On an Optimal Stopping Problem of an Insider

0.09 MB·English

by  Erhan Bayraktar

ON AN OPTIMAL STOPPING PROBLEM OF AN INSIDER
ERHANBAYRAKTARAND ZHOUZHOU
Abstract. Weconsider theoptimal problem sup E n φi , where T >0is a fixed
3 τ∈Tε,T hPi=1 (τ−εi)+i
1 time horizon, (φi) is progressively measurable with respect to the Brownian filtration, εi ∈
t 0≤t≤T
20 [0,T] is a constant, i=1,... ,n, and Tε,T is the set of stopping times that lie between a constant
ε ∈ [0,T] and T. We solve this problem by conditioning and then using the theory of reflected
n
backward stochastic differential equations (RBSDEs). As a corollary, we provide a solution to
a
J the optimal stopping problem supτ∈T0,T EB(τ−ε)+ recently posed by Shiryaev at the International
0 Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of
2
Mathematics in September2012.

]
R
P 1. Main result
.
h
t Let T > 0 and let {Bt,t ∈ [0,T]} be a Brownian motion defined on a probability space (Ω,F,P)
a
m and let F = {Ft,t ∈ [0,T]} be the natural filtration augmented by the P-null sets of F. Consider
[ the optimal stopping time problem

2 n
v (1) v = sup E φi ,
0 (τ−εi)+
0 τ∈Tε,T "i=1 #
X
1
3 where (φit)0≤t≤T is continuous and progressively measurable, εi ∈ [0,T], i = 1,... ,n, are given
.
1 constants, and Tε,T is the set of stopping times that lie between a constant ε ∈ [0,T] and T.
0 Observe that τ −εi is not a stopping time with respect to F for εi > 0. This can be thought of a
3
1 problem of an insider in which he is allowed to peek εi into the futurefor each payoff before making
:
v his stopping decision. The solution to (1) is described by the following result:
i
X
ar Theorem 1. Assume E sup0≤t≤T(ξt+)2 < ∞, where ξt = ni=1φi(t−εi)+, 0 ≤ t ≤ T. Then the
value defined in (1) can be calculated using a reflected backward stochastic differential equation
(cid:2) (cid:3) P
(RBSDE). More precisely, v = EY , for any ε∈ [0,T], where (Y ) satisfies the RBSDE
ε t 0≤t≤T
T
ξ ≤ Y = ξ − Z dW +(K −K ), 0≤ t ≤ T,
t t T s s T t
(2) Zt
T
(Y −ξ )dK = 0,
t t t
Z0
Date: January 22, 2013.
Keywords and phrases. optimalstoppingproblemofaninsider,ReflectedBackwardStochasticDifferentialEqua-
tions (RBSDEs), conditioning.
ThisresearchwassupportedinpartbytheNationalScienceFoundationundergrantsDMS0906257,DMS0955463,
and DMS 1118673.
1
2
Moreover, there exists an optimal stopping time τˆ described by
τˆ= inf{t ∈ [ε,T] :Y = ξ }.
t t
Proof. For any τ ∈ T ,
ε,T
Eξ = E[E[ξ |F ]]≤ E esssupE[ξ |F ] .
τ τ ε τ ε
τ∈Tε,T
h i
Therefore,
(3) v = sup Eξ ≤ E esssupE[ξ |F ] .
τ τ ε
τ∈Tε,T τ∈Tε,T
h i
By Theorem 5.2 in [4] there exists a unique solution (Y,Z,K) to the RBSDE in (2). Then by
Proposition 2.3 (and its proof) in [4] we have
sup Eξ ≥ Eξ = EY = EY = E esssupE[ξ |F ] .
τ τˆ τˆ ε τ ε
τ∈Tε,T τ∈Tε,T
h i
Along with (3) the last inequality completes the proof. (cid:3)
One should note that the optimal stopping problem we are considering is path dependent (i.e.
not of Markovian type) and therefore one would not be able to write down a classical free boundary
problem corresponding to (4) unless one is willing to consider free boundary problems in the path
dependent PDE framework of [3]. This is the main reason we work with RBSDEs.
Now let’s consider Shiryaev’s problem:
(4) v(ε) = sup EB ,
(τ−ε)+
τ∈T0,T
Remark 1. Observe that for ε > 0, insider’s value defined in (4) is strictly greater than 0 (and
hence does strictly better than a stopper which does not posses the insider information):
2
v(ε) ≥ E max B = (ε∧(T −ε)) >v(0) = 0,
t
0≤t≤ε∧(T−ε) rπ
h i
which shows that there is an incentive for waiting. We also have an upper bound
2T
vε ≤ E max B = .
t
0≤t≤T r π
h i
In fact when ε ∈[T/2,T], vε can be explicitly determined as
2(T −ε)
v(ε) = E max B = , ε ∈ [T/2,T].
t
(cid:20)0≤t≤T−ε (cid:21) r π
and we have a strict lower bound for ε ∈[0,T/2]
2ε
v(ε) > E max B = , ε ∈ [0,T/2].
t
(cid:20)0≤t≤ε (cid:21) r π
For arbitrary values of ε ∈ [0,T] we have the following result as a corollary of Theorem 1:
3
Corollary 2. The value defined in (4) can be calculated using an RBSDE. More precisely, vε = Y
0
almost surely, where (Y ) satisfies the RBSDE (2) with ξ defined as ξ = B , 0≤ t ≤ T.
t 0≤t≤T t (t−ε)+
Moreover, there exists an optimal stopping time τˆ described by
(5) τˆ = inf{t ≥ 0: Y = B } ≥ ε∧(T −ε).
t (t−ε)+
Furthermore, the function ε→ v(ε), ε ∈[0,T], is a continuous function.
Proof. The continuity of ε → v(ε), ε ∈ [0,T] is a direct consequence of the stability of RBSDEs
indicated by Proposition 3.6 in [4]. On the other hand, since
v(ε) = sup EB = sup EB , δ ∈ [0,ε],
(τ−ε)+ (τ−ε)+
τ∈T0,T τ∈Tδ,T
from Theorem 1 we can conclude that v(ε) = EY . The latter implies that (Y ) is a martingale
δ t t∈[0,T]
from on t ∈ [0,ε]. It also shows that the stopping time τ defined by
δ
τ = inf{t ∈ [δ,T] : Y = ξ },
δ t t
is optimal for both the problem sup EB and sup EB , δ ∈ [0,ε]. In fact, one
τ∈T0,T (τ−ε)+ τ∈Tδ,T (τ−ε)+
can observe that for t ∈ [0,ε∧(T −ε)), Y ≥ E[Y |F ] > 0 = ξ , a.s., which implies dK = 0, 0 ≤
t ε t t t
t ≤ ε∧(T −ε), a.s, (so the martingale property of (Y ) for t ∈ [0,ε∧(T −ε)] can also be concluded
t
from this point of view). The martingale property on the other hand implies that stopping time
defined in (5) is no less than ε∧(T −ε) almost surely. (cid:3)
Optimal stopping problems of the type
m
sup E ϕj ,
 τ+δj
τ∈T0,T j=1
X
 
where ϕj’s and progressively measurable δj > 0’s are constants, were considered by [1] and [2]. A
generalization combining these type of pay-offs with the one in (1) will be left for future work.
References
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[2] B. Bruder and H. Pham, Impulse control problem on finite horizon with execution delay, Stochastic Process.
Appl., 119 (2009), pp.1436–1469.
[3] I. Ekren, C. Keller, N. Touzi, and J. Zhang, On Viscosity Solutions of Path Dependent PDEs, ArXiv
e-prints, (2011). To appear in theAnnals of Probability.
[4] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez, Reflected solutions of backward
SDE’s, and related obstacle problems for PDE’s,Ann.Probab., 25 (1997), pp.702–737.
Department of Mathematics, University of Michigan
E-mail address: [email protected]
Department of Mathematics, University of Michigan
E-mail address: [email protected]