Optimal stopping under g -expectation with constraints

Accepted Manuscript Optimal stopping under g-expectation with constraints Helin Wu PII: DOI: Reference:

S0167-6377(12)00167-8 10.1016/j.orl.2012.12.009 OPERES 5654

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Operations Research Letters

Received date: 21 June 2012 Revised date: 17 November 2012 Accepted date: 19 December 2012 Please cite this article as: H. Wu, Optimal stopping under g-expectation with constraints, Operations Research Letters (2013), doi:10.1016/j.orl.2012.12.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Optimal stopping under g-expectation with constraints Helin Wu+ School of Mathematics, Shandong University, Jinan 250100, China E-mail: [email protected]

Abstract In this paper, we study a constrained optimal stopping problem under gΓ -expectation. We use gΓ expectation to evaluate the reward process. Along the classic way of martingale method to solve such problems, in our constrained case, the main difficulty comes from the lack of perfect continuous dependence and strict comparison property of gΓ -solution. We overcome this by additional assumptions on the reward process or convexity on gΓ -expectation, and adopt a new way to get a RCLL modification of the value process by the monotonic limit theorem obtained in S.G.Peng [16]. We also give a short discussion about dynamic programming principle and corresponding variational inequalities. Some examples are given in section 3 to show our ideas and applications.

Keywords: Constrained BSDE, gΓ -expectation, Optimal stopping.

1

Introduction

Optimal stopping problem is always studied under linear expectation induced by a probability. Pricing American options in complete market is a well-known example for it in applications. But in recent years, nonlinear expectations were studied more and more by authors when ambiguity was considered. Among all kinds of them, g-expectation, which was introduced in S.G.Peng [15], is a typical example for it enjoys many nice properties like linear expectation, such as continuous property and strict comparison property as well as time-consistence property. Except for ambiguity, another source of nonlinearity in analysis comes from constraint and this is what we want to consider in this paper. We first recall some but not all papers about optimal stopping problem under nonlinear expectations. In discrete time case, F.Riedel [5] studied the optimal stopping problem with time consistent multiple priors. V.Kr¨atschmer and J.Schoenmakers [20] considered the optimal stopping for more general dynamic utility functionals satisfying nice properties such as time consistency and recursiveness but without strict comparison. In continuous time case, an optimal stopping problem was considered under ambiguity in F.Riedel and X.Cheng [6]. In that paper, the authors developed a theory of Snell envelope under g-expectation which characterizes ambiguity in financial markets. D.Engelage [1] considered optimal stopping problems in uncertain environments for an agent assessing utility by virtue of dynamic variational preferences and gave many interesting examples. E.Bayraktar and S.Yao [2] investigated optimal stopping problem under more general framework. Given a stable family of F-expectations (whose definition can be found in F.Coquet, Y.Hu, J.Memin, and S.G.Peng [4]) {Ei }i∈I defined well on a common domain, the authors considered the optimal problems sup

Ei (Yτ + Hτi )

(1.1)

sup inf Ei (Yτ + Hτi ).

(1.2)

(i,τ )∈I×S0,T

and

τ ∈S0,T i∈I

where S0,T denotes all the stopping times valued on [0, T ] and {(Yt + Hti ), i ∈ I} are model-dependent reward processes. Under the typical non-linear g-expectation induced by BSDE, optimal stopping time problem is closely connected to Reflected Backward Stochastic Differential Equation (shortly RBSDE) studied in N. El.Karoui etc. [12]. But in fact, RBSDE and optimal stopping problem are different problems, they are connected by the coincidences between the Snell envelop of a reward process (Lt ) and the solution of RBSDE with lower barrier (Lt ). However, this fact is not obvious but based on, roughly speaking, the following three results when g(t, x, 0) = 0. Let (Lt ) be a continuous progressive measurable process, then (i) The solution of Reflected BSDE with lower barrier (Lt ) is the same with the smallest g-supersolution above (Lt ). 0+

Corresponding author.

1

(ii) The Snell envelope of (Lt ) under g-expectation is the smallest g-super-martingale above (Lt ). (iii) Under suitable assumptions, a stochastic process is a g-super-solution if and only if it is a g-supermartingale. In our paper, we want to study optimal stopping problem under g-expectation as well as constraints were considered, that is, we consider an optimal stopping problem under gΓ -expectation to optimize sup E0g,φ (Lτ ),

(1.1)

τ ∈S0,T

where E0g,φ (·) is the gΓ -expectation introduced in S.G.Peng and M.Y.Xu [18] via Constrained BSDE and (Lt ) is a reward process satisfying some mild assumptions. Constrained problems are interesting and meaningful in control theory and mathematical finance. R.Buckdahn, M.quincampoix, A.Rascanu [14] considered constrained problem in BSDE with the viability keeping solutions inside a given set K. In J.Cvitanic, I.Karatzas[10, 11], authors considered pricing and hedging problem of claims with constrained portfolios, that is, zt ∈ K for some convex closed set, where zt ∈ Rn represents the portfolio process. Here are some meaningful constraints in mathematic finance. Example 1.1 (Prohibition of short-selling of stocks). Taking K = [0, ∞)n , then the constraint zt ≥ 0 means no short-selling of stocks. Example 1.2 (Incomplete market). Taking K = {(x1 , · · · , xn ) ∈ Rn : xi = 0, i = 1, · · · , d.}, the constraint zti = 0, i = 1, · · · , d. d ≤ n−1 means the first d stocks is unviable for the investor and the market is incomplete.

It seems naturally that one can solve (1.1) by solving Constrained BSDE reflected from below by (Lt ) as considered in S.G.Peng and M.Y.Xu [18]. But to our knowledge, the corresponding work of (i), (ii) and (iii) are very complicated in constrained case and have not been established well. Generally speaking, as mentioned in E.Bayraktar and S.Yao [2], strict comparison theorem and stable property for linear or nonlinear expectations are crucial to get a RCLL modification of value processes in optimal stopping problems. The main difficulty for us in constrained case is that, under gΓ -expectation, there is no strict comparison theorem any more, and it can not be represented by a stable class of g-expectations easily. So we need a new way to work out our problem. We still construct the Snell envelope of the reward process under gΓ -expectation but, instead of by strict comparison proposition, obtain a RCLL modification of it via a new approach by the monotonic limit theorem of BSDE developed in S.G.Peng [16]. Along the classic way used in F.Riedel and X.Cheng [6], to overcome the lack of continuous dependence property of gΓ -solution, we assume gΓ -expectation be convex or the reward process be increasing with time. Our paper is organized as follows. In section 2, we state necessary definitions and facts about Constrained BSDE to continue our work. In section 3, we state our main results and give its proof by a modified method of classic martingale approach. In section 4, we study the dynamic programming principle and variational inequalities.

2

CBSDE and gΓ -expectation

Given a probability space (Ω, F, P ) and Rd -valued Brownian motion (Wt ), we consider a sequence {(Ft ); t ∈ [0, T ]} of filtrations generated by (Wt ) and augmented by P -null sets. We use L2 (FT ) to denote the space of all FT -measurable random variables ξ : Ω → R for which k ξ k2 = E[|ξ|2 ] < +∞, and use HT2 (Rd ) to denote the space of predictable process ϕ : Ω × [0, T ] → Rd for which Z T k ϕ k2 = E[ |ϕ|2 ] < +∞. 0

We denote L∞ (FT ) as the Banach space of all P -essentially bounded real variables on (Ω, FT , P ). For functions ϕ : [0, T ] × R × Rd → R, we assume throughout this paper, (A1)

|ϕ(ω, t, y1 , z1 ) − ϕ(ω, t, y2 , z2 )| ≤ M (|y1 − y2 | + |z1 − z2 |), ∀(y1 , z1 ), (y2 , z2 ) ∈ R × Rd

for some M > 0 and (A2)

ϕ(·, y, z) ∈ HT2 (R) ∀y ∈ R, z ∈ Rd . 2

The backward stochastic differential equation (shortly BSDE ) driven by g(t, y, z) is given by −dyt = g(t, yt , zt )dt − zt∗ dWt ,

(2.1)

where yt ∈ R and zt∗ ∈ Rd is the transpose of a vector. For ξ ∈ L2 (FT ) and g satisfying (A1) and (A2), the existence and uniqueness of adapted solution (y(t), z(t)) of (2.1) was proved in E.Pardoux, S.G.Peng [3]. We call (g, ξ) above as standard parameters. The pair of (yt , zt ) satisfying (2.1) is called as g-solution. In constrained case, however, an increasing RCLL process is always added into (2.1) to put yt upward and, such a triple (yt , zt , Ct ) is called as g-super-solution as defined below. Definition 2.1 (g-super-solution). A g-super-solution of a BSDE associated with a standard parameter (g, ξ) is a vector process (yt , zt , Ct ) satisfying −dyt = g(t, yt , zt )dt + dCt − zt∗ dWt , or being equivalent to yt = ξ +

Z

t

T

g(s, ys , zs )ds −

Z

T

t

zs∗ dWs

yT = ξ,

+

Z

T

dCs ,

(2.2)

(2.20 )

t

where (Ct , t ∈ [0, T ]) is an increasing, adapted, right-continuous process with C0 = 0 and zt∗ is the transpose of zt . In this paper, we formulate constraints like those in S.G.Peng [16]. For a given function φ(t, y, z) : [0, T ] × R × Rd → R+ , we define subsets in [0, T ] × R × Rd as Γt := {(t, y, z)|φ(t, y, z) = 0}, ∀t ∈ [0, T ]. A super-solution (yt , zt , Ct ) is said to satisfy constraints if it is forced in Γt , i.e. (t, yt , zt ) ∈ Γt ,

a.s., a.e.

on

[0, T ] × Ω.

(2.3)

Existence problem of Etg,φ (ξ) for a terminal variable ξ ∈ L2 (FT ) is crucial for Constrained BSDE analysis. In our paper, we assume (Lt ) be bounded in L∞ (FT ) and furthermore (A3)

ϕ(·, y, 0) = 0

∀y ∈ R.

Among all triples satisfying (2.2) and (2.3), we are mainly interested in the smallest one and give it a name gΓ -solution as defined below. Definition 2.2 (gΓ -solution). A g-super-solution (yt , zt , Ct ) is said to be the the minimal solution of a constrained backward differential stochastic equation (shortly CBSDE), given yT = ξ, subjected to the constraint (2.3) if for any other g-super-solution (yt0 , zt0 , Ct0 ) satisfying (2.3) with yT0 = ξ, we have yt ≤ yt0 a.e.,a.s.. We call g,φ the minimal solution as gΓ -solution and denote yt part as Et,T (ξ) and call it as dynamic gΓ -expectation. When both g and φ satisfy assumption (A3), gΓ -solution satisfies time consistence property and we write it simply as Etg,φ (ξ). For any ξ ∈ L2 (FT ), if we denoting Hφ (ξ) as the set of all g-super-solutions (yt , zt , Ct ) satisfying constraint (2.3) with terminal condition yT = ξ, then, according to S.G.Peng [16], the minimal one, namely, the gΓ -solution exists when Hφ (ξ) is not empty. More explicitly, we can obtain the minimal super-solution via a penalization g,φ method, that is, if we let gn := g + nφ, then gn -solutions approximate Et,T (ξ) increasingly. Generally speaking, since the increasing part Ct of gΓ -solution varies against terminal value, it is hard to get a similar priori estimation to prove continuous dependence property of gΓ -solution as done in unconstrained case, but we can still get a continuous property in a semi sense, i.e. a monotonic continuity. Proposition 2.1. Suppose g(t, y, z) and φ(t, y, z) : [0, T ] × R × Rd → R satisfy assumptions (A1), (A2) and (A3), {ξm ∈ L∞ (FT ), m = 1, 2, · · · } is a sequence in L∞ (FT ) converges to ξ ∈ L∞ (FT ) increasingly. When Hφ (ξ) is not empty, then we have lim Etg,φ (ξn ) = Etg,φ (ξ). (2.4) n→∞

Proof. First, it is easy to see that Hφ (ξn ) is not empty since ξn ≤ ξ, a.s and Hφ (ξ) is not empty. In fact, if (yt , zt , Ct ) is a super-solution with terminal value ξ, then (yt0 , zt , Ct0 ) is a super-solution with terminal value ξn , where  Ct when 0 ≤ t < T ; 0 Ct = CT + ξ − ξn when t = T. 3

yt0

=



yt ξn

when 0 ≤ t < T ; when t = T.

If {ξm ∈ L∞ (FT ), m = 1, 2, · · · } is increasing, then

Atm,n := Etgn (ξm )

is increasing with m and n for each fixed t. Together with the continuous property of gn -solutions, we can explode the left hand of (2.4) as lim Etg,φ (ξn ) = ess sup ess sup Atm,n . n→∞

n

m

Change the order of m and n in the above equation, by the penalization method to get the gΓ -solution, we obtain the result of this proposition. Except for the semi continuous property stated above, we can also prove the lower semi-continuity of gΓ solution as a function defined on L∞ (FT ) by the same penalization method. Theorem 2.1. Suppose φ(ω, t, y, z) : Ω × [0, T ] × R × Rd → R+ and g(ω, t, y, z) : Ω × [0, T ] × R × Rd → R satisfy assumptions (A1), (A2) and A(3), then for any k ∈ R, Ak := {ξ ∈ L∞ (FT )|E0g,φ (ξ) ≤ k} is closed in L∞ (FT )-norm. Proof. Under assumptions of A(i), i = 1, 2, 3, gΓ -solution Etg,φ (·) is well defined on L∞ (FT ) as proved in Proposition 2.3 below. Suppose a sequence {ξn , n = 1, 2 · · · } ⊂ Ak converges under norm to some ξ ∈ L∞ (FT ). For any ξn , we take {y0m (ξn ), m = 1, 2, · · · } as Z T Z T m m m m m y0 (ξ) = ξ + (g(ys (ξ), zs , s) + mφ(ys (ξ), zs , s))ds − (zsm )∗ dWs , 0

0

ytm (ξ)

where we use as solution of BSDE to emphasize the variance of terminal value ξ. Since y0m (ξn ) converges increasingly to E0g,φ (ξn ) ≤ k as m → ∞, y0m (ξn ) ≤ k for any n and m. For any fixed m, take gm := g + mφ, by the continuous dependence property of gm -solution, we have y0m (ξn ) → y0m (ξ) as n → ∞ and y0m (ξ) ≤ k is obtained for any m. Again, for the fixed ξ ∈ L∞ (FT ), y0m (ξ) → E0g,φ (ξ) as m → ∞. Thus E0g,φ (ξ) ≤ k and Ak is closed under norm in L∞ (FT ) and Etg,φ (·) is lower-semi continuous on L∞ (FT ). 2 Remark 2.1. The usage of above theorem is to combine it with convex analysis. As we know, when a finite function is convex in a Banach space, then it is continuous on it. In the framework of this paper, if Etg,φ (·) is a convex function, then it is continuous on the whole space L∞ (FT ) in norm sense. It is also easy to prove a weak ( not strictly ) comparison proposition of Etg,φ (·) along the same line.

Proposition 2.2. Under the same assumptions as Proposition 2.1, we have Etg,φ (ξ) ≤ Etg,φ (η) for any ξ, η ∈ L2T (R) with P (η ≥ ξ) = 1 when Hφ (η) is nonempty.

We will assume the reward process be bounded in L∞ (FT ) throughout the paper, the reason why we do that is based on the the following result, which can be obtained with a help of a result in S.G.Peng and M.Y.Xu [18]. Proposition 2.3. Suppose both g and φ satisfy assumptions A(i), i = 1, 2, 3, then the gΓ -solution exists for any ξ ∈ L∞ (FT ) with terminal condition yT = ξ. Proof. In S.G.Peng and M.Y.Xu [18], the authors defined a new subspace of L2T (FT ), L2+,∞ (FT ) := {ξ|ξ ∈ L2 (FT ), ξ + ∈ L∞ (FT )}, and the existence of gΓ -solution with terminal value yT = ξ ∈ L2+,∞ (FT ) was proved under the assumption g(t, y, 0) ≤ L0 + M |y|, ∀y ≥ L0

and

(y, 0) ∈ Γt ,

(2.5)

where L0 and M ≥ 0 are large constants. Since g(t, y, 0) ≡ 0, φ(t, y, 0) ≡ 0 by our assumption, it is obvious that L∞ (FT ) ⊂ L2+,∞ (FT ) and (2.5) holds for any L0 ≥ 0 and M ≥ 0. 2 The following properties of Etg,φ (·) will be helpful in our study, their proofs can be found in S.G.Peng and M.Y.Xu [18] . 4

Proposition 2.4. Suppose g(t, y, z) and φ(t, y, z) : [0, T ] × R × Rd → R satisfy assumptions A(i), i = 1, 2, 3, g,φ then the gΓ -expectation Etg,φ (·) := Et,T (·) satisfies: (i) Self-preserving: Etg,φ (ξ) = ξ for any ξ ∈ L∞ (Ft ). (ii) Time consistency: Esg,φ (Etg,φ (ξ)) = Esg,φ (ξ), (iii) 1-0 law: 1A Etg,φ (ξ) = Etg,φ (1A ξ),

3

0 ≤ s ≤ t ≤ T,

ξ ∈ L∞ (FT ).

∀A ∈ Ft .

Optimal stopping under gΓ -expectation

In this section, we want to find an optimal stopping time which attains the super-mum, sup E0g,φ (Lτ ).

(3.1)

τ ∈S0,T

For simplicity and making E0g,φ (Lτ ) defined well for all stopping times τ , we assume the adapted model-depend reward process {Lt , t ∈ [0, T ]} is nonnegative and uniformly bounded in L∞ (FT ). As usual, we define the value function of the optimal stopping problem under gΓ -expectation as Vt := ess sup Etg,φ (Lτ ),

(3.2)

τ ∈St,T

where St,T is the set of all stopping times valued on [t, T ]. Super-martingale ( sub-martingale, martingale ) under gΓ -expectation is defined as done in S.G.Peng and M.Y.Xu [18]. Definition 3.1. An adapted process (Mt ) satisfying Mt ∈ L∞ (Ft ) for any t ∈ [0, T ] is called a gΓ -supermartingale ( sub-martingale, martingale ) on [0, T ], if for 0 ≤ s ≤ t ≤ T we have Esg,φ (Mt ) ≤ (≥, =)Ms . Just as in classic case, we want to prove the snell envelope (Vt ) defined in (3.2) is a gΓ -super-martingale. However, in this constrained case, there is no perfect continuous property for gΓ -expectation. Fortunately, we observe that only the semi continuous property of gΓ -solution obtained in Proposition 2.1 is enough for our proof. We give the following lemma at first. Lemma 3.1. For all t ≥ 0, the family

{Etg,φ (Lτ ) : τ ∈ St,T }

is upwards directed.

Proof. For any stopping times τ, σ ∈ St,T , we want to prove that there is some stopping time ν ∈ St,T such that Etg,φ (Lν ) = Etg,φ (Lτ ) ∨ Etg,φ (Lσ ). In fact, let

A := {Etg,φ (Lτ ) ≥ Etg,φ (Lσ )},

then ν := τ 1A + σ1Ac is a stopping time satisfying above equation thanks to the property of 1-0 law of gΓ expectation. 2 With the help of this lemma, we now prove Proposition 3.1. Suppose g, φ : [0, T ]×R ×Rd → R satisfy assumptions A(i), i = 1, 2, 3 and (Lt ) is an adapted nonnegative process bounded in L∞ (FT ), then the value function (Vt ) defined by (3.2) is a gΓ -supermartingale. Proof. For any t ≥ 0, the lemma above allows us to choose a sequence {τn (t) ∈ St,T , n = 1, 2, · · · } of stopping times such that Etg,φ (Lτn (t) ) ↑ Vt .

Since Etg,φ (Lτn (t) ) converges to Vt increasingly, by the continuous property from below of Proposition 2.1 and time consistency property (ii) in Proposition 2.4, we have Esg,φ (Vt )

= Esg,φ (ess sup Etg,φ (Lτ )) = Esg,φ ( lim Etg,φ (Lτn (t) )) n→∞

τ ∈St,T

= ≤

lim Esg,φ (Etg,φ (Lτn (t) )) = lim Esg,φ (Lτn (t) )

n→∞

n→∞

ess sup Esg,φ (Lτ ) = Vs τ ∈Ss,T

5

for any 0 ≤ s ≤ t ≤ T. 2 To construct an optimal stopping time, we need to show that there is a RCLL modification of (Vt ). However, in the constrained case, the strict comparison theorem does not hold anymore for gΓ -expectation and the usual way to find a RCLL modification by down-crossing inequality becomes impossible. To get over this difficulty, we take a new way with the help of the monotonic limit theorem obtained in S.G.Peng [16]. Lemma 3.2. Under the assumptions as in Proposition 3.1, the value process (Vt ) defined by (3.2) has a RCLL modification. Proof. Let gn := g + nφ and define the value function (Vn (t)) under gn -expectation as Vn (t) := ess sup Etgn (Lτ ),

(3.3)

τ ∈St,T

for any n, then (Vn (t)) are gn -supermartingales and have RCLL modifications according to Lemma F.1 in F.Riedel and X.Cheng [6] or Lemma 5.2 in F.Coquet, Y.Hu, J.Memin, and S.G.Peng [4]. Since gn ≥ g for all n, (Vn (t)) are also g-supermartingales and converges to (Vt ) increasingly as n → ∞, more explicitly, we expand it as follows Vt

= =

ess sup Etg,φ (Lτ ) = ess sup ess sup Etgn (Lτ ) τ ∈St,T

ess sup ess sup n∈N

τ ∈St,T

τ ∈St,T gn Et (Lτ ) = ess

n∈N

sup Vn (t).

n∈N

Therefore, by theorem 3.6 of S.G.Peng [16], (Vt ) still has a RCLL modification. 2 There is no hard to prove that (Vt ) is the smallest gΓ -supermartingale above the reward process in following sense. Proposition 3.2. (Vt ) is the smallest gΓ -supermartingale with RCLL modification dominating (Lt ). Proof. Suppose (St ) is another RCLL gΓ -supermartingale with St ≥ Lt for all t ∈ [0, T ] a.s.. Choose a sequence of stopping times {τn (t), n = 1, 2, ·} in St,T as in the proof of Proposition 3.1, then Vt = lim Etg,φ (Lτn (t) ) ≤ lim sup Etg,φ (Sτn (t) ) ≤ St . n→∞

n→∞

2 With these in hands, we then go to construct an optimal stopping of problem (3.1) taking care of subtle difference from classic case. For any 0 < λ < 1, 0 ≤ t ≤ T , we define the stopping times τ λ (t) := inf{u ≥ t|Lu ≥ λVu } ∧ T. The next lemma helps us to obtain an optimal stopping time as the limit of a sequence of ones. Lemma 3.3. Under assumptions as in Proposition 3.1 and with the notation introduced above, we have Vt = Etg,φ (Vτ λ (t) ). Proof. Let and

Ut := Etg,φ (Vτ λ (t) ). Un (t) := Etgn (Vτ λ (t) ), n = 1, 2, · · · .

We claim that (Ut ) is a gΓ -supermatingale and hence g-supermartingales. In fact, for 0 ≤ t ≤ t + u ≤ T we have g,φ Etg,φ (Ut+u ) = Etg,φ (Et+u (Vτ λ (t+u) )) = Etg,φ (Vτ λ (t+u) ) ≤ Etg,φ (Vτ λ (t) ) = Ut .

Similarly, {(Un (t)), n = 1, 2, · · · } are gn -supermartingales and hence g-supermartingales. It is obviously that (Un (t)) converges increasingly to (Ut ) and, by theorem 3.6 of S.G.Peng [16] , (Ut ) admits a RCLL modification. Let Rt := λVt + (1 − λ)Ut ,

we claim that Rt ≥ Lt a.s. for all t ∈ [0, T ] at first. In fact, when Lt ≥ λVt , τ λ (t) = t and Ut = Vt , Rt = Vt ≥ Lt ; when Lt < λVt , we have Rt ≥ λVt > Lt by Ut ≥ 0. 6

By the definition of Ut and Rt , Ut ≤ Vt and Rt ≤ Vt are obvious. On the other hand, as we have just proved, (Rt ) is a RCLL gΓ -supermartingale dominating (Lt ), then by Proposition 3.2, Rt ≥ Vt . Combining these facts together, we have Rt = Vt = Ut = Etg,φ (Vτ λ (t) ). τ λ.

2 Now let us back to the definition of stopping times of τ λ (t) at t = 0, which, for simplicity, we denote it as Noting that τ λ is increasing with λ and dominated by the stopping time τ ∗ := inf{t ≥ 0 : Lt = Vt } ∧ T.

(3.4)

For a sequence of real numbers (λn ) ⊂ (0, 1) which converges to 1 increasingly, it is obvious that {τ λn } is increasing and converges to stopping time τ ∗ . We now state our main result. Theorem 3.1. Suppose g, φ : [0, T ] × R × Rd → R satisfy assumptions A(i), i = 1, 2, 3 and (Lt ) is an adapted nonnegative process bounded in L∞ (FT ), then τ ∗ is an optimal stopping for problem (3.1) if (Lt ) is increasing in time or E0g,φ (·) is convex. Proof. When (Lt ) is increasing in t, then by Lemma 3.3, we have V0 = E0g,φ (Vτ λn ) ≤

1 g,φ 1 1 g,φ E (Lτ λn ) ≤ E (Lτ ∗ ) ≤ V0 . λn 0 λn 0 λn

(3.5)

When λn converges increasingly to 1, {τ λn } is increasing and converges to a stopping time τ ∗ which is an optimal choice. If E0g,φ (·) is convex, then by Theorem 2.1 and Remark 2.1, it is continuous on L∞ (FT ) and τ ∗ is proved to be optimal as the same way in the classic case. 2 Remark 3.1. The key point in the proof above is to qualify the second inequality in (3.5). Our assumption is trivial to make use of comparison proposition stated in Proposition 2.2. In this case, the terminal time T is optimal obviously. According to S.G.Peng and M.Y.Xu [18], within our assumptions, Vt can be viewed as the solution of the Reflected BSDE with constraint, and τ ∗ defined by (3.4) is an optimal stopping time and the stopped process (Vt∧τ ∗ ) is a gΓ -martingale. Last in this section, we illustrate our ideas by some examples. The first one is a counter-example to show that the first hitting time is not optimal if our assumptions are violated. Example 3.1. Let g ≡ 0 and Γt = At ∪ Bt , 0 ≤ t ≤ 2, where At := {y : 0 ≤ y ≤ and

2 }, 0 ≤ t ≤ 2, 3

2 Bt := {y : max{ , −2t + 2} ≤ y}, 0 ≤ t ≤ 2. 3

Such constraint is not convex obviously. We give a deterministic reward process Lt := − 21 t + 1, 0 ≤ t ≤ 2 which is not increasing but strict decreasing. The value process is  −2t + 2 when 0 ≤ t ≤ 23 ; Vt = − 12 t + 1 when 23 ≤ t ≤ 2. In this case, it is obvious that the first hitting time τ ∗ ≡ 32 is not optimal since E0g,φ (Lτ ∗ ) = 23 but E0g,φ (Lt ) = 2 for any 0 ≤ t ≤ 23 . The reason for the violation is that the constrained domain is not convex and the reward process is decreasing in time. However, if we modify the constrained area to be convex and keep the reward process be unchanged, then τ ∗ ≡ 32 will be optimal.

7

Example 3.2. Let g ≡ 0 and Γt := {y : max{0, −2t + 2} ≤ y}, 0 ≤ t ≤ 2, and the reward process is the same as in the above example Lt := − 21 t + 1, 0 ≤ t ≤ 2. Under this modified constrained area, the value process is still  −2t + 2 when 0 ≤ t ≤ 32 ; Vt = − 12 t + 1 when 23 ≤ t ≤ 2. A careful calculation shows that the first hitting time τ ∗ ≡

2 3

is now optimal since E0g,φ (Lτ ∗ ) = 2 this time.

The next example shows that if we change the reward process to be increasing and keep the the constrained domain be non-convex as in example 3.1, then the first hitting time is still optimal again. Example 3.3. Let g ≡ 0 and Γt = At ∪ Bt , 0 ≤ t ≤ 2, where At := {y : 0 ≤ y ≤

2 }, 0 ≤ t ≤ 2, 3

and

2 Bt := {y : max{ , −2t + 2} ≤ y}, 0 ≤ t ≤ 2. 3 The constraint is still non-convex but we give the deterministic reward process Lt := t, 0 ≤ t ≤ 2 increasing with time. Be careful for calculation, the value process is not  −2t + 2 when 0 ≤ t ≤ 32 ; Vt = − 12 t + 1 when 23 ≤ t ≤ 2.

anymore but is Vt ≡ 2 and the first hitting time is τ ∗ ≡ 2, which is optimal obviously.

We now come to a familiar example for valuating American options with constrained portfolio considered in I.Karatzas, S.G.Kou [9]. For simplicity, we suppose the market U contains a bond S0 (t) and only one stock S1 (t) modeled by dS0 (t) = S0 (t)r(t)dt, S0 (0) = 1 and

dS1 (t) = S1 (t)[b(t)dt + σdW (t),

S1 (0) = s

respectively, where the interest rate r(t), stock appreciation rates b(t) and the volatility matrix σ(t) are the coefficients of this model. They will be assumed throughout to be progressively measurable, and bounded uniformly in (t, ω) ∈ [0, T ] × Ω ; in addition, σ(t, ω) will be assumed to be invertible, with σ −1 (t, ω) bounded uniformly in (t, ω) ∈ [0, T ] × Ω. Under these assumptions, the relative risk process θ(t) := σ −1 (t)[b(t) − r(t)] is a bounded progressively measurable process. A portfolio process π(t) is a progressively measurable process satisfying Z T |π(t)|2 dt < ∞, a.s.. 0

A cumulative consumption process C(t) is an adapted process with increasing, right continuous paths and C(0) = 0, C(T ) < ∞ a.s.. For any given portfolio/cumulative consumption process pair (π, C) and x ∈ R, corresponding wealth process X(t) is given by the linear equation dX(t) = r(t)X(t)dt + π(t)σ(t)θ(t)dt + π(t)σ(t)dW (t) − dC(t), X(0) = x. Let p(t) =



when X(t) 6= 0; when X(t) = 0.

π(t)/X(t) 0

In J.Cvitanic, I.Karatzas[10], for a given convex closed set K, the authors consider constrained pricing and hedging problems with constraint p(t) ∈ K, a.s. for any t ∈ [0, T ]. We now explain how their formulation can be translate into our framework of constrained BSDE. In fact, for some claim ξ ∈ L2 (FT ), if we take πσ as z and X(t) as y(t), then the wealth process to hedge or price ξ can be rewritten as Z Z T

y(t) = ξ +

t

g(y(s), z(s))ds + CT − Ct − 8

T

t

z(s)dW (s)

(3.6)

with g(y, z) = ry + θz. The constraint put on portfolio process p(t) ∈ K can translated into a constraint function as  z dK ( σ(t)y ) when y 6= 0; φ(t, y, z) = (3.7) 0 when y = 0. where dK (·) is the distance function from K. For K, define the support function

δ(x) := sup (−px) p∈K

and its effective domain

˜ K(x) := {x ∈ R|δ(x) < ∞}.

Let H be the space of progressively measurable processes v(t) which satisfy E

Z

T

0

and

(v 2 (t) + δ(v(t)))dt < ∞

D := {v(t) ∈ H|

sup (t,ω)∈[0,T ]×Ω

|v(t, ω)| < ∞}.

With those notations above, J.Cvitanic, I.Karatzas[10] introduced a family of auxiliary markets {Uv |v ∈ D} to handle the constraints on portfolios. They do this by replacing r(t) with rv (t) = r(t) + δ(v(t)) and b(t) with bv (t) = b(t) + v(t) + δ(v(t)) in the origin market U. Let θv = σ −1 (t)[bv (t) − rv (t)]. Then by their wonderful analysis, they prove that the super-hedging price of a contingent claim ξ is given by E(ξ) := sup E v [γv (T )ξ]

(3.8)

v∈D

where γv (t) := exp[−

Rt 0

(r(s) + δ(v(s)))ds] and E v [·] is the expectation under the equivalent measure dP v = exp{− dP

Z

T

0

θv dW (s) −

1 2

Z

T

0

θv2 ds}.

Keeping a close eye on (3.8), we claim that E(ξ) is nothing but a gΓ -expectation with linear g(t, y, z) = r(t)y + θ(t)z, and constraint function φ given by equation (3.7), that is E(ξ) = E0g,φ (ξ). Within the same framework described above, I.Karatzas1, S.G.Kou [9] consider a problem of super hedging and pricing American options with constrained portfolios p(t) ∈ K. Suppose (B(t)) is an American option satisfying suitable assumptions, the author proved the super-price of (B(t)) is sup sup E[γv (τ )B(τ )].

(3.9)

v∈D τ ∈S0,T

Note that we can change the order of v and τ in the above equation, we can rewrite it as sup sup E[γv (τ )B(τ )] = sup E0g,φ (B(τ )),

τ ∈S0,T v∈D

τ ∈S0,T

it is in fact an optimal stopping problem under gΓ -expectation. Under this framework, we can do some explicit calculation with some useful results in I.Karatzas1, S.G.Kou[9] when the American option taking a special form of B(t) = ϕ(S1 (t)) for some nonnegative continuous function ϕ : (0, ∞) → (0, ∞). The proof of the following Theorem can be found in I.Karatzas1,S.G.Kou[9]. Theorem 3.2. When the market coefficients r(t) ≥ 0, b(t), σ(t) are real constants, the super-price of (B(t)) in (3.9), namely V := sup sup E[γv (τ )ϕ(S1 (τ ))] v∈D τ ∈S0,T

can be calculated by

V = sup E[e−rτ ϕ(S ¯ 1 (τ ))], τ ∈S0,T

where

ϕ(x) ¯ := sup [e−δ(v) ϕ(xe−v )]. ˜ v∈K

9

Example 3.4. We consider a market with constant coefficients, that is r(t) ≡ 0, b(t) ≡ b, σ(t) ≡ σ for real numbers b, σ. Let L(t) := (k − S1 (t))+ for some k ≥ 0, then it is nonnegative and L(t) ≤ k. We take ϕ(x) = (k − x)+ and the constraint as the set K = [0, ∞). The constraint function is given by (3.7), which is equivalent to p(t) ∈ K. Then as argued above, the optimal stopping problem under gΓ -expectation is the same as the super-pricing problem for American put option B(t) := (k − S1 (t))+ . To calculate it, we know  0 when v ∈ [0, ∞); δ(v) = sup (xv) = (3.10) ∞ when v ∈ (−∞, 0), x∈(−∞,0] thus when x ≥ 0,

ϕ(x) ¯ = sup [ϕ(xe−v )] ≡ k, v∈[0,∞)

and by theorem 3.2, we have sup E0g,φ (B(τ )) = sup sup E[γv (τ )B(τ )] = sup E[e−rτ ϕ(S ¯ 1 (τ ))] ≡ k.

τ ∈S0,T

τ ∈S0,T v∈D

(3.11)

τ ∈S0,T

In this example, although we have calculated the maximum value of the optimal stopping problem, there is no information help us to find an optimal stopping time under gΓ -expectation. But by the representation of E0g,φ (ξ) = supv∈D E v [γv (T )ξ], we know that the gΓ -expectation is convex. According to theorem 3.1, one optimal stopping time is given by τ ∗ := inf{t ≥ 0 : X t = (k − S1 (t))+ }

where .

X(t) := ess sup ess sup E v [(k − S1 (τ ))+ ] = ess sup Etg,φ (k − S1 (τ ))+ v∈D

τ ∈St,T

τ ∈St,T

Remark 3.2. An interesting interpretation for the above example can be made. Since (k−S1 (t))+ is a American put option with strike price k. Suppose the writer sell it at price x and use x as his initial money to invest in the market to hedge his risk. But at the same time, short-selling is prohibited, then what is the reasonable price he should ask? Example 3.4 shows that, with such constraint, he has no choice but ask exactly the same strike price k. This seems unreasonable, but it explains no put options will survive in a short-selling prohibited market. Remark 3.3. By the penalization method to obtain the gΓ -solution, Vt can be represented by Vt = ess sup ess sup Etgn (Lτ ) τ ∈St,T

n

which is a stopper and controller problem as a special case considered in I.Karatzas, I.M.Zamfirescub [7, 8] taking n as a control but here it is unbounded.

4

Dynamic programming principle and variational inequalities

In this section, we will give dynamic programming principles for optimal stopping problems. Our arguments are based on results about Dynkin’s game obtained in J.Cvitanic,I.Karatzas [11] and our extension to constrained case. In the unconstrained case, we first prove that the value function of the optimal stopping under g-expectation Vt := ess sup Etg (Lτ ),

(4.1)

τ ∈St,T

studied in F.Riedel and X.Cheng [6], where Lt is a nonnegative, adapted and continuous process, satisfies a dynamic programming principle, that is, Vt = ess sup Etg (Lτ 1(τ ≤σ) + Vσ 1(τ >σ) ),

(4.2)

τ ∈St,T

holds for any σ taking values between t and T . Theorem 4.1. For g satisfying usual assumptions (A1) and (A2) about BSDE, then the dynamic programming principle (4.2) holds for the optimal stopping under g-expectation. 10

Proof. We denote the right side of (4.2) as Mt , since Vt ≥ Lt and they are both RCLL, it is obvious that Etg (Lτ 1(τ ≤σ) + Vσ 1(τ >σ) ) ≤ Etg (Vτ 1(τ ≤σ) + Vσ 1(τ >σ) ) = Etg (Vτ ∧σ ) ≤ Vt for any τ, σ and Mt ≤ Vt holds. On the other hand, if we take Lt and Vt as lower and upper obstacles for a BSDE reflected from below and above respectively, then by the relation between bilateral reflected BSDE and Dynkin’s game obtained in J.Cvitanic; I.Karatzas [11], we have Vt = ess sup ess inf Etg [R(τ, σ)] = ess inf ess sup Etg [R(τ, σ)], τ ∈Tt

σ∈Tt

σ∈Tt

(4.3)

τ ∈Tt

where R(τ, σ) = Lτ 1(τ ≤σ) + Vσ 1(τ >σ) . This implies that Vt ≤ Mt and the dynamic programming principle (4.2) is proved. 2 Specially, in the markovian case, when the reward process is given by a function of solutions of Forward SDEs, namely Ls = ϕ(Xst,x ) for some diffusion process Xst,x evolving as dXst,x = b(s, Xst,x )ds + σ(s, Xst,x )dWs , then the value function defined by

Xtt,x = x,

t ≤ s ≤ T,

V (t, x) := ess sup Etg (h(Xτt,x ))

(4.4)

τ ∈St,T

is deterministic. As we have mentioned in the Introduction section, the value function under g-expectation is same with the solution of Reflected BSDE with lower barrier h(Xst,x ) and same with the minimal solution of Constrained BSDE with constraint φ(s, ω, y, z) = (y − h(Xst,x ))− = 0. Then by S.G.Peng, M.Y.Xu [19], V (t, x) in (4.4) satisfies a variational inequality min{−∂t V − F (t, V, DV, D2 V ), V − h(x)} = 0,

Pn where F (t, V, q, S) = 21 i,j=1 [σσ ∗ ]ij (t, x)Sij + hb(t, x), qi + g(t, V, σ ∗ (t, x)q) for q ∈ Rn , S ∈ Rn × Rn . Of course, by the same skill used in S.G.Peng [17], we can still deduce the variational inequality directly via the dynamic programming principle (4.2) with the help of the generator-representation theorem obtained in P.Briand; F. Coquet etc.[13], but it is redundant. In the constrained case, we secondly prove similar results with additional assumptions. We give an extension of the work in J.Cvitanic; I.Karatzas [11] in the following theorem. Theorem 4.2. Let g and φ satisfy assumptions (A1) and (A2), L(t) and U (t) are nonnegative continuous processes and there is some constant B > 0 such that L(t) ≤ B, U (t) ≤ B for any t ∈ [0, T ]. We consider the Dynkin’s game with lower and upper value function defined by V t := ess sup ess inf Etg,φ [R(τ, σ)],

(4.5)

V t = ess inf ess sup Etg,φ [R(τ, σ)],

(4.6)

τ ∈Tt

σ∈Tt

σ∈Tt

τ ∈Tt

where R(τ, σ) := L(τ )1(τ ≤σ) + U (σ)1(σ<τ ) and Tt are stopping times taking values between t and T . If L(t) is increasing in time, then we have V t = V t . Proof. Let gm := g + mφ and Vm (t) be the gm -solution of BSDE reflected from below and above by L(t) and U (t) respectively, V (t) := limm→∞ Vm (t), then by the limit theorem in S.G.Peng [16], V (t) is right continuous and the stopping times τt∗ = inf{s ≥ t : L(s) = V (s)} ∧ T, σt∗ = inf{s ≥ t : U (s) = V (s)} ∧ T, τt∗ (m) = inf{s ≥ t : L(s) = Vm (s)} ∧ T, σt∗ (m) = inf{s ≥ t : U (s) = Vm (s)} ∧ T

are well defined and L(τt∗ ) = V (τt∗ ), U (σt∗ ) = V (σt∗ ), τt∗ ≥ τt∗ (m), σt∗ ≤ σt∗ (m) for any m. For any n ≤ m, by the comparison theorem of BSDE and the relation between two-sided reflected BSDE and Dynkin’s game obtained in J.Cvitanic; I.Karatzas [11], we have Etgn [Vm (τ ∧ σt∗ )] ≤ Etgm [Vm (τ ∧ σt∗ )] ≤ Etgm [Vm (τt∗ (m) ∧ σt∗ (m))] = Vm (t) 11

(4.7)

for any τ and τt∗ (m), σt∗ (m) defined above. On the other hand, one has Vm (t) = Etgm [Vm (τt∗ (m) ∧ σt∗ (m))] ≤ Etgm [Vm (τt∗ (m) ∧ σ)] ≤ Etg,φ [Vm (τt∗ (m) ∧ σ)]

(4.8)

for any σ taking values in [t, T ]. In (4.8) , we want to show that Etg,φ [Vm (τt∗ (m) ∧ σ)] ≤ Etg,φ [V (τt∗ ∧ σ)].

(4.9)

By the definition of τt∗ (m), m = 1, 2 · · · and τt∗ , Vm (τt∗ (m) ∧ σ) = L(τt∗ (m))1(τt∗ (m)≤σ) + Vm (σ)1(σ<τt∗ (m)) and

V (τt∗ ∧ σ) = L(τt∗ )1(τt∗ ≤σ) + V (σ)1(σ<τt∗ )

So we have, by the increasing assumption in time of L(t), (i) when σ < τt∗ (m) ≤ τt∗ ,

Vm (τt∗ (m) ∧ σ) = Vm (σ) = Vm (τt∗ ∧ σ) ≤ V (τt∗ ∧ σ). (ii) when τt∗ (m) ≤ σ < τt∗ , Vm (τt∗ (m) ∧ σ) = Vm (τt∗ (m)) = L(τt∗ (m)) ≤ L(σ) ≤ V (σ) = V (τt∗ ∧ σ). (iii) when τt∗ (m) ≤ τt∗ ≤ σ, Vm (τt∗ (m) ∧ σ) = Vm (τt∗ (m)) = L(τt∗ (m)) ≤ L(τt∗ ) ≤ V (τt∗ ) = V (τt∗ ∧ σ). First, we take limit in (4.7) and (4.8) as m → ∞, with the help of (4.9) and comparison theorem of gΓ -solutions, we have Etgn [V (τ ∧ σt∗ ] ≤ V (t) (4.10) V (t) ≤ Etg,φ [V (τt∗ ∧ σ)].

(4.11)

Then we take limit in (4.10) as n → ∞, by the penalization method, gΓ -solution is obtained as an increasing limit of gn -solutions, we have Etg,φ [V (τ ∧ σt∗ ] ≤ V (t) (4.12) With (4.11) and (4.12) together, noting the facts that

R(τ ∧ σt∗ ) = L(τ )1(τ ≤σt∗ ) + U (σt∗ )1(σt∗ <τ ) ≤ V (τ ∧ σt∗ ) and

R(τt∗ ∧ σ) = L(τt∗ )1(τt∗ ≤σ) + U (σ)1(σ<τt∗ ) ≥ V (τt∗ ∧ σ),

then by the comparison proposition of gΓ -solution, we know V t ≤ V t and the reverse inequality is obvious, thus we complete our proof. 2 Based on the result stated above, we use the same method to get the dynamic programming principle for optimal stopping with constraint. Theorem 4.3. For g, φ satisfying usual assumptions about BSDE (A1) and (A2), then the dynamic programming principle Vt = ess sup Etg,φ (Lτ 1(τ ≤σ) + Vσ 1(τ >σ) ), τ ∈St,T

holds for the optimal stopping under g-expectation with constraint. In constrained case, the corresponding variational inequality can be obtained by taking Reflection and constraint φ as new comprehensive constraints. Then by S.G.Peng, M.Y.Xu [19], V (t, x) defined by V (t, x) := ess sup Etg,φ (h(Xτt,x )) τ ∈St,T

satisfies a variational inequality min{−∂t V − F (t, V, DV, D2 V ), V − h(x) − φ(t, V, σ ∗ DV } = 0, Pn where F (t, V, q, S) = 21 i,j=1 [σσ ∗ ]ij (t, x)Sij + hb(t, x), qi + g(t, V, σ ∗ (t, x)q). 12

References [1] D.Engelage: Optimal stopping with dynamic variational preferences. Journal of Economic Theory. 14(5), 2042-2074 (2011). [2] E.Bayraktar, S.Yao: Optimal Stopping for Non-linear Expectations. Stochastic Processes and Their Applications. 121(2), 185-264 (2011). [3] E.Pardoux, S.G.Peng: Adapted solution of a backward stochastic differential equation. Systems and control letters. 14, 55-62 (1990). [4] F.Coquet, Y.Hu, J.Memin, and S.G.Peng: Filtration consistent Nonlinear Expectations and Related gExpectations. Probability Theory and Related Fields, 123, 1-27 (2002). [5] F.Riedel: Optimal Stopping with Multiple Priors. Econometrica. 77, 857-908 (2009). [6] F.Riedel, X.Cheng: Optimal Stopping Under Ambiguity in Continuous Time. Working Papers 429, Institute of Mathematical Economics. [7] I.Karatzasa, I.M.Zamfirescub: Game approach to the optimal stopping problem. Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports. 77(5), 401 - 435 (2005). [8] I.Karatzasa, I.M.Zamfirescub: Martingale Approach to Stochastic Control with Discretionary Stopping. Applied Mathematics and Optimization, 53(2), 163-184 (2006). [9] I.Karatzas1, S.G.Kou: Hedging American contingent claims with constrained portfolios. Finance Stochast. 2, 215-258 (1998). [10] J.Cvitanic, I.Karatzas: Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3, 652681 (1993) [11] J.Cvitanic, I.Karatzas: Backward Stochastic Differential Equations with reflection and Dynkin games. The Annals of Probability 4(24), 2024-2056 (1996). [12] N. El Karoui, C.Kapoudjian, E.Pardoux, S.G.Peng and M.C. Quenez: Reflected solutions of Backward SDE’s and related obstacle problems for PDE’s. The Annals of Probability 25(2), 702-737 (1997). [13] P.Briand, F. Coquet, Y.Hu, J.Memin, S.G.Peng : A Converse Comparison theorem for BSDEs and Related Properties of g-expectation. Elect. Comm. in Probab. 5, 101-117 (2000). [14] R.Buckdahn, M.quincampoix, A.Rascanu: Viability property for a backward stochastic differential equation and applications to partial differential equations. Probab.Theory.Relat.Fields 116, 485-504 (2000) [15] S.G.Peng: Backward SDE and related g-expectation, Pitman Res. Notes Math. Ser. Longman, Harlow, 364, (1997). [16] S.G.Peng: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Related Fields. 113, 473-499 (1999). [17] S.G.Peng: A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics and stochastics reports. 38, 119-134 (1992). [18] S.G.Peng, M.Y.Xu: Reflected BSDE with a constraint and its applications in an imcomplete market. Bernoulli. 16(3), 614-640 (2010). [19] S.G.Peng, M.Y.Xu: Constrained BSDE and Viscosity Solutions of Variation Inequalities. arXiv preprint arXiv:0712.0306, (2007). [20] V.Kr¨atschmer, J.Schoenmakers: Representations for optimal stopping under dynamic monetary utility functionals. SFB 649 Discussion Papers, (2009).

13