AND DECREASING AFFINE FUNCTIONS
DIANA DOROBANTU
Our purpose is to study a particular class of optimal stopping problems for Markov processes. We justify the value function convexity and we deduce that there exists a boundary function such that the smallest optimal stopping time is the first time when the Markov process passes over a constant boundary. Moreover, we propose a method to find the optimal boundary.
AMS 2010 Subject Classification: 62L15, 60G40.
Key words: strong Markov process, optimal stopping, Snell envelope, passage time.
1. INTRODUCTION
In this paper we study a particular optimal stopping problem for strong Markov processes. We propose a method to find the optimal stopping time form (it will be the first time when the Markov process passes over a constant boundary), as well as for the calculation of the optimal boundary.
In fact we seek to control a stochastic process V of the formV =veX, where v is a real strictly positive constant and X a strong Markov process.
We consider the following optimal stopping problem:
s(v) = sup
τ∈∆E
e−rτf(Vτ)|V0 =v ,
where r > 0, FtV = σ(Vs, s ≤ t), ∆ is the set of FV-stopping times and f is a Borelian function f(V) = −V +c, c > 0. Our problem is, hence, an optimal stopping problem for Markov processes and linear reward. We justify the convexity of the value function sand we deduce that the optimal strategy consists of stopping when the underlying Markov process crosses a boundary, i.e., the smallest optimal stopping time has the form inf{t≥0 :Vt≤ b}. The main result is given by Theorems 3.1, 3.2, 3.3 and Proposition 3.4 which allow to determine the optimal stopping time form and the optimal boundary function.
REV. ROUMAINE MATH. PURES APPL.,56(2011),4, 283–294
Optimal stopping theory is a subject which often appears in the special- ized literature. For different areas of application or different methods for opti- mal stopping problems see, for example, Peskir and Shiryaev (2003). Among others, Salminen (1985), Leland (1994, 1996, 1998), Duffie and Lando (2001), Dayanik and Karatzas (2003), Decamps and Villeneuve (2007, 2008), Ciss´e et al. (2010) studied optimal stopping problems for continuous Markov processes.
Moreover, there are other authors who used L´evy jumps processes (e.g. Pham (1997), Mordecki (1999), Hilberink and Rogers (2002), Kou and Wang (2004), Dao (2005), Kyprianou (2006), Dorobantu (2007), ...), symmetric Markov pro- cesses (e.g. Zabczyk (1984)) or Markov processes possessed the fragmentation property (Kyprianou and Pardo (2011)) for their models. Sometimes the stu- died problem has the form sup
τ≥0E[e−rτh(Vτ)], other times it is more compli- cated sup
τ≥0E[e−rτh(Vτ, τ)]. Our result completes these studies and the aim of the present paper is to solve a stopping time problem for a more general class of processes (more precisely, Markov processes not necessarily continuous).
Contrary to the usual method, our method avoids long calculations of the integro-differential operators.
This paper is organized as follows: we introduce the optimal stopping problem (Section 2). The following section (Section 3) contains the main re- sults which characterize the optimal stopping time and the optimal boundary.
Section 4 is dedicated to the proofs of Theorems 3.1, 3.2 and 3.3. Section 5 contains some examples.
2. OPTIMAL STOPPING PROBLEM
Let V be a stochastic process on a filtered probability space (Ω,F, (Ft)t≥0,P). Assume that V has the form V =veX where v is a real strictly positive constant and X is a strong Markov process such that X0 = 0. Let FV be the right-continuous complete filtration generated by the process V, FtV =σ(Vs, s≤t). We introduce ∆ the set of FV-stopping times.
From now on,E(·|V0 =v) and P(·|V0 =v) are denotedEv(·) andPv(·).
We consider the following optimal stopping problem:
(1) s(v) = sup
τ∈∆Ev
e−rτf(Vτ) ,
where f is a decreasing linear function,f(v) =−v+c, v >0 andr, c >0.
We suppose that the processX checks the following assumptions:
(H1)P
limt↓0Xt=X0
= 1.
(H2) The process (e−rt+Xt, t≥0) is of class D.
(H3) The support ofXt isRfor all t >0.
We will prove later that assumptions (H2) and (H3) may be replaced by
“There exists q ∈R such that the support of Xt is included in ]− ∞, q] for all t >0.”
Under (H1), (H2) and (H3), we prove that the smallest optimal stopping time of (1) is necessarily of the form inf{t≥0 : Vt≤b} and we compute the optimal boundary. We applied the same method in [13] for L´evy processes and linear functions, but it may be extended to a more general class of processes.
3. MAIN RESULTS
The main results characterize the smallest optimal stopping time of (1).
We show the following:
Theorem3.1. Under (H1), (H2)and(H3), there exists at least an opti- mal stopping time for the problem (1). For any c >0, there existsbc >0 such that the smallest optimal stopping time has the following form
τbc = inf{t≥0 :Vt≤bc}.
We introduce an auxiliary functionsb(v) =Ev[e−rτb(−Vτb+c)],v∈R∗+, b∈]0, c[ whereτb= inf{t≥0 :Vt≤b}. Let us point out that if b∈R+, then sb(·) is not necessarily positive. The condition b∈]0, c[ implies the positivity of sb(·).
Remark. Under the assumptions of Theorem 3.1, there exists Bc such that sBc(·) =s(·).
Remark that we can writes.(·) as a function of Laplace transforms L(x) =E
e−r¯τx|X0 = 0
, G(x) =E
e−r¯τx+Xτx¯ |X0 = 0
where ¯τx= inf{t≥0 :Xt≤x}. Indeed, the functions.(·) can be written as sb(v) =−vG lnbv
+cL lnvb .
The following theorems characterize the value of the optimal threshold Bc as a function ofc,L(·) and G(·).
WhenG is discontinuous atx= 0,Bc is easy to obtain.
Theorem3.2. Under(H1), (H2)and(H3), we suppose that the function G is discontinuous at x= 0. Then the smallest optimal stopping time is τ∗= inf{t≥0 :Vt≤Bc}, where Bc =clim
x↑0 1−L(x) 1−G(x).
When G is continuous at x = 0, Bc is more technical to obtain, but it has the same form.
Theorem3.3. Under(H1), (H2)and(H3), we suppose that the function G is continuous at x= 0. Then we have the following:
1. IfGhas left derivative atx= 0 (sayG0(0−)), thenLhas left derivative at x= 0 (say L0(0−)).
2. If moreover G0(0−)6= 0, thenBc ∈[˜b, c[ where ˜b=c lim
x↑0 1−L(x) 1−G(x). 3. If moreovers˜b(·)is strictly convex on]˜b, ∞[, then the smallest optimal stopping time is τ∗ = inf{t≥0 :Vt≤Bc}, where Bc = ˜b.
Theproofs of Theorems 3.1, 3.2 and 3.3 are given in Section 4.
In the case where X is a continuous Markov process, we have directly the following result (direct consequence of [29], chapter 9.1):
Proposition3.4. IfX is a continuous Markov process, then under(H2) and (H3), Bc =c lim
x↑0 1−L(x) 1−G(x).
4. PROOFS
Theproof of Theorem 3.1 requires several results.
Remark thatsis a (decreasing) convex function because it is the sup of (decreasing) linear functions:
s(v) = sup
τ≥0Ev
e−rτ(−Vτv+c)
= sup
τ≥0E1
e−rτ(−vVτ1+c) .
Remark.Since sis a convex function, then it is continuous.
Under (H2), the process (e−rtf(Vt), t ≥ 0) is of class D. According to Theorem 3.4 of [18], the Snell envelope of this process has the form (e−rts(Vt), t≥0). Theorem 3.3 page 127 of [32], allows us to find the optimal stopping of a problem sup
τ≥0Ev[f(Vτ)] wheref is a measurable function. We easily deduce that this result may be applied to a process having the formt7→e−rtf(Vt). In our case, we cannot apply this result for the problem (1) because the process t7→e−rtf(Vt) does not check the assumptions of Theorem 3.3 page 127 of [32];
that is why we rewrite the function sunder a new form.
Lemma 4.1. For v > 0, let s+(v) = sup
τ∈∆Ev[e−rτ(−Vτ+c)+], where x+= max(x,0). Under (H1), (H2) and(H3), s+(v)>0 and s(v) =s+(v) for every v >0.
Proof. We show that if there exists v0 > 0 such that s(v0) < s+(v0), then there existsv1>0 such thats+(v1) = 0. We prove that this last relation cannot be satisfied.
By construction, for eachv >0,s(v)≤s+(v). Let us suppose that there exists v0>0 such thats(v0)< s+(v0).
Under Assumption (H1), the process V. is right continuous at 0. Since the process Y+ : t → Yt+ = e−rt(−Vt +c)+ takes its values in [0, c], the assumptions of Theorem 3.3 page 127 of [32] are checked for Y+. We denote by f+ the functionf+(v) = (−v+c)+; the stopping time
τ+= inf{u≥0 :f+(Vuv0) =s+(Vuv0)}
is the smallest optimal stopping time of the problem s+(v0) = sup
τ≥0Ev0[e−rτ(−Vτ +c)+].
Using the definition ofsand s+, we have Ev0
e−rτ+f(Vτ+)
≤s(v0)< s+(v0) =Ev0
e−rτ+f+(Vτ+) and consequently
Ev0
e−rτ+(f(Vτ+)−f+(Vτ+))
<0, Pv0({ω:f(Vτ+)<0})>0 and Pv0({ω:s+(Vτ+) = 0})>0.
Thus there existsv1 such thats+(v1) = 0. Then for any stopping timeτ, Pv1-almost surelye−rτf+(Vτ) = 0 and in particular for everyt∈R+,f+(Vt) = 0. This involves thatPv1-almost surelyVt≥cwhich is a contradiction because under assumption (H3), the support ofVtisR∗+. Therefore,s+(v)>0 for every v ∈R∗+ and s(v) =s+(v).
Thanks to Lemma 4.1, the problem (1) can be brought back to an opti- mal stopping problem for an American Put option with strike price c. Such a problem has been studied by many authors whenX is a L´evy process (see for example Gerber and Shiu (1994), Pham (1997), Mordecki (1999), Boyarchenko and Levendorskii (2002), Avram, Chan and Usabel (2002), Chesney and Jean- blanc (2004), Asmussen, Avram and Pistorius (2004), Alili and Kyprianou (2005), Kyprianou (2006)). Next, we use a method close to the one used by Pham (1997). Pham studies an optimal stopping problem for an American Put option with finite time horizon. In his modelXis a L´evy process. He uses integro-differential equations to solve his problem.
Proof of Theorem 3.1. By Lemma 4.1, the problem (1) can be written as sup
τ≥0E(Yτ+). By Theorem 3.3 page 127 of [32], τ∗ = inf{u ≥0 : f+(Vu) = s+(Vu)}is the smallest optimal stopping time. However,s(v) =s+(v)>0 for all v >0, so
τ∗ = inf{u≥0 :f(Vu) =s(Vu)}
is the smallest optimal stopping time.
The function sis upper bounded byc becauseY.+ is upper bounded by c and lim
v↓0s(v) = lim
v↓0f(v) = c. Since s is convex, f linear and f(·) ≤ s(·), then {v > 0 : f(v) = s(v)} is an interval of the form ]0, bc]. This means that the smallest optimal stopping timeτ∗ is also the first entrance time ofV in ]0, bc].
The smallest optimal stopping time is hence a hitting time for the pro- cess V.
Proof of Theorem3.2. Let b∈]0, c[. The functionsb(·) has the form sb(v) =
−v+c if v≤b,
−vG lnvb
+cL lnbv
if v > b.
If the function sb(·) is continuous atb, thenb is solution of (2) −b+c=−bG(0−) +cL(0−).
However, G is discontinuous at x= 0, soG(0−)6= 1 and the equation (2) has only one solution
b∗ =c1− L(0−)
1− G(0−) =c lim
x↑0
1− L(x) 1− G(x).
The function shas the form sBc(·) =s(·) and is convex, thus it is con- tinuous, in particular it is continuous at Bc. We deduce thatBc =b∗.
Proof of Theorem3.3
1. By Remark, there exists Bc such thatsBc(·) =s(·). The function sis convex, therefore the right and left derivatives exist everywhere and
(3) s0(v−)≤s0(v+) for allv∈R∗+,
where s0(v−) and s0(v+) are the left and right derivatives of s atv. In parti- cular, this means that
sBc(v) =−vG
lnBc v
+cL
lnBc
v
=s(v)
has right and left derivatives at v=Bc. SinceG has right and left derivatives atx= 0, then L has also right and left derivatives atx= 0.
2.Let us makev=Bc in (3):
−1≤ −1 +G0(0−)− c Bc
L0(0−).
We deduce that Bc ≥˜b=c LG00(0(0−−)) =c lim
x↑0 1−L(x) 1−G(x).
3.If moreover s˜b(·) is strictly convex on ]˜b, ∞[, then (4) s˜b(v)> f(v) for allv >˜b.
Indeed, the graph off is tangent to the graph ofs˜b(·) in v= ˜b.
Suppose that Bc > ˜b, then f(Bc) = s(Bc) = sBc(Bc) ≥ s˜b(Bc) which contradicts (4).
Remark.
• Assumption (H2) and (H3) may be replaced by “There exists q ∈ R such that the support of Xt is included in ]− ∞, q] for all t >0.”.
Under this assumption, we don’t need to use the intermediate Lemma 4.1 to find the smallest optimal stopping time form. In this case the process (f(Vt), t ≥0) is bounded and Theorem 3.3 page 127 of [32] can be directly applied. The function s is not necessarily continuous, but its continuous ex- tension by linear interpolation is convex and the conclusion of Theorems 3.1, 3.2 and 3.3 are true.
•The conditions of Theorem 3.1 are satisfied whenXhas jumps and no diffusion component (see [29], Chapter 9).
Our results are consistent with existing literature. Recall that our prob- lem can be brought back to an American Put optimal stopping problem for strong Markov processes. Various authors have found that, in the case of a L´evy process, the American Put optimal stopping problem is linked to the first passage problem of the L´evy process. Moreover, the optimal threshold is obtained using continuous or smooth pasting condition. For example, in [1, 5] sufficient or necessary and sufficient conditions for smooth and contin- uous pasting were established for different classes of L´evy processes. To this subject (but for a different optimal stopping problem), see also [24, 29]. The aim of this paper is to solve a little more general problem than the American Put optimal stopping problem, for a more general class of processes.
5. EXAMPLES
The examples presented in this section are some models where Ev(e−rτb+aXτb) is known for allb. Next we consider some particular processes, we check that the assumptions of Theorem 3.2, Theorem 3.3 or Proposition 3.4 are satisfied and we solve the problem (1) in each case. We start with a con- tinuous L´evy process (Brownian motion), then we continue with a double exponential jump-diffusion process, a particular spectrally positive process, the Poisson process and a spectrally negative process. We finish this section with two non L´evy processes.
5.1. L´evy processes
We present the results obtained for some L´evy processes (all calculations can be found with all details in [13]).
• Brownian motion. Let X = W where W is a standard Brownian motion (so (H1) and (H3) are checked). Letr > 12, hence assumption (H2) is checked. Using Remark 8.3 page 96 of [19] which gives the Laplace transform of a hitting time in the case of a Brownian motion,
L(x) =ex
√
2r and G(x) =ex(
√
2r+1), x <0.
We easily proved (Proposition 3.4 or Theorem 3.3) that the smallest optimal stopping time is
τ∗ = inf (
t≥0 : Vt= c√
2r r−12 r(√
2r+ 1) )
.
We find Duffie and Lando’s result [14] for a Brownian motion with drift. Their optimal stopping problem can be easily reduced to (1).
• Double exponential jump-diffusion process. The next model is Kou and Wang’s model (we refer to [20, 21] and [7]). Indeed, we suppose that X is a mixed diffusion-jump process and the jump size is a double exponential distributed random variable Xt = mt +σWt +
Nt
P
i=1
Yi, t ≥ 0, where W is a standard Brownian motion, N is a Poisson process with con- stant positive intensity a, (Yi, i ∈ N) is a sequence of i.i.d. random vari- ables (so (H1) and (H3) are checked). The common density of Y is given by fY(y) =pη1e−η1y1y>0+qη2eη2y1y<0,y∈R,wherep+q= 1, p, q >0,η1 >1 andη2 >0. Moreover, we suppose that (Yi,i∈N),N andW are independent.
Under the conditionr > m+σ22+aE(eY−1) whereE(eY−1) = ηη1p
1−1+ηη2q
2+1−1, the process (e−rt+Xt,t≥0) is of class D.
We easily verify the assumptions of Theorem 3.3 using the following results:
L(x) = ψ2(η2+ψ3) (ψ2−ψ3)η2
e−xψ3 −ψ3(η2+ψ2) (ψ2−ψ3)η2
e−xψ2, G(x) =ex
(η2+ψ3)(ψ2−1)
(ψ2−ψ3)(η2+ 1)e−xψ3+(η2+ψ2)(1−ψ3) (ψ2−ψ3)(η2+ 1)e−xψ2
,
where −∞< ψ3 <−η2 < ψ2<0 are the two negative roots of the equation mψ+σ2
2 ψ2+a η1p
η1−ψ + η2q η2+ψ −1
=r.
The smallest optimal stopping time isτ∗= inf n
t≥0 :Vt≤c(r−ψ(1))ψrη 2ψ3(η2+1)
2(1−ψ2)(1−ψ3)
o .
•Exponential jump-diffusion process.We suppose thatXis a mixed diffusion-jump process and the jump size is a random variable with an expo- nential distribution: Xt = mt+σWt+
Nt
P
i=1
Yi, t≥ 0, where W is a standard Brownian motion, N is a Poisson process with constant positive intensity a, (Yi, i ∈ N) is a sequence of i.i.d.r.v. with an exponential distribution, i.e., the common density of Y is given by fY(y) =η1e−η1y1y>0 where η1 >1 (as- sumptions (H1) and (H3) satisfied). Moreover, we suppose that (Yi,i∈N),N and W are independent. As for the double exponential jump-diffusion process (here we consider q = 0, so p = 1), under the condition r > m+ σ22 + ηa
1−1, (H2) is checked.
The results of [15] allows us to write for every x < 0 the form of the functions x7→ L(x) andx7→ G(x):
L(x) =exλ¯ and G(x) =ex(¯λ+1), whereiλ¯ is the unique solution ofr+−imλ+σ22λ2 −ηaiλ
1−iλ = 0.We verify the conditions of Theorem 3.3 and we conclude that the smallest optimal stopping time is
τ∗ = inf (
t≥0 : Vt≤ c¯λ(r−m− σ22 −ηa
1−1) r(¯λ+ 1)
) .
• Poisson process. Let X = −N where N is a Poisson process with constant positive intensity a. Following Remark 4, Theorem 3.2 may be ap- plied. We easily proved that G is discontinuous atx= 0:
L(x) =1x≥0+X
i≥1
a r+a
i
1i−1<−x≤i,
G(x) =1x≥0+X
i≥1
a e(r+a)
i
1i−1<−x≤i.
Using Theorem 3.2 the optimal stopping time is τ∗ = inf
t≥0 : Vt≤ ce(r−a(e−1−1)) er+ea−a
.
•Spectrally negative processes. Case of an unbounded variation process. The case of spectrally negative processes is more complicated and re- quires more calculations than the other examples (for all detailed calculations see [13]). We present here the example of a spectrally negative process with a non null Gaussian component. Throughout this section, we suppose thatX
is a real-valued L´evy process with no positive jumps. Some authors say that X is spectrally negative. The case when X is either a negative L´evy process with decreasing paths or a deterministic drift are excluded in this sequel. The Laplace transform of such a process exists and has the following form (see [4]
page 187–189 for details): E(eλXt) = etψ(λ). We denote by Φ(0) the largest solution of the equation ψ(λ) = 0. In all cases,ψ : [Φ(0),∞[→ R+ is contin- uous and increasing, it is a bijection and its inverse is Φ : [0,∞[→[Φ(0),∞[:
ψ◦Φ(λ) =λ(λ >0). Under the assumptionr > ψ(1) (equivalent to Φ(r)>1), we may prove that (H2) is checked.
Following [4],
G(x) =Z1(r−ψ(1))(−x)−r−ψ(1)
Φ(r)−1W1(r−ψ(1))(−x), L(x) =Z(r)(−x)− r
Φ(r)W(r)(−x), where
W(q)(x) =eΦ(q)xPΦ(q)
t≥0infXt≥0|X0=x
and
Z(q)(x) = 1 +q Z x
0
W(q)(y)dy.
We may verify the conditions of Theorem 3.3. The smallest optimal stop- ping time is τ∗= infn
t≥0 : Vt≤ c(Φ(r)−1)Φ(r) o . 5.2. Non L´evy processes
We consider here two examples of non L´evy processes.
•Absolute value of Brownian Motion. LetX=−|B|whereB is a Brownian motion. Following Remark 4, we can apply Proposition 3.4. Since
G(x) =ex[ch(−x√
2r)]−1 and L(x) = [ch(−x√ 2r)]−1,
we easily compute the optimal stopping time: τ∗ = inf{t≥0 :Vt= 0}=∞.
• Ornstein Uhlenbeck process. Let X be an Ornstein Uhlenbeck satisfying dXt =−λXtdt+ dBt, X0 = 0 where B is a Brownian motion and λ ∈ R. Let us consider for example that r > 12, then (H2) holds (because X ≤B and under this condition the process (e−rt+Bt, t≥0) is of class D).
Following [33], we get G(x) = exL(x) and L(x) = H−r/λ(0)
H−r/λ(x√
λ), where Hα(·) is the Hermite function. We apply Proposition 3.4 and we find Bc = c
√λH−r/λ0 (0)
√
λH−r/λ0 (0)−H−r/λ(0). Using the properties of Hα(·), we have the optimal
stopping time
τ∗ = inf (
t≥0 : Vt=c Γ λ+r2λ Γ λ+r2λ
+ 1
2√
λΓ 2λr )
.
Acknowledgements.This work was partially supported by the MIRACCLE-GICC Project and the Chaire d’excellence “Generale-Actuariat responsable: gestion des risques naturels et changements climatiques”.
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Received 7 October 2011 University of Lyon
University Lyon 1 ISFA LSAF(EA 2429), France diana.dorobantu@univ-lyon1.fr
