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An optimal stopping problem in a diffusion-type model with delay
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Statistics & Probability Letters 76 (2006) 601–608 www.elsevier.com/locate/stapro
An optimal stopping problem in a diffusion-type model with delay Pavel V. Gapeeva, Markus ReiXb, a
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Str. 65, 117997 Moscow, Russia Institute for Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany
b
Received 3 March 2005; received in revised form 17 August 2005 Available online 30 September 2005
Abstract We present an explicit solution to an optimal stopping problem in a model described by a stochastic delay differential equation with an exponential delay measure. The method of proof is based on reducing the initial problem to a freeboundary problem and solving the latter by means of the smooth-fit condition. The problem can be interpreted as pricing special perpetual average American put options in a diffusion-type model with delay. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60G40; 35R35; 91B70; secondary 60J25; 60J60; 60G44 Keywords: Optimal stopping; Stochastic delay differential equation; Diffusion process; Sufficient statistic; Free-boundary problem; Smooth fit; Girsanov’s theorem; Itoˆ’s formula
1. Introduction The main aim of this paper is to present a solution to the optimal stopping problem (3) for the process X that solves the stochastic differential (1) with an exponential delay measure on an infinite time interval. This problem is related to the option pricing theory in mathematical finance, where the process X can describe the logarithm of the price of a risky asset (e.g. a stock) on a financial market. In that case, the value (3) can be formally interpreted as a fair price of a special perpetual average American put option in a diffusion-type market model with delay. In this model, the dynamics of the price depends on its deviation from the running average over past values. In recent years, several control problems for models described by stochastic delay differential equations were studied. Øksendal and Sulem, 2000 proved maximum principles for certain classes of such models and applied them to solving some problems related to finance. Elsanosi et al., 2000 proved a verification theorem of variational inequality type and applied it to finding explicit solutions for some classes of optimal harvesting Corresponding author.
E-mail addresses: [email protected], [email protected] (P.V. Gapeev), [email protected], [email protected] (M. ReiX). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.09.006
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delay problems. Larssen, 2002 established the dynamic programming principle for stochastic delay differential equations. Larssen and Riesebro, 2003 exhibited certain classes of delayed control problems that can be reduced to ordinary control problems. In this paper, we show how an explicit solution to an optimal stopping problem in a model described by a stochastic delay differential equation can be derived. The paper is organized as follows. In Section 2, using change-of-measure arguments, for the initial problem (3) we construct an equivalent optimal stopping problem for the one-dimensional Markov deviation process. In order to find explicit expressions for the value function and the optimal boundary, we formulate an associated free-boundary problem. In Section 3, we derive a solution to the free-boundary problem, which can be expressed by the confluent hypergeometric function and thus admits a representation in closed form. In Section 4, we verify that the solution of the free-boundary problem turns out to be a solution of the initial optimal stopping problem. The main result of the paper is stated in Theorem 4.1. 2. Formulation of the problem First, let us give a precise description of the diffusion-type model with delay. 2.1. Suppose that on some probability space ðO; F; PÞ there exists a standard Wiener process W ¼ ðW t ÞtX0 and a continuous process X ¼ ðX t Þt2R solving the stochastic differential equation: dX t ¼ ðy2 =2ÞðX t lY t Þ2 dt þ yðX t lY t Þ dW t
for tX0; X 0 ¼ x,
where the process Y ¼ ðY t ÞtX0 is defined by Z 0 Yt ¼ els X tþs ds; X t ¼ X 0t for tp0,
(1)
(2)
1
for some y40, l40, and x 2 R given and fixed. Here X 0t , tp0, is a (deterministic) bounded measurable function. Since the exponential of X is a local martingale, the process X can be thought of describing the logarithm of a (discounted) stock price on a financial market. The goal of this paper is to compute the value: V ¼ sup E½edt ðKelY t eX t Þþ ,
(3)
t
where the supremum is taken over all finite stopping times t of the process X (i.e. stopping times with respect to the natural filtration ðFt ÞtX0 of X), and to determine an optimal stopping time at which the supremum in (3) is attained. The value (3) can be interpreted as an arbitrage-free price of a special average American put option, where K40 and d40 are some given constants. Some other optimal stopping problems for geometric Brownian motion with gain functions containing integrals were solved in Kramkov and Mordecki, 1994 and Peskir and Uys, 2005. Note that a different class of optimal stopping problems can be obtained when the underlying process is Markovian, but at the same time, there is a delay in the available information as in Øksendal, 2004. By differentiation it can be shown that the process Y admits the representation: dY t ¼ Z t dt;
Y 0 ¼ y,
(4)
where the process Z ¼ ðZ t ÞtX0 is defined by Zt ¼ X t lY t
(5)
for all tX0. The process Z defined in (5) expresses the deviation of the logarithm of the present value of the process X from its exponentially weighted average lY . By means of Itoˆ’s formula (see e.g. Liptser and Shiryaev, 1977, Theorem 4.4 or Jacod and Shiryaev, 1987, Theorem I.4.57) it can be shown that the deviation process Z solves the stochastic differential equation: dZ t ¼ ðy2 =2ÞZ t ðZ t þ 2l=y2 Þ dt þ yZt dW t ;
Z 0 ¼ z,
(6)
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which admits the explicit solution: Zt ¼
expðy W t ðl þ y2 =2ÞtÞ Rt 1=z þ ðy =2Þ 0 expðy W s ðl þ y2 =2ÞsÞ ds 2
(7)
for za0 (cf. e.g. Øksendal, 1998, Example 5.15 or Gard, 1988, Chapter IV). From the structure of the solution (7) it follows that the process Z started at some zo0 remains negative and explodes in finite time with positive probability. On the other hand, started at some z40 the solution Z exists globally and remains positive, while started at z ¼ 0 it is trapped at the same point. To avoid degeneracy, we thus further assume that z40. Observe that from the one-to-one correspondence (5) between the processes ðX t ; Y t ÞtX0 and ðZ t ; Y t ÞtX0 , by virtue of (4) it follows that the natural filtration of the process Z coincides with ðFt ÞtX0 , and from (6) and (7) it is seen that the latter coincides with the natural filtration of the process W. Note that Y 0 ¼ y in (4) and Z 0 ¼ z in (6) can be straightforwardly expressed by means of the initial function X 0t , tp0. It therefore follows that the value (3) takes the form: V ¼ sup E½edtþX t ðKeZt 1Þþ ,
(8)
t
where the supremum can equivalently be taken over all finite stopping times of the process Z. Taking into account the multiplicative structure of the gain function in (8), without loss of generality we can further assume that x ¼ 0. e ¼ ðW e t ÞtX0 by 2.2. Let us define the process W Z t e t ¼ Wt W yZs ds,
(9)
0
where Z ¼ ðZ t ÞtX0 is given by (5)–(7). Hence, from (6) it follows that the process Z solves the stochastic differential equation: e t ; Z 0 ¼ z, dZ t ¼ ð3y2 =2ÞZ t ðZt þ 2l=ð3y2 ÞÞ dt þ yZ t dW (10) which admits the explicit solution: e t ðl þ y2 =2ÞtÞ expðyW Zt ¼ R t e s ðl þ y2 =2ÞsÞ ds 1=z þ ð3y2 =2Þ 0 expðyW
(11)
e is a diffusion-type process with respect to the for z40. Substituting the expression (11) into (9), we see that W Wiener process W (cf. e.g. Liptser and Shiryaev, 1977, Section IV.2), and its natural filtration clearly coincides with ðFt ÞtX0 . e Þ the right-hand side of the expression (11). Taking into account the assumption z40 Let us denote by At ðW e Þ and At ðW Þ, we get: and using the continuity of At ðW Z t Z t e Þ dso1 ¼ P P A2s ðW A2s ðW Þ dso1 ¼ 1 (12) 0
0
for all tX0. Then, by means of the result of Liptser and Shiryaev, 1977, Theorem 7.6, we obtain: Z t Z 1 t 2 2 E exp yZs dW s y Z s ds ¼1 2 0 0
(13)
for all tX0. Hence, following the arguments in Shiryaev et al., 1994, Section 7 and Shepp and Shiryaev, 1994, Section 2 (see also Shiryaev, 1999, Chapter VIII, Section 2d), we apply the results of Liptser and Shiryaev, 1977, Theorem 7.1 and Revuz and Yor, 1999, Theorem A.6.1 and conclude that there exists a probability e being locally equivalent to P with respect to the filtration ðFt ÞtX0 and such that its density process measure P is given by: Z t Z e t dPjF 1 t 2 2 ¼ exp yZ s dW s y Z s ds (14) 2 0 dPjFt 0
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for all tX0. Thus, by Girsanov’s theorem (see e.g. Liptser and Shiryaev, 1977, Theorem 6.3 or Øksendal, 1998, e ¼ ðW e t ÞtX0 defined in (9) is a standard Wiener process under the Theorem 8.6.4) it follows that the process W e By using (11) it can be verified that Z ¼ ðZ t ÞtX0 is a time-homogeneous (strong) Markov process measure P. e with respect to its natural filtration, which coincides with ðFt ÞtX0 . under P e t Observe that (14) also implies that for any finite stopping time t with respect to ðFt ÞtX0 the restriction PjF is equivalent to PjFt . Then, using the explicit expressions (1) and (5) as well as the assumption x ¼ 0, we obtain the following representation: e t dPjF ¼ eX t dPjFt
(15)
for all finite stopping times t. It therefore follows that for computing the value (8) we can consider the following optimal stopping problem for the Markov process Z given by V ðzÞ ¼ sup Eez ½edt ðKeZt 1Þþ ,
(16)
t
ez denotes the law of the diffusion process started at the point z40 and solving Eq. (10), and the where P supremum is taken over all finite stopping times of Z. Thus, we may say that the deviation process Z plays the role of a sufficient statistic in the optimal stopping problem (16). We will search for an optimal stopping time in (16) of the following form: t ¼ infftX0jZt pB g,
(17)
where B is the largest number from 0ozp log K such that V ðzÞ ¼ Kez 1. The point B is called an optimal stopping boundary. Note that if Kp1 and z40 then the problem (16) becomes trivial, so that we further assume that K41. 2.3. Standard arguments based on the application of Itoˆ’s formula (see e.g. Øksendal, 1998, Theorem 7.3.3) imply that in this case the infinitesimal generator L of the process Z acts on a function F 2 C 2 ð0; 1Þ like: ðLF ÞðzÞ ¼ ð3y2 =2Þzðz þ 2l=ð3y2 ÞÞF 0 ðzÞ þ ðy2 =2Þz2 F 00 ðzÞ
(18)
for all z40. In order to find the unknown value function V ðzÞ from (16) and the unknown boundary B from (17), we refer to the general theory of optimal stopping problems for continuous time Markov processes (see e.g. Grigelionis and Shiryaev, 1966 and Shiryaev, 1978, Section III.8) and formulate the following freeboundary problem: ðLV ÞðzÞ ¼ dV ðzÞ for z4B,
(19)
V ðBþÞ ¼ KeB 1
ðcontinuous fitÞ,
(20)
V ðzÞ ¼ ðKez 1Þþ
for zoB,
(21)
V ðzÞ4ðKez 1Þþ
for z4B,
(22)
where 0oBp log K and (20) plays the role of an instantaneous-stopping condition. Observe that the superharmonic characterization of the value function (see Dynkin, 1963 and Shiryaev, 1978) implies that V ðzÞ is the smallest function satisfying (19)–(22) with the boundary B . Because of the continuity of the process Z we also assume that the following condition holds: V 0 ðBþÞ ¼ KeB
ðsmooth fitÞ.
3. Solution of the free-boundary problem Let us now derive a solution to the free-boundary problem formulated above.
(23)
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3.1. By means of straightforward calculations it can be checked that Eq. (19) has the general solution: 2l 2l V ðzÞ ¼ C 1 zg U g; 2g 2 ; 3z þ C 2 zg L g; 2g 2 1; 3z , (24) y y where C 1 and C 2 are some arbitrary constants and g is given by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 l 1 l 2 2d þ g¼ þ 2þ þ 2. 2 y 2 y2 y Here Uða; b; zÞ is the confluent hypergeometric function, which admits the integral representation: Z 1 1 Uða; b; zÞ ¼ ezt ta1 ð1 þ tÞba1 dt GðaÞ 0
(25)
(26)
for a40 and b41 (see e.g. Abramovitz and Stegun, 1972, Chapter XIII or Bateman and Erde´lyi, 1953, Chapter VI with a different parametrization), and Lða; b; zÞ is the generalized Laguerre polynomial function defined by Lða; b; zÞ ¼
1 X k¼0
Gða þ b þ 1Þ ð1Þk zk Gða þ b þ 1Þ ¼ F ða; b 1; zÞ Gðb þ k þ 1ÞGða k þ 1Þ Gða þ 1ÞGðb þ 1Þ 1 1 k!
(27)
(see e.g. Abramovitz and Stegun, 1972, Chapter XXII or Bateman and Erde´lyi, 1953, Chapter X), where 1 F 1 is the Kummer confluent hypergeometric function and G denotes the Euler Gamma function. It thus follows that in (24) we have C 2 ¼ 0, since otherwise V ðzÞ ! 1 as z ! 1, which should be excluded due to the obvious fact that the value function (16) is decreasing and bounded for all z40. Hence, imposing conditions (20) and (23) on the function (24), we obtain the following equalities: 2l C 1 Bg U g; 2g 2 ; 3B ¼ KeB 1, (28) y 2l 2l gC 1 Bg1 U g; 2g 2 ; 3B 3gC 1 Bg U g þ 1; 2g 2 þ 1; 3B ¼ KeB . (29) y y By solving Eqs. (28) and (29) it therefore follows that the solution of system (19) and (20)+(23) is given by g z Uðg; 2g 2l=y2 ; 3zÞ B (30) 1Þ V ðz; B Þ ¼ ðKe B Uðg; 2g 2l=y2 ; 3B Þ for all z4B , where B satisfies the transcendental equation: g
3BUðg þ 1; 2g 2l=y2 þ 1; 3BÞ KBeB g¼ . 2 KeB 1 Uðg; 2g 2l=y ; 3BÞ
(31)
3.2. In order to prove the existence and uniqueness of the solution of Eq. (31) on the interval ð0; log KÞ, let us denote by GðBÞ the left-hand side and by HðBÞ the right-hand side of Eq. (31). Then HðBÞ is a strictly increasing function on ð0; log KÞ with Hð0þÞ ¼ 0 and Hðlog KÞ ¼ 1. Thus, if we deduce that GðBÞ is a decreasing function on ð0; log KÞ such that Gð0þÞ40, then we will be able to conclude that there exists a unique solution B of Eq. (31) on the interval ð0; log KÞ. To prove Gð0þÞ40, let us note that by applying the change-of-variable formula to (26) it follows that: Z z1b 1 u a1 e u ðu þ zÞba1 du, (32) Uða; b; zÞ ¼ GðaÞ 0 which directly implies: lim z#0
zUða þ 1; b þ 1; zÞ GðaÞGðbÞ b1 ¼ ¼ Uða; b; zÞ Gða þ 1ÞGðb 1Þ a
(33)
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for any b4a þ 141 fixed. Hence, inserting a ¼ g and b ¼ 2g 2l=y2 as well as z ¼ 3B into (33), for the lefthand side of (31) we get: lim GðBÞ ¼ lim g B#0
B#0
3B Uðg þ 1; 2g 2l=y2 þ 1; 3BÞ 2l g ¼ g 2 140. 2 Uðg; 2g 2l=y ; 3BÞ y
To derive the monotonicity of GðBÞ, let us observe that from representation (32) it follows that: R 1 u a R1 ba1 du uðu þ zÞf z ðuÞ du zUða þ 1; b þ 1; zÞ 0 e u ðu þ zÞ ¼ R1 , ¼ 0R 1 ba1 u a1 Uða; b; zÞ a a 0 e u ðu þ zÞ du 0 ðu þ zÞf z ðuÞ du
(34)
(35)
where f z ðuÞ ¼ CðzÞeu ua1 ðu þ zÞba2 , uX0, is a probability density with some normalizing constant CðzÞ for any b4a þ 141 fixed. Then, applying the Cauchy–Schwarz or Jensen inequality and taking into account the fact that b a 140, we obtain: d zUða þ 1; b þ 1; zÞ dz Uða; b; zÞ R1 R1 R1 R1 ðb a 1Þð 0 uf z ðuÞ du 0 ðu þ zÞf z ðuÞ du 0 uðu þ zÞf z ðuÞ du 0 f z ðuÞ duÞ ¼ R1 ða 0 ðu þ zÞf z ðuÞ duÞ2 R1 R1 R1 R1 ðb a 1Þð 0 u2 f z ðuÞ du þ z 0 uf z ðuÞ du 0 uðu þ zÞf z ðuÞ du 0 f z ðuÞ duÞ p ¼ 0. ð36Þ R1 ða 0 ðu þ zÞf z ðuÞ duÞ2 Thus, setting a ¼ g and b ¼ 2g 2l=y2 as well as z ¼ 3B, we may conclude that GðBÞ, being the left-hand side of (31), is decreasing on ð0; log KÞ. This completes the proof of uniqueness. 3.3. So far, we have seen that V ðzÞ ¼ V ðz; B Þ satisfies Eq. (19), and conditions (20) and (23) hold with B ¼ B . Let us now show that inequality (22) is also satisfied. For this, we take logarithms on both sides of (22) and observe that, in view of equality (20) and the fact that V ðz; B Þ is positive, it suffices to verify the inequality: d d log V ðz; B Þ4 logðKez 1Þ (37) dz dz for B ozo log K. By using the definition of V ðz; B Þ in (30) and (31), it is straightforward to see that inequality (37) is equivalent to: g
3zUðg þ 1; 2g 2l=y2 þ 1; 3zÞ Kzez go Kez 1 Uðg; 2g 2l=y2 ; 3zÞ
(38)
for B ozo log K. Thus, following the arguments above and using the monotonicity properties of the functions GðzÞ and HðzÞ (coinciding with the left-hand and right-hand sides of (38)) on the interval ð0; log KÞ, we may conclude that inequality (38) holds for B ozo log K. The latter fact directly implies that (22) is satisfied with V ðzÞ ¼ V ðz; B Þ and B ¼ B . 4. Main result and proof We are now in a position to formulate and prove the main assertion of the paper. Theorem 4.1. Let the process Z be given by (10) and (11) with z40 and assume that K41. Then the value function of the problem (16) takes the form: ( V ðz; B Þ if z4B ; V ðzÞ ¼ (39) Kez 1 if zpB and the optimal stopping time t has the structure (17), where the function V ðz; B Þ is given by (30) and the boundary B is the unique solution of the transcendental Eq. (31).
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Proof. It remains to show that the function (39) coincides with the value function (16) and that the stopping time t from (17) with the boundary B specified above is optimal. Let us denote by V ðzÞ the right-hand side of the expression (39). It follows by construction from the previous section that the function V ðzÞ solves the system (19)–(22), and condition (23) is satisfied. Thus, applying Itoˆ’s formula to edt V ðZ t Þ, we obtain: Z t e t, edt V ðZ t Þ ¼ V ðzÞ þ eds ðLV dV ÞðZs Þ ds þ M (40) 0
e t ÞtX0 defined by where the process ðM Z t es et ¼ eds V 0 ðZ s ÞyZ s dW M
(41)
0
ez . Observe that the time spent by the process Z at the boundary B is is a continuous local martingale under P of Lebesgue measure zero, which allows to extend ðLV dV ÞðzÞ arbitrarily to z ¼ B . Due to the properties (20)–(23), a Taylor expansion shows that V 00 ðB ÞpV 00 ðB þÞ holds, which by the form of the generator in (18) directly implies that ðLV dV ÞðB ÞpðLV dV ÞðB þÞ ¼ 0. Moreover, it can be checked that: d ðLV dV ÞðzÞ ¼ ðLV dV ÞðzÞ þ ð4y2 þ lÞKez þ d (42) dz for all 0ozoB , from where we may conclude that ðLV dV ÞðzÞ is increasing and thus negative on ð0; B Þ. This together with (19) yields ðLV dV ÞðzÞp0 for all z40. From expression (40) it therefore follows that the inequalities: et edt ðKeZt 1Þþ pedt V ðZ t ÞpV ðzÞ þ M
(43)
hold for any finite stopping time t of the process Z started at z40. Let ðsn Þn2N be an arbitrary localizing e t ÞtX0 . Taking in (43) the expectation with respect to the measure sequence of stopping times for the process ðM e Pz , by means of the optional sampling theorem (see e.g. Jacod and Shiryaev, 1987, Theorem I.1.39 or Revuz and Yor, 1999, Theorem II.3.2), we get: ez ½edðt^sn Þ V ðZ t^sn ÞpV ðzÞ þ E ez ½ M e t^sn ¼ V ðzÞ ez ½edðt^sn Þ ðKeZt^sn 1Þþ pE E
(44)
for all z40. Hence, letting n ! 1 and using Fatou’s lemma, we obtain that for any finite stopping time t the inequalities: ez ½edt V ðZ t ÞpV ðzÞ ez ½edt ðKeZt 1Þþ pE E
(45)
are satisfied for all z40. By virtue of the fact that the function V ðzÞ together with the boundary B satisfy the system (19)–(22), by the structure of the stopping time t in (17) and from the expression (40) it follows that the equality: e t ^sn edðt ^sn Þ V ðZ t ^sn Þ ¼ V ðzÞ þ M
(46)
holds. Then, using expression (43) and the fact that the function V ðzÞ is decreasing, we infer the inequalities: e t ^sn pV ðB ^ zÞ V ðzÞ V ðzÞpM
(47)
ez -a.s.), and the latter fact implies for all z40. Note that from (11) it follows that Z t tends to zero as t ! 1 (P ez ½t o1 ¼ 1 for all z40. Hence, letting n ! 1 in (46) and using that for the stopping time (17) we have P conditions (20) and (21) as well as the property V ðB ^ zÞo1, we can apply the Lebesgue dominated convergence theorem to obtain the equality: ez ½edt ðKeZt 1Þþ ¼ V ðzÞ E for all z40, which together with (45) directly implies the desired assertion.
(48) &
Remark 4.2. Let us briefly consider the dependence of the solution on the deviation parameterl, which reflects the impact of the delay. For this, let us denote by V ðz; lÞ the value function from (16) and by B ðlÞ the optimal stopping boundary from (17), where we underline the dependence on l. Then, by the comparison
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theorem for stochastic differential equations applied to (10) and by the structure of the value function (16) it follows that V ðz; lÞ increases in l. Hence, a simple comparison argument yields that B ðlÞ decreases in l. The intuition behind these properties is that the deviation Z is likely to be much smaller when the weighted average lY is mainly taken from recent values (i.e. when l is large). In that case, the solution X of Eq. (1) converges to zero more slowly, and we should await a lower optimal deviation level B ðlÞ before exercising the option in view of the discounted payoff in (8). Acknowledgements This paper was written during the time when the first author was visiting WeierstraX Institute for Applied Analysis and Stochastics (WIAS) Berlin and he is thankful for the hospitality. Financial support from the DFG-Sonderforschungsbereich 649 ‘‘Economic Risk’’ at Humboldt University of Berlin and the Foundation of Berlin Parliament is gratefully acknowledged. References Abramovitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards. Wiley, New York. Bateman, H., Erde´lyi, A., 1953. Higher Transcendental Functions. McGraw-Hill, New York. Dynkin, E.B., 1963. The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl. 4, 627–629. Elsanosi, I., Øksendal, B., Sulem, A., 2000. Some solvable stochastic control problems with delay. Stochastic Stochastic Rep. 71 (1–2), 69–89. Gard, T.C., 1988. Introduction to Stochastic Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 114. Dekker, New York. Grigelionis, B.I., Shiryaev, A.N., 1966. On Stefan’s problem and optimal stopping rules for Markov processes. Theory Probab. Appl. 11 (4), 541–558. Jacod, J., Shiryaev, A.N., 1987. Limit Theorems for Stochastic Processes. Springer, Berlin. Kramkov, D.O., Mordecki, E., 1994. Integral option. Theory Probab. Appl. 39 (1), 201–211. Larssen, B., 2002. Dynamic programming in stochastic control of systems with delay. Stochastic Stochastic Rep. 74 (3–4), 651–673. Larssen, B., Riesebro, N.H., 2003. When are HJB-equations in stochastic control of delay systems finite dimensional? Stochastic Anal. Appl. 21 (3), 643–671. Liptser, R.S., Shiryaev, A.N., 1977. Statistics of Random Processes I. Springer, Berlin. Øksendal, B., 1998. Stochastic Differential Equations. An Introduction with Applications. Springer, Berlin. Øksendal, B., 2004. Optimal stopping with delayed information. Preprint No. 23, Pure Mathematics, Department of Mathematics, University of Oslo, 12pp. Øksendal, B., Sulem, A., 2000. A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In: Menaldi, J.M., Rofman, E., Sulem, A. (Eds.), Optimal Control and Partial Differential Equations—Innovations and Applications. IOS Press, Amsterdam, pp. 64–79. Peskir, G., Uys, N., 2005. On Asian options of American type. In: Kyprianou, A.E., Schoutens, W., Wilmott, P. (Eds.), Exotic Option Pricing and Advanced Le´vy Models. Wiley, Chichester, to appear. Revuz, D., Yor, M., 1999. Continuous Martingales and Brownian Motion. Springer, Berlin. Shepp, L.A., Shiryaev, A.N., 1994. A new look at the pricing of Russian options. Theory Probab. Appl. 39 (1), 103–119. Shiryaev, A.N., 1978. Optimal Stopping Rules. Springer, Berlin. Shiryaev, A.N., 1999. Essentials of Stochastic Finance. World Scientific, Singapore. Shiryaev, A.N., Kabanov, Y.M., Kramkov, D.O., Melnikov, A.V., 1994. On the pricing of options of European and American types, II. Continuous time. Theory Probab. Appl. 39 (1), 61–102.
Statistics & Probability Letters 76 (2006) 601–608 www.elsevier.com/locate/stapro
An optimal stopping problem in a diffusion-type model with delay Pavel V. Gapeeva, Markus ReiXb, a
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Str. 65, 117997 Moscow, Russia Institute for Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany
b
Received 3 March 2005; received in revised form 17 August 2005 Available online 30 September 2005
Abstract We present an explicit solution to an optimal stopping problem in a model described by a stochastic delay differential equation with an exponential delay measure. The method of proof is based on reducing the initial problem to a freeboundary problem and solving the latter by means of the smooth-fit condition. The problem can be interpreted as pricing special perpetual average American put options in a diffusion-type model with delay. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60G40; 35R35; 91B70; secondary 60J25; 60J60; 60G44 Keywords: Optimal stopping; Stochastic delay differential equation; Diffusion process; Sufficient statistic; Free-boundary problem; Smooth fit; Girsanov’s theorem; Itoˆ’s formula
1. Introduction The main aim of this paper is to present a solution to the optimal stopping problem (3) for the process X that solves the stochastic differential (1) with an exponential delay measure on an infinite time interval. This problem is related to the option pricing theory in mathematical finance, where the process X can describe the logarithm of the price of a risky asset (e.g. a stock) on a financial market. In that case, the value (3) can be formally interpreted as a fair price of a special perpetual average American put option in a diffusion-type market model with delay. In this model, the dynamics of the price depends on its deviation from the running average over past values. In recent years, several control problems for models described by stochastic delay differential equations were studied. Øksendal and Sulem, 2000 proved maximum principles for certain classes of such models and applied them to solving some problems related to finance. Elsanosi et al., 2000 proved a verification theorem of variational inequality type and applied it to finding explicit solutions for some classes of optimal harvesting Corresponding author.
E-mail addresses: [email protected], [email protected] (P.V. Gapeev), [email protected], [email protected] (M. ReiX). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.09.006
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delay problems. Larssen, 2002 established the dynamic programming principle for stochastic delay differential equations. Larssen and Riesebro, 2003 exhibited certain classes of delayed control problems that can be reduced to ordinary control problems. In this paper, we show how an explicit solution to an optimal stopping problem in a model described by a stochastic delay differential equation can be derived. The paper is organized as follows. In Section 2, using change-of-measure arguments, for the initial problem (3) we construct an equivalent optimal stopping problem for the one-dimensional Markov deviation process. In order to find explicit expressions for the value function and the optimal boundary, we formulate an associated free-boundary problem. In Section 3, we derive a solution to the free-boundary problem, which can be expressed by the confluent hypergeometric function and thus admits a representation in closed form. In Section 4, we verify that the solution of the free-boundary problem turns out to be a solution of the initial optimal stopping problem. The main result of the paper is stated in Theorem 4.1. 2. Formulation of the problem First, let us give a precise description of the diffusion-type model with delay. 2.1. Suppose that on some probability space ðO; F; PÞ there exists a standard Wiener process W ¼ ðW t ÞtX0 and a continuous process X ¼ ðX t Þt2R solving the stochastic differential equation: dX t ¼ ðy2 =2ÞðX t lY t Þ2 dt þ yðX t lY t Þ dW t
for tX0; X 0 ¼ x,
where the process Y ¼ ðY t ÞtX0 is defined by Z 0 Yt ¼ els X tþs ds; X t ¼ X 0t for tp0,
(1)
(2)
1
for some y40, l40, and x 2 R given and fixed. Here X 0t , tp0, is a (deterministic) bounded measurable function. Since the exponential of X is a local martingale, the process X can be thought of describing the logarithm of a (discounted) stock price on a financial market. The goal of this paper is to compute the value: V ¼ sup E½edt ðKelY t eX t Þþ ,
(3)
t
where the supremum is taken over all finite stopping times t of the process X (i.e. stopping times with respect to the natural filtration ðFt ÞtX0 of X), and to determine an optimal stopping time at which the supremum in (3) is attained. The value (3) can be interpreted as an arbitrage-free price of a special average American put option, where K40 and d40 are some given constants. Some other optimal stopping problems for geometric Brownian motion with gain functions containing integrals were solved in Kramkov and Mordecki, 1994 and Peskir and Uys, 2005. Note that a different class of optimal stopping problems can be obtained when the underlying process is Markovian, but at the same time, there is a delay in the available information as in Øksendal, 2004. By differentiation it can be shown that the process Y admits the representation: dY t ¼ Z t dt;
Y 0 ¼ y,
(4)
where the process Z ¼ ðZ t ÞtX0 is defined by Zt ¼ X t lY t
(5)
for all tX0. The process Z defined in (5) expresses the deviation of the logarithm of the present value of the process X from its exponentially weighted average lY . By means of Itoˆ’s formula (see e.g. Liptser and Shiryaev, 1977, Theorem 4.4 or Jacod and Shiryaev, 1987, Theorem I.4.57) it can be shown that the deviation process Z solves the stochastic differential equation: dZ t ¼ ðy2 =2ÞZ t ðZ t þ 2l=y2 Þ dt þ yZt dW t ;
Z 0 ¼ z,
(6)
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which admits the explicit solution: Zt ¼
expðy W t ðl þ y2 =2ÞtÞ Rt 1=z þ ðy =2Þ 0 expðy W s ðl þ y2 =2ÞsÞ ds 2
(7)
for za0 (cf. e.g. Øksendal, 1998, Example 5.15 or Gard, 1988, Chapter IV). From the structure of the solution (7) it follows that the process Z started at some zo0 remains negative and explodes in finite time with positive probability. On the other hand, started at some z40 the solution Z exists globally and remains positive, while started at z ¼ 0 it is trapped at the same point. To avoid degeneracy, we thus further assume that z40. Observe that from the one-to-one correspondence (5) between the processes ðX t ; Y t ÞtX0 and ðZ t ; Y t ÞtX0 , by virtue of (4) it follows that the natural filtration of the process Z coincides with ðFt ÞtX0 , and from (6) and (7) it is seen that the latter coincides with the natural filtration of the process W. Note that Y 0 ¼ y in (4) and Z 0 ¼ z in (6) can be straightforwardly expressed by means of the initial function X 0t , tp0. It therefore follows that the value (3) takes the form: V ¼ sup E½edtþX t ðKeZt 1Þþ ,
(8)
t
where the supremum can equivalently be taken over all finite stopping times of the process Z. Taking into account the multiplicative structure of the gain function in (8), without loss of generality we can further assume that x ¼ 0. e ¼ ðW e t ÞtX0 by 2.2. Let us define the process W Z t e t ¼ Wt W yZs ds,
(9)
0
where Z ¼ ðZ t ÞtX0 is given by (5)–(7). Hence, from (6) it follows that the process Z solves the stochastic differential equation: e t ; Z 0 ¼ z, dZ t ¼ ð3y2 =2ÞZ t ðZt þ 2l=ð3y2 ÞÞ dt þ yZ t dW (10) which admits the explicit solution: e t ðl þ y2 =2ÞtÞ expðyW Zt ¼ R t e s ðl þ y2 =2ÞsÞ ds 1=z þ ð3y2 =2Þ 0 expðyW
(11)
e is a diffusion-type process with respect to the for z40. Substituting the expression (11) into (9), we see that W Wiener process W (cf. e.g. Liptser and Shiryaev, 1977, Section IV.2), and its natural filtration clearly coincides with ðFt ÞtX0 . e Þ the right-hand side of the expression (11). Taking into account the assumption z40 Let us denote by At ðW e Þ and At ðW Þ, we get: and using the continuity of At ðW Z t Z t e Þ dso1 ¼ P P A2s ðW A2s ðW Þ dso1 ¼ 1 (12) 0
0
for all tX0. Then, by means of the result of Liptser and Shiryaev, 1977, Theorem 7.6, we obtain: Z t Z 1 t 2 2 E exp yZs dW s y Z s ds ¼1 2 0 0
(13)
for all tX0. Hence, following the arguments in Shiryaev et al., 1994, Section 7 and Shepp and Shiryaev, 1994, Section 2 (see also Shiryaev, 1999, Chapter VIII, Section 2d), we apply the results of Liptser and Shiryaev, 1977, Theorem 7.1 and Revuz and Yor, 1999, Theorem A.6.1 and conclude that there exists a probability e being locally equivalent to P with respect to the filtration ðFt ÞtX0 and such that its density process measure P is given by: Z t Z e t dPjF 1 t 2 2 ¼ exp yZ s dW s y Z s ds (14) 2 0 dPjFt 0
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for all tX0. Thus, by Girsanov’s theorem (see e.g. Liptser and Shiryaev, 1977, Theorem 6.3 or Øksendal, 1998, e ¼ ðW e t ÞtX0 defined in (9) is a standard Wiener process under the Theorem 8.6.4) it follows that the process W e By using (11) it can be verified that Z ¼ ðZ t ÞtX0 is a time-homogeneous (strong) Markov process measure P. e with respect to its natural filtration, which coincides with ðFt ÞtX0 . under P e t Observe that (14) also implies that for any finite stopping time t with respect to ðFt ÞtX0 the restriction PjF is equivalent to PjFt . Then, using the explicit expressions (1) and (5) as well as the assumption x ¼ 0, we obtain the following representation: e t dPjF ¼ eX t dPjFt
(15)
for all finite stopping times t. It therefore follows that for computing the value (8) we can consider the following optimal stopping problem for the Markov process Z given by V ðzÞ ¼ sup Eez ½edt ðKeZt 1Þþ ,
(16)
t
ez denotes the law of the diffusion process started at the point z40 and solving Eq. (10), and the where P supremum is taken over all finite stopping times of Z. Thus, we may say that the deviation process Z plays the role of a sufficient statistic in the optimal stopping problem (16). We will search for an optimal stopping time in (16) of the following form: t ¼ infftX0jZt pB g,
(17)
where B is the largest number from 0ozp log K such that V ðzÞ ¼ Kez 1. The point B is called an optimal stopping boundary. Note that if Kp1 and z40 then the problem (16) becomes trivial, so that we further assume that K41. 2.3. Standard arguments based on the application of Itoˆ’s formula (see e.g. Øksendal, 1998, Theorem 7.3.3) imply that in this case the infinitesimal generator L of the process Z acts on a function F 2 C 2 ð0; 1Þ like: ðLF ÞðzÞ ¼ ð3y2 =2Þzðz þ 2l=ð3y2 ÞÞF 0 ðzÞ þ ðy2 =2Þz2 F 00 ðzÞ
(18)
for all z40. In order to find the unknown value function V ðzÞ from (16) and the unknown boundary B from (17), we refer to the general theory of optimal stopping problems for continuous time Markov processes (see e.g. Grigelionis and Shiryaev, 1966 and Shiryaev, 1978, Section III.8) and formulate the following freeboundary problem: ðLV ÞðzÞ ¼ dV ðzÞ for z4B,
(19)
V ðBþÞ ¼ KeB 1
ðcontinuous fitÞ,
(20)
V ðzÞ ¼ ðKez 1Þþ
for zoB,
(21)
V ðzÞ4ðKez 1Þþ
for z4B,
(22)
where 0oBp log K and (20) plays the role of an instantaneous-stopping condition. Observe that the superharmonic characterization of the value function (see Dynkin, 1963 and Shiryaev, 1978) implies that V ðzÞ is the smallest function satisfying (19)–(22) with the boundary B . Because of the continuity of the process Z we also assume that the following condition holds: V 0 ðBþÞ ¼ KeB
ðsmooth fitÞ.
3. Solution of the free-boundary problem Let us now derive a solution to the free-boundary problem formulated above.
(23)
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3.1. By means of straightforward calculations it can be checked that Eq. (19) has the general solution: 2l 2l V ðzÞ ¼ C 1 zg U g; 2g 2 ; 3z þ C 2 zg L g; 2g 2 1; 3z , (24) y y where C 1 and C 2 are some arbitrary constants and g is given by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 l 1 l 2 2d þ g¼ þ 2þ þ 2. 2 y 2 y2 y Here Uða; b; zÞ is the confluent hypergeometric function, which admits the integral representation: Z 1 1 Uða; b; zÞ ¼ ezt ta1 ð1 þ tÞba1 dt GðaÞ 0
(25)
(26)
for a40 and b41 (see e.g. Abramovitz and Stegun, 1972, Chapter XIII or Bateman and Erde´lyi, 1953, Chapter VI with a different parametrization), and Lða; b; zÞ is the generalized Laguerre polynomial function defined by Lða; b; zÞ ¼
1 X k¼0
Gða þ b þ 1Þ ð1Þk zk Gða þ b þ 1Þ ¼ F ða; b 1; zÞ Gðb þ k þ 1ÞGða k þ 1Þ Gða þ 1ÞGðb þ 1Þ 1 1 k!
(27)
(see e.g. Abramovitz and Stegun, 1972, Chapter XXII or Bateman and Erde´lyi, 1953, Chapter X), where 1 F 1 is the Kummer confluent hypergeometric function and G denotes the Euler Gamma function. It thus follows that in (24) we have C 2 ¼ 0, since otherwise V ðzÞ ! 1 as z ! 1, which should be excluded due to the obvious fact that the value function (16) is decreasing and bounded for all z40. Hence, imposing conditions (20) and (23) on the function (24), we obtain the following equalities: 2l C 1 Bg U g; 2g 2 ; 3B ¼ KeB 1, (28) y 2l 2l gC 1 Bg1 U g; 2g 2 ; 3B 3gC 1 Bg U g þ 1; 2g 2 þ 1; 3B ¼ KeB . (29) y y By solving Eqs. (28) and (29) it therefore follows that the solution of system (19) and (20)+(23) is given by g z Uðg; 2g 2l=y2 ; 3zÞ B (30) 1Þ V ðz; B Þ ¼ ðKe B Uðg; 2g 2l=y2 ; 3B Þ for all z4B , where B satisfies the transcendental equation: g
3BUðg þ 1; 2g 2l=y2 þ 1; 3BÞ KBeB g¼ . 2 KeB 1 Uðg; 2g 2l=y ; 3BÞ
(31)
3.2. In order to prove the existence and uniqueness of the solution of Eq. (31) on the interval ð0; log KÞ, let us denote by GðBÞ the left-hand side and by HðBÞ the right-hand side of Eq. (31). Then HðBÞ is a strictly increasing function on ð0; log KÞ with Hð0þÞ ¼ 0 and Hðlog KÞ ¼ 1. Thus, if we deduce that GðBÞ is a decreasing function on ð0; log KÞ such that Gð0þÞ40, then we will be able to conclude that there exists a unique solution B of Eq. (31) on the interval ð0; log KÞ. To prove Gð0þÞ40, let us note that by applying the change-of-variable formula to (26) it follows that: Z z1b 1 u a1 e u ðu þ zÞba1 du, (32) Uða; b; zÞ ¼ GðaÞ 0 which directly implies: lim z#0
zUða þ 1; b þ 1; zÞ GðaÞGðbÞ b1 ¼ ¼ Uða; b; zÞ Gða þ 1ÞGðb 1Þ a
(33)
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for any b4a þ 141 fixed. Hence, inserting a ¼ g and b ¼ 2g 2l=y2 as well as z ¼ 3B into (33), for the lefthand side of (31) we get: lim GðBÞ ¼ lim g B#0
B#0
3B Uðg þ 1; 2g 2l=y2 þ 1; 3BÞ 2l g ¼ g 2 140. 2 Uðg; 2g 2l=y ; 3BÞ y
To derive the monotonicity of GðBÞ, let us observe that from representation (32) it follows that: R 1 u a R1 ba1 du uðu þ zÞf z ðuÞ du zUða þ 1; b þ 1; zÞ 0 e u ðu þ zÞ ¼ R1 , ¼ 0R 1 ba1 u a1 Uða; b; zÞ a a 0 e u ðu þ zÞ du 0 ðu þ zÞf z ðuÞ du
(34)
(35)
where f z ðuÞ ¼ CðzÞeu ua1 ðu þ zÞba2 , uX0, is a probability density with some normalizing constant CðzÞ for any b4a þ 141 fixed. Then, applying the Cauchy–Schwarz or Jensen inequality and taking into account the fact that b a 140, we obtain: d zUða þ 1; b þ 1; zÞ dz Uða; b; zÞ R1 R1 R1 R1 ðb a 1Þð 0 uf z ðuÞ du 0 ðu þ zÞf z ðuÞ du 0 uðu þ zÞf z ðuÞ du 0 f z ðuÞ duÞ ¼ R1 ða 0 ðu þ zÞf z ðuÞ duÞ2 R1 R1 R1 R1 ðb a 1Þð 0 u2 f z ðuÞ du þ z 0 uf z ðuÞ du 0 uðu þ zÞf z ðuÞ du 0 f z ðuÞ duÞ p ¼ 0. ð36Þ R1 ða 0 ðu þ zÞf z ðuÞ duÞ2 Thus, setting a ¼ g and b ¼ 2g 2l=y2 as well as z ¼ 3B, we may conclude that GðBÞ, being the left-hand side of (31), is decreasing on ð0; log KÞ. This completes the proof of uniqueness. 3.3. So far, we have seen that V ðzÞ ¼ V ðz; B Þ satisfies Eq. (19), and conditions (20) and (23) hold with B ¼ B . Let us now show that inequality (22) is also satisfied. For this, we take logarithms on both sides of (22) and observe that, in view of equality (20) and the fact that V ðz; B Þ is positive, it suffices to verify the inequality: d d log V ðz; B Þ4 logðKez 1Þ (37) dz dz for B ozo log K. By using the definition of V ðz; B Þ in (30) and (31), it is straightforward to see that inequality (37) is equivalent to: g
3zUðg þ 1; 2g 2l=y2 þ 1; 3zÞ Kzez go Kez 1 Uðg; 2g 2l=y2 ; 3zÞ
(38)
for B ozo log K. Thus, following the arguments above and using the monotonicity properties of the functions GðzÞ and HðzÞ (coinciding with the left-hand and right-hand sides of (38)) on the interval ð0; log KÞ, we may conclude that inequality (38) holds for B ozo log K. The latter fact directly implies that (22) is satisfied with V ðzÞ ¼ V ðz; B Þ and B ¼ B . 4. Main result and proof We are now in a position to formulate and prove the main assertion of the paper. Theorem 4.1. Let the process Z be given by (10) and (11) with z40 and assume that K41. Then the value function of the problem (16) takes the form: ( V ðz; B Þ if z4B ; V ðzÞ ¼ (39) Kez 1 if zpB and the optimal stopping time t has the structure (17), where the function V ðz; B Þ is given by (30) and the boundary B is the unique solution of the transcendental Eq. (31).
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Proof. It remains to show that the function (39) coincides with the value function (16) and that the stopping time t from (17) with the boundary B specified above is optimal. Let us denote by V ðzÞ the right-hand side of the expression (39). It follows by construction from the previous section that the function V ðzÞ solves the system (19)–(22), and condition (23) is satisfied. Thus, applying Itoˆ’s formula to edt V ðZ t Þ, we obtain: Z t e t, edt V ðZ t Þ ¼ V ðzÞ þ eds ðLV dV ÞðZs Þ ds þ M (40) 0
e t ÞtX0 defined by where the process ðM Z t es et ¼ eds V 0 ðZ s ÞyZ s dW M
(41)
0
ez . Observe that the time spent by the process Z at the boundary B is is a continuous local martingale under P of Lebesgue measure zero, which allows to extend ðLV dV ÞðzÞ arbitrarily to z ¼ B . Due to the properties (20)–(23), a Taylor expansion shows that V 00 ðB ÞpV 00 ðB þÞ holds, which by the form of the generator in (18) directly implies that ðLV dV ÞðB ÞpðLV dV ÞðB þÞ ¼ 0. Moreover, it can be checked that: d ðLV dV ÞðzÞ ¼ ðLV dV ÞðzÞ þ ð4y2 þ lÞKez þ d (42) dz for all 0ozoB , from where we may conclude that ðLV dV ÞðzÞ is increasing and thus negative on ð0; B Þ. This together with (19) yields ðLV dV ÞðzÞp0 for all z40. From expression (40) it therefore follows that the inequalities: et edt ðKeZt 1Þþ pedt V ðZ t ÞpV ðzÞ þ M
(43)
hold for any finite stopping time t of the process Z started at z40. Let ðsn Þn2N be an arbitrary localizing e t ÞtX0 . Taking in (43) the expectation with respect to the measure sequence of stopping times for the process ðM e Pz , by means of the optional sampling theorem (see e.g. Jacod and Shiryaev, 1987, Theorem I.1.39 or Revuz and Yor, 1999, Theorem II.3.2), we get: ez ½edðt^sn Þ V ðZ t^sn ÞpV ðzÞ þ E ez ½ M e t^sn ¼ V ðzÞ ez ½edðt^sn Þ ðKeZt^sn 1Þþ pE E
(44)
for all z40. Hence, letting n ! 1 and using Fatou’s lemma, we obtain that for any finite stopping time t the inequalities: ez ½edt V ðZ t ÞpV ðzÞ ez ½edt ðKeZt 1Þþ pE E
(45)
are satisfied for all z40. By virtue of the fact that the function V ðzÞ together with the boundary B satisfy the system (19)–(22), by the structure of the stopping time t in (17) and from the expression (40) it follows that the equality: e t ^sn edðt ^sn Þ V ðZ t ^sn Þ ¼ V ðzÞ þ M
(46)
holds. Then, using expression (43) and the fact that the function V ðzÞ is decreasing, we infer the inequalities: e t ^sn pV ðB ^ zÞ V ðzÞ V ðzÞpM
(47)
ez -a.s.), and the latter fact implies for all z40. Note that from (11) it follows that Z t tends to zero as t ! 1 (P ez ½t o1 ¼ 1 for all z40. Hence, letting n ! 1 in (46) and using that for the stopping time (17) we have P conditions (20) and (21) as well as the property V ðB ^ zÞo1, we can apply the Lebesgue dominated convergence theorem to obtain the equality: ez ½edt ðKeZt 1Þþ ¼ V ðzÞ E for all z40, which together with (45) directly implies the desired assertion.
(48) &
Remark 4.2. Let us briefly consider the dependence of the solution on the deviation parameterl, which reflects the impact of the delay. For this, let us denote by V ðz; lÞ the value function from (16) and by B ðlÞ the optimal stopping boundary from (17), where we underline the dependence on l. Then, by the comparison
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theorem for stochastic differential equations applied to (10) and by the structure of the value function (16) it follows that V ðz; lÞ increases in l. Hence, a simple comparison argument yields that B ðlÞ decreases in l. The intuition behind these properties is that the deviation Z is likely to be much smaller when the weighted average lY is mainly taken from recent values (i.e. when l is large). In that case, the solution X of Eq. (1) converges to zero more slowly, and we should await a lower optimal deviation level B ðlÞ before exercising the option in view of the discounted payoff in (8). Acknowledgements This paper was written during the time when the first author was visiting WeierstraX Institute for Applied Analysis and Stochastics (WIAS) Berlin and he is thankful for the hospitality. Financial support from the DFG-Sonderforschungsbereich 649 ‘‘Economic Risk’’ at Humboldt University of Berlin and the Foundation of Berlin Parliament is gratefully acknowledged. References Abramovitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards. Wiley, New York. Bateman, H., Erde´lyi, A., 1953. Higher Transcendental Functions. McGraw-Hill, New York. Dynkin, E.B., 1963. The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl. 4, 627–629. Elsanosi, I., Øksendal, B., Sulem, A., 2000. Some solvable stochastic control problems with delay. Stochastic Stochastic Rep. 71 (1–2), 69–89. Gard, T.C., 1988. Introduction to Stochastic Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 114. Dekker, New York. Grigelionis, B.I., Shiryaev, A.N., 1966. On Stefan’s problem and optimal stopping rules for Markov processes. Theory Probab. Appl. 11 (4), 541–558. Jacod, J., Shiryaev, A.N., 1987. Limit Theorems for Stochastic Processes. Springer, Berlin. Kramkov, D.O., Mordecki, E., 1994. Integral option. Theory Probab. Appl. 39 (1), 201–211. Larssen, B., 2002. Dynamic programming in stochastic control of systems with delay. Stochastic Stochastic Rep. 74 (3–4), 651–673. Larssen, B., Riesebro, N.H., 2003. When are HJB-equations in stochastic control of delay systems finite dimensional? Stochastic Anal. Appl. 21 (3), 643–671. Liptser, R.S., Shiryaev, A.N., 1977. Statistics of Random Processes I. Springer, Berlin. Øksendal, B., 1998. Stochastic Differential Equations. An Introduction with Applications. Springer, Berlin. Øksendal, B., 2004. Optimal stopping with delayed information. Preprint No. 23, Pure Mathematics, Department of Mathematics, University of Oslo, 12pp. Øksendal, B., Sulem, A., 2000. A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In: Menaldi, J.M., Rofman, E., Sulem, A. (Eds.), Optimal Control and Partial Differential Equations—Innovations and Applications. IOS Press, Amsterdam, pp. 64–79. Peskir, G., Uys, N., 2005. On Asian options of American type. In: Kyprianou, A.E., Schoutens, W., Wilmott, P. (Eds.), Exotic Option Pricing and Advanced Le´vy Models. Wiley, Chichester, to appear. Revuz, D., Yor, M., 1999. Continuous Martingales and Brownian Motion. Springer, Berlin. Shepp, L.A., Shiryaev, A.N., 1994. A new look at the pricing of Russian options. Theory Probab. Appl. 39 (1), 103–119. Shiryaev, A.N., 1978. Optimal Stopping Rules. Springer, Berlin. Shiryaev, A.N., 1999. Essentials of Stochastic Finance. World Scientific, Singapore. Shiryaev, A.N., Kabanov, Y.M., Kramkov, D.O., Melnikov, A.V., 1994. On the pricing of options of European and American types, II. Continuous time. Theory Probab. Appl. 39 (1), 61–102.