MAXIMUM PROCESS PROBLEMS IN OPTIMAL CONTROL THEORY

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1 MAXIMUM PROCESS PROBLEMS IN OPTIMAL CONTROL THEORY GORAN PESKIR Receied 16 January 24 and in reised form 23 June 24 Gien a standard Brownian motion B t ) t and the equation of motion dx t = t dt+ 2dBt,wesetS t = max s t X s and consider the optimal control problem sup ES τ cτ), where c> and the supremum is taken oer all admissible controls satisfying t [µ,µ 1 ]forallt up to τ = inf{t> X t / l,l 1 )} with µ < <µ 1 and l < <l 1 gien and fixed. The following control is proed to be optimal: pull as hard as possible, that is, t = µ if X t <g S t ), and push as hard as possible, that is, t = µ 1 if X t >g S t ), where s g s) is a switching cure that is determined explicitly as the unique solution to a nonlinear differential equation). The solution found demonstrates that the problem formulations based on a maximum functional can be successfully included in optimal control theory calculus of ariations) in addition to the classic problem formulations due to Lagrange, Mayer, and Bolza. 1. Introduction Stochastic control theory deals with three basic problem formulations which were inherited from classical calculus of ariations cf. [4, pages 25-26]). Gien the equation of motion dx t = µ X t,u t ) dt + σ Xt,u t ) dbt, 1.1) where B t ) t is standard Brownian motion, consider the optimal control problem inf u E x τd L X t,u t ) dt + M XτD ) ), 1.2) where the infimum is taken oer all admissible controls u = u t ) t applied before the exit time τ D = inf{t> X t / C} for some open set C = D c and the process X t ) t starts at x under P x.ifm andl, problem 1.2) issaidtobelagrange formulated. If L andm, problem 1.2)issaidtobeMayer formulated. If both L andm, problem 1.2)is said to be Bolza formulated. Copyright 25 Applied Mathematics and Stochastic Analysis 25:1 25) DOI: /JAMSA.25.77

2 78 Maximum process and optimal control The Lagrange formulation goes back to the 18th century, the Mayer formulation originated in the 19th century, and the Bolza formulation [2] was introduced in We refer to [1, pages ] and the references therein for a historical account of the Lagrange, Mayer, and Bolza problems. Although the three problem formulations are formally known to be equialent see, e.g., [1, pages ], [4, pages 25 26]), this fact is rarely proed to be essential when soling a concrete problem. Setting Z t = LX t,u t )orz t = MX t ), and focusing upon the sample path t Z t for t [,τ D ], we see that the three problem formulations measure the performance associated with a control u by means of the following two functionals: τd Z t dt, Z τd, 1.3) where the first one represents the surface area below or aboe) the sample path, and the second one represents the sample path terminal alue. In addition to these two functionals, it is suggested by elementary geometric considerations that the maximal alue of the sample path max Z t 1.4) t τ D proides yet another performance measure which, to a certain extent, is more sensitie than the preious two. Clearly, a sample path can hae a small integral but still a large maximum, while a large maximum cannot be detected by the terminal alue either. The main purpose of the present paper is to show that the problem formulations based on a maximum functional can be successfully added to optimal control theory calculus of ariations). This is done by formulating a specific problem of this type Section 2)and soling it in a closed form Section 3). The result suggests a number of new aenues for further research upon extending the Bolza formulation 1.2) to optimize the following expression: E x max K ) τd X t,u t + L ) ) ) X t,u t dt + M XτD, 1.5) t τ D wheresomeofthemapsk, L,andM may also be identically zero. Optimal stopping problems for the maximum process hae been studied by a number of authors in the 199 s see, e.g., [6, 3, 1, 8, 5]) and the subject seems to be wellunderstood now. 2. Formulation of the problem Consider a process X = X t ) t soling the stochastic differential equation s.d.e.) dx t = t dt + 2dB t, 2.1)

3 Goran Peskir 79 where B = B t ) t is standard Brownian motion, and associate with X the maximum process ) S t = max X r s 2.2) r t so that X = x and S = s under P x,s,wherex s. Introduce the exit time τ = inf { t> X t / l,l 1 )}, 2.3) where l < <l 1 are gien and fixed, and so let c> in the sequel. The optimal control problem to be examined in this paper is formulated as follows: Jx,s):= supe x,s Sτ cτ ), 2.4) where the supremum is taken oer all admissible controls satisfying t [µ,µ 1 ]for all t τ with some µ < <µ 1 gien and fixed. By admissible we mean that the s.d.e. 2.1) can be soled in Itô s sense either strongly or weakly). Since t is required to be uniformly bounded, it is well known that a weak solution unique in law) always exists under a measurability condition see, e.g., [9, page 155]), where t may depend on theentiresamplepathr X r up to time t. Moreoer,if t = X t ) for some bounded) measurable function, then a strong solution pathwise unique) also exists see, e.g., [9, pages ]). The optimal control problem 2.4) has some interesting interpretations. Equation 2.1) may be iewed as describing the motion of a Brownian particle subject to a fluctuating force Ḃ t ) that is under the influence of a slowly arying) external force t see [7,pages 53 78]).The objectie in 2.4) is therefore to determine an optimum of the externalforcethatoneneedstoexertupontheparticlesoastomakeitsmaximalheight at the time of exit as large as possible in the course of time needed for the same exit to happen as short as possible. Clearly, the interpretation and objectie just described carry oer to many other problems where 2.1) plays a role. It appears intuitiely clear that the optimal control should be of the following bangbang type: at each time either push or pull as hard as possible so as to reach either l 1 or l as soon as possible. The solution of the problem presented in the next section confirms this guess and makes the statement precise in analytic terms. It is also apparent that at each time t we need to keep track of both X t and S t so that the problem is inherently two dimensional. 3. Solution of the problem 1) In the setting of the preious section, consider the optimal control problem 2.4). Note that X t = X t,s t ) is a two-dimensional process with the state space S ={x,s) R 2 x s} that changes increases) in the second coordinate only after hitting the diagonal x = s in R 2.Off the diagonal, the process X = X t ) t changes only in the first coordinate and thus may be identified with X. Moreoer,if t = X t ) for some bounded) measurable function in 2.1), then X is a Marko process. The later feedback controls are sufficient

4 8 Maximum process and optimal control to be considered under fairly general hypotheses see, e.g., [4, pages ]), and this fact will also be proed below. The infinitesimalgeneratorof X may be therefore formally described as follows: L X = L X in x<s, 3.1) = s atx = s, 3.2) where L X is the infinitesimal generator of X. This means that the infinitesimal generator of X is acting on a space of C 2 functions f on S satisfying lim x s f/ s)x,s) =. The formal description 3.1)+3.2) appears in[8, pages ], where the latter fact is also erified. The condition of normal reflection 3.2) was used for the first time in [3] in the case of Bessel processes it was also noted in [6, page 181] in the case of a Bessel process of dimension one). 2) Assuming for a moment that the supremum in 2.4) is attained at some feedback control, and making use of the formal description of the infinitesimal generator 3.1)+3.2), we are naturally led to formulate the following HJB system: sup L X J) c ) = inc for x<s, 3.3) J s x,s) x=s = normal reflection), 3.4) Jx,s) x=l+ = s instantaneous stopping), 3.5) Jx,s) x=l1 = s instantaneous stopping), 3.6) for l <s<l 1, where the infinitesimal generator of X gien is expressed by L X = x + 2 x 2 3.7) and we set C ={x,s) R 2 l <x s<l 1 }.Ourmaineffort in the sequel will be directed to soling the system 3.3) 3.6)in a closed form. More explicitly, the HJB equation 3.3) with J = Jx,s) reads as follows: sup Jx + J xx c ) = 3.8) so that we may expect a bang-bang solution depending on the sign of J x.ifj x <, then t = µ,andifj x >, then t = µ 1. The equation J x = defines an optimal switching cure s g s), and the main task in the sequel will be to determine it explicitly. Further heuristic considerations based on the bang-bang principle just stated when close to l apply µ so to exit at l, and when close to l 1 apply µ 1 so to exit at l 1 )suggest to partition C into the following three subsets modulo two cures x = g s) ands = s

5 l 1 l x l 1 Goran Peskir 81 C 1 C 2 s g s) s x S C 3 l Figure 3.1. The case of small c in problem 2.4). to be found): C 1 = { x,s) R 2 } l <x<g s), s <s<l 1, 3.9) C 2 = { x,s) R 2 } g s) <x s, s <s<l 1, 3.1) C 3 = { x,s) R 2 } l <x s<s, 3.11) where s is a unique point in l,l 1 ) satisfying In addition to 3.11), we also set g s ) = l. 3.12) g l1 ) = x 3.13) to denote another point in l,l 1 ) playing a role. We refer to Figure 3.1 to obtain a better geometric understanding of 3.9) 3.13). [In Figure 3.1 the state space triangle) of the process X t,s t )from2.1)+2.2) splits into three regions. In the region C 1,theoptimal control t equals µ, and in the region C 2 C 3 the optimal control t equals µ 1. The switching cure s g s) isdeterminedastheuniquesolutionofthedifferential equation 3.26) satisfying g l 1 ) = x,wherex l,l 1 ) is the unique solution of the transcendental equation 3.25). In this specific case, we took l = µ = 1, l 1 = µ 1 = 1, and c = 2. It turns out that x =, s = ands = The point s is a singularity point at which dg /ds = + and g takes a finite alue.)]

6 82 Maximum process and optimal control 3) We construct a solution to the system 3.3) 3.6) in three steps. In the first two steps, we determine C 1 C 2 together with a boundary cure s g s) separating C 1 from C 2. Step 1. Consider the HJB equation in C 1 to be found. The general solution of 3.14)isgienby µ J x + J xx c = 3.14) Jx,s) = 1 µ a s)e µx + c µ x + b s), 3.15) where a s)andb s) are some undetermined functions of s. Using 3.5), we can eliminate b s)from3.15) and this yields Jx,s) = 1 µ a s) e µx e µl) + c µ x l ) + s. 3.16) Soling J x x,s) = forx gies x = g s) as a candidate for the switching cure, and also that a s) can be expressed in terms of g s)asfollows: Inserting this back into 3.16)gies J 1) x,s) = a s) = c µ e µg s). 3.17) c µ ) 2 e µg s) e µx e µl) + c µ x l ) + s 3.18) as a candidate for the alue function 2.4)whenx,s) C 1. Step 2. Consider the HJB equation in C 2 to be found. The general solution of 3.19)isgienby µ 1 J x + J xx c = 3.19) Jx,s) = 1 µ 1 a 1 s)e µ1x + c µ 1 x + b 1 s), 3.2) where a 1 s) andb 1 s) are some undetermined functions of s. Soling J x x,s) = forx gies x = g s) as a candidate for the switching cure, and also that a 1 s) canbeexpressed in terms of g s)asfollows: Inserting this back into 3.2)gies J 2) x,s) = a 1 s) = c µ 1 e µ1g s). 3.21) c µ1 ) 2 e µ1g s) x) + c µ 1 x + b 1 s). 3.22)

7 Goran Peskir 83 The two functions 3.18)and3.22) must coincide at the switching cure, that is, J 1) x,s) x=g s)+ = J2) x,s) x=g s) 3.23) giing a closed expression for b 1 s), which after being inserted back into 3.22)yields J 2) x,s) = c ) 2 e µ1g s) x) c ) 2 e µg s) l) µ1 µ c µ 1 g s) x ) + c µ g s) l ) + s + c µ ) 2 c µ1 ) ) as a candidate for the alue function 2.4)whenx,s) C 2. The condition 3.6) withx = s = l 1 can now be used to determine a unique x l,l 1 ) satisfying 3.13). It follows from the expression 3.24)aboethatx soles and is uniquely determined by) the following transcendental equation: 1 ) 2 1 e µ ) 1x l 1) 1 ) 2 1 e µ ) x l ) ) l1 x l ) =. 3.25) µ1 µ µ 1 µ µ 1 µ It may be noted that this equation, and therefore x as well, does not depend on c. Finally, applying the condition 3.4) to the expression 3.24), we obtain a differential equation for the switching cure s g s) that takes the following form: g s) = 1 c/µ1 ) 1 e µ 1gs) s) ) c/µ ) 1 e µ gs) l ) ) 3.26) for all s s,l 1 ), where s <l 1 happens to be a singularity point at which dg/ds = +. Equation 3.26) is soled backwards under the initial condition 3.13), where x l,l 1 ) is found by soling 3.25). The switching cure s g s) is a unique solution of 3.26) obtained in this way. It can also be erified that 3.12)holdsforsomes s,l 1 ). 4) It turns out that the solution of 3.26) satisfying 3.13) can hit the diagonal in R 2 at apoints l,l 1 ) taken to be closest to x,ifc c for some large c to be determined below. When this happens, the construction of the solution becomes more complicated, and the solution of 3.26) fors s,s )isofnouse.wethusfirsttreatthesimpler case when the solution of 3.26) stays below the diagonal the case of small c), and this case is then followed by the more complicated case when the solution of 3.26) hits the diagonal the case of large c). Step 3 the case of small c). Haing characterized the cure s g s)for[s,l 1 ], we hae obtained a candidate 3.18)+3.24) for the alue function 2.4) whenx,s) C 1 C 2.It remains to determine Jx,s)forx,s) C 3, and this is what we do in the final step. As clearly the control µ 1 should be applied, consider the HJB equation µ 1 J x + J xx c = 3.27) in C 3 gien and fixed. The general solution of 3.27)isgienby Jx,s) = 1 µ 1 a 2 s)e µ1x + c µ 1 x + b 2 s), 3.28)

8 84 Maximum process and optimal control where a 2 s) andb 2 s) are some undetermined functions of s. Using the condition 3.5), we can eliminate b 2 s)from3.28) and this yields J 3) x,s) = 1 µ 1 a 2 s) e µ1x e µ1l) + c µ 1 x l ) + s. 3.29) Applying the condition 3.4) to the expression 3.29), we find that a 2 s) should sole µ a 1 2s) = e µ1s 3.3) e µ1l for l <s<s. This equation can be soled explicitly and this gies a 2 s) = e µ1l µ 1 s +log e µ1s e µ1l)) + d, 3.31) where d is an undetermined constant. To determine d, we may use the fact that the maps 3.24)and3.29)mustcoincideats,s ), that is, J 2) x,s) x=s=s = J 3) x,s) x=s=s, 3.32) thelatterbeingknownexplicitlydueto3.12). With this d we can then rewrite 3.31)as follows: a 2 s) = e µ µ1l 1 s s ) e µ ) 1s e log µ1l c ). 3.33) e µ1s e µ1l µ1 Inserting this expression back into 3.29), we obtain a candidate for the alue function 2.4)whenx,s) C 3, thus completing the construction of a solution to the system 3.3) 3.6). Steps 3 5 the case of large c). In this case, the solution s g s) of3.26) satisfying 3.13) hits the diagonal in R 2 at some s l,l 1 ). The set C 1 from 3.9) naturally splits into the following three subsets modulo two cures): C 1,1 = { x,s) R 2 } l <x<g s), s <s<l 1, 3.34) C 1,2 = { x,s) R 2 l <x<s, s <s<s }, 3.35) C 1,3 = { x,s) R 2 l <x<h s), s <s<s }, 3.36) where s,themaps h s), and s in this context will soon be defined. Similarly, the set C 2 from 3.1) naturally splits into the following two subsets modulo one cure): C 2,1 = { x,s) R 2 } g s) <x s, s <s<l 1, 3.37) C 2,2 = { x,s) R 2 h s) <x s, s <s<s }. 3.38) The set C 3 from 3.11) remains the same, howeer, with a new alue for s to be introduced. We refer to Figure 3.2 to obtain a better geometric understanding of 3.34) 3.38). [In Figure 3.2 the state space triangle) of the process X t,s t )from2.1)+2.2) splits into six regions. In the region C 1,1 C 1,2 C 1,3,theoptimalcontrolt equals

9 Goran Peskir 85 l 1 l x l 1 C 2,1 C 1,1 s g s) s x s h s) s C 1,3 s s C 1,2 l C 2,2 C 3 Figure 3.2. The case of large c in problem 2.4). µ, and in the region C 2,1 C 2,2 C 3,theoptimalcontrolt equals µ 1. The switching cure s g s) is determined as the unique solution of the differential equation 3.26) satisfying g l 1 ) = x,wherex l,l 1 ) is the unique solution of the transcendental equation 3.25). The switching cure s h s) differs from the solution s g s) of 3.26) and is determined to meet the condition 3.4) as explained in the text. In this specific case, we took l = µ = 1, l 1 = µ 1 = 1, and c = 3.8. It turns out that x =, s = , s =.82273, and s = For comparison, we state that the smaller of two) s satisfying g s ) = s equals , that s satisfying g s ) = l equals , and that s = The point s is a singularity point at which dg /ds = + and g takes a finite alue.)] It is clear from the construction aboe that in C 1,1 the alue function 2.4)isgienby 3.18), and that in C 2,1 the alue function 2.4) isgienby3.24), where s g s) isthe solution of 3.26) satisfying 3.13). For the points s<s close to s ) we can no longer make use of the solution g s), and the expression 3.18), as clearly the condition 3.4) fails to hold. We need, moreoer, to apply the control µ instead of µ 1, and thus Step 3 presented aboe must be modified. The HJB equation 3.27) is considered with µ in place of µ 1, and this again using 3.5) leads to the expression 3.29)withµ in place of µ 1 : J 1,2) x,s) = 1 µ a 3 s) e µx e µl) + c µ x l ) + s 3.39)

10 86 Maximum process and optimal control as well as 3.3)anditssolution3.31). To determine d in 3.31), we may use that J 1,2) x,s) x=s=s = J i,1) x,s) x=s=s 3.4) for i either 1 or 2, where we set J 1,1) = J 1) and J 2,1) = J 2) with J 1) from 3.18) and J 2) from 3.24). The latter expression in 3.4) is known explicitly due to the fact that g s ) = s. With this d,wecanthenrewriteananalogueof3.33)asfollows: a 3 s) = e µ µl s s ) )) e µ s e log µl c e µs e µ l e µs. 3.41) µ Inserting this expression back into 3.39), we obtain a candidate for the alue function 2.4)whenx,s) C 1,2. Moreoer, the equation J x 1,2) = determines an optimal switching cure h s) = 1 log µ ) µ c a 3s) 3.42) and the points s l,s )ands l,s ) appearing in 3.35)+3.36) and3.38) are determined by the following identities: ) h s = s, 3.43) ) h s = l. 3.44) Note that 3.44) is reminiscent of 3.12), and so is h s)ofg s). Howeer, the two functions are different for s<s. It is clear that the alue function 2.4) is also gien by the formula 3.39) forx,s) C 1,3. To determine the alue function 2.4) inc 2,2, where clearly the control µ 1 is to be applied, we can use the result of Step 2 aboe which leads to the following analogue of 3.24)aboe: J 2,2) x,s) = c ) 2 e µ1h s) x) c ) 2 e µh s) l) µ1 µ c µ 1 h s) x ) + c µ h s) l ) + s + c µ ) 2 c µ1 ) ) for x,s) C 2,2. Finally, in C 3 we should also apply the control µ 1, and thus the result of Step 3 the case of small c ) aboe can be used. Due to 3.44), the expressions 3.29)+3.33)carryoer unchanged to the present case. It must be kept in mind, howeer, that s satisfies 3.44) and not 3.12). This completes the construction of a solution to the system 3.3) 3.6). 5) In this way, we hae obtained a candidate for the alue function 2.4) whenx,s) C in both cases of small and large c. The preceding considerations can now be summarized as follows. Theorem 3.1. In the setting of 2.1) 2.3), consider the optimal control problem 2.4), where the supremum is taken oer all admissible controls as explained.

11 Goran Peskir 87 1) In the case of small c in the sense that the solution of 3.26) stays below the diagonal in R 2, the alue function J = Jx,s) soles the system 3.3) 3.6) and is explicitly gien by 3.18)inC 1 from 3.9), by 3.24)inC 2 from 3.1), and by 3.29)+3.33) in C 3 from 3.11) with s from 3.12), where s g s) is the unique solution of the differential equation 3.26) for s s,l 1 ) satisfying g l 1 ) = x,andx l,l 1 ) is the unique solution of the transcendental equation 3.25). The alues of J at the two boundary cures in C not mentioned are determined by continuity.) 2) In the case of large c in the sense that the solution of 3.26) hits the diagonal in R 2, the alue function J = Jx,s) soles the system 3.3) 3.6) and is explicitly gien by 3.18) inc 1,1 from 3.34), by 3.24) inc 2,1 from 3.37), by 3.39)+3.41) in C 1,2 C 1,3 from 3.35)+3.36), by3.45) inc 2,2 from 3.38), and by 3.29)+3.33) in C 3 from 3.11) with s from 3.44), where s g s) is the unique solution of the differential equation 3.26) for s s,l 1 ) satisfying g l 1 ) = x with x l,l 1 ) as aboe, and s h s) is gien by 3.42)+3.41). ThealuesofJ at the four boundary cures in C not mentioned are determined by continuity.) 3) The optimal control in problem 2.4) is described as follows: pull as hard as possible, that is, t = µ if X t <g S t ),and pushashardaspossible, thatis,t = µ 1 if X t >g S t ),bothaslongast τ,whereτ is gien in 2.3). This description holds in the case of large c by modifying the solution g s) of 3.26) fors<s as follows: g s):= s +1if s s,s ) and g s):= h s) for s s,s ). In both cases of small and large c, formally set g s):= l for s l,s ).Thecontrolt for X t = g S t ) is arbitrary and has no effect on optimality of the result.) To erify the statements, denote the candidate function by Jx,s) forx,s) C. Then, by construction, it is clear that x,s) Jx,s)isC 2 outside the set {g s),s) s <s<l 1 } in C, andx Jx,s) isc 1 at gs) whens is fixed. Actually, the latter mapping is also C 2 as is easily seen from 3.14)and3.19).) Thus, Itô s formula can be applied to the process JX t,s t ) with any admissible,andunderp x,s in this way we get J ) t ) t ) X t,s t = Jx,s)+ J x Xr,S r dxr + J s Xr,S r dsr + 1 t ) J xx Xr,S r d X,X r 3.46) 2 t ) t = Jx,s)+ 2Jx Xr,S r dbr + L ) ) ) X J) Xr,S r dr, where the integral with respect to ds r is zero, since the increment S r equals zero off the diagonal in R 2, and at the diagonal we hae 3.4). Since X τ equals either l or l 1,weseeby3.5)+3.6)thatJX τ,s τ ) = S τ, and thus 3.46) implies the following identity: S τ = Jx,s)+M τ + τ L X ) J) ) Xr,S r ) dr, 3.47) where M t = t 2Jx X r,s r )db r is a continuous local) martingale. Since x,s) J x x,s)is uniformly bounded in C, it follows by the optional sampling theorem that E x,s M τ ) =.

12 88 Maximum process and optimal control Note that there is no restriction to consider only those controls for which E x,s τ) < so that E x,s τ) < as well.) On the other hand, by 3.3) weknowthatl XJ))x,s) c for all x,s) C, with equality if =, and thus 3.47)yields ) τ Jx,s) = E x,s S τ L ) ) ) X J) Xr,S r dr E x,s Sτ cτ ) 3.48) with equality if =. This establishes that the candidate function equals the alue function, and that the optimal control equals, thus completing the proof. Acknowledgment The author was supported by Centre for Mathematical Physics and Stochastics funded by the Danish National Research Foundation). References [1] G. A. Bliss,Lectures on the Calculus of Variations, Uniersity of Chicago Press, Illinois, [2] O. Bolza, Über den Anormalen Fall beim Lagrangeschen und Mayerschen Problem mit gemischten Bedingungen und ariablen Endpunkten, Math. Ann ), German). [3] L. E. Dubins, L. A. Shepp, and A. N. Shiryae, Optimal stopping rules and maximal inequalities for Bessel processes, Theory Probab. Appl ), no. 2, [4] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, [5] S.E.Graersen,G.Peskir,andA.N.Shiryae,Stopping Brownian motion without anticipation ascloseaspossibletoitsultimatemaximum, Theory Probab. Appl ), no. 1, [6] S. D. Jacka, Optimal stopping and best constants for Doob-like inequalities. I. The case p = 1, Ann. Probab ), no. 4, [7] E. Nelson, Dynamical Theories of Brownian Motion, Princeton Uniersity Press, New Jersey, [8] G. Peskir, Optimal stopping of the maximum process: the maximality principle, Ann. Probab ), no. 4, [9] L. C. G. Rogers and D. Williams, Diffusions, Marko Processes, and Martingales. Vol. 2. ItôCalculus, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, New York, [1] L. Shepp and A. N. Shiryae, The Russian option: reduced regret, Ann. Appl. Probab ), no. 3, Goran Peskir: Department of Mathematical Sciences, Uniersity of Aarhus, Ny Munkegade, 8 Aarhus, Denmark address: goran@imf.au.dk

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