mild
assump ions, we can show ha o any 𝔽X-p og essi e p ocess (ξ ) ∈[0,T ], we can ind
a sequence (ℓn)n∈ℕo linea unc ionals on he s a e space o he signa u e such ha
lim
n→∞E[︃∫︂T
0(ξ −⟨X<∞
0, ,ℓ
n⟩)2d ]︃=0;(1.2)
see Co olla y 2.9 below o de ails. This esul is in ma ked con as o he s an-
da d uni e sal app oxima ion esul o signa u es as usually o mula ed, which only
p o ides uni o m con e gence on compac subse s o he pa h space. Le us men-
ion wo ela ed wo ks on global app oxima ion wi h signa u es. Fi s , in Cuchie o
e al. [20], he au ho s s udy global signa u e app oxima ions based on a e sion o
he S one–Weie s ass esul o so-called weigh ed spaces;see[20, Theo em 3.6].
Compa ed o ou heo y, while hey do no equi e a obus e sion o he signa u e,
such weigh ed spaces need o be c a ed ca e ully. Secondly, in he ecen wo k by
Alai a i and Schell [1], he au ho s ob ain (independen ly om us) a simila global
Lp-app oxima ion esul o obus signa u es o bounded a ia ion pa hs, see [1,
984 C. Baye e al.
P oposi ion 4.5], bu based on a mono one class a he han a S one–Weie s ass a -
gumen .
Re u ning o he op imal s opping p oblem (1.1), we gene alise in Sec . 3 wo
s anda d echniques om he Ma ko ian o he non-Ma ko ian case by using signa-
u es, namely he Longs a –Schwa z algo i hm [33] and Roge s’ dual ma ingale
me hod [37]. Deno ing by Y he Snell en elope o he op imal s opping p oblem, see
Sec . 3.2 below o mo e de ails, he Longs a –Schwa z algo i hm is based on he
dynamic p og amming p inciple, ha is,
Y =max(Z ,E[Y +Δ |ℱX
]).
I Xis a Ma ko p ocess, hen E[Y +Δ |ℱX
]=E[Y +Δ |X ], which can be e -
icien ly compu ed by using eg ession (leas -squa es Mon e Ca lo). In he non-
Ma ko ian case, an applica ion o he global app oxima ion esul in Theo em 2.8,
i.e., he con e gence in (1.2), shows ha (unde minimal assump ions) a Longs a –
Schwa z algo i hm con e ges when he condi ional expec a ion is app oxima ed by
linea unc ionals o he signa u e, ha is,
↦→ E[Y +Δ |ℱX
]≈⟨X<∞
0, ,ℓ⟩;
see P oposi ion 3.3.
Rega ding he dual me hod, we ely on Roge s’ cha ac e isa ion ha
y0=in
M∈ℳ2
0
E[︂sup
∈[0,T ](Z −M )]︂,
whe e he in is aken o e all squa e-in eg able ma ingales Ms a ing a 0. I he
unde lying il a ion is B ownian, such ma ingales can be w i en as s ochas ic in-
eg als wi h espec o a B ownian mo ion W, ha is, M =∫︁
0ξsdWs o some
𝔽X-p og essi e p ocess ξ. The app oxima ion esul in Theo em 2.8, i.e., he con-
e gence in (1.2), sugges s o app oxima e he in eg and by linea unc ionals o he
signa u e, ha is,
↦→ ξ ≈⟨X<∞
0, ,ℓ⟩,
and we p o e con e gence a e aking he in imum o e all linea unc ionals ℓ, i.e.,
y0=in
ℓE[︃sup
∈[0,T ](︃Z −∫︂
0⟨X<∞
0,s ,ℓ⟩dWs)︃]︃;(1.3)
see P oposi ion 3.8. Fo nume ically sol ing he dual p oblem (1.3), we ca y ou a
sample a e age app oxima ion (SAA) wi h espec o he coe icien s o he linea
unc ional o he signa u e. Fo a Ma ko ian en i onmen , a ela ed SAA p ocedu e
was ea lie p oposed in Desai e al. [22] and ecen ly e ined in Belomes ny and
Schoenmake s [11] and Belomes ny e al. [10] by using a sui able andomisa ion.
An impo an ea u e o he SAA me hod is ha i elies on non-nes ed Mon e Ca lo
simula ion and hus is e y as in compa ison o he classical nes ed Mon e Ca lo
me hod by Ande sen and B oadie [2].
P imal and dual op imal s opping wi h signa u es 985
Fo bo h he Longs a –Schwa z and dual signa u e me hods, we also p o e con-
e gence o he ini e-sample app oxima ions when he numbe o samples g ows o
in ini y; see P oposi ions 3.4 and 3.10. I is wo h no ing ha a e independen e-
simula ions, he Longs a –Schwa z algo i hm yields lowe -biased, whe eas he dual
me hod gi es uppe -biased alues o he op imal s opping p oblem (1.1), and hus
applying bo h me hods p oduces con idence in e als o he ue alue o y0.
Finally, in Sec . 4, we p o ide i s nume ical examples based on he p imal
and dual signa u e-based app oaches in wo non-Ma ko ian amewo ks. Fi s , in
Sec . 4.1, we s udy he ask o op imally s opping ac ional B ownian mo ion o a
wide ange o Hu s pa ame e s H∈(0,1), ep esen ing he canonical choice o a
s a e p ocess ou side o he Ma ko egime. The same p oblem was al eady s udied in
Becke e al. [8] and la e in Baye e al. [5], and we compa e ou lowe ( esp. uppe )
bounds wi h he esul s he ein. Secondly, in Sec . 4.2, we conside he p oblem o
compu ing Ame ican op ion p ices in he ough Be gomi model (see Baye e al. [4]),
and we compa e ou p ice in e als wi h Baye e al. [7] ( esp. Goudenege e al. [29]),
whe e lowe bounds we e compu ed in he same model.
1.1 No a ion
Fo d,K ∈ℕ, we de ine he so-called ex ended enso algeb a and he K-s ep un-
ca ion he eo by
T(︁(ℝd))︁=∏︂
k≥0
(ℝd)⊗k,T
≤K(ℝd)=
K
∏︂
k=0
(ℝd)⊗k,
whe e we use he con en ion (ℝd)⊗0=ℝ. Fo mo e de ails, including na u al op-
e a ions such as sum +and p oduc ⋆on hese spaces, see o ins ance F iz and
Vic oi [26, Sec . 7.2.1]. Fo any wo d w=i1... i
n o some n∈ℕwi h le -
e s i1,...,i
n∈{1,...,d}, we de ine he deg ee o was he leng h o he wo d,
ha is, deg(w) =n, and deno e by ∅ he emp y wo d wi h deg(∅)=0. Mo eo e ,
o a∈T((ℝd)), we deno e by ⟨a,w⟩ he elemen o a(n) ∈(ℝd)⊗nco esponding
o he basis elemen ei1⊗···⊗ein, whe e {e1,...,e
d}is he s anda d basis o ℝd.
Deno ing by 𝒲d he linea span o wo ds, he pai ing abo e can be ex ended linea ly
o ⟨·,·⟩:T((ℝd))×𝒲d→ℝ. Fo an elemen ℓ∈𝒲d, i.e., ℓ=λ1w1+···+λnwn
o some wo ds w1,...,w
nand scala s λ1,...,λ
n∈ℝ, we de ine he deg ee o ℓby
deg(ℓ) := max1≤i≤ndeg(wi), and o K∈ℕ, we deno e by 𝒲d
≤K⊆𝒲d he subse
o elemen s ℓwi h deg(ℓ) ≤K. Fo wo wo ds wand , we deno e by
∃
he shu le
p oduc gi en by
w
∃
∅=∅
∃
w=w,
wi
∃
j := (w
∃
j )i +(wi
∃
)j, i,j ∈{1,...,d},(1.4)
which bilinea ly ex ends o he span 𝒲do wo ds. We u he de ine he ee nilpo en
Lie g oup o e ℝdby
G(︁(ℝd))︁={︁a∈T(︁(ℝd))︁ {0}:⟨a,w⟩⟨a, ⟩=⟨a,w
∃
⟩,∀w, ∈𝒲d}︁;
986 C. Baye e al.
see [26, Chap. 7.5] o de ails.
Fo α∈(0,1), we deno e by Cα([0,T];ℝd) he space o α-Hölde -con inuous
pa hs X, ha is, X:[0,T]→ℝdsuch ha
∥X∥α;[0,T ]=sup
0≤s< ≤T
|X −Xs|
| −s|α<∞,
whe e |·|deno es he Euclidean no m on ℝd. Deno e by Δ2
[0,T ] he s anda d simplex
Δ2
[0,T ]:= {(s, ) ∈[0,T]2:0≤s≤ ≤T}.Fo L∈ℕand any wo-pa ame e
unc ion on he unca ed enso algeb a,
Δ2
[0,T ]∋(s, ) ↦→ Xs, =(1,X(1)
s, ,...,X(L)
s, )∈T≤L(ℝd),
we deno e by ||| ·|||(α,L) he no m gi en by
|||X|||(α,L) := max
1≤ℓ≤L(︃sup
0≤s< ≤T
|X(ℓ)
s, |
| −s|ℓα )︃1/ℓ
.
We deno e by Cα
g([0,T];ℝd) he space o geome ic α-Hölde ough pa hs X alued
in ℝd, which is he ||| ·|||(α,L)-closu e o L-s ep signa u es o Lipschi z-con inuous
pa hs X:[0,T]→ℝd o L=⌊1/α⌋. Mo e p ecisely, o e e y X∈Cα
g,
he e exis s a sequence (Xn)n∈ℕ⊆Lip([0,T];ℝd)such ha |||Xn−X|||(α,L) →0
as n→∞, whe e Xnis he L-s ep signa u e o Xn, ha is,
Xn
s, := (︃∫︂s< 1<···< ℓ< ⊗dXn
1···⊗dXn
ℓ:0≤ℓ≤L)︃∈G≤L(ℝd),
whe e he in eg als a e de ined in he Riemann–S iel jes sense. Fo he es o he
pape , we always ix L=⌊1/α⌋and use he sho e no a ion ||| ·|||α:= ||| ·|||(α,⌊1/α⌋).
Fo any X∈Cα
g, we deno e by X<∞ he ough-pa h signa u e, which is he unique
(up o ee-like equi alence ∼ , see Boediha djo e al. [12] o de ails) pa h om
Lyons’ ex ension heo em (Lyons [34, Theo em 3.7]), ha is,
Δ2
[s, ]:[0,T]∋(s, ) ↦→ X<∞
s, =(1,X(1)
s, ,...,X(L)
s, ,X(L+1)
s, ,...)∈G(ℝd),
such ha
∥X(k)∥kα <∞,∀k≥0,X<∞
s, =X<∞
s,u ⋆X<∞
u, ,s≤u≤ ,
whe e he la e is called Chen’s ela ion. Finally, by conside ing ime-augmen ed
pa hs (ˆ︁
X )=( , X )and hei geome ic ough-pa h li s ˆ︁
X, he signa u e map
becomes unique due o he s ic ly mono one ime componen . We deno e by
ˆ︁
Cα
g([0,T];ℝd+1) he space o geome ic α-Hölde ough-pa h li s o (ˆ︁
X )=( , X ),
X∈Cα([0,T];ℝd). We o en use he sho e no a ion ˆ︁
Cα
gwhen i is clea om he
con ex ha we a e wo king on he ixed ime in e al [0,T].
P imal and dual op imal s opping wi h signa u es 987
2 Global app oxima ion wi h ough-pa h signa u es
In his sec ion, we p esen he heo e ical ounda ion o his pape , which consis s o
a global app oxima ion esul based on obus ough-pa h signa u es.
2.1 The space o s opped ough pa hs
Fo α∈(0,1), we conside an α-Hölde -con inuous pa h X:[0,T]→ℝds a ing a
X0=x0∈ℝdand deno e by Xa geome ic ough-pa h li o he ime-augmen a ion
( , X ), ha is, X∈ˆ︁
Cα
g([0,T];ℝd+1).
Defini ion 2.1 Fo any α∈(0,1)and T>0, he space o s opped ˆ︁
Cα
g-pa hs is
de ined by he disjoin union
Λα
T:= ⋃︂
∈[0,T ]ˆ︁
Cα
g([0, ];ℝd+1).
Mo eo e , we equip he space Λα
Twi h he inal opology induced by he map
ϕ:[0,T]× ˆ︁
Cα
g([0,T];ℝd+1)→Λα
T,ϕ( ,x)=x|[0, ].
The eason o wo k on his space is he ollowing. I Xis a s ochas ic p ocess and
Xdeno es he andom geome ic li o ( , X ), we de ine ℱX
=σ(X0,s :s≤ )
o 0 ≤ ≤T, i.e., he na u al il a ion gene a ed by X. In Lemma 2.4 below, we
show ha any 𝔽X-p og essi e p ocess (A ) ∈[0,T ]can be w i en as A = (X|[0, ]),
whe e is a measu able unc ion on Λα
T. Thus p og essi ely measu able p ocesses
can be hough o as measu able unc ionals on Λα
T, and we discuss app oxima ion
esul s o he la e below. Simila spaces ha e al eady been conside ed in ela ion
wi h unc ional I ô calculus in Con and Fou nié [19] and Dupi e [23], o p- ough
pa hs in Kalsi e al. [31], and mo e ecen ly in ela ion wi h op imal s opping in Baye
e al. [5].
Rema k2.2 One can also in oduce a me ic dΛon he space Λα
T.Le y∈Λα
T, ha is,
he e exis s 0 ≤s≤Tsuch ha y=y|[0,s]∈ˆ︁
Cα
g([0,s]).Now o any ≥s, we can
ex end y|[0,s] o a geome ic ough pa h ˜
y|[0, ]∈ˆ︁
Cα
g([0, ])as ollows. By geome ic-
i y, he e exis s a sequence o smoo h pa hs (u, yn
u)u∈[0,s]such ha he (canonical)
ough-pa h li s con e ge o y|[0,s]. Then we de ine ˜
y|[0, ] o be he ough-pa h limi
o he canonical li o u↦→ (u, yn
u∧s) o 0 ≤u≤ . This cons uc ion can be used
o de ine
dΛ(x|[0, ],y|[0,s]):= x−˜
yα:[0, ]+| −s|,s≤ .
I has been p o ed in [5, Lemma A.1] ha he opology o he me ic space (Λp
T,d
Λ),
whe e Λp
Tdeno es he space o s opped p- ough pa hs (see [5, Sec . 3]), coincides
wi h he inal opology, and he space o s opped geome ic ough pa hs is Polish. A
simila a gumen can be done o he α-Hölde case by eplacing he p- a ia ion no m
by he α-Hölde -no m and using he ac ha ˆ︁
Cα
gis Polish; see F iz and Vic oi [26,
P oposi ion 8.27].
988 C. Baye e al.
Rema k 2.3 Le Xbe a s ochas ic p ocess alued in ˆ︁
Cα
g([0,T];ℝd+1).I isdis-
cussed in [26, Appendix A.1] ha Xcan be ega ded as a andom a iable alued
in ˆ︁
Cα
g([0,T];ℝd+1), and i s law μXis hen a Bo el measu e on he Bo el σ-algeb a
ℬαwi h espec o ||| ·|||α. Mo eo e , de ine he p oduc measu e dμ := d ⊗dμX.
Fo he su jec i e map ϕ:[0,T]× ˆ︁
Cα
g([0,T];ℝd+1)→Λα
Tde ined abo e, we can
de ine he push o wa d measu e ˆ︁μon Λα
T, in symbols ˆ︁μ:= ϕ#μ=μ◦ϕ−1, which
is gi en by
ˆ︁μ(A) := μ(︁ϕ−1(A))︁ o all A∈ℬ(Λα
T).
Conside he space ℍ2o 𝔽X-p og essi e p ocesses Asuch ha
∥A∥2
ℍ2:= E[︃∫︂T
0A2
sds]︃<∞.(2.1)
The ollowing esul jus i ies he conside a ion o he space Λα
T.
Lemma 2.4 Fo any p ocess A∈ℍ2and α∈(0,1), he e exis s a measu able unc-
ion :(Λα
T,ℬ(Λα
T)) →(ℝ,ℬ(ℝ)) such ha A = (X|[0, ])almos e e ywhe e.
P oo Conside he space o elemen a y 𝔽X-p og essi e p ocesses, ha is, p ocesses
o he o m
An
(ω) := ξn
0(ω)1{0}( ) +
mn−1
∑︂
j=1
1( n
j, n
j+1]( )ξn
j(ω), (2.2)
whe e 0 ≤ n
0<··· <
n
mn≤Tand ξn
jis an ℱX
j-measu able, squa e-in eg able
andom a iable. A s anda d esul o he cons uc ion o s ochas ic in eg als shows
ha his space is dense in ℍ2; see e.g. Ka a zas and Sh e e [32, Lemma 3.2.4]. Thus
we can ind Ano he o m (2.2) such ha An→A o almos e e y ( , ω). Since he
andom a iable ξn
jis measu able wi h espec o he σ-algeb a
ℱX
j:= σ(X0,s :s≤ j)=σ(X|[0, j]),
he e exis s by he Doob–Dynkin lemma a Bo el-measu able unc ion
Fn
j:ˆ︁
Cα
g([0,
j];ℝd+1)→ℝ
such ha ξn
j(ω) =Fn
j(X|[0, j](ω)). Then he unc ions
[0,T]× ˆ︁
Cα
g∋( , x)↦→ 1( n
j, n
j+1]( )F n
j(x|[0, j])
a e (ℬ([0,T])⊗ℱX
T)-measu able, and he e o e such is he unc ion
Fn( , x):= Fn
0(x0)1{0}( ) +
mn−1
∑︂
j=1
1( n
j, n
j+1]( )F n
j(x|[0, j]).
P imal and dual op imal s opping wi h signa u es 995
3.1 F amewo k and p oblem o mula ion
Suppose we ha e a comple e il e ed p obabili y space (Ω, ℱ,𝔽=(ℱ ) ∈[0,T ],P)
o some T>0, ul illing he usual condi ions, and ix α∈(0,1). Fo any 𝔽-adap ed
and α-Hölde -con inuous s ochas ic p ocess (X ) ∈[0,T ] aking alues in ℝdwi h
X0=x0, we conside
–X∈ˆ︁
Cα
g, a geome ic α-Hölde ough-pa h li o ( , X )such ha i s law μX
ul ils Assump ion 2.6;
–X<∞, he obus ough-pa h signa u e in oduced in Sec . 2.3;
–(Z ) ∈[0,T ], a eal- alued 𝔽X-adap ed p ocess such ha sup ∈[0,T ]|Z |∈L2.
The op imal s opping p oblem hen eads
y0=sup
τ∈𝒮0
E[Zτ],(3.1)
whe e 𝒮0deno es he se o 𝔽X-s opping- imes alued in [0,T].
Rema k 3.1 No ice ha he amewo k desc ibed abo e is e y gene al in wo ways.
Fi s , we only assume α-Hölde -con inui y o he s a e p ocess X, including in pa -
icula non-Ma ko ian and non-semima ingale egimes which one encoun e s o
ins ance in ough ola ili y models; see Sec . 4.2. Secondly, by conside ing he p o-
jec ion X↦→ ( , X )on o he i s coo dina e, ou amewo k includes o any payo
unc ion ϕ:[0,T]×ℝd→ℝ he mo e common o m o he op imal s opping
p oblem
y0=sup
τ∈𝒮0
E[ϕ(τ,Xτ)].
Rema k 3.2 In gene al, he e is no canonical way o li ing a p ocess ( , X ) o a
andom ough pa h X, and o en ca e ul jus i ica ion is equi ed, o ins ance based
on a ough-pa h e sion o he Kolmogo o c i e ion; see F iz and Hai e [25, The-
o em 3.1]. Howe e , o a big class o p ocesses (e.g. semima ingales, Gaussian
p ocesses, one-dimensional p ocesses), he e a e canonical choices o andom geo-
me ic ough-pa h li s, and we explain below in Sec . 4how o do i . Mo eo e , we
always look a li s such ha 𝔽X=𝔽X, ha is, he op imal s opping p oblem has he
same unde lying in o ma ion when obse ing Xo X.
3.2 P imal op imal s opping wi h signa u es
Fi s , we p esen a me hod o compu e a lowe -biased app oxima ion yL
0≤y0 o
he op imal s opping p oblem (3.1). Mo e p ecisely, we cons uc a eg ession-based
app oach, gene alising he amous algo i hm om Longs a and Schwa z [33], e-
u ning a subop imal exe cise s a egy. Le us i s quickly desc ibe he main idea o
mos eg ession-based app oaches.
Replacing he in e al [0,T]by a ini e g id {0= 0<
1<···<
N=T}, he
disc e e op imal s opping p oblem eads
yN
0=sup
τ∈𝒮N
0
E[Zτ],
996 C. Baye e al.
whe e 𝒮N
nis he se o s opping imes aking alues in { n,...,
N} o n=0,...,N,
wi h espec o he disc e e il a ion 𝔽X,N =(ℱX
n)n=0,...,N . We de ine he disc e e
Snell en elope by
YN
n=esssup
τ∈𝒮N
n
E[Zτ|ℱX
n],0≤n≤N, (3.2)
and one can show ha YNsa is ies he disc e e dynamic p og amming p inciple (DPP)
YN
n=max(Z n,E[YN
n+1|ℱX
n]), n =0,...,N −1;(3.3)
see o ins ance Peski and Shi yae [36, Theo em 1.2]. Now he key idea o mos
eg ession-based app oaches, such as o ins ance Longs a and Schwa z [33], is
ha by assuming ha Xis a Ma ko p ocess, one can choose a sui able amily o
basis unc ions (bk)and apply leas -squa es eg ession o app oxima e
E[YN
n+1|ℱX
n]≈
D
∑︂
k=0
αkbk
n(X n), 0≤n≤N−1,α
k∈ℝ,∀k≤D, (3.4)
and hen make use o he DPP o ecu si ely app oxima e YN
0=yN
0. O cou se, he
app oxima ion o he condi ional expec a ions in (3.4) hea ily elies on he Ma ko
p ope y, and hus one canno expec such an app oxima ion o con e ge in non-
Ma ko ian se ings.
Re u ning o ou amewo k, we need o eplace (3.4) by a sui able app oxima ion
o he condi ional expec a ions
E[YN
n+1|ℱX
n]= n(X|[0, n]), 0≤n≤N−1.
The uni e sali y esul in Theo em 2.8 now sugges s app oxima ing nby a sequence
o linea unc ionals o he obus signa u e, ha is,
n(X|[0, n])≈⟨X<∞
0, n,ℓ⟩,ℓ∈𝒲d+1,
whe e 𝒲d+1is he linea span o wo ds in oduced in Sec . 1.1.
3.2.1 Longs aff–Schwa z wi h signa u es
In his sec ion, we p esen a e sion o he Longs a –Schwa z (LS) algo i hm [33],
using signa u e-based leas -squa es eg ession. A con e gence analysis o he LS
algo i hm was p esen ed in Clémen e al. [18], and combining hei echniques wi h
he uni e sali y o he signa u e allows us o eco e a con e gen algo i hm.
The main idea o he LS algo i hm is o e o mula e he DPP (3.3) o s opping
imes by aking ad an age o he ac ha op imal s opping imes can be exp essed in
e ms o he Snell en elope. Mo e p ecisely, i is p o ed in Peski and Shi yae [36,
Theo em 1.2] ha he s opping imes τn:= min{ m≥ n:YN
m=Z m} o 0 ≤n≤N
a e op imal in (3.2), and hence one ecu si ely de ines
τN= N,
τn= n1{Z n≥E[Zτn+1|ℱX
n]} +τn+11{Z n<E[Zτn+1|ℱX
n]},n=0,...,N −1.
P imal and dual op imal s opping wi h signa u es 997
Now o any unca ion le el K∈ℕ o he signa u e and some ixed
n=0,...,N −1, assume we a e gi en an app oxima ion τK
n+1o τn+1. Then we
app oxima e he condi ional expec a ion E[ZτK
n+1|ℱX
n]by sol ing he minimisa-
ion p oblem
ℓ∗:=ℓ∗,n,K =a gmin
ℓ∈𝒲d+1
≤K
∥ZτK
n+1−⟨X≤K
0, n,ℓ⟩∥L2,n=0,...,N −1.(3.5)
Se ing ψn,K (x)=⟨x≤K,ℓ
∗,n,K ⟩∈Lλ
sig, we de ine he app oxima ing sequence o
s opping imes
τK
N= N,
τK
n= n1{Z n≥ψn,K (X|[0, n])}+τK
n+11{Z n<ψn,K (X|[0, n])},n=0,...,N −1.
The ollowing esul shows con e gence as he dep h o he signa u e goes o in ini y,
and he p oo is gi en in Appendix A.1.
P oposi ion 3.3 Fo all n=0,...,N,we ha e
lim
K→∞E[ZτK
n|ℱX
n]=E[Zτn|ℱX
n]in L2.
In pa icula ,we ha e yK,N
0=max(Z 0,E[ZτK
1])→yN
0as K→∞.
Le us now desc ibe how o nume ically sol e (3.5) by using Mon e Ca lo simu-
la ions. Besides he unca ion o he signa u e a some le el K, we in oduce wo
u he app oxima ions s eps. Fi s , we eplace he signa u e X<∞by some dis-
c e ised e sion X<∞(J ), o example piecewise linea app oxima ion o he i e -
a ed in eg als, on some ine g id s0=0<s
1<··· <s
J=T, such ha
⟨X<∞
0, (J ), ⟩→⟨X<∞
0, , ⟩in L2as J→∞ o all wo ds and ∈[0,T]. Second,
o i=1,...,M i.i.d. sample pa hs o Zand he disc e ised and unca ed signa u e
X≤K=X≤K(J ), assuming ha τK,J
n+1is known, we es ima e ℓ∗by sol ing (3.5) ia
linea leas -squa es eg ession. This yields an es ima o ℓ∗=ℓ∗,n,J,K,M . De ining
ψn,J,K,M (x)=⟨x≤K,ℓ
∗,n,J,K,M ⟩leads o a ecu si e algo i hm o s opping imes,
o i=1,...,M,
τK,J,(i)
N= N,
τK,J,(i)
n= n1{Z(i)
n≥ψn,J,K,M (X(i)|[0, n])}+τK,J,(i)
n+11{Z(i)
n<ψn,J,K,M (X(i)|[0, n])}.(3.6)
Then he ollowing law-o -la ge-numbe s- ype esul holds ue, which almos di-
ec ly ollows om Clémen e al. [18, Theo em 3.2]; see Appendix A.1.
P oposi ion 3.4 Fo ixed K,J,we ha e
lim
M→∞
1
M
M
∑︂
i=1
Z(i)
τK,J,(i)
n=E[ZτK,J
n]a.s.
998 C. Baye e al.
Mo eo e ,se ing yN,K,J,M
0:=max(Z 0,1
M∑︁M
i=1Z(i)
τK,J,(i)
1
),we ha e
lim
K→∞ lim
J→∞ lim
M→∞yK,N,J,M
0=yN
0,
whe e he con e gence wi h espec o Mis almos su e con e gence.
Rema k 3.5 The ecu sion (3.6) o s opping imes, esp. he esul ing linea unc ion-
als o he signa u e ψn,K,M , p o ide a s opping policy o each sample pa h o Z.By
e-simula ing ˜
Mi.i.d. samples o Zand he signa u e X<∞, we can no ice ha he
esul ing es ima o yK,N,J, ˜
M
0is lowe -biased, ha is, yK,N,J, ˜
M
0≤yN
0, since yN
0is
de ined by aking he sup emum o e all possible s opping policies.
3.3 Dual op imal s opping wi h signa u es
In his sec ion, we app oxima e solu ions o he op imal s opping p oblem in i s
dual o mula ion, leading o uppe bounds yU
0≥y0 o (3.1). The dual ep esen-
a ion goes back o Roge s [37], whe e he au ho shows ha unde he assump ion
sup0≤ ≤T|Z |∈L2, he op imal s opping p oblem (3.1) is equi alen o
y0=in
M∈ℳ2
0
E[︂sup
≤T
(Z −M )]︂,(3.7)
whe e ℳ2
0deno es he space o squa e-in eg able 𝔽X-ma ingales s a ing om 0.
Assuming ha 𝔽Xis gene a ed by a B ownian mo ion W, we can p o e he ollowing
equi alen o mula ion o (3.7).
Rema k 3.6 F om a inancial modelling pe spec i e, assuming ha he il a ion is
gene a ed by an m-dimensional B ownian mo ion Wis na u al in he con ex o di -
usi e o ough ola ili y modelling. In his se ing, W ep esen s he d i e o m/2as-
se s (semima ingales) and hei co esponding m/2 a iance p ocesses.
Theo em 3.7 Assume 𝔽Xis gene a ed by an m-dimensional B ownian mo ion W.
Then o all M∈ℳ2
0, he e exis sequences ℓi=(ℓi
n)n∈ℕ⊆𝒲d+1 o i=1,...,m
such ha
∫︂·
0⟨X<∞
0,s ,ℓ
n⟩⊤dWs:=
m
∑︂
i=1∫︂·
0⟨X<∞
0,s ,ℓ
i
n⟩dWi
s−→ M·ucp as n→∞.
In pa icula , he minimisa ion p oblem (3.7)can be equi alen ly o mula ed as
y0=in
ℓ∈(𝒲d+1)mE[︃sup
≤T(︃Z −∫︂
0⟨X<∞
0,s ,ℓ⟩⊤dWs)︃]︃
=in
ℓ1,...,ℓm∈𝒲d+1E[︃sup
≤T(︃Z −
m
∑︂
i=1∫︂
0⟨X<∞
0,s ,ℓ
i⟩dWi
s)︃]︃.(3.8)
P imal and dual op imal s opping wi h signa u es 999
P oo By he ma ingale ep esen a ion heo em, any squa e-in eg able 𝔽X-ma ingale
can be w i en as
M =∫︂
0α⊤
sdWs=
m
∑︂
i=1∫︂
0αi
sdWi
s,
whe e (αs)s∈[0,T ]is 𝔽X-p og essi ely measu able and squa e-in eg able. Mo eo e ,
since M∈ℳ2
0, i ollows ha E[M2
T]=E[∫︁T
0|α |2d ]<∞. F om Co olla y 2.9,
we know ha he e exis sequences ℓi=(ℓi
n)n∈ℕ⊆𝒲d+1 o i=1,...,m such
ha o αi,n
:= ⟨X<∞
0, ,ℓ
i
n⟩,weha e∥αi,n −αi∥ℍ2→0asn→∞. Using Doob’s
inequali y, we in pa icula ha e
E[︃(︃sup
≤T∫︂
0(αn
s−αs)⊤dWs)︃2]︃≤C
m
∑︂
i=1
E[︃∫︂T
0|αi,n
s−αi
s|2d ]︃
=
m
∑︂
i=1∥αi,n −αi∥2
ℍ2−→ 0.
Bu his eadily implies he i s claim, ha is,
∫︂·
0⟨X<∞
0,s ,ℓ
n⟩⊤dWs−→ ∫︂·
0αsdWs=M·ucp.
To show (3.8), since ∫︁·
0⟨X<∞
0,s ,ℓ
n⟩⊤dWsa e clea ly squa e-in eg able 𝔽X-ma in-
gales, we can no ice ha
in
ℓ∈(𝒲d+1)mE[︃sup
≤T(︃Z −∫︂
0⟨X<∞
0,s ,ℓ⟩⊤dWs)︃]︃≥in
M∈ℳ2
0
E[︂sup
≤T
(Z −M )]︂=y0.
On he o he hand, o any ixed squa e-in eg able ma ingale M, we know he e exis
sequences o wo ds ℓi=(ℓi
n)n∈ℕ⊆𝒲d+1such ha
lim
n→∞sup
≤T(︃Z −∫︂
0⟨X<∞
0,s ,ℓ
n⟩⊤dWs)︃=sup
≤T
(Z −M )in L2.
The e o e
E[︂sup
≤T
(Z −M )]︂=lim
n→∞E[︃sup
≤T(︃Z −∫︂
0⟨X<∞
0,s ,ℓ
n⟩⊤dWs)︃]︃
≥in
ℓ∈(𝒲d+1)mE[︃sup
≤T(︃Z −∫︂
0⟨X<∞
0,s ,ℓ⟩⊤dWs)︃]︃.
Taking he in imum o e all M∈ℳ2
0yields he claim. □
Nex , simila ly o he p imal case, we ansla e he minimisa ion p oblem (3.8)
in o a ini e-dimensional op imisa ion p oblem by disc e ising he in e al [0,T]and
1000 C. Baye e al.
unca ing he signa u e o some le el K. Mo e p ecisely, o 0 = 0<···<
N=T
and some K∈ℕ, we educe he minimisa ion p oblem (3.8) o
yK,N
0=in
ℓ∈(𝒲d+1
≤K)m
E[︂max
0≤n≤N(Z n−Mℓ
n)]︂,(3.9)
whe e o any ℓ=(ℓ1,...,ℓ
m)∈(𝒲d+1
≤K)m, we de ine
Mℓ
=∫︂
0⟨X≤K
0,s ,ℓ⟩⊤dWs=
m
∑︂
i=1∫︂
0⟨X≤K
0,s ,ℓ
i⟩dWi
s.
The disc e e e sion o he dual o mula ion (3.7) is gi en by
yN
0=in
M∈ℳ2,N
0
E[︂max
0≤n≤N(Z n−M n)]︂,
whe e ℳ2,N
0deno es he space o disc e e squa e-in eg able 𝔽X,N -ma ingales. The
ollowing esul shows ha he minimisa ion p oblem (3.9) has a solu ion and he
alue con e ges o yN
0as he le el Ko he signa u e goes o in ini y. The p oo can
be ound in Appendix A.2.
P oposi ion 3.8 The e exis s a minimise ℓ⋆ o (3.9)and
|yN
0−yK,N
0|−→0as K→∞.
Rema k3.9 In a inancial con ex , P oposi ions 3.3 and 3.8 ell us ha yK,N
0con e ges
o he Be mudan op ion p ice as K→∞. Mo eo e , he iangle inequali y gi es
|y0−yK,N
0|≤|y0−yN
0|+|yN
0−yK,N
0|,
and hence he ini e-dimensional app oxima ions con e ge o y0as K,N →∞
whene e he Be mudan p ice con e ges o he Ame ican p ice. Fo ou nume ical
examples, we always app oxima e yN
0 o some ixed N, and he e o e we do no
u he in es iga e he la e con e gence he e.
3.3.1 Sample a e age app oxima ion (SAA)
We now p esen a me hod o app oxima e he alue yK,N
0in (3.9) by using Mon e
Ca lo simula ions. This p ocedu e is called sample a e age app oxima ion (SAA)
and we e e o Shapi o [38] o a gene al and ex ensi e s udy o his me hod. Sim-
ila ly o he p imal case, we in oduce wo u he app oxima ion s eps. Fi s , le
0=s0<···<s
J=Tbe a ine disc e isa ion o [0,T]and deno e by Mℓ,J , esp.
X<∞(J ), an app oxima ion o he s ochas ic in eg al ∫︁⟨X<∞
0,s ,ℓ⟩dWs, esp. he sig-
na u e X<∞, by using an Eule scheme. Second, we conside i=1,...,M i.i.d.
P imal and dual op imal s opping wi h signa u es 1001
sample pa hs Z(i),M(i),ℓ,J and eplace he expec a ion in (3.9) by a sample a e age,
leading o he empi ical minimisa ion p oblem
yK,N,J,M
0=in
ℓ∈(𝒲d+1
≤K)m
1
M
M
∑︂
i=1
max
0≤n≤N(Z(i)
n−M(i),ℓ,J
n). (3.10)
The ollowing esul can be deduced om Shapi o [38, Theo em 4] combined wi h
P oposi ion 3.8. We e e o Appendix A.2 o he de ails.
P oposi ion 3.10 Fo Mla ge enough, he e exis s a minimise β⋆ o (3.10), and
lim
K→∞ lim
J→∞ lim
M→∞yK,N,J,M
0=yN
0,
whe e he con e gence wi h espec o Mis almos su e con e gence.
Rema k 3.11 Le us quickly desc ibe how we sol e (3.10) nume ically. Conside he
numbe D:= ∑︁K
k=0(d +1)k, which co esponds o he numbe o en ies o he
K-s ep signa u e. No ice ha o any elemen ℓ∈𝒲d+1
≤K, we ha e he ep esen a-
ion ℓ=λ1w1+···+λDwD, whe e w1,...,w
Da e all possible wo ds o leng h
a mos K. Since ⟨X≤K
0, ,ℓ⟩=∑︁D
=1λ ⟨X≤K
0, ,w
⟩, he minimisa ion (3.10) has he
equi alen o mula ion
yK,N,J,M
0=in
λ∈(ℝD)m
1
M
M
∑︂
i=1
max
0≤n≤N(︃Z(i)
n−
D
∑︂
=1
λ M(i),w ,J
n)︃.
As desc ibed in Desai e al. [22], he abo e minimisa ion p oblem is equi alen o he
linea p og am
min
x∈ℝM+D
1
M
M
∑︂
j=1
xjsubjec o Ax ≥b,
whe e A∈ℝM(N+1)×(M+D) wi h A=[A1,...,A
M]⊤, and Ax ≥b ep esen s he
cons ain s
xi≥Z(i)
n−
D
∑︂
=1
M(i),w ,J
n,i=1,...,M,n=0,...,N.
Rema k 3.12 A solu ion ℓ⋆ o (3.10) yields he squa e-in eg able 𝔽X,N -ma in-
gale Mℓ⋆, and by e-simula ing ˜
Mi.i.d. samples o Z, he B ownian mo ion Wand
he signa u e X<∞, we can no ice ha he esul ing es ima o yK,N,J, ˜
M
0is uppe -
biased, ha is, yK,N,J, ˜
M
0≥yN
0, since he la e is de ined by aking he in imum o e
all squa e-in eg able 𝔽X,N -ma ingales.
1002 C. Baye e al.
4 Nume ical examples
In his sec ion, we s udy wo non-Ma ko ian op imal s opping p oblems and es
ou me hods o app oxima e lowe and uppe bounds o he op imal s opping alue.
The de ails o he implemen a ions and examples can be ound a h ps://gi hub.com/
lucapelizza i/Op imal_S opping_wi h_signa u es.
Rema k 4.1 In all he nume ical expe imen s pe o med below, we did no obse e
any signi ican di e ence when using he no malised signa u e (choosing he same
no malisa ion he au ho s in oduce in Che y e and Obe hause [17, Sec . 3]) and
when using he s anda d, non-no malised signa u e. Since he enso no malisa ion
inc eases he complexi y o he algo i hms, all he esul s p esen ed he e we e ob-
ained using he s anda d signa u e.
4.1 Op imal s opping o ac ional B ownian mo ion
We s a wi h he ask o op imally s opping a ac ional B ownian mo ion ( Bm),
which ep esen s he canonical choice o a amewo k lea ing he Ma ko and semi-
ma ingale egimes. Recall ha an Bm wi h Hu s pa ame e H∈(0,1)is he unique
con inuous Gaussian p ocess (XH
) ∈[0,T ]wi h
E[XH
]=0,∀ ≥0,
E[XH
sXH
]=1
2(|s|2H+| |2H−| −s|2H), ∀s, ≥0;
see e.g. F iz and Hai e [25, Chap. 9] o mo e de ails. We wish o app oxima e he
alue
yH
0=sup
τ∈𝒮0
E[XH
τ],H∈(0,1), (4.1)
om below and abo e. This example has al eady been s udied in Becke e al. [8,
Sec . 4.3] as well as in Baye e al. [5, Sec . 8.1], and we compa e he esul s below.
Since XHis one-dimensional and α-Hölde -con inuous o any α<H, i s (scala )
ough-pa h li is gi en by
(︃1,XH
s, ,1
2(XH
s, )2,..., 1
L!(XH
s, )L)︃∈Cα
g([0,T];ℝ),
whe e L=⌊1
α⌋. We can ex end i o a geome ic ough-pa h li XH∈ˆ︁
Cα
g([0,T];ℝ2)
o he ime-augmen a ion ( , XH
), as o ins ance desc ibed in [5, Example 2.4]. To
nume ically sol e (4.1), we eplace he con inuous- ime in e al [0,T]by some ini e
g id poin s 0 = 0<
1<···<
N=T. Below we compa e ou esul s wi h [8,
Sec . 4.3], whe e he au ho s chose N=100. Be o e doing so, an impo an ema k
abou he di e ence o ou p oblem o mula ion is in o de .
P imal and dual op imal s opping wi h signa u es 1003
Rema k 4.2 In [8], he au ho s li XH o a 100-dimensional Ma ko p ocess o he
o m ˆ︁
X k=(XH
k,...,XH
1,0,...,0)∈ℝ100, and hey conside he co esponding
disc e e (!) il a ion ˆ︁
ℱk=σ(XH
k,...,XH
1),k=0,...,100. No ice ha his di e s
om ou se ing as we conside he bigge il a ion ℱk=σ(XH
s:s≤ k)(see
Sec s. 3.2 and 3.3) ha con ains he whole pas o XH, no only he in o ma ion
a he pas exe cise da es. Thus in gene al yH
0domina es he lowe bounds om [8],
simply because ou il a ion con ains mo e s opping imes. Simila ly, he ( e y sha p)
uppe bounds in [8] we e ob ained by using a nes ed Mon e Ca lo app oach, which
cons uc s (ˆ︁
ℱk)-ma ingales ha a e no ma ingales in ou il a ion, and hus hei
uppe bounds a e no necessa ily uppe bounds o (4.1).
In Table 1, we p esen in e als o he op imal s opping alues yH
0 o Hu s pa-
ame e s H∈{0.01,0.05,...,0.95}, whe e he lowe ( esp. uppe ) bounds we e
app oxima ed by using Longs a –Schwa z wi h signa u es, esp. he SAA app oach
desc ibed in Sec . 3. We unca e he signa u e a he le el K=6 and apply he p i-
mal app oach using M=106samples o bo h he eg ession and he e-simula ion,
and o he dual app oach, we choose M=15’000 o sol e he linea p og am
om Rema k 3.11 and e-simula e wi h M=105samples. In he i s column, we
choose he ime disc e isa ion o he signa u e equal o he numbe o exe cise da es
as J=N=100. While he lowe bounds a e e y close, ou uppe bounds exceed
hose om [8]. This obse a ion ma ches wi h he commen s made in Rema k 4.2 as
Table 1 In e als o op imal
s opping o Bm: we show
H↦→yH
0wi h N=100
exe cise da es and disc e isa ion
J=100 (le column), J=500
(middle column), and in e als
om [8] ( igh column). The
o e all Mon e Ca lo e o is
below 0.003
HJ=100 J=500 Becke e al. [8]
0.01 [1.518,1.645][1.545,1.631][1.517,1.52]
0.05 [1.293,1.396][1.318,1.382][1.292,1.294]
0.1 [1.045,1.129][1.065,1.117][1.048,1.05]
0.15 [0.83,0.901][0.847,0.895][0.838,0.84]
0.2 [0.654,0.706][0.663,0.698][0.657,0.659]
0.25 [0.507,0.538][0.510,0.533][0.501,0.505]
0.3 [0.363,0.396][0.371,0.392][0.368,0.371]
0.35 [0.248,0.272][0.255,0.270][0.254,0.257]
0.4 [0.153,0.168][0.155,0.165][0.154,0.158]
0.45 [0.069,0.077][0.068,0.076][0.066,0.075]
0.5 [−0.001,0][−0.002,0][0,0.005]
0.55 [0.061,0.071][0.060,0.066][0.057,0.065]
0.6 [0.112,0.133][0.112,0.124][0.115,0.119]
0.65 [0.163,0.187][0.163,0.175][0.163,0.166]
0.7 [0.203,0.234][0.205,0.220][0.206,0.208]
0.75 [0.242,0.273][0.240,0.260][0,242,0.245]
0.8 [0.275,0.306][0.281,0.298][0.276,0.279]
0.85 [0.306,0.335][0.301,0.324][0.307,0.31]
0.9 [0.331,0.357][0.337,0.356][0.335,0.339]
0.95 [0.367,0.381][0.366,0.381][0.365,0.367]
1004 C. Baye e al.
we conside he il a ionˆ︁
𝔽in his case o he lowe bounds; bu ou uppe bounds a e
by cons uc ion uppe bounds o he con inuous p oblem wi h il a ion 𝔽, and he
con inuous ma ingale is app oxima ed only a he exe cise da es. By inc easing he
disc e isa ion o J=500 and he eby adding in o ma ion o he il a ion be ween
exe cise da es, one can see o small H(≤0.2) ha he lowe bounds exceed he
in e als om [8], showing ha e en o N=100 poin s in [0,1], he in o ma ion
be ween exe cise da es is ele an o op imally s opping he Bm.
4.2 Ame ican op ions in ough ola ili y models
The second example we p esen is he p oblem o p icing Ame ican op ions in ough
ola ili y models. Mo e p ecisely, we conside he one-dimensional asse -p ice model
X0=x0,dX
= X d +X (ρdW +√︂1−ρ2dB ), 0< ≤T,
whe e Wand Ba e wo independen B ownian mo ions, he ola ili y ( ) ∈[0,T ]is
an 𝔽W-adap ed con inuous p ocess, ρ∈[−1,1]and >0 is he in e es a e. Fo any
payo unc ion ϕ:[0,T]×ℝ→ℝ, we wan o app oxima e he op imal s opping
p oblem
y0=sup
τ∈𝒮0
E[e− τϕ(τ,Xτ)],(4.2)
whe e 𝒮0is he se o 𝔽:= (𝔽W∨𝔽B)-s opping imes alued in [0,T].I iswo h
no ing ha ou me hod does no depend on he speci ica ion o , and as soon as we
can sample om (X, ), we can apply he me hod o app oxima e alues o Ame ican
op ions.
In he ollowing, we ocus on he ough Be gomi model (see Baye e al. [4]), ha
is, we speci y he ola ili y as
=ξ0ℰ(︃η∫︂
0( −s)H−1
2dWs)︃,
whe e ℰdeno es he s ochas ic exponen ial, and we conside he pa ame e s
x0=100, =0.05, η=1.9, ρ=−0.9, ξ0=0.09. Mo eo e , we conside pu
op ions wi h ϕ( ,x) =(K −x)+ o di e en s ikes K∈{70,80,...,120}, wi h
ma u i y T=1 and N=12 exe cise da es. Thus we w i e (4.2) as he disc e e
op imal s opping p oblem
yN
0=sup
τ∈𝒮N
0
E[e− τ(K −Xτ)+],
whe e 𝒮N
0is desc ibed a he beginning o Sec . 3.2. Mo eo e , o some ine g id
s0=0<s
1<··· <s
J=1, J∈ℕ, and ixed signa u e le el Kand sample
size M, we deno e by yLS
0 he alue yK,N,J,M
0de ined in P oposi ion 3.4 and by
ySAA
0 he alue yK,N,J,M
0de ined in (3.10). We compa e ou esul s wi h he lowe
bounds ob ained in Baye e al. [7] o H=0.07, esp. in Goudenege e al. [29] wi h
H=0.07 and H=0.8.
P imal and dual op imal s opping wi h signa u es 1011
age app oxima ion o y0is hen gi en by
yM
0=min
x∈𝒳FM(x). (A.7)
The ollowing esul p o ides su icien condi ions o he con e gence yM
0→y0
as M→∞, and a mo e gene al e sion can be ound in [38, Theo em 4].
Theo em A.2 Suppose ha
(1) Fis measu able and x↦→ F(x,η)is lowe semicon inuous o all η∈ℝd;
(2) x↦→ F(x,η)is con ex o almos e e y η∈ℝd;
(3) 𝒳is closed and con ex;
(4) (x):= E[F(x,ξ)]is lowe semicon inuous and (x)<∞ o all x∈𝒳;
(5) he se So solu ions o (A.6)is non-emp y and bounded.
Then yM
0→y0as M→∞.
Simila ly as in he p oo o P oposi ion 3.4, o a ixed unca ion le el K∈ℕ,
we deno e by D he numbe o componen s o he unca ed signa u e, which is gi en
by D=∑︁K
k=0(d +1)k. Now e e y ℓ∈𝒲d+1
≤Kis o he o m ℓ=∑︁D
i=1βiwi o
some β∈ℝD, whe e w1,...,w
Da e all wo ds o leng h a mos K. In pa icula ,
o e e y ℓ∈(𝒲d+1
≤K)m, he e is a β∈ℝm×Dand ou usual no a ion eads
Mℓ
=∫︂
0⟨X≤K
0,s ,ℓ⟩⊤dWs=
D
∑︂
j=1
(βj)⊤∫︂
0⟨X≤K
0,s ,w
j⟩dWs,
so ha ins ead o minimising o e ℓ, we can minimise o e β. Le us now o mula e
he minimisa ion p oblem in P oposi ion 3.10 in he language o Theo em A.2 as
E[︂max
0≤n≤N(Z n−Mℓ
n)]︂=E[︃max
0≤n≤N(︃Z n−
m
∑︂
i=1
D
∑︂
j=1
βij ∫︂
0⟨X≤K
0,s ,w
j⟩dWi
s)︃]︃
=E[︂max
0≤n≤N(Z n−𝜷⊤M n)]︂,
whe e we iden i y β∈ℝm×D∼
=ℝmD and M is he (mD)-dimensional ec o
(︃∫︂
0⟨X≤K
0,s ,w
j⟩dWi
s:1≤i≤m, 1≤j≤D)︃.
Finally, de ining he andom ec o
ξ:= (Z 0,M⊤
0,...,Z
N,M⊤
N)∈ℝ(D+1)(N+1)m,
we no ice ha
in
ℓ∈(𝒲d+1
≤K)m
E[︂max
0≤n≤N(Z n−Mℓ
n)]︂=min
x∈ℝmD E[F(x,ξ)],(A.8)
1012 C. Baye e al.
whe e
F(x,η) := max
0≤n≤N(︃ηn(mD+1)−x⊤⎛
⎜
⎝
ηn(mD+1)+1
.
.
.
ηn(mD+1)+1+mD
⎞
⎟
⎠)︃.(A.9)
The e o e, se ing 𝒳=ℝDm and d=(D +1)(N +1)m, he minimisa ion p oblem
o yK,N,J,M
0in P oposi ion 3.10 can simply be w i en in he SAA o mula ion (A.7),
whe e addi ionally he s ochas ic in eg als in Ma e eplaced by he disc e ised e -
sions MJ.
P oo o P oposi ion 3.10 Fi s , i is possible o ew i e he p oo o Lemma A.1
o FMins ead o he expec a ion, o show ha he e exis s a minimise ℓ⋆ o (3.10).
Mo eo e , deno e by yK,N,J
0 he minimisa ion p oblem (A.8), whe e we eplace Mℓ
by he disc e ised e sion Mℓ,J desc ibed in Sec . 3.3.1. By he same easoning as
in P oposi ion 3.8, we can show he limi limK→∞ limJ→∞ yK,N,J
0=yN
0.Now
o ixed K,J,N, we a e le wi h showing almos su e con e gence o yK,N,J,M
0
o yK,N,J
0. Bu his can be deduced om Theo em A.2 i we can show ha (1)–(5)
hold ue o ou Fin (A.9). Clea ly, Fis measu able and i is easy o see ha
x↦→ F(x,η) is con inuous and con ex; hus (1) and (2) eadily ollow. Mo eo e ,
(3) holds ue, and in o de o show (4), se (x) =E[F(x,ξ)]and no ice ha o
x1,x
2∈𝒳,weha e
| (x
1)− (x
2)|≤E[︂max
0≤n≤N(︁(x1−x2)⊤MJ
n))︁]︂≤|x1−x2|E[︂max
0≤n≤N|MJ
n|]︂,
whe e we simply applied he iangle and Cauchy–Schwa z inequali ies. Since we
ha e Mℓ∈L2 o all ℓ, an applica ion o Doob’s inequali y shows o he igh -
hand side ha E[max0≤n≤N|MJ
n|] <∞, and he e o e (4) ollows. Finally, non-
emp yness o S ollows om Lemma A.1, and he p oo o he la e e eals ha
E[F(x,ξ)]→∞as |x|→∞. Thus Smus be bounded, which inishes he p oo .
□
Acknowledgemen s The au ho s would like o hank C. Cuchie o and S. B eneis o aluable ema ks and
help ul discussions abou he global app oxima ion in Sec . 2.
Funding in o ma ion Open Access unding enabled and o ganized by P ojek DEAL.
Decla a ions
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P imal and dual op imal s opping wi h signa u es 1013
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and ins i u ional a ilia ions.
