Discrete-Type Approximations for Non-Markovian Optimal Stopping Problems: Part I 3

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10, 11]). More recent approaches based on path-dependent PDEs (see e.g. [18]) yield differential representations of the Snell envelope (even in the fully nonlinear case) under pathwise regularity on the obstacles related to the reflected BSDE. We also emphasize the recent representation result by [20] for the Snell envelope in terms of true BSDEs defined on an enlarged probability space, containing a jump part and involving sign constraints. We observe the use of reflected BSDEs and path-dependent PDEs to solve concretely the optimal stopping problem beyond the Markovian case seems to be a very hard task. The key difficulty is that solving an optimal stopping time problem for a fully non- Markovian state is essentially a stochastic optimal control problem driven by an infinite-dimensional state. Optimal stopping time problems based on reward path-dependent functionals of fully non-Markovian states arise in many applications and it remains a challenging task to design concrete ways of solving it. One typical example is the problem of pricing American options written on stochastic volatility models (X, V ) where the volatility variable V is a functional of fractional Brownian motion (FBM) with long-range dependence (see e.g [15] and other references therein) or short-range dependence as recently observed by many works (see e.g. [4, 22, 21] and other references therein). In this case, one cannot interpret (X, V ) as an “augmented” finite-dimensional Markov process due to non-trivial cor- relations associated with FBM and hence a concrete solution of the related optimal stopping problem is very challenging. We stress even optimal stopping problems based on finite-dimensional “augmented” Markovian-type models (X, V ) (e.g CIR-type models driven by Brownian motion) are not easy to solve it (see e.g [31, 1]). In practice, the volatility V is not directly observed so that it has to be approximated. If the volatility is estimated with accuracy, then standard Markovian methods can be fairly implemented to solve the control problem. Otherwise, the optimal stopping problem is non-Markovian. The goal of this paper is to present a systematic method of constructing continuation regions and optimal values for optimal stopping problems based on arbitrary continuous reward processes Z. We are particularly interested in the non-Markovian case, i.e., Z = F (X) where F is a path-dependent functional of an underlying non-Markovian continuous process X which is adapted to a filtration generated by a d-dimensional Brownian motion B. In our philosophy, the reward process Z is viewed as a generic non-anticipative functional of B, i.e. (1.2) B 7→ Z(B) rather than X and it may be possibly based on different kinds of states not reducible to vectors of Markov processes: solutions of path-dependent SDEs, processes involving fractional Brownian motions and other singular functionals of Brownian motions. The methodology is based on Leão and Ohashi [27] and Leão, Ohashi and Simas [28] who show a very strong continuity/stability of a large class of Wiener functionals (1.2) w.r.t continuous-time D k random walk structures driven by suitable waiting times Tn ; n, k ≥ 1 (see (2.1) and (2.2)) which encode the evolution of the Brownian motion at small scales ǫk ↓ 0 as k → +∞. Here, ǫk = ϕ(k) for a strictly decreasing function with inverse ξ. The main idea is to reduce the dimension by discretizing the Brownian filtration at a given time t by

k k k k (1.3) Ft := Dj ∩{Tj ≤ t

k k k k k (1.4) Aj := ∆T1 , η1 ,..., ∆Tj , ηj taking values on a cartesian product of a ﬁnite-dimensional discret e space (involving suitable signs k D (ηj )j≥1 of ) times [0,T ]. In contrast to previous discretization schemes, our method has to be k k interpreted as a space-ﬁltration discretization scheme: (i) at random intervals Tn ≤ t

Brownian motion lies on open subsets with dyadic end points of the form (ℓ − 1)ǫk, (ℓ + 1)ǫk for Z k k k k k k ℓ ∈ and −n ≤ ℓ ≤ n, (ii) Ft has the important property that Ft ∩{Tn ≤ t

2. Preliminaries Throughout this article, we are going to fix a d-dimensional Brownian motion B = {B1,...,Bd} on the usual stochastic basis (Ω, F, P), where Ω is the space C([0, +∞); Rd) := {f : [0, +∞) → Rd continuous}, equipped with the usual topology of uniform convergence on compact intervals. P is the Wiener measure on Ω such that P{B(0) = 0} = 1 and F := (Ft)t≥0 is the usual P-augmentation of the natural filtration generated by the Brownian motion. In the sequel, we recall the basic structure of the theory presented by the works [27, 28] and we refer the reader to these works for all unexplained points. 2 For a fixed positive sequence {ǫk; k ≥ 1} such that k≥1 ǫk < +∞, and for each j = 1,...,d, we k,j define T0 := 0 a.s. and we set P

k,j k,j j j k,j (2.1) Tn := inf Tn−1

∞ k,j k,j A (t) := ǫk σ 11 k,j ; t ≥ 0, n {Tn ≤t} n=1 X where

1; if Bj (T k,j ) − Bj (T k,j ) > 0 σk,j := n n−1 n −1; if Bj (T k,j ) − Bj (T k,j ) < 0, n n−1 Fk,j k,j for k,n ≥ 1 and j = 1,...,d. Let := {Ft ; t ≥ 0} be the natural filtration generated by {Ak,j (t); t ≥ 0}. The multi-dimensional filtration generated by Ak is naturally characterized as follows. Fk k k k,1 k,2 k,d Let := {Ft ;0 ≤ t < ∞} be the product filtration given by Ft := Ft ⊗Ft ⊗···⊗Ft for k t ≥ 0. Let T := {Tm; k,m ≥ 0} be the order statistics obtained from the family of random variables k,j k {Tℓ ; ℓ, k ≥ 0; j =1,...,d}. That is, we set T0 := 0,

k k,j k k,j k,j k (2.2) T1 := inf T1 ,Tn := inf Tm ; Tm ≥ Tn−1 1≤j≤d 1≤j≤d n o m≥1 n o for n ≥ 1. The structure D := {T , Ak,j ;1 ≤ j ≤ d, k ≥ 1} is a discrete-type skeleton for the Brownian motion in the language of [28] (see Def. 2.1 in [28]). Throughout this article, we set

k k k k k ¯ k k (2.3) ∆Tn := Tn − Tn−1, N (t) := max{n; Tn ≤ t}, tk := max{Tn ; Tn ≤ t}; t ≥ 0, and k,j k 1; if ∆A (Tn ) > 0 k,j k,j k ηn := −1; if ∆A (Tn ) < 0  k,j k  0; if ∆A (Tn )=0, k k,1 k,d for n ≥ 1. Let us denote ηn := ηn ,...,ηn , d I k k k k k := (i1 ,...,id); iℓ ∈ {−1, 0, 1} ∀ℓ ∈{1,...,d} and |ij | =1 j=1 n X o DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 5

S I S Sn and k := (0, +∞) × k for k,n ≥ 1. The n-fold Cartesian product of k is denoted by k and a Sn k k ˜k k ˜k Sn k ˜k I generic element of k will be denoted by bn := (s1 , i1,...,sn, in) ∈ k where (sr , ir ) ∈ (0, +∞) × k for 1 ≤ r ≤ n. We also denote

j k k tj := sℓ , ℓ X=1 k k j for a given list (s1 ,...,sj ) ∈ (0, +∞) where j ≥ 1. The driving noise in our methodology is given by the following discrete-time process

k k k k k Sn (2.4) An := ∆T1 , η1 ,..., ∆Tn , ηn ∈ k a.s. k k −1 Sn P Sk One should notice that F k = (An) (B( )) up to -null sets, where B( n) is the Borel σ-algebra Tn k Sn generated by k ; n ≥ 1.

Transition Probabilities. The law of the system will evolve according to the following probability measure deﬁned on

Pk P k Sn n(E) := {An ∈ E}; E ∈B( k ),n ≥ 1. Pk Pk Sr−n Pk Sn By the very deﬁnition, n(·) = r (·× k ) for any r>n ≥ 1. By construction r ( k × ·) is a S Pk regular measure and B( k) is countably generated, then it is well-known there exists ( n-a.s. unique) k Sr−n Sn a disintegration νn,r : B( k ) × k → [0, 1] which realizes

Pk k k k k k Pk k r (D)= 11D(bn, qn,r)νn,r(dqn,r|bn) n(dbn), Sn Sr−n Z k Z k Sr k k for every D ∈ B( k), where qn,r is the projection of br onto the last (r − n) components, i.e., k k ˜k k ˜k k k ˜k k ˜k Sr k Pk qn,r = (sn+1, in+1,...,sr , ir ) for a list br = (s1 , i1,...,sr , ir ) ∈ k. Sometimes, we denote ν0,r := r k k for r ≥ 1. If r = n + 1, we denote νn+1 := νn,n+1. For an explicit formula for this conditional probability, we refer the reader to [30]. Let Bp(F) be the space of c`adl`ag F-adapted processes Y such that

p E p kY kBp := kY k∞ < ∞ where kY k∞ := sup0≤t≤T |Y (t)|, 1 ≤ p< ∞ and 0

k p Deﬁnition 2.1. We say that Y = (X )k≥1, D is an imbedded discrete structure for X ∈ B (F) if Xk is a sequence of Fk-adapted pure jump processes of the form

∞ k k k (2.5) X (t)= X (T )11 k k ;0 ≤ t ≤ T, n {Tn ≤t

k (2.6) lim kX − XkBp =0 k→+∞ for some p ≥ 1. 6 DORIVALLEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

3. Construction of an imbedded structure for S and near optimal stopping times

For t ≤ T , we denote Tt(F) as the set of all F-stopping times τ such that t ≤ τ ≤ T a.s.. For n ≥ 0, F F k F we denote by Tk,n( ) := Tt( ) for t = Tn . To shorten notation, we set Tk,n := Tk,n( ). Throughout this article, we assume that reward process Z is an F-adapted continuous process and it satisﬁes the integrability regularity condition

p (A1) kZkBp < ∞ ∀p ≥ 1.

For a given reward process Z, let S be the Snell envelope associated with Z

S(t) := ess sup E [Z(τ) |Ft] , 0 ≤ t ≤ T. τ∈Tt(F) We assume S satisﬁes the following integrability condition:

p (A2) kSkBp < ∞ ∀p ≥ 1.

Assumptions (A1-A2) are not essential in the sense that the range of integrability can be relaxed. In order to simplify the exposition, we then assume (A1-A2) hold true. Since the optimal stopping time problem at hand takes place on the compact set [0,T ], it is crucial to know the correct number of periods in our discretization scheme. For this purpose, let ⌈x⌉ be the smallest natural number bigger or equal to x ≥ 0. We then denote

−2 e(k,T ) := d⌈ǫk T ⌉; k ≥ 1. Lemma 3.1. For any 0

k k,i k,i − |Td ǫ 2T − T |≤ max | max T −2 − T |, | min T −2 − T | a.s. ⌈ k ⌉ 1≤i≤d ⌈ǫk T ⌉ 1≤i≤d ⌈ǫk T ⌉ for every k ≥ 1. We then apply Lemman 2.2 in [25] to conclude the proof. o Due to this result, we will reduce the analysis to the deterministic number of periods e(k,T ). We k,m Fk denote Dn as the set of all -stopping times of the form

m k τ = T 11 k , i {τ=Ti } i=n k X k where {τ = Ti ; i = n,...,m} is a partition of Ω and 0 ≤ n ≤ m. Let us denote Dn,T := {η ∧ T ; η ∈ k,∞ k,∞ Fk Dn }. We observe that D0 is the set of all -totally inaccessible stopping times. Let {Zk; k ≥ 1} be a sequence of pure jump processes of the form (2.5) and let {Sk; k ≥ 1} be the associated value process given by

∞ k k k (3.1) S (t) := S (T )1 k k k ;0 ≤ t ≤ T, n {Tn ≤t∧Te(k,T )

k k k k k k Y,k,p k ∆S (Tn+1) k S (T ) := ess sup E Z (τ ∧ T ) F k and U S(T ) := E F k . n Tn n 2 Tn τ Dk,e(k,T ) " ǫk # ∈ n h i

DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 7

The following result shows the fundamental role played by the imbedded discrete structure driven by a skeleton D := {T , Ak,j ;1 ≤ j ≤ d, k ≥ 1}: It enables to lift the discrete structure into the Brownian ﬁltration without loosing Fk-adaptedness in the optimization problem. We postpone the proof of Proposition 3.1 to Section 4. Proposition 3.1. Let Zk be a pure jump process of the form (2.5). For each n ≥ 0 and k ≥ 1, we have

E k k E k k ess sup Z τ |FT k = esssup Z τ |FT k a.s. k n τ∈T n τ∈Dn,T k,n h i h i 3.1. ǫ-optimal stopping times. The dynamic programming algorithm allows us to deﬁne the stopping and continuation regions as follows

k Si Zk k Vk k S(i, k) := bi ∈ k; i (bi )= i (bi ) (stopping region) n o k Si Vk k Zk k D(i, k) := bi ∈ k; i (bi ) > i (bi ) (continuation region) Vk n Sn R Zk Sn Ro where 0 ≤ i ≤ e(k,T ) and n : k → and n : k → are Borel functions which realize

k k Vk k k k Zk k (3.2) S (Tn )= n(An) a.s and Z (Tn ∧ T )= n(An) a.s; n =0,...,e(k,T ). k k k k,m k m Let Y (i) := Z (Ti ∧ T ); i ≥ 0. and we set Jn as the set of all (F k )i=0-stopping times taking Ti values on {n,n +1,...,m} for a given 0 ≤ n

k k k k (3.4) ess sup E Y (η)|F k =ess sup E Z (τ ∧ T )|F k a.s. Tn Tn k,e(k,T ) k,e(k,T ) η∈Jn h i τ∈Dn h i for each 0 ≤ n ≤ e(k,T ). The dynamic programming principle can be written as

k τe(k,T ) := e(k,T ) (3.5) k k τ := j11 k + τ 11 k c ;0 ≤ j ≤ e(k,T ) − 1 ( j Gj j+1 (Gj ) where

k Zk k E Zk k k Gj := j (Aj ) ≥ τ k (Aτ k ) Aj ;0 ≤ j ≤ e(k,T ) − 1 ( j+1 j+1 ) h i k k k Sj R and τ = τ0 a.s. The sequence of functions Uj : k →

k k k E Zk k k k (3.6) bj 7→ Uj (bj ) := k (A k ) Aj = bj ;0 ≤ j ≤ e(k,T ) − 1 τj+1 τj+1 are called continuation values and theyh play a key role in thei obtention of the optimal value.

The value functional which gives the best payoﬀ can be reconstructed by means of the dynamic programming principle over the e(k,T )-steps which provides 8 DORIVALLEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

E k Zk E Vk k sup Y (η) = max 0 (0); 1 (A1 ) , η∈Jk,e(k,T ) 0 n o E Vk k E Zk k where 1 (A1 ) = k (A k ) . Moreover, τ1 τ1 h i E k k E k k E k k E k k (3.7) Y (τ ) = Z (Tτ k ∧ T ) = sup Z (τ ∧ Te(k,T )) = sup Z (τ ∧ Te(k,T )) k τ∈T (F) τ∈D0,T 0 where the last fundamental equality in identity (3.7) is due to Proposition 3.1. Let us denote

k E k k (3.8) V0 := sup Z (τ ∧ Te(k,T )) ; k ≥ 1. k τ∈D0,T We are now able to state the main results of this article. The proof of Theorem 3.1 is postponed to Section 4. Theorem 3.1. Let S be the Snell envelope associated to a reward process Z satisfying assumptions (A1-A2). Let {Zk; k ≥ 1} be a sequence of pure jump processes of the form (2.5) and let {Sk; k ≥ 1} k be the associated value process given by (3.1). If Z = (Z )k≥1, D is an imbedded discrete structure for Z where (2.6) holds for p > 1, then S = (Sk) , D is an imbedded discrete structure for S, k≥1 where lim E sup |Sk(t) − S(t)| =0. k→+∞ 0≤t≤T Moreover, {Sk; k ≥ 1} is the unique sequence of pure jump processes of the form (2.5) which satisﬁes the following variational inequality

Y,k,p k k k k k (3.9) max U S(Ti ); Z (Ti ∧ T ) − S (Ti ) = 0 i = e(k,T ) − 1,..., 0, a.s. n k k o k k S (Te(k,T )) = Z (Te(k,T ) ∧ T ) a.s. k Theorem 3.2. Let Z = (Z )k≥1, D be an imbedded discrete structure for the reward process Z and let τ k be the optimal stopping time given by (3.3). Then, T k ∧ T is an ǫ-optimal stopping time in the τ k Brownian ﬁltration, i.e., for a given ǫ> 0,

E E k sup Z(η) − ǫ< Z(Tτ k ∧ T ) η∈T0(F) for every k suﬃciently large. Moreover,

E k E k k (3.10) sup Z(τ) − V0 ≤ kZ (· ∧ Te(k,T )) − Zk∞ → 0 τ∈T (F) 0 as k → +∞. Proof. The imbedded discrete structure property, the path-continuity of Z and Lemma 3.1 yield E k k kZ (· ∧ Te(k,T )) − Zk∞ → 0 as k → +∞. This shows that

E k k E E k k (3.11) sup Z (τ ∧ Te(k,T )) − sup Z(τ) ≤ sup |Z (t ∧ Te(k,T )) − Z(t)| → 0 τ∈T (F) τ∈T (F) 0≤t≤T 0 0 as k → +∞ . By (3.7) (see also Proposition 3.1), (3.8) and (3.4), we conclude (3.10). Now, by using (3.7) and (3.11), for a given ǫ> 0, we have DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 9

E E k k E k k (3.12) sup Z(τ) − ǫ< sup Z (τ ∧ Te(k,T ) ∧ T ) = Z (Tτ k ∧ T ) τ∈T0(F) τ∈T0(F) k E k k for every k sufficiently large. The imbedded discrete structure property of Z yields Z (Tτ k ∧T ) < k k E Z(T ∧ T ) + ǫ for every k sufficiently large. This implies sup F E Z(τ) − 2ǫ< E Z(T ∧ T ) τ k τ∈T0( ) τ k for every k sufficiently large. It remains to show T k is an F-stopping time. Clearly, T k :Ω → R is τ k τ k + F-measurable. We claim that

k ′ k k ′ Tτ k(ω)(ω) ≤ t and ω|[0,t] = ω |[0,t] then Tτ k(ω)(ω)= Tτ k(ω′)(ω ). k k k Recall τ (ω) = min 0 ≤ j ≤ e(k,T ); Aj (ω) ∈ S(j, k) and we notice that Tτ k(ω)(ω) ≤ t means that k all information thatn we need to compute Tτ k(ω)(ω) lieso on the Brownian path ω|[0,t]. This can be easily seen by the fact that each continuous time random walk Ak,j ; j =1,...,d only (possibly) jumps at the k ′ k stopping times (Tn )n≥0. In this case, if ω|[0,t] = ω |[0,t] and Tτ k(ω)(ω) ≤ t , then we necessarily have k k ′ Tτ k(ω)(ω) = Tτ k(ω′)(ω ). From Galmarino’s test (see [16] Nos. IV 94-103, pp 145-152), we conclude k that Tτ k is an (Ft)t≥0-stopping time, where (Ft)t≥0 is the raw filtration generated by the Brownian k F motion on the Wiener space. This shows that Tτ k ∧ T is an -stopping time and (3.12) allows us to conclude the proof.e e Remark 3.1. The importance of imbedded discrete structures is already apparent in Theorem 3.2: k e(k,T ) Maximization along the discrete type filtration (F k ) is essentially equivalent to maximization Tn n=0 along the Brownian filtration for k large enough. More importantly, the imbedded structure allows us to reduce dimensionality of the optimal stopping problem (see Section 5.3 and Example 5.1).

4. Proofs of Proposition 3.1 and Theorem 3.1 In order to prove convergence, we need a subtle pathwise argument on the conditional expectations involved in the optimization problem. For a given pure jump process of the form (2.5), there exists a k Sℓ R list of Borel functions hℓ : k → which realizes

∞ k k k Z (t)= h (A )1 k k a.s.;0 ≤ t ≤ T. ℓ ℓ {Tℓ ≤t

hk (bk ); if tk ≤ T Zk (bk )= n n n n n hk(bk); if T

r k k k (4.1) τ = Tℓ gℓ (Aℓ ) a.s ℓ n X= k k where g (A )= 1 k ; n ≤ ℓ ≤ r. The action space in our optimization problem is given by ℓ ℓ {τ=Tℓ }

j j j A := x = (x1,...,xj ) ∈ N ; xℓ =1 ; j > 1. ℓ n X=1 o Ar−n+1 Sr The elements of × k will be denoted by

k,n,r k k k o := an,...,ar , br . 10 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

k Sℓ k,gk Sr For a given list of Borel functions gℓ : k →{0, 1} realizing (4.1), we deﬁne the map Ξn,r : k → Ar−n+1 Sr × k given by

k,gk k k k k k k k Sr Ξn,r (br ) := gn(bn),...,gr (br ), br ; br ∈ k. k r k k k k Ar−n+1 We observe that we may choose (gℓ )ℓ=n in such way that (gn(bn),...,gr (br )) ∈ for every k Sr k Ar−n+1 Sr R br ∈ k. Moreover, there exists an explicit map γn,r : × k → such that

k k k,gk k (4.2) Z (τ ∧ T )= γn,r Ξn,r Ar a.s., where

r r r k k,n,r k k 1 k 1 k k k 1 1 k 1 k γ (o ) := h (b ) {1}(a ) k (b )+ h (b ) k k {1}(a ) k (b ) . n,r ℓ ℓ ℓ {tℓ ≤T } ℓ ℓ ℓ {tℓ ≤T T } j ℓ n ℓ n j=n ! X= X= X We should also compute a pathwise representation of Zk(τ ∧ T ) but for a slightly more general class of stopping times. Let Tk,n,r be the set of all F-stopping times of the form

r e k η = T 1 k ℓ {η=Tℓ } ℓ n X= k k,r where {η = T }∈F k ; n ≤ ℓ ≤ r. Of course, D ⊂ Tk,n,r for every 0 ≤ n ≤ r and k ≥ 1. Recall ℓ Tℓ n k that Tn < ∞ a.s for every n ≥ 0. In this case, it is known (see e.g. Corollary 3.22 in [23]) that k −1 (Φ ) O = F k , where O is the optional σ-algebra one Ω × R+ and j Tj

k k ∗ Φj (ω) := ω,Tj (ω) ; ω ∈ Ω , j ≥ 1, P ∗ k where (Ω ) = 1. To keep notation simple, we choose a version of Φj deﬁned everywhere and with a k slight abuse of notation we write it as Φj . Based on this fact, for a given η ∈ Tk,n,r there exists a list k R a Borel functions ϕℓ :Ω × + →{0, 1} which realizes e r k k k (4.3) η = Tℓ ϕℓ (Φℓ ) a.s. ℓ n X= k R k,ϕk For a list of Borel functions ϕℓ : Ω × + → {0, 1} realizing (4.3), we then deﬁne the map Ξn,r : Rr−n+1 Ar−n+1 Sr Ω × + → × k given by

k,ϕk k k k k k k Ξn,r (ω, xn,...,xr, br ) := ϕn Φn(ω, xn) ,...,ϕr Φr (ω, xr) , br , Rr−n+1 k Sr for ω ∈ Ω, (xn,...,xr) ∈ + and br ∈ k. By construction, we have

k k k,ϕk k (4.4) Z (η ∧ T )= γn,r Ξn,r Jn,r a.s., k k k k where Jn,r := Id,Tn ,...,Tr , Ar and Id:Ω → Ω is the identity map. In the sequel, Hk : B(Ω × Rr−n+1) × Sr → [0, 1] is the disintegration of P ◦ J k w.r.t Pk and νk n,r + k n,r r n,r Pk Pk is the disintegration of r w.r.t n for r>n ≥ 1. Recall the notation introduced in Section 2. k k,r Lemma 4.1. Let Z be a pure jump process of the form (2.5). For each τ ∈ Dn and η ∈ Tk,n,r, for 0 ≤ n < r, we have e E k k k k,gk k k k k k Z (τ ∧ T )|FT k = γn,r Ξn,r An, qn,r νn,r(dqn,r|An) a.s n Sj Z k h i DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 11 and

k k k k,ϕk k k k k k k k k E Z (η∧T )|F k = γ Ξ ω, xn,...,xr, A , q H (dωdxn . . . dxr|A , q )ν (dq |A ), Tn n,r n,r n n,r n,r n n,r n,r n,r n Sj Ω×Rj h i Z k Z + k r k r a.s for j = r − n, where (gℓ )ℓ=n and (ϕℓ )ℓ=n are Borel functions associated to τ and η, respectively. Proof. The proof is a direct consequence of the representations (4.2) and (4.4), so we omit the details.

We are now able to prove Proposition 3.1 which is the main ingredient in the proof of Theorem 3.1 and the estimate (3.10) in Theorem 3.2.

Proof of Proposition 3.1. For the moment, let us ﬁx r>n. We claim

E k k E k k (4.5) esssup Z τ ∧ T |FT k = esssup Z τ ∧ T |FT k a.s., e n n τ∈Tk,n,r τ∈T k,n,r h i h i F k k where T k,n,r := {τ is an − stopping time; Tn ≤ τ ≤ Tr a,s}. For a given τ ∈ T k,n,r, let us deﬁne F k k the -stopping time Q(τ) = min{Tp ; Tp > τ} and

r+1 k η(τ)= T 1 k ∈ Tk,n,r. ℓ−1 {Q(τ)=Tℓ } ℓ n X= P k k We observe that {η(τ)= Tn−1} = 0. Moreover, since Z hase c`adl`ag paths, we have

Zk(τ ∧ T )= Zk (Q(τ) ∧ T ) − = Zk η(τ) ∧ T a.s. This proves (4.5). We now claim that

E k k E k k (4.6) esssup Z τ ∧ T |FT k = esssup Z τ ∧ T |FT k a.s. e n k,r n τ∈Tk,n,r τ∈Dn h i h i In order to check (4.6), we make use of Lemma 4.1 as follows. Let us deﬁne the following objects:

k k k k k k Fn,r(bn) := max Gn,r(an,...,ar , bn), k k Ar−n+1 (an,...,ar )∈ where k k k k k k k k k k k k Gn,r(an,...,ar , bn) := γn,r an,...,ar , bn, qn,r νn,r(dqn,r|bn), Sr−n Z k and

k k k k k k Fn,r(bn) := max Gn,r(an,...,ar , bn), k k Ar−n+1 (an,...,ar )∈ where e e k k k k k k k k k k k k k k k G (a ,...,a , b ) := γ a ,...,a , b , q H (dωdxn . . . dxr|b , q )ν (dq |b ), n,r n r n − − n,r n r n n,r n,r n n,r n,r n,r n Sr n Ω×Rr n+1 Z k Z + e k Sk k k Ar−n+1 Sr−n Sr−n+1 for each bn ∈ n and (an,...,ar ) ∈ . Since for each E ∈ B( k )) and F ∈ B(Ω × k ), the disintegrations

k k k k k k bn 7→ νn,r(E|bn) and br 7→ Hn,r(F |br ) k k are Borel functions, then we can safely state that (see Prop 7.29 in [6]) both Gn,r and Gn,r are Borel functions. Therefore, Lemma 4.1 yields e 12 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

k k,gk k k k k k,ϕk k k k G (Prn ◦ Ξ (A )) = E Z (τ ∧ T )|F k , G Prn ◦ Ξ (A ) = E Z (η ∧ T )|F k n,r n,r n Tn n,r n,r n Tn k h i h i a.s for each τ ∈ Dn,r and η ∈ Tk,n,r, where e

k,gk k k k k k k Pren ◦ Ξn,r (bn) := gn(bn),...,gr (br ), bn k,gk k Ar−n+1 Sn is the projection of Ξn,r (br ) onto × k and

k,ϕk k k k k k k Prn ◦ Ξn,r (bn) := ϕn Φn(ω, xn) ,...,ϕr Φr (ω, xr) , bn

k,ϕk k Ar−n+1 Sn is the projection of Ξn,r (br ) onto × k . By construction,

k k k k F (A ) = esssup E Z τ ∧ T |F k a.s n,r n Tn τ Dk,r ∈ n h i and k k E k k Fn,r(An) = esssup Z τ ∧ T |FT k a.s. e n τ∈Tk,n,r h i k k e we notice that Fn,r = Fn,r and hence the claim (4.6) holds true. The argument outlined above actually shows that (4.6) holds when r =+∞, namely e E k k E k k (4.7) esssup Z τ ∧ T |FT k = esssup Z τ ∧ T |FT k a.s e n k,∞ n τ∈Tk,n τ∈Dn h i h i where Tk,n is the set of all F-stopping times of the form

∞ e k τ = T 1 k ℓ {τ=Tℓ } ℓ=n k X and {τ = T }∈F k ; ℓ ≥ n. This holds due to the fact that the set of sequences of natural numbers ℓ Tℓ ∞ ∞ (xi)i=n such that j=n xj = 1 is countable. Lastly, the argument to prove (4.5) actually shows that P E k k E k k (4.8) esssup Z τ ∧ T |FT k = esssup Z τ ∧ T |FT k a.s., e n n τ∈Tk,n τ∈T k,n h i h i F k where T k,n := {τ is an − stopping time; Tn ≤ τ ≤ +∞ a.s.}. Identities (4.7) and (4.8) allow us to conclude the proof.

Proof of Theorem 3.1: The variational inequality (3.9) is a straightforward consequence of the classical discrete time dynamic programming principle so we shall omit this proof. Let us check the convergence. At ﬁrst, we observe the Snell envelope S has continuous paths. Indeed, by assumption (A1), the reward process is a class D regular process and hence by applying Th 2.3.5 in [26] and assumption (A2), the associated Snell envelope is a D regular process as well. Therefore, the Snell envelope S has continuous paths under assumptions (A1-A2). Let us deﬁne

∞ k k k δ S(t) := E S(T )|F k 1 k k ;0 ≤ t ≤ T. n Tn {Tn ≤t

lim E sup |δkS(t) − S(t)| =0. k→+∞ 0≤t≤T DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 13

One can easily check that {E Z(τ) |F k ; τ ∈ T } has the lattice property (see e.g. Def. 1.1.2 in Tn k,n [26]). Therefore, the tower property of the conditional expectation and Prop. 1.1.4 in [26] yield

∞ ∞ k E k k E k δ S(t)= S(Tn ) |FT k 11{T k≤t

∞ k k k k S (t) = ess sup E Z (τ ∧ T ) |F k 11 k k e(k,T ) T {Tn ≤t

p k k p k k k E sup |S (t) − δ S(t)| ≤ E sup E sup |Z(t) − Z (t ∧ T )| F k 11 k e(k,T ) T {Tn ≤T } 0≤t≤T n≥0 0≤t≤T n h i

E k k p (4.9) ≤ C sup |Z(t) − Z (t ∧ Te(k,T ))| 0≤t≤T as k → ∞ for some p > 1 due to the imbedded discrete property. In order to prove the right-hand side of (4.9) vanishes as k → ∞ we just notice that Z has continuous paths and a simple triangle inequality argument jointly with Lemma 3.1 allows us to conclude the proof.

5. Examples of non-Markovian optimal stopping problems In this section, we show how Theorem 3.2 can be applied to concrete non-Markovian optimal stopping problems. To simplify the presentation, we set d = 1. Throughout this section, we make use of the following notation. Let D([0,t]; R) be the linear space of R-valued c`adl`ag paths on [0,t] and let Λ := {(t,ω(· ∧ t)); t ∈ [0,T ]; ω ∈ D([0,T ]; R)} be the set of stopped paths endowed with the distance

′ ′ ′ ′ ′ β dβ((t,ω); (t ,ω )) := sup |ω(u ∧ t) − ω (u ∧ t )| + |t − t | , 0≤u≤T where 0 <β ≤ 1. We say that G is a non-anticipative functional if it is a Borel mapping and

G(t,ω)= G(t,ω(· ∧ t)); (t,ω) ∈ [0,T ] × D([0,T ]; R). The coeﬃcients of the SDEs will satisfy the following regularity conditions:

Assumption I: The non-anticipative mappings α :Λ → R and σ :Λ → R are Lipschitz continuous, i.e., there exists a constant KLip > 0 such that

′ ′ ′ ′ ′ ′ |α(t,ω) − α(t ,ω )| + |σ(t,ω) − σ(t ,ω )|≤ KLipdθ (t,ω); (t ,ω ) ′ ′ R for every t,t ∈ [0,T ] and ω,ω ∈ D([0,T ]; ), where 0 <θ ≤ 1.

Assumption II: The reward process is given by

(5.1) Z(t)= F (t,X) 14 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO where X satisﬁes (5.2) or (5.13) and F :Λ → R is a non-anticipative Lipschitz functional, i.e., there exists constant kF k such that

′ ′ ′ ′ |F (t,ω) − F (t ,ω )|≤kF kdθ (t,ω); (t ,ω ) ′ ′ R for every t,t ∈ [0,T ], ω,ω ∈ D([0,T ]; ), where 0 <θ ≤ 1. 5.1. Non-Markovian SDE driven by Brownian motion. The underlying state process is the following SDE

t t (5.2) X(t)= x + α(s,X)ds + σ(s,X)dB(s); 0 ≤ t ≤ T, Z0 Z0 with a given initial condition X(0) = x ∈ R. One can easily check by routine arguments that the SDE (5.2) admits a strong solution in Bp(F) for every p ≥ 1. The natural candidate for an imbedded k discrete structure (Z )k≥1, D w.r.t Z is given by

∞ k k k (5.3) Z (t) := F T ,X 1 k k ;0 ≤ t ≤ T n {Tn ≤t

k k k k k k k k k k k (5.4) X (Tn ) := X (Tn−1)+ α Tn−1,X ∆Tn + σ(Tn−1,X ∆A (Tn ) k ∞ k k 1 for 1 ≤ n ≤ e(k,T ) where X (t)= X (T ) T k t T k 0; |B(t)| =1} (see e.g [9]) given by

1 I∗(x) = sup λx − ln ; x< 1. λ<0 " cosh( 2|λ|)!#

k Proposition 5.1. If Assumptions I-II hold true for θ =1p/2 and X satisﬁes (5.2), then (Z )k≥1, D given by (5.3) is an imbedded discrete structure for Z. More importantly, for each β ∈ (0, 1) and ζ > 1, there exists a constant C which depends on α, σ, β and ζ such that

k E 2 2β −1 −2 ∗ −1 (5.5) V0 − sup [Z(τ)] ≤ C ǫk + exp − ζ ǫk I 1 − δ + δ ln(2δ ) τ∈T0(F) for every k ≥ 1 and δ ∈ (0, 1). Proof. Let us ﬁx 0 <β < 1 and ζ > 1. A direct application of Corollary 6.1 in [29] jointly with Assumptions I-II and Lemma 2.2 in [28] yields k 2 2 2β (5.6) kZ − ZkB2 ≤kF k C(α, σ, β)ǫk ; k ≥ 1, for a constant C(α, σ, β). In the sequel, C is a constant which may diﬀer from line to line. Triangle inequality yields

E k k 2 E k k k 2 E k 2 k k (5.7) kZ (· ∧ Te(k,1)) − Zk∞ ≤ 2 kZ (· ∧ Te(k,1)) − Z k∞ +2 kZ − Zk∞ =: I1 + I2 . k By (3.10) in Theorem 3.2, we only need to estimate I1 . We observe k 2E k k k k 21 E k (5.8) I ≤ CkF k max |X (T ) − X (Tp )| T k < + C |1 − T |; k ≥ 1. 1 k k k e(k,1) { e(k,1) 1} e(k,1) T

k k k We set Ek = {Te(k,1) 2, we get

k 2 Tp k k (5.9) E max α s¯ ,X ds 1 k ≤ Ck1 − T k ζ ; k ≥ 1, k Ek∩Gp e(k,1) L p≥1 T k Z e(k,T ) k k k k for a constant C which depends on α. Here, G p = {Te(k,1) 0 and k ≥ 1, let us denote

1 k k k k 2 k k k k Ek,δ := {Te(k,1) δ}, Ek,δ := {Te(k,1) 0, we split

k 2 k 2 Tp Tp k 1 k 1 max σ(¯sk, X )dB(s) E ∩Gk = max σ(¯sk, X )dB(s) 1 k 1 k p 1 Ek,δ∩Gp p≥ ZT k p≥ ZT k e(k,1) e(k,1)

k 2 Tp k + max σ(¯sk, X )dB(s) 1 2 k 1 Ek,δ∩Gp p≥ ZT k e(k,1)

k k =: J1 (δ)+ J2 (δ).

By the additivity of the stochastic integral, we shall apply Burkholder-Davis-Gundy and H¨older inequalities jointly with Lemma 6.2 in [29] to get a constant C which depends on σ such that

1 1 E k P k k ζ P k ζ (5.10) J1 (δ) ≤ C {δ + Te k,

k for any ζ > 2, k ≥ 1 and δ ∈ (0, 1). In order to estimate J2 (δ), we need to work with the modulus of continuity of the stochastic integral

t k mk(h) := sup σ(s,X )dB(s) t,s∈[0,1],|t−s|≤h s Z k 2 for h> 0. We notice J2 (δ) ≤ mk(δ) a.s for every k ≥ 1 and δ > 0. By applying Th 1 in [19] jointly with Assumption I and Lemma 6.2 in [29], we arrive at the following estimate

2 (5.11) EJ k(δ) ≤ Cδ ln 2 δ for every k ≥ 1 and δ > 0, where C is a constant which only depends on (α, σ). Finally, since −2 −2 k ǫk k 2 e(k, 1) = ǫk and T −2 = i=1 ∆Tn is an iid sum of random variables with mean ǫk, we shall use ǫk classical Large Deviations techniquesP to ﬁnd for each q ≥ 1, a constant C (only depending on q) such that

P k −2 ∗ E k q 2q (5.12) {δ + T −2 < 1}≤ exp − ǫk I (1 − δ) , |T −2 − 1| ≤ Cǫk ǫk ǫk for every k ≥ 1 and δ ∈ (0, 1). Now, we just need to use (5.12) into (5.8), (5.9) and (5.10). By observing the estimates (5.6) and (5.7), we then conclude the proof. 16 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

5.2. Non-Markovian SDE driven by Fractional Brownian motion. In this section, we analyze Theorem 3.2 when the reward process Z = F (X) is driven by the following SDE

t (5.13) X(t)= x + α(s,X)ds + BH (t) Z0 where α is a bounded non-anticipative functional satisfying Assumption I and BH is the one-dimensional 1 fractional Brownian motion (henceforth abbreviated by FBM) over the interval [0,T ] for 2

t BH (t)= ρH (t,s)B(s)ds;0 ≤ t ≤ T, Z0 1 t 1 3 2 −H H− 2 H− 2 where ρH (t,s) := ∂sKH (t,s) and KH (t,s) := dH s s u (u − s) du;0

t¯k k k BH (t) := ρH (t¯k,s)A (s)ds;0 ≤ t ≤ T, Z0 where we recall t¯k is given by (2.3). Due to singularity of the kernel ρH jointly with appearance of k k random times and the piecewise constant process A , it is not obvious BH fulﬁlls this requirement. The proof of the next result is postponed to the Appendix.

1 1 1 Proposition 5.2. If 2

E k ¯ 1−2λ sup |BH (tk) − BH (t)|≤ Cǫk 0≤t≤T for every k ≥ 1.

k The natural candidate for an imbedded discrete structure (Z )k≥1, D w.r.t (5.1) is given by ∞ k k k (5.14) Z (t) := F T ,X 1 k k ;0 ≤ t ≤ T n {Tn ≤t

k k k k k k k k k X (Tn ) := X (Tn−1)+ α Tn−1,X ∆Tn + ∆BH (Tn ) for 1 ≤ n ≤ e(k,T ). We are now able to state the main result of this section. To simplify the presentation, we set T = 1.

1 Proposition 5.3. If Assumptions I-II hold for θ = 1, X satisﬁes (5.13) with 2

k E 1−2λ (5.15) V0 − sup [Z(τ)] ≤ Cǫk τ∈T0(F) for every k ≥ 1. DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 17

1 1 Proof. Let us ﬁx H − 2 <λ< 2 . By repeating the same argument presented in Proposition 6.2 in [29] jointly with the Lipschitz property of F , one can easily check there exists a constant C(α,β,H,λ) such that k 1−2λ (5.16) kZ − ZkB1 ≤kF kC(α,β,H,λ)ǫk for every k ≥ 1 and β ∈ (0, 1). By arguing just like the proof of Proposition 5.1, we have

k E 1−2λ k V0 − sup [Z(τ)] ≤kF kC(α,β,H,λ)ǫk + kF k sup |X(t ∧ Te(k,1)) − X(t)| τ∈T0(F) 0≤t≤1

for every k ≥ 1. We observe

(H−ǫ) |BH (t) − BH (s)|≤ C|t − s| Gǫ a.s. for every t,s ∈ [0, 1]. Therefore, k k (H−ǫ) sup |BH (t) − BH (t ∧ Te(k,1))|≤ C|1 − Te(k,1)| Gǫ a.s. 0≤t≤1 By using (5.12) and Cauchy-Schwarz’s inequality, for every 0 <γ

E k 2γ kBH (· ∧ Te(k,1)) − BH k∞ ≤ Cǫk . 1 Since, we shall take γ + λ> 2 , we then conclude the proof. 5.3. How to compute optimal values ? At this point, it is instructive to end this section with some guidelines on how to concretely produce optimal values in a given (possibly non-Markovian) optimal stopping time problem over a a given interval [0,T ]. For simplicity of exposition, the dimension of the underlying Brownian motion will be set equal to d = 1. At first, we recall the key objects of our k methodology are given by (3.5), (3.6) and (3.8), where each random element An induces an image k Pk Sn probability measure ρn := n on k ; n ≥ 1 where ρ0 is just the Dirac concentrated on 0. Zk Zk Zk Let = { n; n =0,...,e(k,T )} be a list of Borel functions which realizes (3.2). Recall must be interpreted as the payoff functional composed with a pathwise version of an imbedded discrete structure (see Definition 2.1) for a given state which is typically an Euler-type approximation driven k Zk Sn R 2 Sn k by A . See [7, 29] for further details. We assume that n : k → ∈ L ( k ,ρn) for every n = k k 0,...,e(k,T ). Let us now select a subset {Uj ; j = 0,...e(k,T ) − 1} of functions such that Uj ∈ 2 Sj k L ( k,ρj ) for each j =0,...,e(k,T ) − 1. For each choice of functions, we set inductively b b k τe(k,T ) := e(k,T ) (5.17) k k τj := j11 bk + τj+111 bk c ;0 ≤ j ≤ e(k,T ) − 1, ( Gj (Gj ) b k Zk k k k k k k where Gj := { j (Aj ) ≥ Uj (Ab j )};0 ≤ j ≤ e(k,T )−1 and τ = τ0 . Here, Uj (·) should be interpreted E Zk k k as a suitable approximation of bk (Abk )|Aj = · for each j =0,...,e(k,T ) − 1. τj+1 τj+1 b k b b b b The set {τj ;0 ≤ j ≤ e(k,T )} induces a set of conditional expectations E k k k E Zk k k Y (τj+1)|Aj = bk (Abk ) Aj ; j =0,...,e(k,T ) − 1, b τj+1 τj+1 so that one can postulateh i h i b 18 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

Zk k max 0 (0); U0 (0) as a possible approximation for (3.8). Having said that, we are now able to produce a Longstaﬀ- Schwartz-type Monte-Carlo scheme as demonstrated byb [7] which we brieﬂy outline here. For further details, we refer the reader to [7]. k R k Given any positive integer N, select HN,0 ⊂ . For each j = 1,...,e(k,T ) − 1, select HN,j ⊂ 2 Sj k k L ( k,ρj ). The sets HN,j;0 ≤ j ≤ e(k,T ) (the so-called approximation architectures) possibly depend on N and their choice are dictated by some a priori information that one possibly has about k the continuation values (3.6). From Aℓ ;0 ≤ ℓ ≤ e(k,T ) , generate N independent samples

k k k A0,i, A1,i,..., Ae(k,T ),i; i =1,...,N. This step can be implemented by means of the perfect simulation algorithm developed by [13]. For a concrete implementation in the context of hedging for European-type options, see [8]. For each ℓ =0,...,e(k,T ), let us denote

k k k k k k k AℓN := Aℓ,1, Aℓ,2,..., Aℓ,N ; ... ; Ae(k,T ),1, Ae(k,T ),2,..., Ae(k,T ),N with (e(k,T ) − ℓ + 1)N-factors. k k k k Zk For j = e(k,T ) − 1, we set τj+1 := τj+1(A j N ) := e(k,T ) and generate {(Aj,i, ( bk )i); 1 ≤ i ≤ ( +1) τj+1 Zk Zk k N}, where we deﬁne ( τbk )i := e(k,T )(Ae(k,T ),i); 1 ≤ i ≤ N. We then select j+1 b b N 2 k 1 Zk k (5.18) Uj := arg ming∈Hk ( τbk )i − g(Aj,i) . N,j N j+1 i=1 X bk k One should notice that Uj is a functional of AjN and we assume the existence of a minimizer (see Remark 4.4 in [7]) b k Sj Sj N Se(k,T ) N R (5.19) Uj : k × k × ... × k → k k k k of (5.18) which possibly depend on N. With Uj at hand, we compute τji = τj AjN , the value b i k k k that τj = τj AjN assumes based on the i-th sample according to (3.5). In this case, we set b b b b b Zk Ak ; if τ k = j Zk j j,i ji (5.20) τbk := Zk k k k j i τbk Aτbk ,i ; if τji = τ(j+1)i ( (j+1)i (j+1)i b k where τ(j+1)i = e(k,T ) for 1 ≤ i ≤ N. b b Based on (5.18) and (5.20), we then repeat this procedure inductively j = e(k,T ) − 2,..., 1, 0 until step j =0b to get

k k Zk τji, Uj , bk ;0 ≤ j ≤ e(k,T ), 1 ≤ i ≤ N. τj i For j = 0, we set b b k Zk k k V0(A0N ) := max 0 (0), U0 (A0N ) . The above procedure is consistent with the limitingn optimal stoppingo time problem as demonstrated b b by the following result. It is an immediate consequence of Theorem 4.1 in [7] and Propositions 5.1 and 5.3. In the sequel, vc denotes the Vapnik-Chervonenkis dimension of a subset. DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 19

k 2 Sj k k (H1) For each k ≥ 1, suppose that HN,j ⊂ L ( k,ρj ) and there exists νk such that vc HN,j ≤ νk < +∞ for every j =1,...,e(k,T ) − 1 and for every N ≥ 2. k (H2) For each k ≥ 1, there exists Bk such that sup{kfk∞; f ∈ HN,j} ≤ Bk < +∞ for every j =0,...,e(k,T ) − 1 and N ≥ 2.

By applying Propositions 5.1 and 5.3 and Th 4.1 in [7], we arrive at the following result. Corollary 5.1. Assume (A1-A2-H1-H2) and the hypotheses of Propositions 5.1 and 5.3 hold true. k 2 k Let us assume the architecture spaces HN,j are dense subsets of L (ρj ) for each j =1,...,e(k,T ) − 1, for each positive integer N ≥ 2 and k ≥ 1. Then, for every k ≥ 1 suﬃciently large

k k (5.21) lim |V0 (A0N ) − S(0)| =0 a.s. N→+∞ E where S(0) = supτ∈T0(F) [Z(τ)] and Z isb given by (5.1). In order to compute a precise number of steps in our scheme, we just need to apply Propositions 5.1 and 5.3 combined with Corollary 4.1 in [7]. In this case, similar to the classical Markovian case, k smoothness of the continuation values Uj ; j =0,...,e(k,T ) − 1 is crucial for how to choose approximation spaces to get the most favorable rate of convergence by properly balancing the approximation error and the sample error. Example 5.1. For simplicity of exposition, let X be the state process given in Section 5.2, where the terminal time, the level of discretization ǫk, the drift and the payoﬀ are deﬁned, respectively, by

T =1, ǫk = ϕ(k), α(s, η)= b(η(s)), F :Λ → R; η 7→ F (η) := h(η(1)) for each η ∈ D([0, 1]; R) and s ∈ [0, 1], where b and h are Lipschitz bounded functions and ϕ : [0, ∞) → 2 1 [0, ∞) is a strictly decreasing function (with inverse ξ) such that k≥1 ϕ (k) < ∞. We ﬁx 2 1. We apply Corollary 4.1 in [7] and Propositione 5.3 to statee that

−2 E k k 2+e(k,1)−1 (5.22) |V0(A0N ) − V0 | = O log(N)N ,

k 1−2λ (5.23) b |V0 − S(0)| = O(ǫk ). With the estimates (5.22) and (5.23) at hand, we are now able to infer the amount of work (complexity) to recover the optimal value for a given level of accuracy e. Indeed, let us ﬁx 0 < e1 < 1. Equation (5.23) allows us to ﬁnd the necessary number of steps related to the discretization as follows. We 1 1 1−2λ 1−2λ ∗ 1−2λ observe ǫk ≤ e1 ⇐⇒ k ≥ ξ(e1 ) and with this information at hand, we shall take k = ξ(e1 ). This produces

1 e(k∗, 1) = ϕ2(k∗) l m 20 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

−k number of steps associated with the discretization procedure. For instance, if ϕ(k)=2 , e1 =0.40, ∗ log2 0.40 ∗ 2×1.88 H =0.6, then we shall take λ =0.15 and k = − 0.70 =1.88. This produces e(k , 1) = ⌈2 ⌉ = ∗ 14 number of steps. Of course, as e1 ↓ 0, the number of steps e(k , 1) ↑ +∞, e.g., if e1 = 0.2, then ∗ log2 0.2 ∗ ∗ k = − 0.70 = 3.31, e(k , 1) = 99 and so on. For a given prescribed error 0 < e2 < 1 and k , equation (5.22) allows us to ﬁnd the necessary number N for the Monte-Carlo scheme in such way E k∗ k∗ that |V0(A0N ) − V0 | = O(e2).

b 6. Appendix: Proof of Proposition 5.2 In the sequel, C is a constant which may diﬀer from line to line. Let us denote t (H− 1 ) ( 1 −H) (H− 3 ) (−H− 1 ) (H− 3 ) (H− 1 ) ρH,1(t,s)= t 2 s 2 (t − s) 2 , ρH,2(t,s)= s 2 u 2 (u − s) 2 du Zs for 0

(6.1) |ρH (t,s)|≤ C{ρH,1(t,s)+ ρH,2(t,s)} for 0

N k(T ) H−η k k k (6.2) sup |BH (t¯k) − BH (t)|≤ sup |BH (t¯k) − BH (t¯k)| + kBH kH−η ∆Tn a.s 0≤t≤T 0≤t≤T n=1 _ for every 0 <η

t¯k k H− 1 (6.3) sup ρH,1(t¯k,s)|A (s) − B(s)|ds ≤ ǫkT 2 a.s. 0≤t≤T Z0 ¯ tk ¯ k The delicate component is 0 ρH,2(tk,s)|A (s) − B(s)|ds. Let us ﬁx p> 1 such that R 1 1 1 1 1 (6.4) + > >λ> + H − . 2 p 2 p 2 1 1 so that we must have p> 1−H > 2. Let q> 1 be a conjugate exponent such that (− 2 −H+λ)q+1 > 0. The choice (6.4) implies

1 u p E s−λp|Ak(s) − B(s)|pds < ∞ ( Z0 ) for every 0 ≤ u ≤ T. We observe

t¯k T λ 1 1 3 k −1+ q p H− 2 1 sup ρH,2(t¯k,s)|A (s) − B(s)|ds ≤ C u (dk(u)) u {N k(u)>0}du 0≤t≤T Z0 Z0 T λ 1 1 3 −1+ q p H− 2 1 + C u (dk(u)) u {N k(u)=0}du Z0 where C only depends on H and u −λp k p dk(u) := s |A (s) − B(s)| ds. Z0 DISCRETE-TYPE APPROXIMATIONS FOR NON-MARKOVIAN OPTIMAL STOPPING PROBLEMS: PART I 21

To shorten notation, let us denote k T k k E N (u)+1 −λp k p 1 J (u)= s |A (s) − B(s)| ds {N k(u)>0}. " Z0 # We observe

1 p 1 1 k dk(u) {N k(u)=0} ≤kBkλu {N k(u)=0}, dk(u) {N k(u)>0} ≤ J (u) a.s for every u ∈ [0,T ]. Let us now estimate J k(u) for a given u ∈ (0,T ]. A lengthy but straightforward calculation based on the strong Markov property of the shifted Brownian motion over the stopping k times Tn yields

1 1 1 2(−λ+ 1 ) γp pγ 1 1 E p p 2 E P βp k p (dk(u)) ≤ Cu ǫk + kBkλ {N (u)=0}u for a constant C which depends on (λ, p), where γ,β > 1 are conjugate exponents. Moreover,

1 1 2 P pβ k − pβ pβ {N (u)=0}≤ Cu ǫk ; u> 0,pβ > 2, for a constant C which only depends on pβ. This implies

t¯k 1 2 1 1 k H 1 λ 2(−λ+ ) pβ H− +λ− E ¯ − 2 + 2 2 pβ (6.5) sup ρH,2(tk,s)|A (s) − B(s)|ds ≤ CT ǫk + Cǫk T 0≤t≤T Z0 1 1 by adjusting pβ <λ< 2 . Summing up the steps (6.1), (6.2), (6.3), (6.5) and using Fernique’s theorem, Lemma 2.2 in [28] and H¨older’s inequality, there exists a constant C = C(θ,H,a,b,λ,β,η) such that

2(H−η)− 2 (1−θ) 2 E k ¯ b 1−2λ pβ sup |BH (tk) − BH (t)|≤ C(θ,H,a,b,λ,β,η) ǫk + ǫk + ǫk 0≤t≤T for every a,b > 1 conjugate exponents, θ ∈ (0, 1), (λ, p) satisfying (6.4), β > 1 and η ∈ (0,H). Finally, we can choose (η,θ,b) in such way that 2 1 > 2λ> 2η + (1 − θ) b 2 and since 2H > 1, we then have 1 − 2λ < 2H − 2η + b (1 − θ) . We also can choose (p,β) in such way that 1 1 1 1 1 >λ> − > + H − 2 2 pβ p 2 2 and in this case 1 − 2λ< pβ . This concludes the proof. 22 DORIVAL LEAO,˜ ALBERTO OHASHI, AND FRANCESCO RUSSO

Acknowledgement: Alberto Ohashi and Francesco Russo acknowledge the ﬁnancial support of Math Amsud grant 88887.197425/2018-00. Alberto Ohashi acknowledges ENSTA ParisTech for the hospitality and CNPq-Bolsa de Produtividade de Pesquisa grant 303443/2018-9. The authors would like to thank the Referees for the careful reading and suggestions which considerably improved the presentation of this paper.

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