Aurélien Alfonsi
email: alfonsi@cermics.enpc.fr

Benjamin Jourdain Cermics

A Call-Put Duality for Perpetual American Options

Keywords: Perpetual American options, Dupire's formula, Call-Put Duality, Calibration of volatility, Optimal stopping

 that in models with time-homogeneous local volatility functions and constant interest and dividend rates, the European Put prices are transformed into European Call prices by the simultaneous exchanges of the interest and dividend rates and of the strike and spot price of the underlying. This paper investigates such a Call Put duality for perpetual American options. It turns out that the perpetual American Put price is equal to the perpetual American Call price in a model where, in addition to the previous exchanges between the spot price and the strike and between the interest and dividend rates, the local volatility function is modified. We prove that equality of the dual volatility functions only holds in the standard Black-Scholes model with constant volatility. Thanks to these duality results, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price x 0 . The knowledge of the Put (resp. Call) prices for all strikes enables to recover the local volatility function on the interval (0, x 0 ) (resp. (x 0 , +∞)).

Introduction

In a model with local volatility function ς(t, x), interest rate r and dividend rate δ dS x t = ς(t, S x t )S x t dW t + (rδ)S x t dt, t ≥ 0

S x 0 = x (1)
the initial price h(T, y) = E e -rT (y -S x T ) + of the European Put option considered as a function of the maturity T > 0 and the Strike y > 0 solves Dupire's partial differential equation [START_REF] Dupire | Pricing with a smile[END_REF] :

∂ T h(T, y) = ς 2 (T,y)y 2 2
∂ 2 yy h(T, y) + (δr)y∂ y h(T, y)δh(T, y), T, y > 0 h(0, y) = (yx) + , y > 0 One easily deduces that the function h(Tt, y) for (t, y) ∈ [0, T ] × R * + satisfies the pricing partial differential equation for the Call option with strike x and maturity T in the model d Sy,T t = ς(Tt, Sy,T t ) Sy,T t dW t + (δr) Sy,T t dt, t ∈ [0, T ] Sy,T

0 = y (2)
with local volatility function ς(Tt, y), interest rate δ and dividend rate r. Therefore h(T, y) = E e -δT ( Sy,T Tx) + and one deduces the following Call-Put duality relation which is also a consequence of [START_REF] Andreasen | Put Call Reversal[END_REF] ∀T ≥ 0, ∀x, y > 0, E e -rT (y -S x T ) + = E e -δT ( Sy,T Tx) + .

Since it derives from Dupire's formula, this Call-Put duality equality is closely related to calibration issues. One remarks that in the particular case of a time-homogeneous volatility function (ς(t, x) = σ(x)), then Sy,T t also evolves according to the same time-homogeneous volatility function.

In this work, we are interested in deriving such a Call-Put duality relation in the case of American options and in investigating consequences in terms of calibration. In the Black-Scholes model with constant volatility ς(t, x) = σ, when τ denotes a bounded stopping-time of the natural filtration of the Brownian motion (W t ) t≥0 , one has E e -rτ yxe σWτ +(r-δ-σ 2 2 )τ + = E e -δτ e σWτ -σ 2 2 τ ye -σWτ +(δ-r+ σ 2 2 )τx + = E e -δτ ye -σWτ +(δ-r-σ 2 2 )τx

+
where the second equality follows from Girsanov theorem. Taking the supremum over all stopping-times τ smaller than T one deduces that the price of the American Put option with maturity T is equal to the price of the American Call option with the same maturity up to the simultaneous exchange between the underlying spot price and the strike and between the interest and dividend rates. Extensions of this result when the underlying evolves according to the exponential of a Lévy process have been obtained in [START_REF] Fajardo | Put-Call Duality and Symmetry[END_REF]. Let us also mention that another kind of duality has been investigated in [START_REF] Peskir | A note on the call-put parity and a call-put duality[END_REF]. But, to our knowledge, no study has been devoted to the case of models with local volatility functions like [START_REF] Andreasen | Put Call Reversal[END_REF].

In the present paper, we consider the case of perpetual (T = +∞) American options in models with time-homogeneous local volatility functions ς(t, x) = σ(x). In the first part, we recover well-known properties of the perpetual American call and put pricing functions by extending an approach recently developed by Beibel and Lerche [START_REF] Beibel | A New Look at Optimal Stopping Problems related to Mathematical Finance[END_REF] in the Black-Scholes case. This makes the paper self-contained. In the second part, we introduce the framework used in the remaining of the paper. In the third part of the paper, we consider the exercise boundaries as functions of the strike variable and characterize them as the unique solutions of some non-autonomous ordinary differential equations. The fourth part is dedicated to our main result. We prove that the perpetual American Put prices are equal to the perpetual American Call prices in a model where, in addition to the exchanges between the spot price of the underlying and the strike and between the interest and dividend rates, the volatility function is modified. We also derive an expression of this modified volatility function. Notice that in the European case presented above, timehomogeneous volatility functions are not modified. The fifth part addresses calibration issues. It turns out that for a given initial value x 0 > 0 of the underlying one recovers the restriction of the time-homogeneous volatility function σ(x) to (0, x 0 ] (resp. [x 0 , +∞)) from the perpetual Put (resp. Call) prices for all strikes. In the last part, we show that at least when δ < r, in the class of volatility functions analytic in a neighbourhood of the origin, the only ones invariant by our duality result are the constants. This means that the case of the standard Black-Scholes model presented above is very specific.

Perpetual American put and call pricing

We consider a constant interest spot-rate r that is assumed to be nonnegative and an asset S t which pays a constant dividend rate δ ≥ 0 and is driven by a homogeneous volatility function σ : R * + → R * + that satisfies the following hypothesis. Hypothesis (H vol ): σ is continuous on R * + and there are 0 < σ < σ < +∞ such that: ∀x > 0, σ ≤ σ(x) ≤ σ.

In other words, S t is assumed to follow under the risk-neutral measure the SDE:

dS t = S t ((r -δ)dt + σ(S t )dW t ). ( 3
)
With the assumption made on σ, we know that for any initial condition x ∈ R * + , there is a unique solution in the sense of probability law (see for example Theorem 5.15 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF], using a log transformation) denoted by (S x t , t ≥ 0). Moreover, Theorem 4.20 ensures that the strong Markov property holds for (S x t , t ≥ 0). Under that model, we denote by

P σ (x, y) = sup τ ∈T 0,∞ E e -rτ (y -S x τ ) + and C σ (x, y) = sup τ ∈T 0,∞ E e -rτ (S x τ -y) +
respectively the prices of the American perpetual put and call options with strike y > 0 and spot x. Here, T 0,∞ simply denotes the set of the stopping times with respect to the natural filtration of (S x t , t ≥ 0). Since e -rt S x t = x exp -δt

+ t 0 σ(S x u )dW u -1 2 t 0 σ 2 (S x u )du and t 0 σ 2 (S x u )du ≥ σ 2 t → t→+∞ +∞, it follows from the Dubins-Schwarz theorem that e -rt S x t → t→+∞ 0 a.s. ( 4
)
As a consequence, e -rt (y -S x t ) + → t→+∞ 1 {r=0} y and e -rt (S x ty) + → t→+∞ 0 a.s. On {τ = ∞}, we thus set e -rτ (y -S x τ ) + = 1 {r=0} y and e -rτ (S x τy)

+ = 0. ( 5
)
Let us consider the second-order ordinary differential equation

1 2 σ 2 (x)x 2 f ′′ (x) + (r -δ)xf ′ (x) -rf (x) = 0, x > 0. ( 6
)
According to Borodin and Salminen ([3], chap. 2) the functions

∀x > 0, f ↑ (x) = E[e -rτ x 1 ], if x ≤ 1 1/E[e -rτ 1 x ], if x > 1 and f ↓ (x) = 1/E[e -rτ 1 x ], if x ≤ 1 E[e -rτ x 1 ], if x > 1 (7)
where for x, y > 0, τ x y = inf{t ≥ 0, S x t = y} ( inf ∅ = +∞ ), are the unique solutions (up to a multiplicative constant) that are positive and respectively increasing and decreasing. The volatility function σ being continuous, [START_REF] Fajardo | Put-Call Duality and Symmetry[END_REF] ensures that these functions are C 2 on R * + .

Remark 1.1. It is easy using the strong Markov property to get:

∀x, y > 0, E[e -rτ x y ] = f ↑ (x)/f ↑ (y), if x ≤ y f ↓ (x)/f ↓ (y), if x ≥ y.
Remark 1.2. Assuming r > 0, one has (see Borodin and Salminen [START_REF] Borodin | Handbook of Brownian Motion -Facts and Formulae[END_REF])

lim x→0 f ↓ (x) = +∞, lim x→+∞ f ↓ (x) = 0 lim x→0 f ↑ (x) = 0, lim x→+∞ f ↑ (x) = +∞.
The function f ↓ (resp. f ↑ ) is thus, up to a multiplicative constant, the unique solution to [START_REF] El Karoui | Robustness of the Black and Scholes Formula[END_REF] such that lim

x→+∞ f (x) = 0 (resp. lim x→0 f (x) = 0).
Remark 1.3. In the case δ = 0, we have the analytical solutions :

f ↓ (x) = ϕ(x) ϕ(1) where ϕ(x) = x +∞ x 1 v 2 exp - v 1 2r uσ 2 (u) du dv, f ↑ (x) = x. Indeed, since f (x) = x is solution of 1 2 σ 2 (x)x 2 f ′′ (x) + rxf ′ (x) -rf (x) = 0, we search a general solution that can be written f (x) = x f (x). This leads to 1 2 σ 2 (x)x f ′′ (x) + (r + σ 2 (x)) f ′ (x) = 0 and then f ′ (x) = C 1 x 2 exp - x 1 2r uσ 2 (u) du . Therefore, ∃C 1 , C 2 ∈ R, f (x) = C 2 x + C 1 x +∞ x 1 v 2 exp - v 1 2r uσ 2 (u) du dv.
Now, we are in position to show the existence of an optimal stopping time and give the call and put prices. Let us mention here that the problem of perpetual optimal stopping is treated in the paper of Dayanik and Karatzas [START_REF] Dayanik | On the optimal stopping problem for onedimensional diffusions[END_REF], for a general payoff function and an underlying evolving according to a general one-dimensional time homogeneous diffusion process. Villeneuve [START_REF] Villeneuve | On the Threshold strategies for Optimal Stopping arising in Real Option Theory[END_REF] considers a model where the constant dividend rate δ in ( 3) is replaced by a function δ(S t ) and gives sufficient condition on the payoff function ensuring that a threshold strategy is optimal. Here, we give a direct proof that generalizes the approach developed by Beibel and Lerche [START_REF] Beibel | A New Look at Optimal Stopping Problems related to Mathematical Finance[END_REF] in the Black-Scholes case.

Theorem 1.4. Assume r > 0. For any strike y > 0, there is a unique x * σ (y) ∈ (0, y)∩(0, r δ y] such that τ P x = inf{t ≥ 0, S x t ≤ x * σ (y)} (convention inf ∅ = +∞) is an optimal stopping time for the put and:

∀x ≤ x * σ (y), P σ (x, y) = (y -x) + , ∀x > x * σ (y), P σ (x, y) = y -x * σ (y) f ↓ (x * σ (y)) f ↓ (x) > (y -x) + . (8)
In addition, we have f ′ ↓ (x * σ (y)) < 0 and:

x * σ (y) -y = f ↓ (x * σ (y)) f ′ ↓ (x * σ (y)) . ( 9
)
Last, the smooth-fit principle holds: ∂ x P σ (x * σ (y), y) = -1. Remark 1.5. If r = 0, ∀x, y > 0, P σ (x, y) = y since for any stopping time τ , (y -S x τ ) + ≤ y and equality holds for τ = +∞ by [START_REF] Dupire | Pricing with a smile[END_REF].

Proof. Let us define:

∀z > 0, h(z) = (y -z) + f ↓ (z) and h * = sup z>0 h(z).
Since the function h is continuous such that h(y) = 0 and h(0+

) = 0 (Remark 1.2), x * σ (y) = sup{z > 0, h(z) = h * } belongs to (0, y) and is such that h(x * σ (y)) = h * . Since the function h is C 2 on (0, y), we have h ′ (x * σ (y)) = 0 and h ′′ (x * σ (y)) ≤ 0. These conditions give easily f ↓ (x * σ (y)) + (y -x * σ (y))f ′ ↓ (x * σ (y)) = 0 and f ′′ ↓ (x * σ (y)) ≥ 0. Since f ↓ is positive and x * σ (y) < y, we have f ′ ↓ (x * σ (y)
) < 0 and deduce [START_REF] Hobson | Volatility Misspecification, Option Pricing and Superreplication via Coupling[END_REF]. The second order condition and equation [START_REF] El Karoui | Robustness of the Black and Scholes Formula[END_REF]

then give x * σ (y)(r -δ)f ′ ↓ (x * σ (y)) -rf ↓ (x * σ (y)) ≤ 0 and so ry -δx * σ (y) ≥ 0.
Now let us check the optimality of τ P x and consider τ ∈ T 0,∞ . By Fatou's lemma and Doob's optional sampling theorem, we have

E[e -rτ (y -S x τ ) + ] ≤ lim inf t→+∞ E e -rτ ∧t (y -S x τ ∧t ) + = lim inf t→+∞ E e -rτ ∧t f ↓ (S x τ ∧t )h(S x τ ∧t ) ≤ h(x * σ (y))lim inf t→+∞ E e -rτ ∧t f ↓ (S x τ ∧t ) ≤ h(x * σ (y))f ↓ (x) since e -rt f ↓ (S x t ) = f ↓ (x) + t 0 e -ru σ(S x u )S x u f ′ ↓ (S x u
)dW u is a nonnegative local martingale and therefore a supermartingale. If x ≥ x * σ (y), we have using Remark 1.1:

E[e -rτ P x (y -S x τ P x ) + ] = E[e -rτ x x * σ (y) (y -S x τ x
x * σ (y)

) + ] = (y -x * σ (y))E[e -rτ x x * σ (y) ] = h(x * σ (y))f ↓ (x)
and τ P x is optimal for x ≥ x * σ (y). Since

x * σ (y) = sup{z > 0, h(z) = h * }, we have (y -x) + = h(x)f ↓ (x) < f ↓ (x)h(x *
σ (y)) for x > x * σ (y), and finally deduces (8) for x ≥ x * σ (y). We consider now the complementary case x ∈ (0, x * σ (y)), and set τ ∈ T 0,∞ . Using the strong Markov property and the optimality result when the initial spot is x * σ (y), we get ) + ] ≤ (yx). Now, we state the similar result for the call prices.

E[e -rτ (y -S x τ ) + ] ≤ E[e -rτ ∧τ x x * σ (y) (y -S x τ ∧τ x x * σ (y) ) + ]. On {t < τ x x * σ (y) }, we have S x t < x * σ (y), de -rt (y -S x t ) = e -rt (
Theorem 1.6. Assume δ > 0. For any strike y > 0, there is a unique Υ * σ (y) ∈ (y, ∞) ∩ [ r δ y, +∞) such that τ C x = inf{t ≥ 0, S x t ≥ Υ * σ (y)} is an optimal stopping time for the call and:

∀x ≥ Υ * σ (y), C σ (x, y) = (x-y) + , ∀x < Υ * σ (y), C σ (x, y) = Υ * σ (y) -y f ↑ (Υ * σ (y)) f ↑ (x) > (x-y) + . ( 10
)
In addition, we have f ′ ↑ (Υ * σ (y)) > 0 and:

Υ * σ (y) -y = f ↑ (Υ * σ (y)) f ′ ↑ (Υ * σ (y)) . ( 11
)
Last, the smooth-fit principle holds:

∂ x C σ (Υ * (y), y) = 1.
Remark 1.7. If δ = 0, ∀x, y > 0, C σ (x, y) = x. Indeed, the Call-Put parity E[e -rt (S x ty) + ] = xye -rt + E[e -rt (y -S x t ) + ] gives the convergence to x in both cases r > 0 and r = 0 when t → +∞. Now, thanks to the Fatou lemma, we have for τ ∈ T 0,∞ :

E [e -rτ (S x τ -y) + ] ≤ lim inf t→+∞ E [e -rτ ∧t (S x τ ∧t -y) + ] ≤ lim inf t→+∞ E [e -rτ ∧t S x τ ∧t ] = x.
Proof. The proof works as for the put, and we just hint the differences. We define

∀z > 0, h(z) = (z -y) + f ↑ (z) and h * = sup z>0 h(z).
Let us admit for a while that lim z→+∞ h(z) = 0. Then, since h(y) = 0 and h is continuous, h reaches its maximum in Υ * σ (y) = inf{z > 0, h(z) = h * }, and Υ * σ (y) ∈ (y, ∞). This gives [START_REF] Martini | Propagation of Convexity by Markovian and Martingalian Semigroups[END_REF]. We then consider the case x ≤ Υ * σ (y) and show that τ C x = τ x Υ * σ (y) is optimal. Note that in the special case r = 0, we have to use Proposition 1.8 which is stated below. Finally, we prove that τ C

x is optimal when x > Υ * σ (y) using that δΥ * σ (y)ry ≥ 0. Now, let us check that lim z→+∞ h(z) = 0. In the case r = 0, it is straightforward using the explicit form given in Proposition 1.8 below that f

↑ (x) ≥ 1 1+ 2δ σ 2
x 1+2δ/σ 2 -1 for x ≥ 1, and we have then lim z→+∞ h(z) = 0. When r > 0, Itô's Formula gives

de -rt (S 1 t ) 1+a = e -rt (S 1 t ) 1+a (a + 1)σ(S 1 t )dW t + [a(r + (a + 1)σ 2 (S 1 t )/2) -(a + 1)δ]dt .
When a > 0, the drift term is bounded from above by a(r + (a + 1)σ 2 /2) -(a + 1)δ and we can find a > 0 such that this bound is negative since a(r

+ (a + 1)σ 2 /2) -(a + 1)δ → a→0 -δ < 0. Then, for x ≥ 1, we have E[e -rτ 1 x ∧t (S 1 τ 1
x ∧t ) 1+a ] ≤ 1 thanks to Doob's optional sampling theorem. The Fatou lemma gives then E[e -rτ 1

x (S 1 τ 1

x ) 1+a ] ≤ 1, and therefore we get

f ↑ (x) = 1/E[e -rτ 1
x ] ≥ x 1+a . This shows lim z→+∞ h(z) = 0.

Proposition 1.8. In the case r = 0, the unique nonincreasing and increasing solution of (6) starting from 1 in 1 are respectively:

f ↓ (x) = 1, f ↑ (x) = ψ(x) ψ(1) where ψ(x) = x 0 exp v 1 2δ uσ 2 (u) du dv.
Moreover, we have f ↑ (x) = 1/P(τ 1 x < +∞) for x ≥ 1 and f ↑ (x) = P(τ x 1 < +∞) for x ∈ (0, 1].

Proof. When r = 0, the differential equation 1 2 σ 2 (y)y 2 f ′′ (y) -δyf ′ (y) = 0 is easy to integrate: f ′ (y) = C 3 exp y 1 2δ uσ 2 (u) du and then g(y) = C 4 + C 3 y 0 exp v 1 2δ uσ 2 (u) du dv for C 3 , C 4 ∈ R. For x ≥ 1, f ↑ (S 1 τ 1
x ∧t ) is a bounded martingale that converges almost surely to f ↑ (x)1 {τ 1

x <+∞} thanks to (4) and lim x→0 f ↑ (x) = 0. Therefore f ↑ (x)P(τ 1 x < +∞) = 1, and the proof is the same for x ∈ (0, 1].

To conclude this section, we state a comparison result which will enables us to compare x * σ (y) and Υ * σ (y) with the exercise boundaries obtained in the Black-Scholes model with constant volatility σ (resp. σ) where σ (resp. σ) bounds the function σ from below (resp. above). Proposition 1.9. Let us consider two volatility functions σ 1 and σ 2 such that ∀x > 0, σ 1 (x) ≤ σ 2 (x) and that satisfy (H vol ). We also assume that either f ′′ ↓,σ 1 or f ′′ ↓,σ 2 (resp. either f ′′ ↑,σ 1 or f ′′ ↑,σ 2 ) are nonnegative functions and r > 0 (resp. δ > 0). Then, we have

∀x, y > 0, P σ 1 (x, y) ≤ P σ 2 (x, y) (resp. C σ 1 (x, y) ≤ C σ 2 (x, y))
and we can compare the exercise boundaries:

∀y > 0, x * σ 1 (y) ≥ x * σ 2 (y) (resp. Υ * σ 1 (y) ≤ Υ * σ 2 (y)).
Here and in the proof below, we add in the notation for each mathematical object the volatility function to which it refers. El Karoui and al. [START_REF] El Karoui | Robustness of the Black and Scholes Formula[END_REF] and Hobson [START_REF] Hobson | Volatility Misspecification, Option Pricing and Superreplication via Coupling[END_REF] prove that for a convex payoff function, the price of an American option with finite maturity is a convex function of the underlying spot price. They deduce monotonicity with respect to the local volatility function. Their results imply at the same time the convexity assumption made in the above proposition and its conclusion. In this paper, we prefer to give autonomous proofs of these results in our simple framework. And we will first use proposition 1.9 to compare with the Black-Scholes case where convexity is obvious. We can then deduce (Lemma 3.1) that f ′′ ↓ and f ′′ ↑ are positive for any σ satisfying (H vol ). Proof. Let us consider for example the put case with f ′′ ↓,σ 1 ≥ 0. Let x ≥ z > 0. Ito's formula gives:

de -rt f ↓,σ 1 (S x,σ 2 t ) = e -rt f ′ ↓,σ 1 (S x,σ 2 t )σ 2 (S x,σ 2 t )S x,σ 2 t dW t + e -rt σ 2 2 (S x,σ 2 t ) 2 (S x,σ 2 t ) 2 f ′′ ↓,σ 1 (S x,σ 2 t ) +(r -δ)S x,σ 2 t f ′ ↓,σ 1 (S x,σ 2 t ) -rf ↓,σ 1 (S x,σ 2 t ) dt.
The term between brackets is nonnegative since we have σ 1 ≤ σ 2 and f ↓,σ 1 is a convex function solving [START_REF] El Karoui | Robustness of the Black and Scholes Formula[END_REF]. Therefore we get

E[e -rνn∧τ x,σ 2 z f ↓,σ 1 (S x,σ 2 νn∧τ x,σ 2 z
)] ≥ f ↓,σ 1 (x) using the optional sampling theorem, where

ν n = inf{t ≥ 0, S x,σ 2 t ≥ n} ∧ n. Since ν n → a.s. +∞ and f ↓,σ 1 (S x,σ 2 νn∧τ x,σ 2 z ) is bounded by f ↓,σ 1 (z), Lebesgue's dominated convergence theorem gives then E[e -rτ x,σ 2 z ] ≥ f ↓,σ 1 (x)/f ↓,σ 1 (z) and so E[e -rτ x,σ 2 z ] ≥ E[e -rτ x,σ 1 z ] using Remark 1.1. The same conclusion holds when f ↓,σ 2 is convex by estimating E[e -rτ x,σ 1 z f ↓,σ 2 (S x,σ 1 τ x,σ 1 z )]. We then get the result: if P σ 1 (x, y) > (x -y) + , P σ 1 (x, y) = (y -x * σ 1 (y))E[e -rτ x,σ 1 x * σ 1 (y) ] ≤ (y - x * σ 1 (y))E[e -rτ x,σ 2 x * σ 1 (y) ] = E[(y -S x,σ 2 τ x,σ 2 x * σ 1 (y) ) + e -rτ x,σ 2 x * σ 1 (y) ] ≤ P σ 2 (x, y). Now we just observe that {x > 0, P σ 1 (x, y) > (x -y) + } ⊂ {x > 0, P σ 2 (x, y) > (x -y) + } and thus x * σ 1 (y) = inf{x > 0, P σ 1 (x, y) > (x -y) + } ≥ inf{x > 0, P σ 2 (x, y) > (x -y) + } = x * σ 2 (y).

Framework and notations

We will present in this section the framework that we will consider in all the paper. To clarify the duality, we will use names that implicitly refer either to the primal (or "real") world, or to the dual world. This denomination has no mathematical meaning since, as we will see, there are no difference between them. On the contrary, from a financial point of view, natural variables such as the interest rate, the dividend rate have their true meaning in the primal world, while in the dual world they interchange their role. This is the reason why we also name the primal world "real" world.

The primal ("real") world

The primal world is the framework we just have described. The spot interest rate r is constant and nonnegative, and S t is an asset which pays a constant dividend rate δ ≥ 0 and is driven by a homogeneous volatility function σ : R * + → R * + that satisfies (H vol ) under a risk-neutral measure. The prices of the perpetual American put and call are respectively denoted by P σ (x, y) and C σ (x, y), and their exercise boundary by x * σ and Υ * σ .

The dual world

In the dual world, δ plays the role of the interest rate and r of the dividend rate; x plays the role of the strike and y is the spot value of the share. Let η : R * + → R * + be an homogeneous volatility function that is also assumed to satisfy (H vol ). We consider then (S y t , t ≥ 0) the solution of dS t = S t ((δr)dt + η(S t )dW t ) that starts from y at time 0. Under that model, we denote respectively by

p η (y, x) = sup τ ∈T 0,∞ E e -δτ (x -S y τ ) + and c η (y, x) = sup τ ∈T 0,∞ E e -δτ (S y τ -x) +
the prices of the perpetual put and call with strike x > 0 and spot y. We can define, as in the primal world, g ↓ and g ↑ as the unique decreasing (non-increasing when δ = 0) and increasing positive solution to

1 2 η 2 (x)x 2 g ′′ (x) + (δ -r)xg ′ (x) -δg(x) = 0, (12)
and we name the exercise boundaries ξ * η (x) < x and y * η (x) > x that are respectively associated with the put p η (y, x) and the call c η (y, x).

Notations

The aim of this paper is to put in evidence a duality relation and interpret put (resp. call) prices in the primal world as call (resp. put) prices in the dual world for a specific volatility function η = σ (resp. η = σ). When r = 0 (resp. δ = 0), this is trivial since P σ (x, y) = c η (y, x) = y (resp. C σ (x, y) = p η (y, x) = x) but not really fruitful, and we take thus the following convention in the sequel. Convention 2.1. We will always assume r > 0 (resp. δ > 0) to state properties on P σ and c η (resp. C σ and p η ).

Both worlds being mathematically equivalent, we will work with the put price in the primal world and the call price in the dual world in order not to do the things twice. Following the Convention 2.1, we will consider a positive interest rate. Let us then denote from now:

f = f ↓ and g = g ↑ ,
and define α(y)

= y-x * σ (y) f (x * σ (y)) and β(x) = y * η (x)-x g(y * η (x))
. The functions α and β are positive functions and it follows from the present section that :

∀y > 0, ∀x ≥ x * σ (y), P σ (x, y) = α(y)f (x) (13) ∀x > 0, ∀y ≤ y * η (x), c η (y, x) = β(x)g(y). (14)
That product form will play an important role for the duality. Let us finally introduce notations relative the the Black-Scholes model. We define for ς > 0

a(ς) = δ -r + ς 2 /2 -(δ -r + ς 2 /2) 2 + 2rς 2 ς 2 < 0, b(ς) = r -δ + ς 2 /2 + (δ -r -ς 2 /2) 2 + 2δς 2 ς 2 = 1 -a(ς) > 1,
and we can easily check that f (x) = x a(ς) (resp. g = x b(ς) ) when σ(x) = ς (resp. η(x) = ς).

In that case, the unique solution to (9) (resp. [START_REF] Martini | Propagation of Convexity by Markovian and Martingalian Semigroups[END_REF]) is:

x * ς (y) = a(ς) a(ς) -1 y (resp. y * ς (x) = b(ς) b(ς) -1 x). (15)
With Proposition 1.9, we deduce very useful estimations on the exercise boundaries.

Lemma 2.2. If, ∀x > 0, σ ≤ σ(x) ≤ σ and σ ≤ η(x) ≤ σ, then we have :

a(σ) a(σ) -1 y ≤ x * σ (y) ≤ a(σ) a(σ) -1 y with a(σ) a(σ) -1 < min(1, r/δ) (16) b(σ) b(σ) -1 x ≤ y * η (x) ≤ b(σ) b(σ) -1 x with max(1, δ/r) < b(σ) b(σ) -1 . ( 17
)
Proof. It is straightforward from Proposition 1.9 and (15) to get the bound on the exercise boundaries since x → x γ is convex for γ ∈]0, 1[. We have to show that for ς > 0, a(ς) a(ς)-1 < min(1, r/δ) and b(ς) b(ς)-1 > min(1, δ/r). Since a(ς) < 0 and b(ς) > 1, we get that a(ς)/(a(ς) -1) ∈ (0, 1) and b(ς)/(b(ς)-1) ∈ (1, +∞). We can also check that a(ς)/(a(ς)-1) is a root of the polynomial

Q(X) = δX 2 -(r+δ+ς 2 /2)X +r. As Q(x) = 0 ⇐⇒ σ 2 x = 2(1-x)(r-δx)
and since a(ς)/(a(ς) -1) ∈ (0, 1), we then deduce that a(ς)/(a(ς) -1) < r/δ. In the same way, we have b(ς)/(b(ς) -1) > δ/r.

ODE for the exercise boundary

We have seen previously that the exercise boundaries satisfy

x * σ (y) -y = f (x * σ (y))/f ′ (x * σ (y)) (resp. y * η (x) -x = g(y * η (x))/g ′ (y * η (x))). ( 18
)
We will soon prove that for fixed y > 0 (resp. x > 0), (18) admits a unique solution x * σ (y) (resp. y * η (x)).

Lemma 3.1. The function f ′ (resp. g ′ ) is negative (resp. positive) and f ′′ (resp. g ′′ ) is positive on (0, +∞). Moreover, the boundaries x * σ (y) and y * η (x) are respectively the unique solution to y -

x + f (x)/f ′ (x) = 0 and y -x -g(x)/g ′ (x) = 0. Last, x * σ (y), α(y), y * η (x) and β(x) are C 1 functions on R * + .
Remark 3.2. Positivity of f ′′ and g ′′ and ( 13) and ( 14) imply positivity of ∂ 2 x P σ (x, y) and ∂ 2 y c η (y, x) in the continuation regions.

Differentiating (18) with respect to y (resp. x), one obtains

1 = (x * σ ) ′ (y) f (x * σ (y))f ′′ (x * σ (y)) f ′ (x * σ (y)) 2 (resp. 1 = (y * η ) ′ (y) g(y * η (y))g ′′ (y * η (y)) g ′ (y * η (y)) 2
). Using (18) and equation ( 6) (resp. ( 12)) one deduces the following result (see equation ( 22) below). Proposition 3.3. Let us assume that the volatility functions σ and η satisfy (H vol ). Then, the boundaries x * σ (y) and y * η (x) satisfy the following ODEs:

(x * σ ) ′ (y) = x * σ (y) 2 σ(x * σ (y)) 2 2(y -x * σ (y))(ry -δx * σ (y)) , ( 19
) (y * η ) ′ (x) = η 2 (y * η (x))y * η (x) 2 2(y * η (x) -x)(ry * η (x) -δx) . ( 20
)
Proof of Lemma 3.1. We only give the proof in the put case, the argument being similar for the call. By [START_REF] El Karoui | Robustness of the Black and Scholes Formula[END_REF], for x > 0, f ′′ (x) has the same sign as h(x) = rf (x) + (δr)xf ′ (x).

If for some x > 0, f ′ (x) = 0, then since f is positive, f ′′ (x) > 0. Therefore x is a local minimum point of f which contradicts the decreasing property of this function. Hence f ′ is a negative function. When δ ≤ r, h and therefore f ′′ are positive functions. When δ > r, we remark that if

f ′′ (x) = 0 then h ′ (x) = δf ′ (x) < 0.
Since the continuous function f ′′ and h have the same sign, this implies that

∀x > inf{z > 0 : f ′′ (z) ≤ 0}, f ′′ (x) < 0. ( 21
)
Now for y > 0, by ( 6) then (18) , we have

x * σ (y) 2 σ(x * σ (y)) 2 2 f ′′ (x * σ (y)) f ′ (x * σ (y)) = r f (x * σ (y)) f ′ (x * σ (y)) -(r -δ)x * σ (y) = δx * σ (y) -ry. (22)
By ( 16), the right-hand-side is negative and moreover lim y→+∞ x * σ (y) = +∞. Hence sup{z > 0 : f ′′ (z) > 0} = +∞ and with (21), we conclude that f ′′ > 0.

According to (18), F (x * σ (y), y) = 0 where

F (x, y) = y -x + f (x)/f ′ (x).
The function F is C 1 on (0, +∞) × (0, +∞) and such that

∀x, y > 0, ∂ x F (x, y) = -f (x)f ′′ (x)/f ′ (x) 2 < 0.
Therefore for fixed y > 0, x * (y) is the unique solution to F (x, y) = 0. Moreover, y → x * (y) is C 1 by the implicit function theorem. Last, one deduces from (18) that α(y) is a C 1 function.

The positivity of f ′′ and g ′′ gives the following result.

Corollary 3.4. The comparison result stated in Proposition 1.9 holds for any σ 1 ≤ σ 2 satisfying (H vol ).

Let us now give a uniqueness result for the ODEs ( 19) and (20).

Proposition 3.5. There is only one solution x * σ of (19) (resp. y * η of (20)) defined on (0, +∞) that satisfies ∀y > 0,

c 1 y ≤ x * σ (y) ≤ c 2 y with 0 < c 1 ≤ c 2 < min(1, r/δ) (resp. ∀x > 0, d 1 x ≤ y * η (x) ≤ d 2 x with d 1 > max(1, δ/r)).
Proof. Let us first remark that the uniqueness result for (19) is equivalent to the uniqueness result for (20). Indeed, it is easy to see that x * σ (y) is solution of (19) if and only if ŷ(x) := 1/(x * σ (1/x)) is solution of (20) with the volatility function η(x) = σ(1/x). This new volatility also satisfies (H vol ). Moreover,

d 1 x ≤ ŷ(x) ≤ d 2 x with d 1 > max(1, δ/r) if, and only if 0 ≤ c 1 y ≤ x * σ (y) ≤ c 2 y with 0 < c 1 ≤ c 2 < min(1, r/δ).
Let us suppose then that there are two solutions of (20), y 1 (x) and y 2 (x), that are defined on R + and satisfy d 2 x ≥ y j (x) ≥ d 1 x for some d 2 > d 1 > max(1, δ/r). Since y ′ 1 (x) > 0 for x > 0, y 1 is invertible and we have:

d dx y -1 1 (y 2 (x)) = y 2 (x) 2 η(y 2 (x)) 2 2(y 2 (x) -x)(ry 2 (x) -δx) 2(y 2 (x) -y -1 1 (y 2 (x)))(ry 2 (x) -δy -1 1 (y 2 (x))) y 2 (x) 2 η(y 2 (x)) 2 = (y 2 (x) -y -1 1 (y 2 (x)))(ry 2 (x) -δy -1 1 (y 2 (x)) (y 2 (x) -x)(ry 2 (x) -δx) .
Thus, the function ψ(x) = y -1 1 (y 2 (x))/x solves

ψ ′ (x) = 1 x y 2 (x) -ψ(x)x y 2 (x) -x × ry 2 (x) -δxψ(x) ry 2 (x) -δx -ψ(x) = 1 x 1 - ψ(x) -1 y 2 (x)/x -1 1 - ψ(x) -1 ry 2 (x)/(δx) -1 -ψ(x) . ( 23
)
The estimation d 2 x ≥ y j (x) ≥ d 1 x for j ∈ {1, 2} with d 1 > max(1, δ/r) implies that :

∃A > 0, ∀x > 0, 1/A ≤ ψ(x) ≤ A, ( 24
) ∀x > 0, ψ(x) < min y 2 (x) x , ry 2 (x) δx , y 2 (x) x -1 > 0 and ry 2 (x) δx -1 > 0. ( 25
)
Since local uniqueness holds for (23) by the Cauchy Lipschitz theorem, the only solution ϕ such that ϕ(1) = 1 is the constant ϕ ≡ 1. Therefore checking that (24) does not hold for solutions ϕ satisfying (25) and such that ϕ(1) = 1 is enough to conclude that ψ ≡ 1.

Let ϕ be a solution to (23) satisfying (25). If ϕ(1) > 1, by local uniqueness for (23), for all x ∈ R * + , ϕ(x) > 1. By (25), one deduces that for all

x ∈ R * + , ϕ ′ (x) < 1-ϕ(x)
x < 0. Therefore, ϕ ′ (x) ≤ (1ϕ(1))/x for x ∈ (0, 1], and we have

ϕ(x) ≥ ϕ(1) + (1 -ϕ(1)) ln(x) → x→0 +∞
which is contradictory to (24). In the same manner, if ϕ(1) < 1, ϕ(x) < 1 for x ∈ R * + and ϕ is strictly increasing. In particular, for

x ≤ 1, ϕ ′ (x) ≥ (1 -ϕ(1))/x and therefore ϕ(1) -ϕ(x) ≥ (1 -ϕ(1)) ln(1/x) → x→0
+∞ and this yields another contradiction.

Corollary 3.6. Let us denote C = {f ∈ C 1 (R * + ), s.t. f (0) = 0, ∃0 < a < b, ∀x ≥ 0, a ≤ f ′ (x) ≤ b}. The application σ → x * σ (resp. η → y * η ) is one-to-one between the set {σ ∈ C(R * + ) that satisfies (H vol )} and the set of function Cx = {x ∈ C, s.t. ∃0 < c 1 ≤ c 2 < min(1, r/δ), ∀y > 0, c 1 y ≤ x(y) ≤ c 2 y} (resp. Cy = {y ∈ C, s.t. ∃ max(1, δ/r) < d 1 ≤ d 2 , ∀x > 0, d 1 x ≤ y(x) ≤ d 2 x}.) Proof.
σ(x * (y)) = 2(y -x * (y))(ry -δx * (y))x * ′ (y) x * (y) (26)
is well defined thanks to the hypothesis made on x * . As x * σ satisfies (16) and solves the same ODE (19) as x * , we have x * ≡ x * σ using Proposition 3.5. The proof for η → y * η is the same and gives incidentally the expression of η in function of the exercise boundary y * (x):

η(y * (x)) = 2(y * (x) -x)(ry * (x) -δx)y * ′ (x) y * (x) . ( 27
)
4 The call-put duality

This section is devoted to the key result of the paper : for related local volatility functions σ and η, we can interpret a put price in the primal world as a call price in the dual world.

The main result

Theorem 4.1 (Duality). The following conditions are equivalent:

1. ∀x, y > 0, P σ (x, y) = c η (y, x).

(28)

2.

x * σ and y * η are reciprocal functions: We have on the one hand P σ (x, y) = yx on {(x, y), x ≤ x * σ (y)} and P σ (x, y) > yx outside, and on the other hand c η (y, x) = yx on {(x, y), y ≥ y * η (x)} and c η (y, x) > yx outside. The duality relation (28) imposes then that {(x, y), x ≤ x * σ (y)} = {(x, y), y ≥ y * η (x)} and so y * η (x * σ (y)) = y. 2 =⇒ 3, 4 : Taking the derivative of the last relation, we get thanks to (19) and (20) 16)), one concludes by Proposition 3.5.

∀x > 0, x * σ (y * η (x)) = x. 3. η ≡ σ where σ(y) = 2(y -x * σ (y))(ry -δx * σ (y)) yx * σ (y)σ(x * σ (y)) . ( 29
)
4. σ ≡ η ˜where η ˜(x) = 2(y * η (x) -x)(ry * η (x) -δx) y * η (x)xη(y * η (x)) . ( 30
)
x * σ (y) 2 σ(x * σ (y)) 2 2(y-x * σ (y))(ry-δx * σ (y)) η 2 (
2 =⇒ 1 : The equality (28) is clear in the exercise region since {(x, y), x ≤ x * σ (y)} = {(x, y), y ≥ y * η (x)}. Let us check that it also holds in the continuation region. Using the product form [START_REF] Villeneuve | On the Threshold strategies for Optimal Stopping arising in Real Option Theory[END_REF], and the smooth-fit principle (Theorem 1.6) we get for all y ∈ R *

+ y -x * σ (y) = β(x * σ (y))g(y) 1 = -β(x * σ (y))g ′ (y).
Differentiating the first equality with respect to y, one gets 1-x * σ (y) ′ = x * σ (y) ′ β ′ (x * σ (y))g(y)+ β(x * σ (y))g ′ (y), which combined with the second equality gives

-1 = β ′ (x * σ (y))g(y).
Dividing by the first equality and using (18), one deduces

β ′ β (x * σ (y)) = f ′ f (x * σ (y)). Since x * σ : R * + → R * + is a bijection, there is a constant C = 0 such that β ≡ Cf . Since ∀y > 0, α(y)f (x * σ (y)) = y -x * σ (y) = β(x * σ (y))g(y)
, one has α ≡ g/C. From ( 13) and ( 14), one concludes that (28) holds.

In this proof, we have shown that α is proportional to g, and so there is a constant C > 0 such that P σ (x, y) = Cf (x)g(y) for x ≤ x * σ (y). In the Black-Scholes' case, we have P σ (x, y) = Cx a(σ) y b(σ) for x ≤ x * σ (y), and we are able to calculate C using the boundary condition P σ (x * σ (y), y) = yx * σ (y). We get that C = b(σ) -b(σ) /(-a(σ)) a(σ) . We retrieve then the already known analytical formulae for the put prices (e.g. Gerber and Shiu [START_REF] Gerber | Martingale Approach to Pricing Perpetual American Options[END_REF]).

An analytic example of dual volatility functions

σ(y) = 2(y -x * σ (y))(ry -δx * σ (y))y * ′ (x * σ (y)) y .
Let us consider the family of exercise boundaries

y * (x) = x x + a bx + c
where a, b, c are positive constants such that max(c/a, b) < min(1, r/δ) (condition ensuring y * ∈ Cy ). Since y * (x) ′ = (bx 2 + 2cx + ac)/(bx + c) 2 , one has

σ(x) = 2 ((r -δb)x + ra -δc)((1 -b)x + a -c) bx 2 + 2cx + ac , x > 0.
Moreover, the function x * σ (y) is the only positive root of the polynomial function: X 2 + X(aby)cy, that is:

x * σ (y) = 1 2 by -a + (by -a) 2 + 4cy
and

∀y > 0, σ(y) = 2(y -x * σ (y))(ry -δx * σ (y))(bx * σ (y) 2 + 2cx * σ (y) + ac) y(bx * σ (y) + c) .
This example enables us to check numerically the duality. We have plotted in Figure 1, the prices of an American put P σ (T, x, y) in the primal world for the local volatility σ(x) and an American call c σ (T, y, x) in the dual world for the local volatility σ(x) as functions of the maturity T . These prices have been computed using a finite difference method. We can see at T = 10 that the limit value is quite reached and both prices are equal. The plots are nonetheless distinct which means that the duality does not hold for finite maturities. We have also plotted, in function of T , C σ (T, x, y) in the primal world and p σ (T, y, x) in the dual world to check numerically whether the volatility function σ is such as C σ (x, y) = p σ (y, x).

As we can see, the curves do not seem to converge toward the same limit when T is large. This means that the volatility function σ such that ∀x, y > 0, C σ (x, y) = p σ(y, x) (obtained from σ as η ˜is obtained from η but with exchange of r and δ) is different from σ.

5 Consequence of the duality : A (theoretical) method of calibration for the volatility σ(x)

In that section, we will put in evidence the importance of the duality within the calibration scope. We suppose for this that we are on a (virtual) market where are traded perpetual securities, and where the short interest rate r and the dividend rate δ can be observed.

This means that we know the price of the share x 0 , and the market quotes on that share the perpetual American puts and calls for all strikes K > 0. We name respectively p(K) and c(K) these prices and denote:

X = sup{K > 0, c(K) = x 0 -K} and Y = inf{K > 0, p(K) = K -x 0 }. ( 31
)
We will first suppose that the put and call prices derive from a time-homogeneous local volatility model before relaxing this assumption.

The calibration procedure

Let us assume that there is a volatility function σ satisfying (H vol ) such that for all K > 0, p(K) = P σ (x 0 , K) and c(K) = C σ (x 0 , K). The following proposition says that these prices characterize σ and its proof gives a constructive way to retrieve the volatility function from the prices.

Proposition 5.1. Let us consider x 0 > 0. The map

σ → ((P σ (x 0 , K), C σ (x 0 , K)), K > 0)
is one-to-one on the set of volatility functions satisfying (H vol ).

Proof. We first consider the put case. The differential equation satisfied by the put prices in the continuation region makes only appear the values and the derivatives in x, K being fixed. Hence, we cannot exploit directly the prices. But the duality relation enables to get a differential equation in the strike variable. Thanks to the Duality Theorem, we have P σ (x 0 , K) = c σ (K, x 0 ) for some σ satisfying (H vol ). It is then easy to calibrate σ(.). Indeed, one has K 2 σ(K) 2 2 p ′′ (K) + K(δr)p ′ (K)δp(K) = 0 for K < Y = y * σ (x 0 ). Since the differential equation is valid only for K < Y , we only get σ on (0, Y ] by continuity:

∀K ≤ Y, σ(K) = 1 K 2(δp(K) + K(r -δ)p ′ (K)) p ′′ (K) which is well defined since p ′′ (K) = ∂ 2 K c σ (K, x) > 0 (Remark 3.2)
. Then, we can calculate the exercise boundary y * σ(x), for x ∈ (0, x 0 ], solving (20) supplemented with the final condition y * σ (x 0 ) = Y backward. This step only requires the knowledge of σ only on the interval (0, Y ]. Finally, we can recover the desired volatility σ(x) for x ≤ x 0 thanks to (29):

∀x ∈ (0, x 0 ], σ(x) = 2(y * σ(x) -x)(ry * σ(x) -δx) xy * σ (x)σ(y * σ(x)) . ( 32
)
Now let us consider the calibration to the call prices. This relies on the same principle, but we have to be careful because the Duality Theorem is stated given to the call interest rate δ and dividend rate r. So we have to interchange these variables when we apply that theorem. There is a function σ satisfying (H vol ) such that : ∀K > 0, C σ (x 0 , K) = p σ (K, x 0 ). We have

1 2 K 2 σ(x) 2 c ′′ (K) + (δ -r)Kc ′ (K) -δc(K) = 0 for K > X = ξ * σ (x 0 ). Thus, we get ∀K ≥ X, σ(K) = 1 K 2(δc(K) + K(r -δ)c ′ (K)) c ′′ (K)
which is well defined for analogous reasons. We can then obtain as before the exercise boundary solving (19) forward

∀y ≥ x 0 , ξ * σ(y) ′ = ξ * σ(y) 2 σ(ξ * σ(y)) 2 2(y -ξ * σ (y))(δy -rξ * σ(y))
, ξ * σ(x 0 ) = X and we finally get the volatility σ(y) for y ≥ x 0 using the Duality Theorem. More precisely, we interchange r and δ in (29) to get

σ(y) = 2(y -ξ * σ(y))(δy -rξ * σ(y)) yξ * σ(y)σ(ξ * σ(y)) . ( 33
)
This calibration method, although being theoretical, sheds light on a striking and interesting result: the perpetual American put prices only give the restriction of σ(x) to (0, x 0 ] and the call prices only the restriction of σ(x) to [x 0 , +∞). This has the following economical interpretation : long-term American put prices mainly give information on the downward volatility while long-term American call prices give information on the upward volatility. This dichotomy is remarkable. In comparison, according to Dupire's formula [START_REF] Dupire | Pricing with a smile[END_REF], there is no such phenomenon for European options : the knowledge of the call prices gives the whole local volatility surface, not only one part. In other words, the European call and put prices give the same information on the volatility while the perpetual American call and put prices give complementary information.

Thus, one may think that the perpetual American call and put prices only depend on a part of the volatility curve. This is precised by the Proposition below that gives necessary and sufficient conditions on the volatility functions to observe the same put prices (resp. call prices). Proposition 5.2. Let us consider x 0 > 0 and σ 1 (.), σ 2 (.) two volatility functions satisfying (H vol ). Then, the following properties are equivalent:

(i) ∀y > 0, P σ 1 (x 0 , y) = P σ 2 (x 0 , y) (resp. ∀y > 0, C σ 1 (x 0 , y) = C σ 2 (x 0 , y)) (ii) ∀y ≤ y * σ2 (x 0 ), σ1 (y) = σ2 (y). (resp. ∀x ≥ ξ * σ1 (x 0 ), σ1 (x) = σ2 (x)
where σj denotes the local volatility function such that ∀x, y > 0, C σ j (x, y) = p σj (y, x).)

(iii) ∀x ∈ (0, x 0 ], σ 1 (x) = σ 2 (x) and y * σ1 (x 0 ) = y * σ2 (x 0 ). (resp. ∀x ∈ [x 0 , +∞), σ 1 (x) = σ 2 (x) and ξ * σ1 (x 0 ) = ξ * σ2 (x 0 ).) (iv) ∀x ∈ (0, x 0 ], σ 1 (x) = σ 2 (x) and f ′ ↓,σ 1 (x 0 ) f ↓,σ 1 (x 0 ) = f ′ ↓,σ 2 (x 0 ) f ↓,σ 2 (x 0 ) . (resp. ∀x ∈ [x 0 , +∞), σ 1 (x) = σ 2 (x) and f ′ ↑,σ 1 (x 0 ) f ↑,σ 1 (x 0 ) = f ′ ↑,σ 2 (x 0 ) f ↑,σ 2 (x 0 ) .) (v) f ↓,σ 1 and f ↓,σ 2 (resp. f ↑,σ 1 and f ↑,σ 2 ) are proportional on (0, x 0 ] (resp. [x 0 , +∞)). (vi) ∀x ≤ x 0 , ∀y > 0, P σ 1 (x, y) = P σ 2 (x, y) (resp. ∀x ≥ x 0 , ∀y > 0, C σ 1 (x, y) = C σ 2 (x, y)).
Remark 5.3.

• Among these many conditions, let us remark that condition (ii) on the dual volatility is much simpler than condition (iii) on the primal volatility since the latter requires the equality of the dual exercise boundaries at x 0 .

• When δ = 0, according to Remark 1.3, in the put case, condition (iv) also writes ∀x ∈ (0,

x 0 ], σ 1 (x) = σ 2 (x) and +∞ x 0 1 v 2 exp - v x 0 2r uσ 2 1 (u) du dv = +∞ x 0 1 v 2 exp - v x 0 2r uσ 2 2 (u)
du dv.

• Since, by definition of f ↓,σ j (resp. f ↑,σ j ) and the strong Markov property

∀0 < z ≤ x, E[e -rτ x σ j ,z ] = f ↓,σ j (x)/f ↓,σ j (z) (resp. ∀0 < x ≤ z, E[e -rτ x σ j ,z ] = f ↑,σ j (x)/f ↑,σ j (z)), the probabilistic counterpart of assertion (v) is ∀0 < z ≤ x ≤ x 0 , E[e -rτ x σ 1 ,z ] = E[e -rτ x σ 2 ,z ] (resp. ∀x 0 ≤ x ≤ z, E[e -rτ x σ 1 ,z ] = E[e -rτ x σ 2 ,z ]).
Proof. We consider for example the put case. (i) =⇒ (ii) : See the proof of Theorem 5.1.

(ii) =⇒ (iii) : Let us define ψ(x) = (y * σ1 ) -1 (y * σ2 (x))/x. We can show as in the proof of Proposition 3.5 that ψ(x 0 ) = 1 and then ψ ≡ 1 on (0, x 0 ], otherwise it would go to 0 or +∞ when x → 0, which is not possible thanks to (17). We get then ∀x ∈ (0, x 0 ], σ 1 (x) = σ 2 (x) using (30) that express σ j in function of y * σj and σj , j ∈ {1, 2}. (iii) =⇒ (iv) : Thanks to (18) and Theorem 4.1, we have

f ′ ↓,σ 1 (x 0 ) f ↓,σ 1 (x 0 ) = -1 y * σ1 (x 0 )-x 0 = -1 y * σ2 (x 0 )-x 0 = f ′ ↓,σ 2 (x 0 ) f ↓,σ 2 (x 0 ) . (iv) =⇒ (v) : The set of solutions to 1 2 σ 2 1 (x)x 2 f ′′ (x) + (r -δ)xf ′ (x) -rf (x) = 0 on (0, x 0 ]
is a two-dimensional vector space, but thanks to the relation

f ′ ↓,σ 1 (x 0 ) f ↓,σ 1 (x 0 ) = f ′ ↓,σ 2 (x 0 ) f ↓,σ 2 (x 0 ) , f ↓,σ 1 and f ↓,σ 2 are proportional on (0, x 0 ] : ∀x ≤ x 0 , f ↓,σ 1 (x) = f ↓,σ 1 (x 0 ) f ↓,σ 2 (x 0 ) f ↓,σ 2 (x). (34)
(v) =⇒ (vi) : The proportionality implies that ∀x ∈ (0, x 0 ], x) , and then (y * σ1 (x)x) -1 = (y * σ2 (x)x) -1 using (18) and Theorem 4.1. Therefore ∀x ∈ (0, x 0 ], y * σ1 (x) = y * σ2 (x).

f ↓,σ 1 (x) ′ f ↓,σ 1 (x) = f ↓,σ 2 (x) ′ f ↓,σ 2 (
We have [START_REF] Peskir | A note on the call-put parity and a call-put duality[END_REF], and obtain from (34) that

α σ 1 (y * σ1 (x))f ↓,σ 1 (x) = α σ 2 (y * σ2 (x))f ↓,σ 2 (x) using
∀x ≤ x 0 , ∀y ≤ y * σ1 (x 0 ), α σ 1 (y) = f ↓,σ 2 (x 0 ) f ↓,σ 1 (x 0 ) α σ 2 (y) = f ↓,σ 2 (x) f ↓,σ 1 (x) α σ 2 (y). (35)
Thus, we deduce from ( 13), ( 34) and (35) the equality of the put prices for the low strikes ∀x ≤ x 0 , ∀y ≤ y * σ1 (x), P σ 1 (x, y) = P σ 2 (x, y).

For y > y * σ1 (x) = y * σ2 (x), the equality is clear since both prices are equal to yx. (vi) =⇒ (i) : clear.

Let us observe that the point (ii) of the last proposition allows to exhibit different volatility functions with analytic expressions that give the same put (or call) prices. Let us consider the same family as in subsection 4.2 coming from the call exercise boundary y * 1 (x) = x x+a bx+c (assuming a, b, c > 0 and max(c/a, b) < min(1, r/δ)). For x 0 > 0, we introduce the exercise boundary:

y * 2 (x) = y * 1 (x) for x ≤ x 0 and y * 2 (x) = y * 1 (x 0 ) + (y * 1 ) ′ (x 0 )(x -x 0 ) for x ≥ x 0 .
The condition (y * 2 ∈ Cy ) is satisfied provided that (y * 1 ) ′ (x 0 ) > max(1, δ/r). This is automatically ensured by the assumptions made on a, b, c since (y * 1 )

′ (x 0 ) = (bx 2 0 + 2cx 0 + ac)/(b 2 x 2 0 + 2bcx 0 + c 2 ).
That family is such that σ1 (y) = σ2 (y) for y ≤ y * 2 (x 0 ). We can then calculate σ 2 as in subsection 4.2 using the relation

σ 2 (x) = √ 2(ry * 2 (x)-δx)(y * 2 (x)-x) x √ y * 2 (x) ′
. This gives σ 2 (x) = σ(x) for x ≤ x 0 and for x ≥ x 0 ,

σ 2 (x) = 2 [(r(y * 1 ) ′ (x 0 ) -δ)x + r(y * 1 (x 0 ) -x 0 (y * 1 ) ′ (x 0 ))][((y * 1 ) ′ (x 0 ) -1)x + y * 1 (x 0 ) -x 0 (y * 1 ) ′ (x 0 )] x 2 (y * 1 ) ′ (x 0 )
.

In Figure 2, we have plotted the same example as in Figure 1 (x = 0.5 and y = 0.4), adding the graph of T → P σ 2 (T, x, y). The volatility function σ 2 has been calculated with the formula above with x 0 = 0.5. According to Proposition 5.2 and the Duality, the three prices are equal when T is large. In the second example (x = 3 and y = 1), we still observe that P σ (T, x, y) and c σ (T, y, x) converge toward the same value when T is large. On the contrary, the limit price of P σ 2 (T, x, y) is significantly different. To observe the same price, we should have taken, according to Proposition 5.2, x 0 ≥ 3.

Calibration to "real" call and put prices

In that subsection, we address some problems that arise if one tries to apply the calibration procedure when the prices p(K) and c(K) do not derive from a time-homogeneous model. We assume however that they are smooth functions of the strike K, and focus for example on the calibration to put prices. Firstly, let us observe that the arbitrage-free theory allows to define a dual volatility as previously by (0, Y ]:

∀K < Y, η p (K) = 1 K 2(δp(K) + K(r -δ)p ′ (K)) p ′′ (K) . (36)
Indeed, the payoff convexity in K ensures the positivity of p ′′ (K) and the arbitrage-free assumption ensures that δp(K) + K(rδ)p(K) is nonnegative, so that the square-root is well defined. Let us prove the last point and suppose the contrary (i.e. ∃y > 0 such that d dy e δy p(e (r-δ)y ) < 0) to exhibit an arbitrage opportunity. In that case, there is z > y such that e δy p(e (r-δ)y ) > e δz p(e (r-δ)z ). We then sell one put with strike e (r-δ)y and buy e δ(z-y) puts with strike e (r-δ)z . This initial transaction generates a positive flow. The hedging works as follows: naming τ the time at which the put sold is exercised, we have to pay e (r-δ)y -S τ . In other words, we receive one share and borrow e (r-δ)y in cash. We keep this position until time τ + zy. At this time, we have exactly e δ(z-y) shares and puts with strike e (r-δ)z . Thus, we obtain at least e δ(z-y) e (r-δ)z = e (r-δ)y e r(z-y) and we cancel the debt. The next proposition gives sufficient conditions that allow to construct an homogeneous volatility which is consistent to the observed prices.

Proposition 5.4. Let us assume that

K ∈ R * + → p(K) is a C 1 function, C 2 on R * + -{Y } with Y = inf{K > 0 : p(K) = K -x 0 } < +∞.
Let us also assume that η p defined by (36) is bounded from below and above by two positive constants and admits a left-hand limit in Y . Then, if we extend η p in any continuous function on (0, +∞) satisfying (H vol ) still denoted by η p , we have ∀K > 0, P η p ˜(x 0 , K) = p(K). Notice that once we choose the extended function η p , we obtain η p ˜by first solving (20) on R * + starting from x 0 with the condition y * ηp (x 0 ) = Y and then using (30). Proof. The functions K → p(K) and K → c ηp (K, x 0 ) solve [START_REF] Myneni | The pricing of the American option[END_REF]. Since we have 0 ≤ p(K) ≤ K for arbitrage-free reasons, both functions go to 0 when K → 0. Thanks to Remark 1.2, they are proportional to g ↑ and therefore there is λ > 0 such that: For the call case, everything works in the same manner, but we need to assume moreover that c(K) → 0 when K → +∞. This is a rather natural hypothesis that plays the same role as p(K) → 0 when K → 0. Proposition 5.5. Let us assume that K ∈ R * + → c(K) is a C 1 function, C 2 on R * + -{X} with X = sup{K > 0, c(K) = x 0 -K} > 0 and lim K→+∞ c(K) = 0. Let us also assume that η c defined by

∀K ≤ Y, p(K) = λc ηp (K, x 0 ).
∀K > X, η c (K) = 1 K 2(δc(K) + K(r -δ)c ′ (K)) c ′′ (K)
is bounded from below and above by two positive constants and admits a right-hand limit in X. Then, if we extend η c in any continuous function on (0, +∞) satisfying (H vol ) still denoted by η c , we have ∀K > 0, C ηc ˇ(x 0 , K) = c(K) where η c ˇis obtained from η c like σ from σ in the end of the proof of Proposition 5.1. Therefore, we are able to find volatility functions that give exactly the put prices and others that give exactly the call prices. Now, the natural question is whether one can find a volatility function σ that is consistent to both the put and call prices. According to Proposition 5.2, all the volatility functions η p ˜(resp. η c ˇ) giving the put (resp. call) prices coincide on (0, x 0 ) (resp. (x 0 , +∞)). The only volatility function possibly giving both the put and call prices is

σ(x) =    η p ˜(x) if x < x 0 η c ˇ(x) if x > x 0 .
We deduce from Proposition 5.2 :

Proposition 5.6. Assume that η p ˜(x - 0 ) = η c ˇ(x + 0 ). Then, ∀K > 0, p(K) = P σ (x 0 , K) and c(K) = C σ (x 0 , K) iff x * σ (Y ) = x 0 and Υ * σ (X) = x 0 .
6 The Black-Scholes model: the unique model invariant through this duality

The purpose of that section is to put in evidence the particular role played by the Black-Scholes' model for the perpetual American call-put duality. We have recalled in the introduction that constant volatility functions are invariant by the duality. We have also mentioned that for the European case, the call-put duality holds for all maturities without any change of the volatility function. Here, on the contrary, we are going to prove that if the duality holds for the perpetual American options with the same volatility: ∀x, y > 0 P σ (x, y) = c σ (y, x)

then, under some technical assumptions, necessarily σ(.) is a constant function.

Proposition 6.1. Let us consider a positive interest rate r and a nonnegative dividend rate δ < r. We suppose that the volatility function σ satisfies (H vol ), and is analytic in a neighborhood of 0, i.e.

∃ρ > 0, ∀x ∈ [0, ρ), σ(x) = ∞ k=0 σ k x k . (38)
Then, (37) holds if and only if ∀x ≥ 0, σ(x) = σ 0 .

We have already shown in the introduction that (37) holds in the Black-Scholes' case. So we only have to prove the necessary condition. We decompose the proof into the three following lemmas. Lemma 6.2. Let us consider a volatility function that satisfies (H vol ). If the dual volatility function σ is analytic in a neighborhood of 0, then the boundaries x * σ and y * σ are also analytic in a neighborhood of 0. Lemma 6.3. Let us suppose that σ satisfies (H vol ) and is analytic in a neighborhood of 0. Let us assume moreover that r > δ. If the equality (37) holds, σ is constant in a neighborhood of 0: ∃ρ > 0, ∀y ∈ [0, ρ], σ(y) = σ 0 . Lemma 6.4. Let us suppose that σ is a constant function on [0, ρ] for ρ > 0 satisfying (H vol ) and (37). Then, σ is constant on R + (and x * σ and y * σ are linear functions).

Proof of Lemma 6.2. Let us first show that x * σ is analytic in 0. Thanks to the relation (18), we have

g(y * σ (x)) g ′ (y * σ (x)) = y * σ (x) -x, and therefore g(y) g ′ (y) = y -x * σ (y). Thus, x * σ (y) is analytic in 0 iff φ(y) = g(y)
g ′ (y) is analytic in 0. Using the relation [START_REF] Myneni | The pricing of the American option[END_REF] and

φ ′ = 1 -g ′′ g ′ φ, we get that φ is solution of φ ′ (y) = 1 + 2 σ2 (y) (δ -r)φ(y)/y -δ(φ(y)/y) 2 . ( 39
)
Notice that φ(y) = yx * σ (y) and ( 16) imply that if φ is analytic in 0 then the coefficient of order 0 in its expansion vanishes and the coefficient of order 1 belongs to (0, 1).

To complete the proof we are first going to check that if ψ(y) = ∞ k=1 φ k y k with φ 1 ∈ (0, 1) solves (39) in a neighborhood of 0 then φ ≡ ψ in this neighborhood. Then we will prove existence of such an analytic solution ψ. We have ψ(0) = 0, and the function ψ being analytic with φ 1 = 0, its zeros are isolated points. There is therefore a neighborhood of 0, (0, 2ǫ) where ψ does not vanish. Let us consider γ a solution of γ ′ -1 ψ γ = 0 starting from γ(ǫ) = 0 in ǫ : γ(x) = γ(ǫ) exp

x ǫ 1 ψ(u) du . Since ψ solves (39), it is not hard to check that γ is solution of [START_REF] Myneni | The pricing of the American option[END_REF] with η = σ. The limit condition γ(x) → Thus we have γ(y) = cg(y) with c = 0 and ψ(y) = g(y)/g ′ (y) = φ(y) = yx * σ (y). We can then write x * σ (y) = (1φ 1 )y -∞ k=2 φ k y k in the neighborhood of 0 with 1φ 1 > 0. It is well-known that in that case, the reciprocal function y * σ is also analytic in 0. Let us turn to the existence of ψ. Since σ 0 ≥ σ > 0, y → 2 σ2 (y) is an analytic function in the neighborhood of 0. Thus, there is ρ 0 > 0 and a 0 > 0 such that

∀y ∈ [0, ρ 0 ], 2 σ2 (y) = ∞ k=0 a k y k and ∞ k=0 |a k |ρ k 0 < ∞.
The analytic function k≥1 φ k y k solves (39) if and only if

∞ k=0 (k + 1)φ k+1 y k = 1 + (δ -r) ∞ k=0 i+j=k a i φ j+1 y k -δ ∞ k=0 i+j+l=k a i φ j+1 φ l+1 y k .
Identifying the terms of order 0, we get that φ 1 solves P (φ 1 ) = 0 where P (x) = δa 0 x 2 + (1 -(δr)a 0 )x -1. Since P (0) = -1 < 0 and P (1) = ra 0 > 0, the polynomial P admits a unique root on (0, 1) and we choose φ 1 equal to this root. Then, by identification of the terms with order k, we define the sequence (φ k ) k≥1 inductively by

φ k+1 = (δ -r) i+j=k,j =k a i φ j+1 -δ i+j+l=k,j =k,l =k a i φ j+1 φ l+1 k + 1 + (r -δ)a 0 + 2δa 0 φ 1 .
This ratio is well defined since (rδ)a 0 + 2δa 0 φ 1 = δa 0 φ 1 + 1/φ 1 -1 > 0.

We still have to check that the series k≥1 φ k y k is defined in a neighborhood of 0. To do so, we are going to show that there is ρ > 0 such that the sequence (|φ k |ρ k ) k≥1 is bounded.

We have for 1 ≤ k ≤ n:

|φ k+1 |ρ k ≤ |δ -r| k-1 j=0 |a k-j |ρ k-j |φ j+1 |ρ j + δ k i=0    j+l=k-i j =k,l =k |φ j+1 |ρ j |φ l+1 |ρ l    |a i |ρ i k + 1 .
Let us suppose that for 1 ≤ j < k, |φ j+1 |ρ j ≤ 1/(j + 1). Then,

|φ k+1 |ρ k ≤ |δ -r|ρ k j=1 |a j |ρ j-1 + δ k i=0 j+l=k-i 1 j+1 1 l+1 |a i |ρ i k + 1 .
We remark that j+l=k-i

1 j+1 1 l+1 = 1 k-i+2 j+l=k-i 1 j+1 + 1 l+1 ≤ 2 ln(k-i+1)+1 k-i+2
, and we finally get:

|φ k+1 |ρ k ≤ 2δ|a 0 | ln(k+1)+1 k+2 + ρ(|δ -r| + 2δ) k j=1 |a j |ρ j-1 k + 1 (40) since ln(k-i+1)+1 k-i+2 ≤ 1. Let us now consider k 0 such that ∀k ≥ k 0 , 2δ|a 0 | ln(k+1)+1
k+2 < 1/2. Now, we chose ρ ∈ (0, ρ 0 ) small enough such that ∀k ≤ k 0 , |φ k+1 |ρ k ≤ 1/(k + 1) and ρ(|δ -r| + 2δ) ∞ j=1 |a j |ρ j-1 < 1/2. Then we get by induction from (40) that ∀k ≥ k 0 , |φ k+1 |ρ k ≤ 1/(k + 1).

Proof of Lemma 6.3. On the one hand, thanks to the assumption, σ = σ is analytic in 0, and therefore x * σ is analytic in 0 thanks to Lemma 6.2:

∃ρ > 0, ∀y ∈ [0, ρ), x * σ (y) = ∞ i=1
x i y i and σ(y) = ∞ i=0 σ i y i .

On the other hand, it is not hard then to deduce from (29), σ = σ and the differential equation (19) that

x * σ (y) ′ = 2(yx * σ (y))(ryδx * σ (y)) y 2 σ(y) 2 .

(41)

From Corollary 3.6 and (26), we get

x * σ (y) ′ = (yx * σ (y))(ryδx * σ (y))((x * σ ) -1 ) ′ (y) ((x * σ ) -1 (y)y)(r(x * σ ) -1 (y)δy) .

(42)

Now, we consider n = inf{i ≥ 2, x i = 0} and suppose it finite. We can get easily that:

x * σ (y) = x 1 y + x n y n + . . . x * σ (y) ′ = x 1 + nx n y n-1 + . . . (1x 1 )(rδx 1 ) +

x n

x n 1 ((r + δ)x 1 -2r)y n-1 + . . .

The right hand side of (42) has then the following expansion:

x 1 1 + x n (1 -x 1 )(r -δx 1 ) 2δx 1 -(r + δ) + 2r x n 1 - r + δ x n-1 1 y n-1 - nx n x n 1 y n-1 + . . .
The equality of the terms of order n -1 in (42) then leads to:

nx n x n-1 1 = x n (1 -x 1 )(r -δx 1 )
2δx n+1

In the case δ = 0 this gives n(1 + x n-1 1 ) = x n-1 1 + • • • + x 1 + 2 which is not possible because x 1 ∈ (0, 1). When 0 < δ < r, we denote α = r/δ > 1 and rewrite (43):

n(1 + x n-1 1 )(α -x 1 ) = -2x n 1 + (α -1)x n-1 1 + • • • + (α -1)x 1 + 2α = α -x n 1 + (α -x 1 ) 1 -x n 1 1 -x 1 .
Therefore, n(1

+ x n-1 1 ) = α-x n 1 α-x 1 + 1-x n 1 1-x 1 < 2 1-x n 1 1-x 1 because β → β-x n 1
β-x 1 is decreasing on (1, α) (x n 1 < x 1 ). To show that this is impossible, we consider P n (x) = n(1 + x n-1 ) -2 n-1 k=0 x k . We have P n (1) = 0 and for x < 1, P ′ n (x) = n(n-1)x n-2 -2 n-1 k=1 kx k-1 = 2 n-1 k=1 k(x n-2x k-1 ) < 0. Thus P n is positive on [0, 1) and P n (x 1 ) > 0 which is a contradiction.

Conclusions and further developments

Addressing Call-Put duality for American options with finite maturity in models with timedependent local volatility functions like (1) would be of great interest. For the perpetual case treated in this paper, we could take advantage of a very nice feature : in the continuation region, the price of the option writes as the product of a function of the underlying spot price by another function of the strike price. Unfortunately, this product property no longer holds in the general case.

Next, according to our numerical experiments (see figure 2), American Put and Call prices computed in infinite maturity dual models may differ for finite maturities. This means that in the case of a time-homogeneous primal local volatility function ς(t, x) = σ(x), if there exists a dual local volatility function for some finite maturity T , then this volatility function is either time-dependent or depends on the maturity T . On the contrary, in the European case presented in the introduction, time-homogeneous volatility functions are preserved by the duality.

Let us nevertheless conclude on an encouraging remark. Let P (T, x, y) denote the initial price of the American Put option with maturity T and strike y in the model (1) and x * (T, y) stand for the corresponding exercise boundary such that P (T, x, y) = (yx) + if and only if x ≤ x * (t, y). Then the smooth-fit principle writes P (T, x * (T, y), y) = yx * (T, y) ∂ x P (T, x * (T, y), y) = -1 .

Differentiating the former equality with respect to y yields ∂ x P (T, x * (T, y), y)∂ y x * (T, y) + ∂ y P (T, x * (T, y), y) = 1 -∂ y x * (T, y).

With the second equality, one deduces that ∂ y P (T, x * (T, y), y) = 1. Therefore the smoothfit principle automatically holds for the dual Call option if there exists any.

  By (30) and (20), if y * ∈ Cy (where Cy is defined in Corollary 3.6), then the reciprocal function of y * is the put exercise boundary x * σ associated to the local volatility functionσ(x) = 2(ry * (x)δx)(y * (x)x) x y * (x) ′ .Now by (27), y * is the call exercise boundary associated with the dual volatility function :

Figure 1 :

 1 Figure 1: P σ (T, x, y) and c σ (T, y, x), and C σ (T, x, y) and p σ(T , y, x) as functions of the maturity T for x = 0.5, y = 0.4, r = 0.2, δ = 0.1 and the volatility parameters (a, b, c) = (1, 0.4, 0.1).

4 x

 4 = 3 and y = 1

Figure 2 :

 2 Figure 2: P σ (T, x, y) (solid line), P σ 2 (T, x, y) (dashed line with crosses) and c σ (T, y, x) (dashed line) in function of the time T for a = 1, b = 0.4, c = 0.1, x 0 = 0.5, r = 0.2 and δ = 0.1.

The C 1

 1 assumption made on p ensures p(Y ) = Yx 0 and p ′ (Y ) = 1. This gives g ↑ (Y )/g ′ ↑ (Y ) = Yx 0 and therefore Y = y * ηp (x 0 ) using Lemma 3.1. Thus, c ηp (Y, x 0 ) = Yx 0 = p(Y ) and λ = 1. One concludes with Theorem 4.1.

x→0 0 ( 1 φ 1 u and so x ǫ 1 ψ

 011 cf. Remark 1.2, still valid for g ↑ when δ = 0) is satisfied since we have 1 ψ(u) ∼ u→0

(x * σ ) - 1 (y) = 1 x 1 y -xn x n+1 1 y 1 ( 1 -nxn x n 1 y

 11111 n + . . . ((x * σ ) -1 ) ′ (y) = 1x n-1 ) + . . . and then(1x * σ (y)/y)(rδx * σ (y)/y) = (1x 1 )(rδx 1 ) + x n (2δx 1 -(r + δ))y n-1 + . . . (x * σ ) -1 (y) y -

1 -(r + δ)x n 1 -

 11 (r + δ)x 1 + 2rnx n .Since x n = 0 and with a simplification we get n(1 + x n-1

Proof of Lemma 6 . 4 . 2 0 2 (

 6422 It is easy to get from (19) and σ = σ thatx * σ (y) ′ = x * σ (y)σ(x * σ (y)) yσ(y) . (44)We have σ(x) = σ 0 for x ∈ [0, ρ]. Since x * σ (y) solves (44) and x * σ (y) ≤ y, x * σ (y) ′ = x * σ (y)/y on [0, ρ]. Therefore, x * σ (y) = x 1 y for y ∈ [0, ρ]. Thanks to (19), x 1 is the unique root in (0, min(1, r/δ)) of x 1 σ 2 0 = 2(1x 1 )(rδx 1 ). Now let us observe that (19) gives for y ∈ (0, y * σ(ρ)], x * σ (y) ′ =x * σ (y) 2 σ y-x * σ (y))(ry-δx * σ (y)) with x * σ (ρ) = x 1 ρ. Since y → x 1 y solves this ODE, for which local uniqueness holds thanks to the Cauchy Lipschitz theorem, we then have x * σ (y) = x 1 y on [ρ, y * σ (ρ)] and so y * σ(ρ) = (x * σ ) -1 (ρ) = ρ/x 1 . Then, (44) gives σ 0 /σ(y) = 1 on [ρ, ρ/x 1 ]. Thus, we prove by induction on n that x * σ (y) = x 1 y and σ(y) = σ 0 for y ∈ [0, ρ/(x 1 ) n ]. This shows the desired result.

  If σ is a continuous function satisfying (H vol ), by (19) and (16), x * σ belongs to Cx . The one to one property is easy to get. If x * σ 1 ≡ x * σ 2 with σ 1 and σ 2 satisfying (H vol ), the ODE (19) ensures that σ 2 1 (x

* σ 1 (y)) = σ 2 2 (x * σ 2 (y)) for y > 0. Therefore σ 1 ≡ σ 2 .

Let us check the onto property and consider x * (y) ∈ Cx . The function σ defined by

Acknowledgements.

We thank Damien Lamberton (Univ. Marne-la-vallée) and Mihail Zervos (King's College) for interesting discussions. We also thank Alexander Schied (TU Berlin) for pointing out the work of Beibel and Lerche [2] to us and Antonino Zanette (University of Udine) for providing us with the routine that calculates American option prices.


Simon Schmidt
email: simon.schmidt@ujf-grenoble.fr

The 3/5-conjecture for weakly S(K 1,3 )-free forests

Keywords: domination game; 3/5-conjecture AMS Subj. Class.: 05C57, 91A43, 05C69

The 3/5-conjecture for the domination game states that the game domination numbers of an isolate-free graph G on n vertices are bounded as follows: γ g (G) ≤ 3n 5 and γ ′ g (G) ≤ 3n+2 5 . Recent progress have been done on the subject and the conjecture is now proved for graphs with minimum degree at least 2. One powerful tool, introduced by Bujtás is the so-called greedy strategy for Dominator. In particular, using this strategy, she has proved the conjecture for isolate-free forests without leafs at distance 4. In this paper, we improve this strategy to extend the result to the larger class of weakly S(K 1,3 )-free forests, where a weakly S(K 1,3 )-free forest F is an isolate-free forest without induced S(K 1,3 ), whose leafs are leafs of F as well.

Introduction

The domination game, introduced five years ago in [START_REF] Brešar | Domination game and an imagination strategy[END_REF] is played on an arbitrary graph G by two players, Dominator and Staller. They alternately choose a vertex from G such that at least one previously undominated vertex becomes dominated. The game ends when no move is possible. Dominator aims to end the game as soon as possible, while Staller wants to prolong it. By D-game (resp. S-game) we mean a game in which Dominator (resp. Staller) plays first. Assuming that both players play optimally, the D-game domination number γ g (G) (resp. the S-game domination number γ ′ g (G)) of a graph G, denotes the total number of chosen vertices during D-game (resp. S-game) on G. There is already a flourishing amount of works on the subject, see for various examples [START_REF] Košmrlj | Realizations of the game domination number[END_REF][START_REF] Dorbec | The domination game played on unions of graphs[END_REF][START_REF] Brešar | Complexity of the game domination problem[END_REF][START_REF] Brešar | Domination game: Effect of edgeand vertex-removal[END_REF][START_REF] Brešar | How long one can bluff in the domination game?[END_REF][START_REF] Nadjafi-Arani | Characterisation of forests with trivial game domination numbers[END_REF]. The most outstanding conjecture concerning this game is probably the so-called 3/5-conjecture. (For related developments concerning the total domination game see [START_REF] Henning | Total version of the domination game, Graphs Combin[END_REF][START_REF] Henning | The 4/5 upper bound on the game total domination number[END_REF].) Conjecture 1.1 (Kinnersley, West and Zamani, [START_REF] Kinnersley | Extremal problems for game domination number[END_REF]) If G is an isolate-free graph on n vertices, then γ g (G) ≤ 3n 5 and γ ′ g (G) ≤ 3n + 2 5 .

If true, the bounds are known to be tight, even if we restrict ourself to forests. See [START_REF] Brešar | Domination game: extremal families of graphs for the 3/5-conjectures[END_REF] for a study of the structure of graphs reaching these bounds. Recently Bujtás has made a breakthrough by introducing a powerful greedy strategy for Dominator [START_REF] Cs | Domination game on trees without leaves at distance four, Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications[END_REF]. In particular, using this technique, she has proved the conjecture for a large subclass of forests [START_REF] Cs | Domination game on trees without leaves at distance four, Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications[END_REF][START_REF] Cs | Domination game on forests[END_REF]. In that case, the upper bound for S-game turns two be even better. This greedy strategy could also be applied to graphs with minimum degree at least 3.

In [START_REF] Cs | On the game domination number of graphs with given minimum degree[END_REF], the conjecture is proved for this class of graph. Even more recently, Henning and Kinnersley [START_REF] Henning | Bounds on the game domination number[END_REF] established the truth of the 3/5-conjecture over the class of graphs with minimum degree at least 2. Hence the 3/5-conjecture remains open only for graphs with pendant vertices. In this paper, we prove the 3/5 conjecture for a larger class of forests, which contains the forests without leafs at distance 4. Even if forests deserve interest by themselves, we emphasize that the general conjecture cannot be easily reduced to the one on forests. There are actually graphs which have greater domination number than any of their spanning trees [START_REF] Brešar | Domination game played on trees and spanning subgraphs[END_REF].

Let S(K 1,3 ) be the graph obtained by subdividing once all the edges of the star K 1,3 (see Figure 1). A weakly S(K 1,3 )-free forest is a forest which does not contain:

• isolated vertices and • induced S(K 1,3 ) whose leafs are leafs in the original forest. The class of weakly S(K 1,3 )-free forests obviously contains the isolate-free forests without leafs at distances 4. For this class of forests we are able to extend Bujtás greedy strategy to prove the following more general theorem.

Theorem 1.3 If F is a weakly S(K 1,3 )-free forest on n vertices, then γ g (F ) ≤ 3n 5 and γ ′ g (F ) ≤ 3n + 2 5 .
The upper bounds are tight in both cases. For example, by [START_REF] Brešar | Domination game: extremal families of graphs for the 3/5-conjectures[END_REF]Theorem 3.7], trees from Figure 2 (a) reach the bound for D-game. For S-game, we easily verify that forests from Figure 2 (b) attain the bound as well.

x n

x 1

x 2 (a)

y n y 1 y 2 y 0 (b)
Figure 2: Extremal weakly S(K 1,3 )-free forests

In the second section of the article, we are going to prove the upper bounds of our main theorem. The first subsection describes Bujtás' greedy strategy and introduces an improvement of it, the so-called breaking-P 5 strategy. The second subsection deals with the proof of the main theorem for D-game. Finally, in the last subsection, we explain briefly how to modify the proof to extend the result to S-game.

Proof of the main theorem

We start by proving the upper bound for D-game. All the results of the two following subsections are stated assuming that D-game is played. In the last subsection, we explain how to modify the proof to get the desired upper bound for S-game.

The greedy strategy and the breaking-P 5 strategy

First we recall Bujtás' greedy strategy for Dominator and give definitions and background needed to understand it. Our strategy will be compatible with the greedy one. In fact, it is only a refinement of the first phase of this strategy. Let F be an isolate-free forest on n vertices. All vertices will be colored with one of the following three colors: white (W ), blue (B) and red (R). In addition to colors, each vertex of F has also a value. It is this value which ables to define a greedy like strategy. The coloring and the values will change all along the game. At the end of turn k ≥ 0, the color and value of a vertex are defined as follows.

• A vertex is white and its value is 3 if it is not already dominated.

• A vertex is blue and its value is 2 if it is dominated, but at least one of its neighbors is not.

• A vertex is red and its value is 0 if N [v] is entirely dominated.

Note that, when a vertex is selected by a player, it is always going to turn red during the current turn. We denote respectively by W k (F ), B k (F ) and R k (F ), the set of white, blue and red vertices at the end of turn k ≥ 0. When F will be clear from the context, we will omit it in the notations. At the end of turn k ≥ 0, we define the residual forest F k as the forest obtained from F by deleting all the red vertices and all the edges between two blue vertices (note that F 0 = F ). Since all the legal moves are either white or blue vertices and all the blue vertices are already dominated, the game after k moves is the same played on the forest F or played on the residual forest F k . The following statement proved in [START_REF] Cs | Domination game on forests[END_REF] is particularly useful. In particular, the first item ensures that any white leaf in a residual forest is also a leaf in the original forest. In other words, no white leaf is created during the game. Lemma 2.1 Let F be an isolate-free forest and k ≥ 0.

1. If v belongs to W k , then v has the same neighborhood in F k as in F .

If F has no isolated vertex, then neither has F k .

For k ≥ 0, the value of the residual forest F k is the sum of the values of all its vertices. We denote this value by p(F k ). By definition, p(F ) = 3n. At each turn, the value of the residual forest decreases. We say that a player seizes s points during turn k ≥ 1, when p(F k-1 )p(F k ) = s. In the greedy strategy the game will be divided into four phases described below. Dominator will always start by applying, if possible, Phase 1 of the strategy. For i ∈ {2, 3, 4}, Phase i starts at Dominator's turn only if Phase i -1 is no longer possible. If at this moment, the strategy of Phase i is not applicable, then it will be skipped. When Dominator starts to play in a new phase, he will never go back to a previous one, even if a change in the residual graph might cause this phase applicable again.

Greedy strategy.

• Phase 1. At his turn Dominator seizes at least 7 points and at least two vertices turn red during this turn.

• Phase 2. At his turn Dominator seizes at least 7 points.

• Phase 3. At his turn Dominator seizes at least 6 points. Moreover, he applies the following two rules.

-Dominator selects a vertex which ensures the maximal possible gain achievable at this turn.

-Under the above rule, Dominator always prefers to play a white stem in the residual forest, which has a white leaf neighbor. • All the vertices are in W k , save the center of the path which is in B k .

• Both ends of the path are leafs in F k .

The blue center of a critical P 5 is called a critical center. It is proved in [START_REF] Cs | Domination game on forests[END_REF], that these critical centers could only appear during Phase 1. Since no new white leaf is created along the game, the following observations are also easy to prove.

Observation 2.3 If a vertex in B k is not a critical center at the end of turn k ≥ 1,
then it is not a critical center at the end of any turn k ′ ≥ k.

In order to state precisely the results obtained by Bujtás, we need more definitions related to this phase. Let k * ≥ 0 be the turn at the end of which Phase 1 is over.

If Phase 1 is skipped, it is easy to show that the 3/5-conjecture holds (the forest is actually a disjoint union of edges). For, we assume in the rest of the paper that k * ≥ 1.

The number of critical centers at the end of turn k * will be denoted by c * . For each turn of Phase 1, when Dominator decreases the value of the graph by more than 7, he does better than what it is expected during this phase. It is the same for Staller when she decreases the value by more than 3 points. These bonus points are defined as follows.

Definition 2.4 For k ∈ {1, ..., k * }, set:

e k = p(F k-1 ) -p(F k ) -7 if turn k is due to Dominator, p(F k-1 ) -p(F k ) -3 if it is due to Staller.
The sum of all the bonus points earned during Phase 1 will be denoted by e * , that is

e * = e 1 + • • • + e k * .
In [9, Theorem 2], Bujtás proved the following result.

Proposition 2.5 Let F be an isolate-free forest on n vertices. If Dominator plays D-game according to the greedy strategy, then the number of turns will be at most 3ne * + c * 5 .

By Lemma 2.1, no new white leaf is created all along the game. It follows that for an isolate-free forest F without leafs at distance 4, c * is always equal to 0. Hence, the above proposition shows that the 3/5-conjecture holds for D-game when restricted to this class of forests. For S-game similar arguments show that for this restricted class the upper bound is slightly better than the one announced by the conjecture:

γ ′ g (F ) ≤ 3|V (F )|+1 2 .
We now improve the greedy strategy in order to apply it to the larger class of weakly S(K 1,3 )-free forests. Let F be an isolate-free forest on n vertices. From now on, we assume that F is arbitrarily rooted. More precisely, if F is a forest with m ≥ 1 connected components T 1 ,...,T m , for any i ∈ {1, ..., m} we select some r i ∈ V (T i ) to be the root of the tree T i . For every vertex v ∈ V (T i ), the height of v is defined by

h(v) = d(r i , v).
We define P(F ) as the set of paths of length 4 between two leafs of F . We say that a vertex d ∈ V (F ) is a dangerous center after k ≥ 0 moves, if the followings holds.

• The vertex d is the center of a path P in P(F ).

• All the vertices of P are in W k .

The set of dangerous centers after k ≥ 0 moves will be denoted by D k (F ). When F will be clear from the context, we will only write D k . Because no new white leaf is created in the residual forests along the game (see Lemma 2.1), the two following results clearly hold.

Lemma 2.6 If k ′ ≥ k ≥ 0 then D k ′ ⊆ D k , that is no new dangerous center is created during the game.
Lemma 2.7 If a vertex in W k does not belong to D k , with k ≥ 0, then it could not be a critical center at the end of any turn k ′ > k.

The following lemma is important, because it gives an alternative definition of the class of weakly S(K 1,3 )-free forests.

Lemma 2.8 If F is a weakly S(K 1,3 )-free forest and d is a dangerous center in D 0 (F ), then there are exactly two vertices a 1 , a 2 ∈ N (d), such that any path in P(F ) whose center is d contains a 1 and a 2 .

Proof. Let l 1 a 1 da 2 l 2 be a path in P(F ) whose center is the dangerous center d. Let f be a leaf of a path P in P(F ) whose center is also d. We have to prove that the unique neighbor u of f is either a 1 or a 2 . By way of contradiction, assume that P contains neither a 1 nor a 2 . Then the subtree induced by {l 1 , l 2 , f, a 1 , a 2 , u} is clearly isomorphic to S(K 1,3 ). Since l 1 , l 2 and f are leafs of F , the forest is not weakly S(K 1,3 )-free. A contradiction.

At least one of the two vertices a 1 , a 2 defined in Lemma 2.8 has a height strictly greater than d. Without lost of generality, we always assume that h(a 2 ) > h(d) and we say that a 2 is the P 5 -child of d. We introduce now the breaking-P 5 strategy. The notation a 1 and a 2 will always refer to the two vertices defined in the above lemma. We will say that a 1 is the vertex related to the dangerous center d.

Breaking-P 5 strategy.

• Phase 1.0: if D k-1 is not empty at his turn k ≥ 1, Dominator selects a vertex d ∈ D k-1 with maximum height and he plays the vertex a 1 related to d. To emphasize that d is not the vertex played during this turn, we will always say that d is elected.

• Phase 1.1: at his turn, Dominator gets at least 7 points and at least two vertices turn red.

• Phases 2 to 4: these phases are the same as for the greedy strategy.

The phases follow one another in the same way as for the greedy strategy. We write k * * ≥ 0 for the number of turns in Phase 1.1, c * * for the number of critical centers at the end of turn k * * and e * * = e 1 + • • • + e k * * . If k * * = 0, then we are in the settings of Bujtás' proof. Hence, we assume now that k * * ≥ 1. As we will see in the following proposition, the breaking-P 5 strategy is also a greedy strategy and we have a result similar to Proposition 2.5.

Proposition 2.9 Let F be an isolate-free forest on n vertices. If Dominator plays D-game according to the breaking-P 5 strategy, then the number of turns will be at most 3ne * * + c * * 5 .

Proof. Let d be the dangerous center elected by Dominator according to the rules of Phase 1.0. Let a 1 be the neighbor of d defined in Lemma 2.8. Since Dominator plays according to the breaking-P 5 strategy, he plays a 1 . This vertex and at least one of its white leaf neighbors turn from white to red. Moreover, the elected vertex d turns from white to blue. Hence Dominator sizes at least 7 points, which in turn implies that the breaking-P 5 strategy is a greedy strategy. Applying Proposition 2.

The proof for D-game

We now prove that if F is a weakly S(K 1,3 )-free forest on n vertices, then γ g (F ) ≤ 3n 5 . For this purpose, we introduce two new processes for each turn of the game. These processes come in addition to the coloring and values defined in the greedy strategy and they only concern blue vertices. When turning blue, some vertices will also be highlighted. We denote by H k the set of highlighted vertices at the end of turn k ≥ 1. The highlighting process will only occur at Dominator's turns. Moreover all blue vertices will get either weight 0 or 1. We denote by w k : B k → {0, 1} the weight function at the end of turn k ≥ 1. For v ∈ V (F ) and X, Y ∈ {B, R, W }, we write v : X Y to indicate that the color of v changes from X to Y during the current turn.

Highlighting Process.

• When Dominator elects the dangerous center d ∈ D k-1 at turn k ≥ 1, d is highlighted.

Note that H k ⊆ B k , because the elected vertex always turns blue during the turn where it is elected.

Weighting Process. • Additional process for turn k, k ≥ 1. This process is applied after the two former processes. If the P 5 -child of a vertex d ∈ H k-1 turns from blue to red without being played by one of the player, then d gets a weight of 1.

Before being able to complete the proof, we need some technical definitions and lemmas. Until the end of this subsection, we always assume that Dominator plays according to the breaking-P 5 strategy. Definition 2.10 Let u be a vertex of a rooted forest F . For each turn k ≥ 0, we define the following connected subtrees of F \ R k .

• C k (u) is the largest connected subtree of F \ R k which contains u.

• C + k (u) is the largest connected subtree of F \ R k which contains u and whose vertices have height greater or equal to h(u).

• C - k (u) = C k (u) \ C + k (u)
. Remark that all these three sequences are non increasing with respect to inclusion. The following result follows directly from the definition. Lemma 2.11 Let F be a rooted forest. For all k ≥ 0 and all u, v ∈

V (F ), if v is in C + k (u) (resp. in C - k (u)), then C + k (v) ⊆ C + k (u) (resp. C - k (v) ⊆ C - k (u)).
Lemma 2.12 Let F be a rooted forest and d be a vertex in D 0 (F ), which is highlighted

at turn k ≥ 1. Then C + k ′ (d) ∩ D k ′ = ∅, for all k ′ ≥ k.
Proof. According to the prescribed strategy for Dominator, a dangerous center is highlighted at turn k ≥ 1 only if it has a maximum height among all the dangerous center from D k-1 . In other words, (ii) The vertex d is not the P 5 -child of d ′ .

C + k-1 (d) ∩ D k-1 = {d}. Moreover, C + k ′ (d) ⊆ C + k-1 (d) and D k ′ ⊆ D k-1 , for all k ′ ≥ k. Hence, C + k ′ (d) ∩ D k ′ ⊆ {d}. Finally, since d belongs to B k , it is not in D k ′ . We conclude that C + k ′ (d) ∩ D k ′ = ∅.
Proof. We prove (i) first. Let d be a dangerous center highlighted at turn k. The vertex d has to belong to D k-1 . By definition of the sets of dangerous centers, this implies that its P 5 -child a 2 is in W k-1 . Since d is highlighted at turn k this turn belongs to Dominator and during this turn he plays the vertex a 1 related to d. This vertex is not adjacent to the white vertex a 2 . Hence a 2 remains white at the end of turn k. Moreover, by Lemma 2.12, we have

C + k (d) ∩ D k = ∅.
This implies that Dominator will not play in C + k (d) until the end of Phase 1.0. In other words, Dominator will play all his remaining moves in Phase 1.0 in

C - k (d). But a 2 ∈ W k is not adjacent to any vertex in D - k (d).
Hence, Dominator's further moves in Phase 1.0 will not change the color of a 2 .

We now prove (ii). Assume first that k ′ > k. By (i), the P 5 -child of d ′ belongs to W k ′ . But d has turned blue at turn k. Hence d cannot be the P 5 -child of d ′ . Suppose now that k ′ < k. By Lemma 2.12, C + k-1 (d ′ ) ∩ D k-1 is empty. Therefore the P 5 -child of d ′ does not belong to D k-1 . But to be highlighted at turn k, d has to belongs to D k-1 . We conclude as previously.

We are now ready to state the following proposition, which will directly imply our main result for D-game. Proposition 2.14 Let F be a weakly S(K 1,3 )-free forest. For all k ∈ {0, ..., k * * }, define

S k = k i=1 e i and K k = v∈B k w k (v).
The following statements are true.

• I k : K k ≤ S k . • II k : ∀v ∈ B k \ H k , if w k (v) = 0, then C - k (v) = ∅ and C + k (v) ∩ D k = ∅. • III k : ∀v ∈ B k , if w k (v) = 0,
then v is not a critical center at the end of turn k.

Proof. We will proceed by induction on k. Since B 0 is empty and K 0 = S 0 = 0, the three statements are trivially true for k = 0. Suppose that I k , II k and III k are true for some k ∈ {0, ..., k * * -1}. We denote by b k+1 the number of vertices which turn blue during turn k + 1.

First, we deal with the weighting process for Dominator's turn. We assume that turn k + 1 belongs to Dominator. At least two vertices turn from white to red and Dominator earns 6 points with these two vertices. Hence, e k+1 ≥ b k+1 -1, which implies that S k+1 ≥ S k + b k+1 -1. During this turn the weight of the vertices in B k does not change and only b k+1 -1 new blue vertices get weight 1 (the highlighted vertex gets weight 0). Therefore, K k+1 ≤ K k + b k+1 -1. Using the induction hypothesis, it yields K k+1 ≤ S k+1 . We conclude that I k+1 is true.

Let w be a vertex in B k , such that w k (w) = 0. Since C + k+1 (w)∩D k+1 ⊆ C + k (w)∩D k and C - k+1 (w) ⊆ C - k (w)
, the truth of statement II k+1 is straightforward from the induction hypothesis II k for the old blue vertices whose weight remains 0. In other words, we only have to pay attention to blue vertices, new or old, which get new weight 0 during turn k + 1. Here, the weight of the former blue vertices does not change during this turn and all the new blue vertices which are not in H k+1 get weight 1. Hence II k+1 is trivially true. By Observation 2.3, in order to prove III k+1 , we only need to pay attention to vertices whose weight turns to 0 during this turn. Here, it is the case for an unique vertex, the dangerous center d ∈ D k , which is elected by Dominator and highlighted during turn k + 1. We emphasize that this is the only point of the whole proof where the weakly S(K 1,3 )-free condition is used. Since Dominator follows the breaking-P 5 strategy, it implies that the vertex a 1 (defined in Lemma 2.8) is played by Dominator at turn k + 1. Hence, a 1 belongs to R k+1 . Let now P be a path in P(F ) whose center is d. By Lemma 2.8, P goes through a 1 which is in R k . Therefore P cannot be a critical P 5 in the residual forest F k+1 , which in turn implies that d is not a critical center at turn k + 1. In conclusion III k+1 holds. Now, we prove the heredity of our statements for the weighting process of Staller's turn. We suppose that turn k + 1 belongs to Staller and denote by v the vertex played by her. We have to deal with the five different cases of the weighting process. We start by proving I k+1 . For the two first cases, Staller earns 3 points with the vertex v which turns from white to red. Hence e k+1 ≥ b k+1 and S k+1 ≥ S k + b k+1 .

Case (A). Because all new blue vertices get weight 1 and the weight of former blue vertices does not change, we have K k+1 ≤ K k +b k+1 . By induction hypothesis, K k ≤ S k , so we conclude that W k+1 ≤ S k+1 .

Case (B). Since v is the P 5 -child of the vertex d ∈ H k , it is adjacent to a leaf l of F . Since v belongs to W k , this leaf is also in W k . Hence, at least two vertices, v and l turn from white to red during turn k + 1. It implies that e k+1 ≥ 3 and S k+1 ≥ S k + 3. Since all new blue vertices get weight 0 and d is the only former blue vertex whose weight is increased to 1, we get

K k+1 ≤ K k + 1 ≤ S k + 1 < S k+1 .
For the three last cases, the vertex v played by Staller turns from blue to red and Staller earns 2 points with this vertex. Hence, e k+1 ≥ b k+1 -1 and

S k+1 ≥ S k + b k+1 -1.
Case (C). If w k (v) = 0, then all new blue vertices get weight 0 and K k+1 ≤ K k . By induction, we get K k+1 ≤ S k ≤ S k+1 . Now, if w k (v) = 1, all new blue vertices get weight 1. But v will not be blue anymore, that is to say v ∈ B k+1 . Hence, its weight 1 will not count for K k+1 . Therefore,

K k+1 ≤ K k + b k+1 -1 ≤ S k+1 .
Case (D). As above, the weight of v will not count for K k+1 . Hence

K k+1 ≤ K k + 1 - w k (v). Since v is the P 5 -child of a vertex d ∈ H k ⊆ B k , C - k (v) contains at least d. By Lemma 2.13 (ii), v is not in H k . Hence, the induction hypothesis II k yields w k (v) = 1. In conclusion, K k+1 ≤ K k + 1 -w k (v) ≤ K k ≤ S k ≤ S k+1 .
Case (E). Let a 2 be the P 5 -child of v.

(a) If a 2 ∈ W k , then K k+1 ≤ K k + b k+1 -1, because a 2 gets weight 0. By induction, K k+1 ≤ S k + b k+1 -1 ≤ S k+1 . (b) If a 2 ∈ B k , a 2 gets new weight 0. Therefore, K k+1 ≤ K k + b k+1 -w k (a 2 ). Since v ∈ H k ⊆ B k , C - k (a 2 ) contains v. By Lemma 2.13, a 2 is not in H k . Finally, II k implies that w k (a 2 ) = 1, which in turn implies K k+1 ≤ K k + b k+1 -1 ≤ S k+1 . (c) If a 2 ∈ R k , we have K k+1 ≤ K k + b k+1 -w k (v),
(v) = w k ′ (v) = 1
and we can conclude as before.

We now prove II k+1 and III k+1 . As for Dominator's turn, we only have to pay attention to the blue vertices, whose weight becomes 0 during turn k + 1.

Case (A). None of the new blue vertices gets weight 0 and no former blue vertex weight change from 1 to 0. Then, II k+1 and III k+1 are trivially true.

Case (B). The only vertices whose weight turn to 0 during the turn are the new blue vertices in N (v). Let u be such a vertex. The vertex v is the P 5 -child of a vertex d ∈ H k . Note that because d was already blue at the beginning of turn k + 1, we have d = u. The vertex d is the only neighbor of v whose height is strictly less than the height of v. Hence h(u) > h(v) > h(d). Since v is now red, C - k+1 (u) is empty. The vertex d has been highlighted during a Dominator's turn, say turn k ′ < k + 1. Since v and u are in

W k ′ , u belongs to C + k ′ (d). By Lemma 2.11, C + k+1 (u) ∩ D k+1 ⊆ C + k ′ (u) ∩ D k ′ ⊆ C + k ′ (d) ∩ D k ′
. By Lemma 2.12, the last set of these inclusion list is empty. Thus C + k+1 (u) ∩ D k+1 is also empty. In conclusion II k+1 is proved. Moreover, we have

C + k ′ (d) ∩ D k ′ = ∅ and u ∈ C + k ′ (d). Hence u is not in D k ′ , but it is in W k ′ .
By Lemma 2.7, u is not a critical center at the end of turn k + 1. In conclusion, III k+1 holds.

Case (C). If w k (v) = 1, then no blue vertices, old or new, would see their weight turn to 0. In that case II k+1 and III k+1 are trivially true. Suppose that w k (v) = 0 and let u ∈ N (v) be a new blue vertex. The vertex v is not in

H k . By induction hypothesis II k , C - k (v) = ∅ and C + k (v) ∩ D k = ∅. Hence, u has to belong to C + k (v) \ D k .
By Lemma 2.7, u cannot be a critical center at turn k + 1. That proves III k+1 . By Lemma 2.11,

C + k+1 (u) ⊆ C + k+1 (v). Therefore C + k+1 (u) ∩ D k+1 = ∅. Finally C - k+1 (u) = ∅, because v is red at the end of turn k + 1. In conclusion II k+1 is true.
Case (D). The proof is the same as for Case (B).

Case (E). Let a 2 be the P 5 -child of v. For subcases (a) and (b), this vertex is the only blue vertex which gets weight 0 during this turn. The vertex a 2 belongs to

C + k (v). It implies that C + k+1 (a 2 ) ⊆ C + k (a 2 ) ⊆ C + k (v) and C + k+1 (a 2 ) ∩ D k+1 ⊆ C + k (v) ∩ D k . Since v is an highlighted vertex, C + k (v)
∩D k is empty by Lemma 2.12. Hence C + k+1 (a 2 )∩D k+1 is also empty. Moreover, because v is red at the end of turn k + 1, we have C - k+1 (a 2 ) = ∅. That shows that II k+1 holds. Since v ∈ H k , this vertex has been highlighted during a Dominator's turn, say turn k ′ ≤ k. We have

C + k ′ (v) ∩ D k ′ = ∅.
Hence, a 2 does not belongs to D k ′ . By Lemma 2.13, a 2 belongs to W k ′ . Therefore, applying Lemma 2.7, the vertex a 2 is not a critical center at the end of turn k + 1. In conclusion III k+1 is true. Finally, for subcase (c), no blue vertex get new weight 0, so there is nothing to prove.

Finally, we have to prove that applying the additional weighting process at the end of turn k + 1 does not change the true of statement I k+1 , II k+1 and III k+1 . Since no blue vertex gets weight 0 during this process, II k+1 and III k+1 remain trivially true. Let x 1 , ..., x m be the P 5 -children of some vertices in H k , which turn red without being played by one of the player. After applying this process, the new value of K k+1 will be less or equal to K k+1 + mw k (x 1 ) -• • •w k (x m ). By the P 5 -child definition, C - k (x i ) ∩ H k is non empty, for any i ∈ {1, ..., m}. Moreover, by Lemma 2.13, we have x i ∈ H k . Applying the induction hypothesis II k , we get that w k (x i ) = 1. That proves that the value of K k+1 is not increased by the application of the additional process. We conclude that I k+1 remains true.

Theorem 2.15 If F is a weakly S(K 1,3 )-free forest on n vertices, then γ g (F ) ≤ 3n 5 .

Proof. By Proposition 2.9, we only have to prove that c * * ≤ e * * . Let K k * * and S k * * be defined as in Proposition 2.14. Note that S k * * = e * * . Statement III k * * from Proposition 2.14 implies that c * * ≤ K k * * . Moreover, statement I k * * implies that K k * * ≤ S k * * . We conclude that c * * ≤ e * * .

The proof for S-game

The proof for S-game will proceed in the same way as for D-game, except that now we introduce Phase 0 of the game, which corresponds to the first move of Staller. Hence, Phase 1.1 will now start at turn k = 2. We still write k * * for the last turn of Phase 1.1 and c * * for the number of critical centers at the end of turn k * * . We set e 1 = p(F 0 )p(F 1 ) -5. For k ≥ 2, e i is defined as for D-game. We also set e * * = e 2 + • • • + e k * * . In the proof of [9, Theorem 2], it is proved that, by following the greedy strategy in S-game for an isolate-free forest on n vertices, Dominator can force the game to end in at most 3ne 1e * + c * 5 turns.

Here, e * is the amount of bonus point earned during Phase 1 and c * is the number of critical centers at the end of this same phase. As for D-game, this statement implies the following proposition. Proposition 2.16 Let F be an isolate-free forest on n vertices. If Dominator plays S-game according to the breaking-P 5 strategy, then the number of turns will be at most 3ne 1e * * + c * * 5 .

We can define the weighting and the highlighting process exactly in the same manner as for D-game. We can state a proposition very similar to Proposition 2.14. Only statement I k will change. For this new proposition, we will have K k ≤ S k + 2, for all k ≥ 0. Indeed e 1 is equal to the number of blue vertices minus 2. Since all these blue vertices will get weight 1, we have K 1 ≤ S 1 + 2. Finally, this new proposition will directly prove that c * * ≤ e 1 + e * * + 2. It clearly implies the below theorem.

Theorem 2.17 If F is a weakly S(K 1,3 )-free forest on n vertices, then γ ′ g (F ) ≤ 3n + 2 5 .

Theorem 1 . 2 (

 12 Bujtás 2014) If F is an isolate-free forest on n vertices, without leafs at distance 4, then γ g (F )

Figure 1 :

 1 Figure 1: S(K 1,3 )

  5, we have that D-game will end in at most 3ne * + c * 5 turns. Moreover, any critical center created at turn k > k * * must belong to D k-1 . But D k = ∅, for all k ≥ k * * . Hence, by Lemma 2.7, we have c * = c * * . Finally, it is obvious that e * ≥ e * * . In conclusion, if Dominator follows the breaking-P 5 strategy, D-game ends in at most 3ne * * + c * * 5 turns.

•

  Dominator's turn k, k ≥ 1. All new blue vertices which are not highlighted get weight 1. The new highlighted vertex d gets weight 0. • Staller's turn k, k ≥ 1. She selects vertex v. (A) v : W R and v is not the P 5 -child of a vertex d ∈ H k-1 . All new blue vertices get weight 1. (B) v : W R and v is the P 5 -child of a vertex d ∈ H k-1 . All new blue vertices get weight 0 and the vertex d gets weight 1. (C) v : B R, v ∈ H k-1 and v is not the P 5 -child of a vertex d ∈ H k-1 . All new blue vertices get a weight equal to the weight of v. (D) v : B R, v ∈ H k-1 and v is the P 5 -child of a vertex d ∈ H k-1 . All new blue vertices get weight 0 and d get weight 1. (E) v : B R and v ∈ H k-1 . In particular v is in D 0 . Let a 2 be the P 5 -child of v. (a) If a 2 ∈ W k-1 , all new blue vertices but a 2 get weight 1 and a 2 gets weight 0. (b) If a 2 ∈ B k-1 , all new blue vertices get weight 1 and a 2 gets new weight 0. (c) If a 2 ∈ R k-1 , all new blue vertices get weight 1.

Lemma 2 .

 2 [START_REF] Henning | The 4/5 upper bound on the game total domination number[END_REF] Let F be a rooted forest. Let d and d ′ be two vertices in D 0 (F ) which are respectively highlighted at turns k and k ′ ≥ 1. Then the following hold.(i) The P 5 -child a 2 of the vertex d belongs to W k . Moreover, if a 2 is going to turn red or blue during Phase 1.0, this could be only during a turn of Staller.

•

  Phase 4. At his turn Dominator seizes at least 3 points.

	Roughly speaking, Bujtás' proof is to show that during a turn of Dominator and the
	turn of Staller which follows, the value of the residual forest is in average decreased by
	10 points. It implies directly that after at most 3n/5 moves, the value of the residual
	graph is 0, which means the game is over. But it works only if we do not deal with
	what Bujtás called critical P 5 .
	Definition 2.2 A critical P 5 in F k , with k ≥ 0 is a path of length 4 such that the
	following holds.

  because v does not belong to B k+1 . By Lemma 2.13, we know that a 2 has turned red some moves after the moment v has been highlighted. Moreover, it happened during a Staller's turn. There are two possibilities. First, a 2 has turned red because it has been played by Staller at turn k ′ < k + 1. Case B or Case D of the weighting process for Staller's turn has been applied. Hence, during this turn, v has gotten weight 1. Second, a 2 might have turned red without being played by Staller. The additional weighting process ensures that d has also gotten weight 1. Moreover, the only weighting process, which can change the former weight of a blue vertex into 0, is the second item of Case E for Staller's turn. But, this process is applied only if v is the P 5 -child of another highlighted vertex. Once more, Lemma 2.13 ensures that it is impossible because v is itself highlighted. In all the cases w k

Acknowledgements

The author would like to thank Sandi Klavžar and Csilla Bujtás for helpful discussions about this work. In particular, the nice definition of weakly S(K 1,3 )-free forest owes a great deal to Sandi Klavžar. The research was financed by the ANR-14-CE25-0006 project of the French National Research Agency and by the grant CMIRA Explo'RA Doc from La Région Rhône Alpes.