Optimal Prediction of the Ultimate Maximum of Brownian Motion

Transcription

1 Optimal Prediction of the Ultimate Maximum of Brownian Motion Jesper Lund Pedersen University of Copenhagen At time start to observe a Brownian path. Based upon the information, which is continuously updated through the observation of the path, a stopping time is determined such that the path is as close as possible to its unknown ultimate maximum over a finite time interval. The closeness is measured by a q-mean or by a probability distance. This can be formulated as an optimal stopping problem. The method of proof relies upon a representation of a conditional expectation of the gain process and the principle of smooth fit at a single point.. Introduction The ultimate maximum of a Brownian motion observed during a finite time interval is unknown at any point of time in that interval and only becomes known at the terminal time. At time start to observe the Brownian path. The problem is to determine a stopping time based only on the information accumulated to date that secures that the path is as close as possible to the ultimate maximum over the given time interval. The closeness is measured by a q-mean or by a probability distance in this paper. The problem can be interpreted as predicting the unknown ultimate maximum. This has applications to financial mathematics regarding decisions on anticipated market movements without knowing the exact date of the optimal occurrence. In mathematical terms the problem is formulated the following way. Let B t t be a standard Brownian motion started at zero and let F t t be the natural filtration generated by B t. Denote the maximum process associated with B t bys t =max r t B r.forfixedq>and ε>, the problem is to compute the two value functions..2 V q W ε =inf E S B τ q =sup P S B τ ε where τ is a stopping time of B t, and to find an optimal stopping time in each of the two optimal stopping problems, that is, a stopping time for which the optimum is attained. The measured distances between S and B τ are expressed in terms of.3 Ds x =s x q and Ds x = [,ε] s x which in this paper are called the q-mean distance function and the probability distance function, respectively. The author is supported by a Steno grant from the Danish Natural Science Research Council. 2 Mathematics Subject Classification. Primary 6G4, 6J65. Secondary 6J6. Key words and phrases. Brownian motion, ultimate maximum, optimal stopping, Lévy s distributional theorem, free-boundary problem, smooth fit at a single point, viscosity solution.

2 The gain processes in the two problems are not adapted to the filtration and hence the problem falls outside the class of stopping problem studied in general optimal stopping theory see [8, Chapter 3]. Problem. was initially solved in Graversen, Peskir & Shiryaev [3] in the case of the mean-square distance, that is, Ds x =s x 2. Research in optimal stopping problems of functions of the Brownian motion and its associated maximum process was started by Jacka [4] and later by Dubins, Shepp & Shiryaev [2]. The main aim of the present paper is twofold. First, this paper extends results of [3] which cover the situation of the mean-square distance. Second, the paper introduces a new type of problem based on measuring the distance between S and B τ by a probability distance. Explicit formulas are derived for the value functions and the optimal stopping times are displayed for both types of measuring the distance. The rest of this section is devoted to establishing a basic connection between the two problems and standard finite-horizon optimal stopping problems, which are simpler to work with. The next result with its proof is an extension of the prediction result in [3, Section 3]. Let ϕy = 2π e y2 /2 and Φy = y ϕu du y R denote the density and the distribution functions of a standard normal variable, respectively. Proposition.. Let G :[, R be a Borel function and fix <t<. conditional expectation E GS B t Ft is given by where E GS B t Ft = GSt B t F t S t B t + f t y = 2 y t ϕ t S t B t Gu f t u du y t and F t y =2Φ y are, respectively, the density and the distribution functions of S t. Proof. Fix <t<. The stationary independent increments of B t givethat E GS B t Ft = E GSt {max t r B r B t +B t } B t Ft = E Gs {max t r B r B t +x} x s=s t x=b t Then the = E Gs {S t + x} x s=s t. x=b t Since S t B t, the density and the distribution function of S t are f t and F t and the above expectation for x s is E Gs {S t + x} x = Gs x F t s x+ Substituting this formula in the above equation the result follows. s x Gu f t u du. Remark.2. It is only the stationary independent increments of B t and the finite expectation of S that are used to prove Proposition.. The result can therefore, be extended to processes with these two properties. 2

3 Recalling, the function D in.3, using that EDS B τ = E EDS B τ F τ and then applying Proposition., simple calculations give that problems. and.2 can be written as V q W ε =inf E S τ B τ q +2 τ q/2 Φu /q du S τ B τ / τ q =sup E F τ ε [,ε] S τ B τ. Thus,. and.2 are rewritten as ordinary optimal stopping problems, that is, the gain processes are adapted to the filtration. The above optimal stopping problems can be further simplified. The key fact that B t t and S t B t t are equal in law by Lévy s distributional theorem see [7, Theorem 2.3, Chapter VI], together with facts from general optimal stopping theory, show that the stopping problems are equivalent to evaluating V q W ε =inf E B τ q +2 τ q/2 Φu /q du B τ / τ q =sup E F τ ε [,ε] B τ. These two problems are standard finite-horizon optimal stopping problems and are inherently two-dimensional problems. Remark.3. The idea in [3] to solve problem. with q = 2 is the following; using a stochastic integral representation of the ultimate maximum S and because Ds x =s 2 2sx + x 2 problem. can be rewritten as an equivalent path-dependent integral optimal stopping problem. This approach does only work for this particular D. However, the idea indicates that for other choices of D it will prove useful to condition the gain process on the filtration as is done above. 2. Prediction by a q-mean distance The first main result is contained in the next theorem and is the solution of problem.. For a, b R, let Ma,b,x=+ a x + aa+ b 2! bb+ x2 + denote the Kummer confluent hypergeometric function see [, Chapter 3]. Theorem 2.. Consider the optimal stopping problem. and let q>. The value function is given by V q = e z2 q /2 Hz q / M +q,, z2 q where z q is the unique strictly positive root of the equation H z 2. Hz + z =+qz M 3+q, 3, z M +q,, z and z Hz is given by 2.2 Hz =z q +2 Φu /q du for z. The optimal stopping time is given by see Figure τ =inf{ <t :S t B t z q t }. z q 3

4 .5.5 t t Figure. The left drawing is a computer simulation of a Brownian path B t ω fort [, ]. The right drawing is the maximum process S t ω associated with the Brownian path. z 4 t z q t z 2 z z.5. t Figure 2. A drawing of optimal stopping strategies 2.3 for different values of q>forthe Brownian path from Figure. Numerical computations show that z 4.348, z 2.23, z.5.69 and z..26. Remark 2.2. The case of a general time interval [,T] reduces to the case of the unit interval stated above. Indeed, the Brownian scaling property implies that inf E q S T B τ = T q/2 inf E q S B τ τ T = T q/2 V q. The same argument shows that the optimal stopping time is given by τ =inf{ <t T : S t B t z q T t }. Proof. Let q>begiven and fixed. Recall from Section that problem. is equivalent to the standard finite-horizon optimal stopping problem 2.4 V q =inf E B τ q +2 τ q/2 Φu /q du. B τ / τ q 4

5 The method of deterministic time-change see [6] can be applied due to the form of the gain function in problem 2.4. Let T t = e 2t be the time-change and let Z t t be the time-changed process given by Z t = B Tt / T t = e t B e 2t. ThenZ t is the strong solution of the stochastic differential equation 2.5 dz t = Z t dt + 2 dβ t whichcanbeverifiedbyitô s formula, where β t t is the Brownian motion given by β t = 2 Tt u db u = 2 t Tu db Tu. For more details see [3] and [6]. Therefore, Z t is a diffusion with the infinitesimal generator 2.6 L Z = z z + 2 z 2 for z R. ObservethatZ t is a non-recurrent diffusion, that is, Z t P-a.s. for t. Since the time-change t T t is strictly increasing, if σ is a stopping time of Z t thenτ = T σ is a stopping time of B t and vice versa. Then by the foregoing facts, it is straight forward that B Tσ q +2 T σ q/2 Φu /q du B T σ / T σ q = e Z qσ σ q +2 Φu /q du = e qσ H Z σ Z σ q where H is given in 2.2. Problem 2.4 reduces to computing V q =inf E e qσ H Z σ σ where the infimum is taken over all stopping times σ of Z t. This is a one-dimensional optimal stopping problem and it can be solved by a standard approach. Therefore, introduce the problem 2.7 V z =inf σ E z e qσ H Z σ for z R, wherez = z under P z and the infimum is taken as above. The first step to solve problem 2.7 is to apply heuristic arguments to establish a candidate for the value function and a candidate for the optimal stopping time. From general optimal stopping theory, the stopping time 2.8 σ =inf{ t>: Z t z q } should be optimal where z q > is the optimal stopping point to be found. Due to the stochastic differential equation 2.5 and the domain of continuous observation z q,z q the value function V should be even. To compute the value function z V z and to determine z q in view of 2.7 and 2.8 it is natural to formulate the following system 2.9 L Z V z =qvz for z z q,z q 2. V ±z q =Hz q instantaneous stopping 2. V ±z q =±H z q smooth fit with L Z as in 2.6. The system is a free-boundary Stefan problem. 5

6 The general solution to 2.9 is V z =C e z2 /2 M +q,, z2 + C ze z2 /2 M 2+q, 3, z where C and C 2 are unknown constants. As noted above the value function should be even so that C 2 = and hence 2.2 V z =C e z2 /2 M +q,, z for some C to be found. The two conditions 2. and 2. determine z q and C uniquely. Note that M a,b,x= a M + a,+b,x and taking log on both sides of 2.2 and using b 2. and 2., elementary calculations give that z q is the strictly positive root of equation 2. and C = e z2 q /2 Hz q /M +q,, q z2. These heuristic arguments give that the candidate for the value function is Hz q e M +q,, z2 z2 q z 2 / V z = M +q, 2 2, if z <z q 2 z2 q H z if z z q and the candidate for the optimal stopping time is σ zq =inf{ t>: Z t z q }. It remains to verify that z V z coincides with the value function z V z given in 2.7 and that σ zq is an optimal stopping time. Note that σ zq < P-a.s. since Z t is non-recurrent andinfactσ zq has finite expectation and therefore, E z σ zq < as well. The latter will be used later in the proof. The function z V z isc 2 everywhere except at ±z q where it is C and the Lebesgue measure of those t>forwhichz t = ±z q is zero. Then Itô-Tanaka formula and 2.5 yield that 2.3 e qt V Z t =V z+m t + where M t = 2 t t e qu L Z qv Z u { Zu z q} du e qu V Z u dβ u is a continuous local martingale and hence e qt V Z t V z +M t for all t sincel Z qv z for z z q.letσ be any stopping time for Z t and choose a localization sequence {γ n } of bounded stopping times of M t. Then E z e qσ γ n H Z σ γn E z e qσ γ n V Z σ γn V z+e z Mσ γn = V z for all n where the first inequality follows from V z H z for all z. Letting n and using Fatou s lemma and finally taking the infimum over all stopping times shows that 2.4 V z V z is valid. To prove equality in 2.4 and that σ zq 2.5 V z =E z e qσ z q H Zσz q. is optimal it is enough to verify that 6

7 Equation 2.3 yields that e qσ zq Hzq =e qσ zq V Zσz q =V z+m σ z q and taking expectation on both sides implies that V z =Hz q E z e qσ zq because Ez M σz q =. Indeed by Burkholder-Davis-Gundy inequality and the fact that σz q E z e qu V Z u 2 du C Ez σzq < it follows that E z Mσz q =,wherec is a constant see also Remark 2.5 below. The conclusion is that V q = V = e z2 q /2 Hz q / M +q,, q z2. Transforming σ zq back to problem., τ from 2.3 is optimal. Remark 2.3. If <q, the value function is given by V q = E S q = 2q /π Γ +q 2 and an optimal stopping time is τ. For <q<, the calculations to compute 2.7 are similar to the case q>and the details will be omitted. The following formulas are valid. The value function in 2.7 is V z =2 q /2 π Γ +q 2+q Γ /4 D 2 2 e z2 +q z+d +q z where /4 D +q z = e z2 u q e zu u2 /2 du Γ + q is the parabolic cylinder function see [, Chapter 9]. The optimal stopping time 2.7 is given by τ =inf{ t>: Z t =}. For q =, problem. is trivial since for any stopping time τ, the optional sampling theorem implies that ES B τ =ES. Therefore, any stopping time is optimal. A result from general optimal stopping theory states that 2.3 is the minimal optimal stopping time for problem., that is, if τ # is another optimal stopping time for. then τ τ # P-a.s. One would expect that the minimal optimal stopping time τ q converges to the minimal optimal stopping time τ forq, but this is not true in this case. Indeed, z q z + for q wherez q is the unique strictly positive root of 2.. Thus τ q inf{ <t :S t B t z + t } >τ for q. This is illustrated in Figure and 2 where z q is numerically calculated for different values of q. Remark 2.4. If Ds x =s x 2, the following problem is equivalent to problem. inf Var S B τ = V 2 ES 2 = V 2 2/π.. In this case B τ is the optimal predictor estimator of the ultimate maximum S that minimise the variance of the error. 7

8 Remark 2.5. The argument to verify 2.5 extends to a more general setting and leads to the following explicit formulas for the Laplace transform of σ zq =inf{ t> : Z t z q }. For λ>, define the function l λ z =E z e λσ zq. General Markov process theory gives that z l λ z solves 2.9 with q = λ and satisfies l λ ±z q =. The argument quoted above gives E e M +λ,, z2 z2 q z2 / z e λσ z q = M +λ,, if z <z q z2 q if z z q for λ>. Since B t t and S t B t t are equal in law, the stopping time τ from 2.3 is distributed as the stopping time τ =inf{ t : B t z q t }. This observation together with Brownian scaling and time-change, shows that E τ λ/2 = E e λσ zq. Thus, it is not difficult to get that E τ λ/2 = e z2/2/ q M +λ,, q z2. For the special cases λ =2andλ = 4 the formula reads E τ = and E τ +zq 2 2 = +2zq z4 q and then it is easy to calculate Eτ = z2 q 2zq 4 and Varτ +zq 2 = + zq zq 2 + zq 4. The expectation and variance of τ are also calculated in [3] by a different method. 3. Prediction by a probability distance Recall from Section that problem.2 is equivalent to the finite-horizon optimal stopping problem W ε =sup E F τ ε [,ε] B τ. The method of time-change does not simplify this problem. Therefore, introduce the problem 3. W t, x = sup t E x F t τ ε [,ε] B τ for t andx R, whereb = x under P x and τ is a stopping time of B t. Denote the first passage time of the reflected Brownian motion by 3.2 τ ε =inf{ t>: B t = ε }. For any stopping time τ t, let τ be the first passage time after time τ to the level ε of B t, i.e. τ =inf{ t>τ : B t = ε }. τ is a stopping time of B t and in view of the gain function F t ε [,ε] x itisclearthate x F t τ ε [,ε] B τ E x F t τ t ε [,ε] B τ t. The conclusion to draw is that in problem 3. it is only optimal to stop if B τ = ε on the set {τ < t}. Therefore, the principle of smooth fit is not satisfied in the usual sense, that is, the value function W is not C at all points of the boundary of the domain of continued observation see [8, Chapter 3.8]. This is due to the discontinuous gain function. 8

9 In order to establish a candidate for the solution for problem 3., general theory combined with Brownian scaling property prompts that the strategy to defer stopping until the remaining time is t t ε andthenstopthefirsttimethat B t isequaltoε, thatis, τ =inf{ t ε <s t : B s = ε } should be optimal where t ε is to be found. The crucial point to establish a candidate for the optimal strategy is how to determine t ε. A priori this is not clear. By virtue of the gain function is continuous in the time variable t together with the guessed shape of the domain of continued observation an intuitive argument leads to that the value function should be smooth at the point t ε,ε. This intuitive argument is denoted the principle of smooth fit at a single point. The principle provides a method to determine t ε and is the key idea in the approach to solve problem 3.. To do this, determine t ε, define the function by 3.3 W ε t, x =E x F t τε tε [,ε] B τε t for t, x [, ] R where τ ε is given in 3.2. In general x W ε t, x isonlycontinuousat x = ε. Lett ε be the point in the time interval [, ] where x W ε t ε,x is differentiable at x = ε. If x W ε t, x is not differentiable at x = ε for any t [, ] then set t ε =. Hence Brownian scaling property implies that t ε = ε/ε 2 where ε > satisfies that x W ε,x is differentiable at x = ε. The hitting density h x ε of τ ε in 3.2 see [2, Section 6] is given by h x εy = k= x ε y 3/2 k x +2k +ε x ε ϕ y y 3/2 x +2k +ε ϕ y if x<ε if x>ε for y> and it is then possible to derive an equation for ε. Indeed, computing the function W ε t, x using the density h x ε givesthat 3.4 W ε t, x = t + 2 F t y ε k= k x +2k +ε y 3/2 k sgn x +2k +ε Φ k= x +2k +ε ϕ dy y x +2k +ε t for x<εand 3.5 W ε t, x = t F t y ε x ε y 3/2 x ε ϕ dy y 9

10 for x>ε. More calculations give that where Δ ε t = lim x ε Δ ε + t = lim x ε W ε x W ε x t Λ ε F t y ε t = y 3/2 Λ ε 2 t = 2 t k= =2Λε t+2λ ε 2 t Δ ε + t 2 2π t t = lim F t y ε z y 3/2 k k= 2k +ε k ϕ. t Therefore, ε is the unique positive root of the equation 3.6 Δ ε + + =Λ ε + Λ ε 2. 2π z2 z ϕ y dy y 2k +ε2 2k +ε ϕ dy y y Numerical calculation gives that ε.7. So the candidate for the optimal strategy is 3.7 τ =inf{ t ε <s t : B s = ε } and the associated candidate for the value function is W t, x =E x F t τ ε [,ε] B τ. If t t ε then τ = τ ε t and hence W t, x =W ε t, x. If t<t ε, strong Markov property and Itô s formula imply that W t, ε =E ε W ε t ε,b ε t t = W ε t ε = W ε t ε where this last equality following from,ε+e ε t,ε E ε t ε t ε t W ε L B W ε t ε,b u { Bu ε} du W ε t t ε,b u du L B W ε t, x+ t, x = t for t<and x ε which is a fact from general Markov process theory. It was used that the infinitesimal generator of B t isgivenby Hence L B = 2 2 x 2. W ε t t, ε t=t W = t, ε = F t ε ε ε t t=t t t=t ε and W t, x issmoothatthepointt, x =t ε,ε. In other words, the candidate for the value function satisfies the principle of smooth fit at a single point.

11 The proposed solution to problem 3. tranformed back to the initially problem.2 is stated in the following theorem. Theorem 3.. Consider the optimal stopping problem.2 and let ε>. Let t ε be the point in the time interval [, ] satisfying that x W ε t ε,x is differentiable at x = ε where W ε t, x is given in 3.4 and 3.5. If x W ε t, x is not differentiable at x = ε for all t [, ] then set t ε =.Thent ε = ε/ε 2 where ε is the positive root of equation 3.6 numerical calculation gives that ε.7. i If t ε ii If t ε =, then the value function is given by ε = 2ε Φ y W ε k= >, then the value function is given by W ε = 2 t ε k 2k + y 3/2 W ε t ε,y y ϕ dy. t ε 2k +ε ϕ dy. y In both cases the optimal strategy is to defer stopping until the remaining time is t ε then stop the first time that S t B t is equal to ε see Figure 3, that is, τ =inf{ t ε <t :S t B t = ε } inf =. and.2 ε t Figure 3. A drawing of optimal stopping strategies for different values of ε for the Brownian path from Figure. Remark 3.2. The case of a general time interval [,T] reduces to the case of the unit interval stated above. Indeed, the Brownian scaling property implies that sup τ T P S T B τ ε =sup P S B τ ε/ T = W ε/ T and the optimal strategy is given by τ = inf{ t T < t T : S t B t = ε } where t T = T ε/ε 2.

12 Proof. The idea to showing that the proposed solution is correct is to apply the verification theorem in [5, Theorem 3.]. The verification theorem is based on viscosity solution techniques and requires a continuous gain function. Therefore, an approximation of F t ε [,ε] x is necessary. Some notation from [5] will be adapted. Let ε> be given and fixed. As a starting point, let S =, R and for t, x S define the continuous gain function G n t, x =F t ε n e x ε and the associated value function 3.8 W n t, x = sup E x Gn t + τ,b τ t for n and the supremum is taken over all stopping times τ t of B t. Note that t can take negative values and when needed in the sequel, the previously defined functions can easily be extended. The gain function G n is chosen such that G n t, x =F t ε for x <ε and G n t, x F t ε [,ε] x forn. Furthermore, 3.9 LB + Gn t, x > t for x ε and hence it is only optimal to stop if B τ = ε on {τ < t}. This is similar to problem 3. and the optimal strategies must have the same structure. In a similar vein as for problem 3., define the function J n ε t, x =E x Gn t + τ ε t,b τε t where τ ε is given in 3.2. Observe that J n ε dominates the gain function G n as can be seen from Itô s formula and 3.9. Let t n < be the point where x J n ε t n,x is differentiable at x = ε. Sett = ε/ε 2 which is the point where x W ε t,x is differentiable at x = ε note that t is negative when ε>ε. Since J n ε dominates W ε and J n ε t, x =W ε t, x for x ε, thent n >t and t n t for n. The heuristic argument, the principle of smooth fit at a single point, gives that the candidate for the optimal strategy in problem 3.8 is τ n =inf{ t n <s t : B s = ε } and the associated candidate for the value function is J n t, x =E x Gn τ n,b τn. The arguments exposed above for t, x W t, x givethatt, x J n t, x issmoothatthe point t, x =t n,ε. By Itô s formula and the definition of t n yields that J n dominates J n ε and hence also dominates G n. As mentioned at the start of the proof, the verification theorem in [5] based on viscosity solution is applied next to establishing the candidates indeed are the solution to problem 3.8. For fixed n, introduce the following variational inequality associated with the problem 3.8 see [5] 3. min { L B t Jt, x,jt, x Gn t, x } = fort, x S 3. Jt, x =G n t, x for t, x S. Let J : S R be a continuous function. J is a viscosity solution of the variational inequality 3. and 3. if the following two conditions hold: 2

13 . J is a viscosity subsolution of 3. and 3., that is, J satisfies 3. and if for any j C 2 S andanyt, x S such that j J and jt, x =Jt, x then 3.2 min { L B t jt, x,jt, x Gn t, x }. 2. J is a viscosity supersolution of 3. and 3., that is, J satisfies 3. and if for any j C 2 S andanyt, x S such that j J and jt, x =Jt, x then 3.3 min { L B t jt, x,jt, x Gn t, x }. The next step is to verify that J n is a viscosity solution of the variational inequality. Subsolution: Since J n t, x =G n t, x fort n t<and x = ε and L B + t J nt, x = otherwise, it is clear that 3.2 is satisfied. Thus J n is a subsolution. Supersolution: Let j C 2 S begivensuchthatj J n and jt, x =J n t, x. As above, if t<t n or x ε it is trivial that 3.3 holds. If t n <t<and x = ε there does not exist any C 2 function j such that jt, ε =J n t, ε andj J n due to J n x t, x x=ε > J n x t, x x=ε+. Hence 3.3 holds trivially. If t, x =t n,ε, then recall that J n is continuous differentiable at t, x =t n,ε. The function J n j has a local minimum at t, x =t n,εandbythe first order conditions j t t n,ε= J n t t n,ε and j x t n,ε= J n x t n,ε and by the second order conditions lim x ε L B J n t n,x L B jt n,ε. Using the fact that L B + t Jn t n,x=for x ε if follows immediately that L B + t jtn,ε and 3.3 is fulfilled. Hence J n is a supersolution. The conclusion is that J n is the viscosity solution of the variational inequality 3. and 3.. Then the verification theorem in [5, Theorem 3.] provides that the candidate J n is the value function of the optimal stopping problem 3.8. Moreover, τ n is the optimal stopping time. Finally, let t =andx =. NoticethatG n τ n,b τn F τ ε [,ε] B τ P-a.s. and by monotone convergence J n, W, for n. It is clear that W, W n, = E G n τ n,b τn for all n. Letting n then W, W, = E F τ ε [,ε] B τ. The conclusion is that the candidate W, in 3.3 is the value function in the optimal stopping problem 3. and τ in 3.7 is the optimal strategy. Since W ε = W, and transforming back the optimal strategy to the initially problem.2 proves the theorem. Example 3.3. Numerical computations of the value function as a function of the error ε is presented in Figure 4. From the figure one sees that if, for example, ε =.75 then W.75 = sup P S B τ and the optimal strategy is illustrated in Figure 3 with t Conversely, if the prediction of the ultimate maximum is done with a given probability, say 95% that is W ε =.95, then the maximal error is ε.2 and the optimal strategy is illustrated in Figure 3 with t ε.9. 3

14 .95 W ε ε Figure 4. A drawing of the value function W ε as a function of the error ε. References [] Abramowicz, M.and Stegun, I.A Handbook of Mathematical Functions. National Bureau of Standards. [2] Dubins, L.E., Shepp, L.A. and Shiryaev, A.N Optimal stopping rules and maximal inequalities for Bessel processes. Theory Probab. Appl [3] Graversen, S.E., Peskir, G. and Shiryaev, A.N. 2. Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Probab. Appl [4] Jacka, S.D. 99. Optimal stopping and best constants for Doob-like inequalities I: The case p =. Ann. Probab [5] Øksendal, B.and Reikvam, K Viscosity solutions of optimal stopping problems. Stochastics Stochastics Rep [6] Pedersen, J.L.and Peskir, G. 2. Solving non-linear optimal stopping problems by the method of time-change. Stochastic Anal. Appl [7] Revuz, D.and Yor, M Continuous Martingales and Brownian Motion. Third edition. Springer. [8] Shiryaev, A.N Optimal stopping rules. Springer. Jesper Lund Pedersen Department of Applied Mathematics and Statistics, University of Copenhagen Universitetsparken 5, DK-2 Copenhagen, Denmark jesper@stat.ku.dk jesper 4