Optimal stopping of a mean reverting diffusion: minimizing the relative distance to the maximum

Transcription

1 Optimal stopping of a mean reverting diffusion: minimiing the relative distance to the maimum Romuald ELIE CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine and CREST elie@ceremade.dauphine.fr Gilles-Edouard ESPINOSA Department of Mathematics ETH Zurich gilles-edouard.espinosa@math.eth.ch February 211 Abstract Considering a diffusion X mean reverting to and starting at X >, we study the control problem inf θ E f X θ sup s [,τ] X s, where f is a given function and τ is the net random time where the diffusion X crosses ero. Our motivation is the obtention of optimal selling rules related to the minimiation of the relative distance between a stopped mean reverting portfolio and its upcoming maimum. We provide a verification result for this stochastic control problem and derive the solution for different criteria f. For a power utility type criterion f : y y λ with λ >, instantaneous stopping is always optimal. On the contrary, for a relative quadratic error criterion, f : y 1 y 2, selling is optimal as soon as the process X crosses a specified function ϕ of its running maimum X. As in [5] and [8], the inverse of ϕ identifies as the maimal solution of a highly non linear ordinary differential equation. These results reinforce the idea that optimal prediction problems of similar type lead easily to solutions of different nature. Nevertheless, we observe numerically that the continuation region for the relative quadratic error criterion is very small, so that the optimal selling strategy is close to immediate stopping. Key words: optimal stopping, optimal prediction, running maimum, free boundary PDE, verification, mean reverting diffusion MSC Classification 2: 6G4, 91B28, 35R35 1

2 1 Introction At a first glance, trying to stop a one-dimensional process as close as possible to its ultimate maimum may be viewed as a hopeless ambition. Graversen, Peskir and Shiryaev were the first authors who tackled successfully this challenging problem. Considering a one dimensional Brownian motion B on the time interval [, 1], they solve in [6] the optimal stopping problem inf θ E [ B θ B1 2], where B1 denotes the maimum of the process B at time 1 and θ is any stopping time smaller than 1. Stopping is optimal as soon as the drawdown of the Brownian motion, i.e. the gap between its current maimum and its value, goes below the function t c 1 t, for a specified constant c. Urusov [1] observes that this strategy leads to a good approimation of the last time τ where the Brownian motion reaches its maimum, since it also solves the problem inf θ E [ θ τ ]. For a drifted Brownian motion, this property is no longer satisfied, and Du Toit and Peskir [2, 3] characterie both solutions of these problems. Once again, stopping is optimal as soon as the drawdown of the drifted Brownian motion enters a time-to-horion dependent region. Considering instead a geometric Brownian motion S t t 1, several authors Shiryaev, Xu and Zhou [9], Du Toit and Peskir [4] or Dai, Jin, Zhong and Zhou [1] tried to minimie the relative distance between the stopped process S τ and its ultimate maimum. In particular they solve the problem sup θ E [S θ /S1 ]. Their purpose is of course the obtention of an optimal selling rule of the stock S as close as possible to its ultimate maimum. As pointed out in [4], the formulation in terms of ratio between the stopped process and its maimum has the effect of stripping away the monetary value of the stock, focusing only on the underlying randomness. Using either probabilistic or deterministic methods, the common interpretation of the solution derived in these papers is that one should "sell bad stocks and keep good ones". Indeed, introcing the "goodness inde" α of the stock as the ratio between its ecess return rate and its square volatility rate, the optimal strategy appears to be of "bang-bang" type: one should immediately sell the stock if α 1/2 and keep it until maturity otherwise. Focusing also on the problem inf θ E [S 1 /S θ], Du Toit and Peskir [4] observe that one should sell immediately if α <, keep until the end if α > 1 and stop as soon as the ratio S /S hits a specified deterministic function of time in the intermediate case. It is worth noticing that these two optimal prediction problems of similar type offer therefore different optimal selling strategies for a large range of parameter set. Of course, the only consideration of stocks with Black Scholes type dynamics is unrealistic and limitative. A recent paper of Espinosa and Toui [5] allows for the consideration of more general diffusion dynamics and, as a by proct, requires to focus on a stationary version of this problem. Studying an even more realistic infinite time horion problem and a mean reverting diffusion portfolio X with general dynamics starting at X >, they treat the problem: inf θ E [ Xτ X θ 2] where τ is the first time where X hits ero and θ is any 2

3 stopping time smaller than τ. They solve eplicitly this problem as a free boundary problem and obtain that one should sell the portfolio whenever the running maimum X and the drawdown X X are both large enough. In a similar framework, the purpose of this paper is to minimie the relative distance between a stopped mean reverting positive scalar diffusion and its upcoming maimum when it reaches ero. The consideration of the ratio of the stopped process with respect to its upcoming maimum allows to capture the scale of the prices themselves and we solve the two following problems [ ] λ Xθ V 1 = sup E θ X τ, for λ >, and V 2 = inf θ [ X ] E τ X 2 θ, where τ is the first time where X hits and θ is any stopping time smaller than τ. For the first problem V 1, we prove that the optimal stopping strategy consists in liquidating the portfolio immediately. This conclusion is in accordance with the results of [1, 4, 9] since keeping the stock until maturity is obviously irrelevant in our framework. Hence, the bang-bang type strategy also occurs for general mean reverting diffusion dynamics and for any relative power utility type criterion λ >. Conversely for V 2, when minimiing the relative quadratic distance between the process and its ultimate maimum, the optimal selling time is the first time where the process X goes below a specified function ϕ of its running maimum X. Similarly to [8] and [5], this function ϕ or more precisely its inverse is characteried as the "biggest" solution of an ordinary differential equation and can be easily approimated numerically. X τ As already observed by Du Toit and Peskir [4], our results confirm that optimal stopping prediction problems of similar nature can lead to different types of optimal solution. Predicting the maimum of a process is really intricate and the corresponding optimal strategy strongly depends on the criterion choice. However, we shall temper a bit this conclusion since, in our case, numerical eperiments provided in the paper show that the function ϕ is close to the identity function. Hence, even if immediate stopping is not optimal, a portfolio manager will not wait long until the drawdown of his portfolio X X goes below X ϕx. The paper is organied as follows. The net section provides the set up of the problem and derives preliminary properties. Section 3 is dedicated to the obtention of a general verification theorem allowing to treat the first and the second optimal stopping problems at once. Sections 4 and 5 tackle successively the power utility type criterion and the quadratic distance one. In both cases, the value function solution is presented and discussed at the beginning of the section, numerical results are presented, and the technical proofs are postponed to the end of it. 3

4 2 Problem formulation 2.1 The optimiation problem of interest Let W be a scalar Brownian motion on the complete probability space Ω, F, P, and denote by F = {F t, t } the corresponding augmented canonical filtration. Let X be a diffusion process given by the following dynamics: dx t = µx t dt + σx t dw t, t, 2.1 together with an initial data := X >, where µ and σ are Lipschit continuous functions. We will assume that the process X mean-reverts towards the origin in the sense that: µ, for. 2.2 We denote by τ := inf{t, X t = } the first time where the process X hits the origin, T the set of F-stopping times θ such that θ τ a.s and X. := sup s. X s the running maimum of X. We consider the following optimiation problem: V := inf E f Xθ, 2.3 θ T where f is a continuous function on [, 1], C 1 on, 1], and such that, there eist two constants A > and η > satisfying X τ f A η 2, < In order to use dynamic programming techniques, we introce as usual the process Z defined by Z t := Xt for a given >, and define the corresponding value function associated to the optimiation problem 2.3 : V, := inf E Xθ, f,,, 2.5 θ T where is defined by and corresponds to the domain where X, Z lies. Z τ := {,, and > }, 2.6 Remark 2.1 Of course, we aim at considering cases where the function f is not increasing, since otherwise a straightforward optimal strategy consists in waiting until time τ. Remark 2.2 Notice that the definition of differs from the one in [5]. Notice also that contrary to [5], the problem is not invariant by translation. More precisely, if one considers the criterion inf θ f b+x θ b+z τ, the problem might not be well-defined for b < since we can have b + Z τ =. For b > or > b if b < however, the problem makes sense and could be studied in a similar fashion, but is not a particular case of what we do here. 4

5 Defining the reward function g from immediate stopping g, := E, f,,, 2.7 we may re-write this problem in the standard form of an optimal stopping problem Z τ V, = inf θ T E, gx θ, Z θ,, Assumptions and first properties Let introce the so-called scale function S defined, for, by S := u ep αrdr, with α := 2µ σ Remark 2.3 Since the process X mean reverts towards, the function α is non negative. Therefore, the scale function S is increasing, conve and dominates the Identity function. By construction, S satisfies S = αs and is related to the law of Z τ via the estimate P, [Z τ u] = P [Xτ u] 1 u = 1 S 1 u,,, u >. Su Using the scale function S, the reward function g rewrites as + S f g, = f 1 S S which is well-defined since f is continuous on [, 1] so that we have S u f u Su 2 f Via an integration by part, we dece g, = f S u S u, Su 2,, 2.1 S u Su 2 = f S,,. f u u 2,,, > Su Observe that the previous integral is well defined since, combining Remark 2.3 with estimate 2.4, we compute f u u 2 Su A u η 2 1 u 3 = A η 2 u 1+η, < u. If f is C 1 on [, 1], since g, = f, 2.11 also holds true for = and >. In this paper, we aim at considering a general framework including the classical types of mean reverting processes. In particular, we intend to treat the following diffusion dynamics: Brownian motion with negative drift: α constant, positive and S = eα 1 α ; 5

6 Co Ingersol Ross process: α = α +b and S = e α +b αb b with α > and b > ; Ornstein-Uhlenbeck process: α = α and S = e α2 /2. For this purpose, we impose on α similar but less restrictive conditions as in [5] and work under the following standing Assumption: α = 2µ σ 2 :, R is a C 2 positive non-decreasing concave function with α =.2.12 Remark 2.4 As in [5], we observe for later use that the restriction 2.12 implies in particular that the function 2S αs 2 is a non negative increasing function. Remark 2.5 One can wonder if the present problem can be solved from [5] using the following change of variable: Y := ln1 + X, since Y would also be a process meanreverting towards if X is. We claim that this is not the case. Indeed, we can observe that f 1+X θ 1+Z τ = f ep ln1+x θ ln1+z τ, and define l : f ep. First, as briefly eplained in Remark 2.2, the problem considered here where b = cannot be deced from the one with b = 1. Moreover, for the functions f that we intend to study, f : λ or f : , the conveity of l required in [5] is not satisfied. Finally, if X is for eample an Ornstein-Uhlenbeck process, one can compute that the function α Y associated to Y is of the form α Y : y 2αe 2y e y + 1, which is conve on R +, and therefore does not satisfy the assumptions of [5]. 3 A PDE verification argument This section is devoted to the obtention of a PDE characteriation for the solution of the control problem of interest 2.5. We first derive the corresponding HJB equation and then provide a verification theorem. 3.1 The corresponding dynamic programming equation The linear second order Dynkin operator associated to the diffusion 2.1 is simply given by L : v v αv, with α = 2µ σ 2, for. By construction, observe that the scale function S satisfies in particular LS =. Since the value function of interest V rewrites as the solution of a classical optimal stopping control problem 2.8, we epect V to be solution of the associated dynamic programming equation. Namely, V should be a solution of the Hamilton Jacobi Bellman equation: minlv; g v = ; v, = f, v, =,, The first term indicates that V is dominated by the immediate reward function g and that the dynamics of v in the domain are given by the Dynkin operator of the diffusion X. The 6

7 second relation manifests that only immediate stopping is possible whenever the diffusion X has reached. Finally, the last one is the classical Von Neumann condition encountered whenever the diffusion process hits its maimum. As in any optimal stopping problem, the domain of definition of the value function subdivides into two subsets: the stopping region S where immediate stopping is optimal and the continuation region where the optimal strategy consists in waiting until the stochastic process enters the stopping region. The optimal stopping time is the first time where the process arrives in the stopping region, and, in order to obtain a stopping time, we epect the region S to be a closed subset of. Of course, the stopping region is characteried by the relation v = g since g is the reward function from immediate stopping. Depending on the position of, with respect to the region S, we epect the dynamics of 3.13 to rewrite On the stopping region: v, = g,, Lg, ; On the continuation region: v, g,, Lv, = ; Everywhere: v, 1 {=} =. In the net sections of the paper, we ehibit different shapes of stopping and continuation regions depending on the objective function f. We observe that, although the objective functions may appear rather similar, the optimal strategies can be very different. 3.2 The verification theorem As detailed above, we epect the value function V given by 2.5 to be solution of the Hamilton Jacobi Bellman equation The solution of this problem is intimately related to the form of the associated stopping region S. Afterwards, we shall not prove that V is indeed a weak solution of this PDE but instead try to guess a regular solution to the PDE and verify that it satisfies the assumptions of the following verification theorem. Theorem 3.1 Let v be a bounded from below function continuous on and piecewise C 2,1 on \ {,, > }. i If v satisfies Lv, v g as well as v, for >, then v V. ii More precisely, if v, = for > and there eists a closed set S containing the ais {,, > } such that v = g on S, Lv on S \ {,, > }, v g and Lv = on \ S, 3.14 then v = V and θ := inf{t, X t, Z t S} is an optimal stopping time. iii If in addition v < g on \ S, then θ is the "smallest" optimal stopping time, in the sense that θ ν a.s. for any optimal stopping time ν. 7

8 Proof. We prove each assertion separately. i Fi X, Z :=,. Let θ T and define θ n = n θ inf{t ; Z t n or Z t 1 n } for n N. Since X, Z takes value in a compact subset of, a direct application of Itô s formula gives v, = vx θn, Z θn θn LvX t, Z t σx t 2 dt θn v X t, Z t σx t dw t θn v X t, Z t dz t. Combining estimates Lv and v X t, Z t dz t = v Z t, Z t dz t with the fact that X, Z lies in a compact subset of, we dece v, E, vx θn, Z θn Since v g, this leads directly to v, E, gx θn, Z θn = E, E Xθn,Z θn f Xθn Z τ = E, f Xθn Z τ. Clearly as n, θ n θ almost surely. Since X θn /Z τ 1 and f is continuous, Lebesgue s dominated convergence theorem gives: E, fx θn /Z τ n E, fx θ /Z τ, leading to v, V,,,. ii Observe that this framework is more restrictive than the previous one, so that v V on. For, S, we have v, = g, V, by definition of g. We now fi, \ S and prove that v, V,. Let θ := inf{t ; X t, Z t S}. Observe that θ T since S contains the ais {,, }. The regularity of v implies LvX t, Z t = for any t [, θ. As before, we define θn := n θ inf{t ; Z t n or Z t 1 n }, which is a stopping time since S is closed. A very similar computation leads directly to v, = E, vx θ n, Z θ n. Since v is bounded from below and v g f, v is bounded. Therefore the sequence vxθ, Z n θn n is uniformly integrable and we dece that v, = E,vX θ, Z θ. Since X θ, Z θ S and v = g on S, we get Xθ v, = E, gx θ, Z θ = E, f V,. Thus v = V on and θ is an optimal stopping time. Z τ iii For a given,, we argue by contradiction and suppose the eistence of a stopping time ν T satisfying Pν < θ > and V, = E, fx ν /Z τ. 8

9 By assumption, we have V X ν, Z ν < gx ν, Z ν on {τ < θ }, which combined with estimate V g implies Xν V, = E, f = E, gx ν, Z ν > E, V X τ, Z τ V,, Z τ where the last inequality follows from the definition of V. This leads to a contradiction, which guarantees the minimality of θ. Remark 3.1 From the definition of g, one easily checks that g, = for any >. Therefore, in the PDE dynamics 3.14, the Neumann boundary condition v, = is only necessary for, \ S, since it is automatically satisfied otherwise. Remark 3.2 Whenever g is a continuous function on, C 2,1 w.r.t., on \{,, > } and Lg on \ {,, > }, then v = g and S = satisfy the assumptions of Theorem 3.1 ii. In that case, immediate stopping is always optimal. We prove in Proposition 3.1 that the reverse is true. Notice also that epression 2.1 implies that an immediate sufficient condition for g to be in C C 2,1 \ {,, > } is that f is C 2 on, 1]. Proposition 3.1 Assume that g is C on, C 2,1 w.r.t., on \ {,, > }, and that there eists, \ {,, > } such that Lg, <. Then, immediate stopping at, is not optimal or equivalently V, < g,. Proof. Since Lg is continuous at,, there eists a neighborhood U of, in such that Lg, < for any, U. Without loss of generality, we can assume that U is compact in. Let X, Z =,. Since >, there eists θ T such that E, θ > and let define θ 1 := 1 θ inf{t ; X t, Z t U } T. {θ > } = {θ 1 > }, we also have E, θ 1 >. Using Itô s formula, we compute: θ1 g, = gx θ1, Z θ1 LgX u, Z u σx u 2 θ1 g X u, Z u σx u dw u θ1 g X u, Z u dz u. Since From Remark 3.1, g, = for > so that the last term of the previous epression disappears. Since U is compact and E, θ 1 >, taking conditional epectations, we dece that g, > E, gx θ1, Z θ1 V,. In the net sections, we investigate two particular cases of objective functions, for which we ehibit functions v and stopping regions S, which satisfy the assumptions of Theorem 3.1 and are in general non trivial. 9

10 4 The power utility case Let first eamine the case where the function f is given by f : λ λ, for λ >. In other words, we are computing the following value function V λ, := 1 λ λ sup Xθ E,,,, λ > θ T Z τ Consider an investor, whose relative preferences are given by a power utility function and suppose that he detains at time a given portfolio X mean reverting towards. optimal stopping time at which he should liquidate his portfolio is the solution of the previous control problem. With a given finite time horion T, Toit and Peskir [4] as well as Shiryaev, Xu and Zhou [9] investigate the case where X is a Geometric Brownian motion. They conclude that the optimal strategy consists in waiting until time T if the portfolio has promising returns i.e. 1 < 2µ/σ 2 = α, >, with our notations, and sell immediately otherwise. In our mean reverting framework, waiting until the wealth reaches is obviously a non optimal strategy. For a linear utility function λ = 1, we prove in Theorem 4.1 below that immediate stopping is also optimal. Depending on the value of λ, the latter may no longer be the case for the non linear problem Nevertheless, we observe that optimal stopping is still optimal for the practical value function of interest V λ,, for >. 4.1 The particular case where λ 1 For λ 1, we prove hereafter that immediate stopping is always optimal. For λ = 1, these conclusions are therefore in accordance with those of [4, 9] obtained for the case of an eponential Brownian motion on a fied time horion. The A direct application of estimate 2.11 proves that the reward function g λ associated to problem 4.16 is given by g λ, = λ λ λ + λ S,,, λ >. Suu1+λ The net theorem indicates that the framework of Remark 3.2 holds for λ 1, so that g λ coincides with the value function on. Theorem 4.1 For λ 1, immediate stopping is optimal for problem 4.16, so that V λ, = g λ,,,, < λ 1. Proof. For any λ > and, with >, we compute g λ, = λ 1 λ + {λλ 1 S + λ S } Suu 1+λ. 1

11 Differentiating one more time and using the relation LS =, we get, g λ, = 1 λ λ 2 λ + {λλ 1λ 2 S + 2λ λ 1 + λ αs } Suu 1+λ, for any λ 1 and <. Combining the previous estimates, we dece that 1 Lg λ, = λ 2 [α + 1 λ] λ S λ Su u 1+λ + 2 λ 1 S λ Suu 1+λ, 4.17 for any λ 1 and <. In the case, where λ = 1, we get similarly 1 Lg 1, = α S u 2 + 2S Su u 2,, Su Furthermore, since S is increasing, we have S λ Su u 1+λ S λ Su u 1+λ λ u 1+λ = 1,,, λ >. λ Plugging this estimate in 4.17 and 4.18, we see that Lg λ on \ {,, > } for any λ 1. As detailed in Remark 3.2, since g is C on and C 2,1 on \ {,, > }, we dece that V λ = g λ on and consequently immediate stopping is optimal for any λ Construction of the solution when λ > 1 We now turn to the more interesting and intricate case where λ > 1. Then, the function Lg λ is still given by epression 4.17 and we observe that: Lg λ, 1 λ λ 2 λ <, for any > and λ > 1. Therefore, Lg λ is not non negative on and Proposition 3.1 ensures that the associated continuation region is non empty. Since immediate stopping shall not be optimal close to the ais {, ; > }, we epect to have a stopping region of the form S λ := {, ; ϕ λ }. Hence, our objective is to find functions v λ and ϕ λ satisfying Lv λ, = for < < ϕ λ and,, 4.19 v λ, = g λ, and Lg λ, for ϕ λ and,, 4.2 v λ, = for >, 4.21 v λ, = for > Since we look for regular solutions, we complement the above system by the continuity and the smoothfit conditions v λ ϕ λ, = g λ ϕ λ, and v λ ϕ λ, = g λ ϕ λ,, for >

12 The stopping region S λ will then be defined as S λ := {, ; ϕ λ } {, ; > } Since the optimiation problem of practical interest corresponds to the value of V λ on the diagonal {, ; > }, our main concern here is to find out if ϕ λ equals or not, hence indicating if immediate stopping is always optimal on the diagonal. Surprisingly, we verify hereafter that ϕ λ = so that immediate stopping is the optimal strategy for the practical problem of interest. Due to the dynamics of 5.41 and since LS =, the function v λ must be of the form v λ, = A + BS,, \ S. Combined with the continuity and smooth-fit conditions 4.23, this leads to v, = g λ ϕ λ, + gλ ϕ λ, S ϕ λ [S S ϕλ ],, \ S. The free boundary ϕ λ is then determined by the Dirichlet condition 4.21 and must satisfy: g λ ϕ λ, S ϕ λ = g λ ϕ λ, S ϕ λ,, \ S. The net lemma introces a free boundary function ϕ λ satisfying this required condition. It also provides useful properties of this free boundary function and its technical proof is postponed to Section 4.4. Lemma 4.1 For any λ > 1, the function ϕ λ given by ϕ λ g, :, arg min, [,] S is a well defined increasing C 1 function, satisfying: i ϕ λ <, for any > ; ii ϕ λ maps, onto, y λ, where y λ is the unique non null ero of y ys y λsy. Remark 4.1 Observe that, for any fied > and λ > 1, g λ, /S converges to as goes to, since S dominates the Identity function as pointed out in Remark 2.3. Therefore, the function g λ., /S. is well defined on [, ] for any >. Before providing the value function solution and verifying that it satisfies the requirements of Theorem 3.1, we still need to check that the stopping region S λ associated to ϕ λ is indeed a good candidate, i.e. that the second part of 4.2 holds. This is the purpose of the net lemma, which proof is also postponed to Section 4.4. Lemma 4.2 For any λ > 1, the function Lg λ is non negative on {,, ϕ λ }. 12

13 Given the free boundary ϕ λ defined above and the corresponding stopping region S λ, we are now in position to provide the optimal strategy and value function solutions of the problem Theorem 4.2 For any λ > 1, the value function V λ solution of problem 4.16 is given by V λ, = g λ ϕ λ S, S ϕ λ 1 {<ϕ λ } + g λ, 1 { ϕ λ },, The smallest optimal stopping time associated to this stochastic control problem is given by θ λ := inf {t, X t ϕ λ Z t }, λ > 1. Proof. Let denote by v λ the candidate value function defined by the right-hand side of We shall prove that v λ coincides with the value function 4.16 by checking that it satisfies all the requirements of Theorem 3.1. It is immediate that v λ is bounded from below by because g λ. Since g λ is C 1 on and C 2,1 w.r.t., on \ {,, > }, and ϕ λ is C 1 by Lemma 4.1, v λ is C 2,1 w.r.t., on both \S and S \{,, > }, so that it is piecewise C 2,1 on \{,, > }. By construction, v λ is continuous on and we recall from the definition of ϕ λ that g λ ϕ λ, S ϕ λ S ϕ λ = g λ ϕ λ,, >. Therefore, v λ is C 1 on. The closed stopping region associated to the value function v λ is naturally given by By definition, v λ = g λ on S λ and we dece from Lemma 4.2 that Lg λ on the set S λ \ {,, > }. By construction, we have Lv λ = on \ S λ. For any >, since g λ., /S achieves its minimum at a unique point ϕ λ, we get g λ ϕ λ S, S ϕ λ < g λ,, < ϕ λ, and we dece that v λ < g λ on \ S λ. Finally, since v λ, = g λ, = for any >, all the requirements of ii-iii in Theorem 3.1 are in force, and the proof is complete. 4.3 Properties of the solution We first observe that the two previous cases where λ is above or below 1 seem to be of different natures. However, we prove hereafter that this is not the case and provide via simple arguments the continuity of V λ with respect to the parameter λ. Proposition 4.1 The mapping λ V λ is continuous on,. 13

14 Proof. We fi λ 1 and λ 2 in, such that λ 1 λ 2. First notice that since X. Z τ on [, τ], we necessarily have λ2 λ1 λ 2 V λ 2 Xθ Xθ, = sup E, sup E, θ T Z τ θ T Z τ = λ 1 V λ 1,,, Now, using Jensen s inequality, we observe that [ E, Xθ Z τ λ1 ] λ 2 λ 1 λ2 Xθ E, λ 2 V λ 2,, θ T,,. Z τ Bringing this epression to the power λ 1 /λ 2 and taking the supremum over θ, we dece from 4.26 that [ λ1 V λ 1, ] λ 2 λ 1 λ 2 V λ 2, λ 1 V λ 1,,,. Therefore λ 2 V λ 2 λ 1 V λ 1 whenever λ 2 λ 1 and we dece the continuity of V λ with respect to λ. For λ > 1, Theorem 4.2 indicates that the stopping region S λ associated to problem 4.16 is given by Since ϕ λ =, we see that the stopping region S λ includes in particular the ais {,, > }. Therefore, if an investor detains a portfolio mean reverting to ero and hopes to get close to its upcoming maimum before it reaches ero according to the criterion 4.16, he should liquidate the portfolio immediately. Theorem 4.1 indicates that this is also the case for λ, 1] and these results are in accordance with those of [9] for an eponential Brownian motion on a finite fied horion, since waiting until maturity is irrelevant in our framework. Nevertheless, changing the criterion of interest may lead to value functions where immediate stopping is not optimal on the ais {,, > }. This is eactly the purpose of Section 5. Figure 1 represents the frontier between the stopping and the continuation regions for different values of λ larger than 1 and associated to an Orstein-Uhlenbeck with parameter α = 1 and a CIR-Feller process with parameters α = 1 and b = 1. We first observe that the shape of the free boundary ϕ λ is rather similar in both cases, and we observe indeed this feature for a large range of parameter set. Furthermore, the mapping λ ϕ λ seems to be continuous, property which is easily verified from the definition of ϕ λ. Second, we notice that the free boundary ϕ λ is decreasing with respect to λ. Indeed, arguing as in Part 2. of the proof of Lemma 4.1, one can easily check that the function R + S /S is decreasing starting from 1. Hence, by definition of y λ, the valuation domain [, y λ ] of ϕ λ shrinks monotonically to {} as λ decreases to 1, hence leading to the absence of continuation region for the problem V 1. 14

15 Free Boundary of OU for different lambda 2 1,8 1,6 1,4 1,2 Free Boundary of CIR for different lambda 2 1,8 1,6 1,4 1,2 3 4 Process maimum 2 1 Portfolio maimum 3 2 1,5 1 1,5 2 2,5 3 Process value,5 1 1,5 2 2,5 3 3,5 4 Portfolio value Figure 1: Optimal frontier for an OU α = 1 and a CIR α = 1, b = 1 with different parameter λ Remark 4.2 Considering for eample an Ornstein-Uhlenbeck portfolio X, one verifies easily from their definitions that the free boundary ϕ λ and the value function v λ are continuous with respect to the parameter α R, characteriing the dynamics of the mean reverting portfolio X. Hence, the continuation and stopping regions are not too sensitive to eventual estimation errors of this parameter of interest. 4.4 Proofs of Lemma 4.1 and Lemma 4.2 This section provides successively the proofs of Lemma 4.1 and Lemma 4.2. Proof of Lemma 4.1 Fi λ > 1. Let introce the functions m : S λs 2 1 S and λ λ l : Suu 1+λ, so that the derivative of the function of interest rewrites [ ] g, = λ 1 {m l}, S λ, A useful estimate. We will use several times the following epansion as : αs S Indeed, recalling that LS = and integrating by parts, we compute: S = S1 + 1 αus u = S1 + S αu α S 1 α1 1 1/α us u S α, 15

16 since S as and 2.12 implies that 1/α as. 1. Definition of ϕ λ. In order to justify that ϕ λ is well defined, we study separately the functions m and l. We observe first that the function l is negative, increasing and, according to 4.28, satisfies l λ λ αu u 1+λ S u λ S, 4.29 where the second equivalence comes from the following computation: λ λ [ αu u 1+λ S u = λ λ ] λ1 + λ λ u 1+λ S + u u 2+λ S, >. u We now turn to the study of m and compute, for any >, m = {λ α} λs 3 S 2 M, with M : S λ α S. Differentiating one more time, we obtain M 2 S [ λ 2 1 = λ α 2 2 ] α 2 + 2α, >. Since λ > 1 while α is non-negative, increasing and concave, the term in between brackets is decreasing. Furthermore M = λ 1 > and, since α α for >, we get M S λ α 2 [ λ α 2 + 2α ]. Thus M is first increasing and then decreasing. Furthermore, estimate 4.28 implies that ] 2α M [1 S = λ + 1 α λ α λ α S S. Since M =, we dece that m is first increasing and then decreasing. Then we have as, m 1 λ λ, and, using 4.28, m α λ λs >, for sufficiently large. 4.3 Since the function l is negative, we dece that, for any >, there is a unique point in,, denoted ϕ λ, such that m ϕ λ = l, and is the unique minimum of g, /S on [,. This point is also the unique solution of g, S g, S = for any fied. The implicit functions theorem implies that ϕ λ is C 1 on,. We prove hereafter that ϕ λ <, for any >, so that ϕ λ corresponds to the definition given in the statement of the lemma. 16

17 2. ϕ λ <, for any >. For any >, since m l is first negative and then positive, on,, the property ϕ λ < will be a direct consequence of the estimate m l >, that we prove now. First observe that the derivative of h : [m l] λ is given by h = 1 + α 2 λ S 3 S n, >, with n : S 1 + α S. Hence h has the same sign as n and, differentiating one more time, we compute [ n = S α 1 + α + 2α + ] α 1 + α 2 = α α 1 + α 2 S, for any >. Since α is concave and non-negative, we have α α for >, and, plugging this estimate in the previous epression, we obtain n 1 α α 2 S, >. Hence, n is non-increasing starting from n =, and therefore h is also non-increasing on,. Furthermore, we know from 4.29 and 4.3 that h = [m l] λ > for sufficiently large, so that we have m l >, for any >. 3. ϕ λ is increasing and valued in [, y λ ]. Recall that m ϕ λ = l and l is increasing and negative. Since m is also increasing when it is negative, we dece that ϕ λ is increasing. Since after crossing ero, the function m remains positive, ϕ must be smaller than the point where m crosses ero, for any >. By definition of m, this point y λ is implicitly defined by y λ S y λ = λsy λ. Therefore ϕ λ. y λ and, since l, we even have ϕ λ y λ. Proof of Lemma 4.2 Proof. We fi λ > 1 and recall from estimate 4.17 in the proof of Theorem 4.1 that Lg λ is given by Lg λ, = λ 2 [α + 1 λ] 1 λ S λ Su u 1+λ + 2 λ 1 S λ Suu 1+λ, 4.32 for any <. Since S is increasing, we first observe that Lg λ,. for any > such that α + 1 λ. Denoting by λ the unique point of R + defined implicitly by we dece that Lg λ,. for any λ. λ α λ = λ 1, 17

18 It remains to treat the case where < λ and we compute Lgλ, = λ λ 2 { [λ 1 α]s S 2S 1+λ }, <. S For any fied, λ, the previous epression in between brackets is increasing with respect to, negative for = and positive for large enough. Hence, for any, λ, Lg,. is first decreasing, then increasing and Lg, goes to as goes to infinity. Denoting by γ λ the inverse of ϕ λ, we dece that Lg,, for any γ λ, if and only if Lg, γ λ, for any fied, λ. Since ϕ λ and hence γ λ are increasing, it therefore only remains to verify that Lg., γ. on, λ. We recall from the proof of Lemma 4.1 that γ λ is defined implicitly by γ λ λ[γ λ ] λ Suu 1+λ = 1 S S λs 2, < < λ. For a given, λ, plugging this estimate into 4.32, we dece 1 S S λs 2 Lg, γ λ λ 1 S = α + 1 λ λ[γ λ ] λ S + which after simplifications leads to 2 λ 1 S [γ λ ] λ, Lg, γ λ = α λλ 1 S 2 λs 2 [γ λ ] λ h, with h : S α λ S. In order to get the sign of Lg., γ λ., we look for the sign of h and compute [ h = S α 1 + λ + α + 2α + ] α 1 + λ + α 2 = S [ λ λ + α α α ] S [ λ λ + α 2 1 α 2], < λ. since α α for >, e to the concavity of α. By definition of λ, we dece that h is non-decreasing on, λ. But h = and therefore Lg., γ. on, λ, which concludes the proof. 5 Minimiation of the relative quadratic error Let us now consider the case where f : Therefore, we are computing the following value function V, := 1 2 inf θ T E, 1 X 2 θ,, Z τ 18

19 With such a criterion, the investor tries to minimie the epected value of the squared relative error between the value of the stopped process and the maimal value of the process up to τ. In other words he wants to minimie the epectation of [Z τ X θ /Z τ ] 2, whereas in the previous section, λ = 2 would correspond to the minimiation of 1 X θ /Z τ 2, which is not as natural. In contrast with the previous optimal stopping problem 4.16, we prove that stopping immediately even for = is not optimal in general. This suggests that the nature of the stopping region closely depends on the criterion of interest. 5.1 Construction of the solution From 2.11, we compute the corresponding reward function: g, = S 1 u u 2,,. Su In view of Proposition 3.1, we would require Lg, in order for some, to be in the stopping region. Let us first compute g, = [S + S ] g, = [2 + α]s for any,. Combining these estimates, we dece u 2 Su [2S + 2 S ] u 2 Su [2S + 4S + 2 αs ] u 3 Su, u 3 Su, Lg, = 1 2 [1 + α ] + [2S αs] u 2 Su [S + 2S 2 αs] u 3,, Su In view of Theorem 3.1 i, if Lg on, then immediate stopping is optimal, v = g and the problem is trivial. However, the net result gives sufficient conditions such that it is not the case. Consider the following condition: α 2 2α < Remark 5.1 Notice that 5.35 will be satisfied for an Ornstein-Uhlenbeck process as well as a CIR-Feller process with positive "mean", for which we respectively have α = α and α = α +b respectively, α and b being positive constants. More generally, as soon as α =, 5.35 is satisfied. However, for a drifted Brownian motion or a degenerated CIR-Feller process with "mean" equal to, 5.35 does not hold true. Proposition 5.1 Assume that 5.35 is satisfied. Then there eists an open subset of on which Lg <. 19

20 Proof. Using the asymptotic epansions from Proposition 5.4 in Section 5.4, we compute for close to : Lg, = [2S αs] u 2 Su [S + 2 2S αs] u 3 Su = 2 + α + O α 2 α2 2α ln + oln α2 + O α O = α2 2α 6 ln + oln. Since ln when, we see that if 5.35 holds, then Lg, < for in a neighborhood of, so that we have the result by continuity of Lg. In view of Proposition 3.1, an immediate consequence of Proposition 5.1 is that stopping immediately is not optimal in general, even for initial conditions such that =, which is the practical case of interest. Hence, the optimal strategy shall be very different from the one in the power utility case. Since we do not have Lg on the entire space but we can eercise only in that region, we first need to study the set and we define similarly: Γ + := {,, Lg, }, 5.36 Γ := {,, Lg, } In fact, observe that 5.34 rewrites as: Lg, = α 2 + [2S αs] 1 2 u u 2 Su S 2 u 3,, Su By Remark 2.4, we have 2S αs 2 and therefore each of the three terms above are positive if 2, and so Lg, > for 2 and,, 5.39 which implies that {,, 2} IntΓ +. Moreover we have the following result, which proof is given in Section 5.4 below. Lemma 5.1 For any >, there eists δ, 2 such that Lg,. is increasing on [, δ and decreasing on δ, 2]. 2

21 In view of 5.39, we can define the following function on R + \ {}: Γ := inf{, Lg, }. 5.4 Lemma 5.1 and 5.39 imply that, if > Γ, then Lg, >, while if, Γ, then Lg, <. We also dece that Γ > implies Lg, Γ =. Notice that Γ is continuous, and, from 5.39, we also know that Γ < 2. The net result provides the main properties of Γ: it is increasing and equal to the Identity function for sufficiently large. Again the proof is postponed to Section 5.4. Proposition 5.2 We have the two following properties: i Γ is increasing on, + ; ii Denoting Γ := sup{ ; Γ > }, we get Γ <. Notice that Γ + implies directly Γ >. Now that we have a better understanding of the set Γ +, we epect to have a stopping region of the form {, ; γ} and, our objective is then to find functions v and γ, satisfying the following free-boundary problem: Lv, = for < < γ and,, 5.41 v, = g, and Lg, for γ and,, 5.42 v, = 1 2 for >, 5.43 v, = for > In order to allow for the application of Itô s formula, the verification step requires a value function which is C 1, and piecewise C 2,1 with respect to,. Therefore, as in the previous section, we complement the above system by the continuity and the smooth-fit conditions v, γ = g, γ and v, γ = g, γ, for > The stopping region S will then be defined as: S := {, ; γ} {, ; > } First by 5.41, on the continuation region, v is of the form: v, = A + BS,, \ S. Then, on the interval where γ is one-to-one, the continuity and smoothfit conditions 5.45 imply that v, = gγ 1, + g γ 1, S γ 1 [S S γ 1 ],, \ S. 21

22 Finally, the Neumann condition 5.44, implies that we epect the boundary γ to satisfy on its domain of definition the following ODE: γ = γ 2 Lg, γ γ 1 1 S S γ As in [5], there is no a priori initial condition for this ODE. In the sequel, we take this ODE with no initial condition as a starting point to construct the boundary γ. Notice that this ODE has infinitely many solutions, as the Cauchy-Lipschit condition is locally satisfied whenever 5.47 is complemented with the condition γ = for any < < and 2. We will follow the ideas of [5], however in our case, 5.47 is not well-defined for γ = 2, so that our framework requires to be more cautious. Notice also that we encounter here a similar feature as in Peskir [8]. The following result selects an appropriate solution of 5.47, and its proof is given in Section 5.5. Proposition 5.3 Let IntΓ be non empty. Then, there eists an increasing continuous function γ defined on R + with graph {, γ : > }, such that: i On the set { > : γ > }, γ is a C 1 solution of the ODE 5.47, ii {, γ : > } Γ +, and {, γ : > and γ > } IntΓ +, iii γ = for all Γ. Since γ is increasing, we can define: ϕ := γ Now that we have constructed the free-boundary ϕ, we are able to state the following result. Theorem 5.1 Let IntΓ be non empty, γ be given by Proposition 5.3 and ϕ be defined by Then the value function V solution of problem 5.33 is given, for,, by: g,, if ϕ V, := gϕ, + g ϕ, S S ϕ S ϕ, if > ϕ Moreover, the smallest optimal stopping time associated to this stochastic control problem is given by θ := inf {t, X t ϕz t }. Proof. Let v be defined by 5.49 and recall that S is defined by The result follows from verifying that all the assumptions of Theorem 3.1 ii and iii are satisfied. 1. Regularity of v. We know from Proposition 5.3 that γ and therefore ϕ are continuous and hence v is continuous on by construction. Furthermore, by Proposition 5.3 i and ii together with the dynamics of the ODE 5.47, γ is a C 1 function with positive derivative on the set 22

23 { > ; γ > }. Therefore ϕ is C 1 as well on { > ; ϕ < } so that it is immediate that v is C and piecewise C 2,1 w.r.t.,. Furthermore, since Γ < by Proposition 5.2, \ S is bounded. Since v is continuous and g, v is bounded from below. 2. Dynamics of v. By definition, we have Lv = on \ S. By Proposition 5.3 ii, Lg, γ for >, and we dece from Lemma 5.1 and 5.39 that Lg, for any, such that γ. Hence, 5.39 ensures that Lg on S. It remains to prove that v, = for >. We fi >. If ϕ, since g, =, we have v, = as well. Suppose now that ϕ <. Then, by Proposition 5.3 i, γ satisfies 5.47 in a neighborhood of ϕ, and by Proposition 5.3 ii, Lg ϕ, >, which implies γ ϕ >, so that: ϕ Lgϕ, = 1 2 2ϕ We then compute from the definitions of v and g that 1 1 S ϕ S. S S ϕ v, = g ϕ, + g S ϕ [ 1 = 2 1 2ϕ 1 S ϕ S + ϕ S S ϕ Lgϕ, S ϕ ] S S ϕ + ϕ Lgϕ, S ϕ =. 3. Comparing v and g. Finally, the fact that v g on and v < g on \ S follows from similar arguments as in the proof of Proposition 6.2 in [5] but the demonstration is simpler in our contet since Γ <. For the sake of completeness, we detail this proof. For, such that > ϕ, we compute S S ϕ v, g, = gϕ, + g ϕ, S g,, ϕ and, differentiating twice w.r.t. and using 5.45, we verify that v, g, = S ϕ Lgu, S. 5.5 u Therefore, from Lemma 5.1 and Proposition 5.2 i, for any fied, the function v g, is either decreasing on [ϕ, ], or decreasing on [ϕ, δ and then increasing on δ, ] for a given δ ϕ,. For any >, since vϕ, = gϕ,, we only need to prove that n := v, g, < if ϕ <. Since v, = g, = for >, we compute: n = v, g, = S Lgu, ϕ S u, >. 23

24 We assume the eistence of a fied < Γ such that n and ϕ < and work towards a contradiction. We first observe that necessarily n >. If not, Lgu, ϕ S u implies that Lgu, ϕ S u > for any ϕ,, and 5.5 combined with vϕ, = gϕ, leads to n < which is impossible. Since n is continuous, this implies that n is increasing on any connected subset of {, ϕ < }. Defining a := inf{ > ; ϕ = } Γ <, we get na = va, a ga, a >, which contradicts the definition of v. Therefore, n < for any > such that ϕ < and we dece that v g on and v < g on \ S. 5.2 Properties of the value function Theorem 5.1 and Proposition 5.1 indicate that, at least for processes satisfying 5.35, such as the Ornstein-Uhlenbeck process or the CIR-Feller process, the diagonal {, ; > } is not included in the stopping region S. In other words, it is not always optimal to stop immediately, even when starting from points such that =. Therefore, the form of the solution and the nature of the optimal strategy to apply in order to be as close as possible to the maimum using this criterion is very different from the ones obtained in Section 4 or in [9]. The Ornstein-Uhlenbeck process as well as the CIR-Feller process are two eamples for which the coefficient α satisfies Conditions 2.12 and IntΓ. Indeed we have α = α and α = α +b respectively, where α and b are two positive constants. Therefore, Condition 5.35 is satisfied, ensuring that IntΓ by Proposition 5.1. Hence, Theorem 5.1 can be applied. Figure 2 represents the boundary ϕ for those two processes, with α = 1 for the OU process and α, b =.1,.1 for the CIR process. We observe that the continuation region is in fact pretty small since the free boundary is very close to the diagonal ais. Therefore, even if immediate stopping is not optimal, an investor should not wait long until the process X, X enters the stopping region. Remark 5.2 Similarly to Proposition 7.3 of [5], an homogeneity result can be derived for the OU process, so that the free boundary for any α > can be deced by a change of scale from the one for α = 1. The Brownian motion with negative drift is another eample for which α satisfies Condition However, since α = α > is constant, Condition 5.35 does not hold. Although we did not verify it, numerical computations suggest that Lg on. Finally, we can also consider the case of a Brownian motion. In this case, α =, so that α does not satisfy Condition However, for any,, we can compute from 24

25 Free Boundary of OU Free Boundary of CIR,5,3 Process maimum,4,3,2,1 Process maimum,2,1,1,2,3,4,5 Process value,1,2,3 Process value Figure 2: Optimal frontier for an OU α = 1 and a CIR α =.1, b = that Lg, = 2 on. Since the proofs of Theorem 3.1 and Remark 3.2 do 3 not require Condition 2.12, we dece that immediate stopping is always optimal. Remark 5.3 Let α be associated to an Ornstein-Uhlenbeck or a CIR process and hence be parametried by a possibly bi-dimensional parameter set a. Since the parameter set a may be badly estimated, let consider a sequence of parameter set a n converging to a and denote by α n the corresponding sequence of functions. Then, S n, g n and all their derivatives converge respectively to S, g and their derivatives in the sense of the uniform norm on the compact sets. Moreover, Γ n converges to Γ in the same sense so that for n sufficiently large, IntΓ n. ODE 5.47 also depends continuously on a n, so that n defined by 5.63 converges to, and γ n given by Proposition 5.3 converges pointwise to γ. Since γ n is increasing for any n and γ is continuous, Dini s theorem implies that the convergence is uniform on any compact set of R + \{}. Let us prove that ϕ n converges to ϕ in the same sense. Let y > be fied, we define n := ϕ n y and := ϕy. We shall prove that n. Indeed, since ϕ n y [ y 2, y] for any n, { n; n N} is relatively compact in R + \ {}. Now let be the limit of a subsequence of n. For notational reasons, let us write n, forgetting that it is a subsequence. Since γ n converges uniformly on compact sets of R + \ {}, γ n n γ. Recalling that γ n n = y for any n, we get γ = y and therefore =. In consequence, n, or in other words, ϕ n converges pointwise to ϕ on R + \{}. Noticing that ϕ n = for any n and ϕ = and using again Dini s theorem, we see that ϕ n converges to ϕ uniformly on the compact sets of R +. This finally implies that V n converges pointwise to V. As a consequence, if one makes a small mistake estimating the parameters of the model, the inced mistake on the free boundary as well as the mistake on the value function will be small as well. 25

26 5.3 Generaliation As in the previous section, we may also consider, for any λ >, the following etension of the previous problem: In that case, 2.1 rewrites V λ, := 1 λ inf E, 1 X λ θ,, θ T Z τ g λ, = 1 λ 1 λ + S 1 λ 1 u,,, λ > u 2 Su If λ = 2, Lg 2 is given by If λ = 1, the control problem has already been solved in Section 4 and Lg 1 is given by For any λ > such that λ {1, 2}, we compute : { 1 Lg λ, = λ λ 2 [ S 2 1 Su u 3 λ 2 λ 2 1 u u 4 ] } λ 3 u + [2S αs] 1 λ 2 λ 1 u λ u u 3 + α Su λ,5.53 for <. In this case, the sign of Lg λ is hardly identifiable analytically, and we shall restrict our analysis to simple remarks and guesses on the solution of the problem Noticing that 1 λ 2 2 u λ 2 1 λ 3 u 3 u = 1 u 4 1 λ 2 2 for < <, we dece from 4.18, 5.39 and 5.53 that Lg λ,, for λ> and 1 λ Therefore, for 1 λ 2, we epect to obtain as for λ = 2 a free boundary γ λ in between the ais {, ; > } and {, λ ; > }. We verify easily as in Proposition 4.1 that λ V λ is continuous and, as epected, we observe a disappearance of the free boundary γ λ for λ = 1. On the other hand, for λ < 1, we observe that Lg, < for small enough and large enough. Indeed, recalling from 4.28 that S αs when, an integration by parts leads to u 2 Su 1. Assuming moreover that α = and plugging 2 S this estimate in 5.53, we get Lg, λ 1 2 <, for any λ < 1. In view of Proposition 3.1, this implies that the stopping region cannot have the same form as the one in the quadratic case λ = 2. It even suggests that the nature of the stopping region could be similar to the one of Section

27 5.4 Proofs of Lemma 5.1 and Proposition 5.2 This section is dedicated to the proofs of Lemma 5.1 and Proposition 5.2, but we first state the asymptotic epansions used in Proposition 5.1. Proposition 5.4 As, we have the following epansions: S = 1 + α + α + α o2 ; S = + α α + α o3 ; αs = α u 2 Su 2 u 3 Su α 2 + 2α = α 2 α2 2α ln + o ln ; 12 = α α2 2α + o 6 α 3 + 4αα + 3α + o 3 ; 1. Proof. As, we directly compute the epansion: S = S + S S3 + o 2 = 1 + α + α + α o2. The eact same reasoning also leads to S = + α α + α o 3 ; 6 αs = α + 2 α 2 + 2α + 3 α 3 + 4αα + 3α + o Using one of the previous estimates, we get u 2 Su = = = u α 2 u + α +α 2 6 u 2 + ou 2 1 α 2 u α + α 2 αu 2 u ou u 3 α 2u 2 + α2 2α 1 + o 12u u = α 2 α2 2α ln + o ln, 12 which is justified since all the non-ero terms go to infinity when. compute 2 u 3 Su = 2 u 4 α u 3 + α2 2α 6u 2 = α α2 2α + o o 1. u 2 u 3 Similarly, we

28 Proof of Lemma 5.1 Differentiating 5.34 w.r.t., we compute Lg, = αs 2S 2 S S S = [2 α 2] 3 S Let us introce α as the unique solution of: + [2 2α]S + 4S 3 S 2 2α S 3,,. S α 2 α α α = If α, then 2 α 2 is negative on [, 2, whereas if > α, then there eists, 2 such that 2 α 2 will be positive on,, ero at and negative on, 2. Let be fied and let us introce F : S S + 2S 2 2 α 2, which is well defined and continuous on [, 2] if < α, on [, 2 if = α and on [, 2] \ { } if > α. Furthermore F is increasing, since we compute on the domain of definition of F : F = S + 4S 2 α 2 2 >. We consider first the case where α. Then F and Lg,. have opposite signs on [, 2. Since F is increasing, F < while F 2 = S2 S >, Lg,. is increasing on [, δ and decreasing on δ, 2], for a certain δ, 2. We now turn to the case where > α. Then F and Lg,. have the same sign on [, and opposite signs on, 2]. Since F is increasing, F >, F + = and F 2 >, we see that again Lg,. is increasing on [, δ and decreasing on δ, 2], for a certain δ, 2, 2. Proof of Proposition 5.2 i Γ is increasing on, + We prove the two assertions separately. We fi > such that Γ >. Then Lg., Γ. = in a neighborhood of, and using the implicit functions theorem, Γ is C 1 in a neighborhood of and we have: Γ Lg, Γ + Lg, Γ = We will prove that Γ >. Denoting m := 2S αs which is increasing and positive, we get combining Lg, Γ = and 5.38: m Γ u 2 u 3 = S Su Γ αγ αγ u 3 Su Γ 2 Γ 2, 28