Joint distribution of a Lévy process and its running supremum

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Joint distribution of a Lévy process and its running supremum

Laure Coutin, Waly Ngom, Monique Pontier

To cite this version:

Laure Coutin, Waly Ngom, Monique Pontier. Joint distribution of a Lévy process and its running supremum. Journal of Applied Probability, Cambridge University press, 2018, 55 (2), pp.488-512.

�10.1017/jpr.2018.32�. �hal-01527939�

Joint distribution of a Lévy process and its running supremum

Laure Coutin

^∗

Waly Ngom

^†

Monique Pontier

^‡

February 28, 2017

Abstract

Let X be a jump-diusion process and X^∗ its running supremum. In this paper, we rst show that for anyt >0,the law of the pair(X_t^∗, Xt)has a density with respect to Lebesgue measure and compute this one. This allows us to show that for anyt >0,the pair formed by the random variableXtand the running supremumX_t^∗ ofX at time tcan be characterized as a solution of a weakly valued-measure partial dierential equation. Then we compute the marginal density ofX_t^∗ for allt >0.

Keywords: Lévy process, partial dierential equation, running supremum process, rst hit- ting time.

A.M.S. Classication: 60G51, 60H20, 60H99.

1 Introduction

Consider a Lévy process (X

t

, t ≥ 0), starting from zero, which is right continuous left limited.

If moreover X is the sum of a drifted Brownian motion and a compound Poisson process, it is called a mixed diusive-jump process. As any Lévy process, X has stationary and independent increments and is characterized by its Laplace transform. The mixed diusive-jump processes and the notion of rst passage time (behavior of certain processes at rst passage time) are very useful and widely studied.

Introducing the running supremum at time t, X

_t^∗

and the rst passage τ

b

of X at level b, the probability

P

(X

t

≥ a, X

_t^∗

≥ b) =

P

(X

t

≥ a, τ

_b

≤ t) for some xed real numbers (a, b), a ≤ b and b > 0, is of great importance, for example, in pricing barrier options while the logarithm of the underlying asset price is modeled by a jump-diusion process. In this idea, Kou and Wang [5] give the explicit expression of the Laplace transform of the joint distribution of the double exponential mixed diusive-jump process and its running supremum.

In [4], Jeanblanc et al. consider the rst passage time by a diusion at a deterministic function h that depends on time and they dene a function of τ

_h

and X which satises the Fokker-Planck Equation.

∗coutin@math.univ-toulouse.fr, IMT.

†ngomwaly@gmail.com, IMT.

‡pontier@math.univ-toulouse.fr, IMT: Institut Mathématique de Toulouse, Université Paul Sabatier, 31062 Toulouse, France.

In [1], it is well noted (Theorem 2.2.9 and Exercise 2.2.10) that the

¹₂

−stable subordinator is the rst passage time of a standard Brownian motion and the inverse Gaussian subordinator is the rst passage time of standard Brownian motion with a drift.

Mark Veillette and Murad S. Taqqu study in [9] the rst passage time of a subordinator D . Since D is in general non-Markovian with non-stationary and non-independent increments, they derive a partial dierential equation for the Laplace transform of the n − time tail distribution

P

(τ

_t1

> s

₁

, · · · , τ

_tn

> s

_n

) where τ

_tk

= inf { s : D

_s

> t

_k

} for a subordinator (D

_s

, s ≥ 0) . With this result, they give a recursive formula for multiple-time moments of the local time of a Markov process in terms of its transition density.

The authors of [2] use a partial dierential equation (PDE) approach to show that the calibration of an implied volatility surface and the pricing of contingent claims can be as simple in mixed diusive-jump framework as it is in a diusion framework.

This work characterizes the law of the pair U

_t

formed by the random variable X

_t

and the running supremum X

_t^∗

of X at time t, with a valued-measure partial dierential equation and gives an explicit expression for the density function of this pair. Then the marginal density of X

_t^∗

is given.

The paper is organized as follows: Section 2 provides the main result. Section 3 gives the density function of the pair formed by the random variable X

_t

and its running supremum X

_t^∗

and Section 4 is devoted to the proof of the main result of Section 2. To nish, one concludes and gives some auxiliary results in Appendix.

2 Valued measure dierential equation for the joint law

We introduce some preliminary concepts for the diusion part: for a standard Brownian motion W and a real number m , let be

X ˜

t

= mt + W

t

, X ˜

_t^∗

= sup

s≤t

X ˜

s

. (1)

In [4] page 147, Jeanblanc et al. show that the pair ( ˜ X

_t^∗

, X ˜

_t

) has a density with respect to Lebesgue measure on

R²

noted p(., .; ˜ t) where

˜

p(b, a; t) = 2(2b − a)

√ 2πt

³

exp [

− (2b − a)

²

2t + ma − m

²

t 2 ]

1{max(0,a)<b}

. (2) In all the following, Φ

_G

means the standard normal Gaussian distribution and one often uses the following:

1 − Φ

_G

(x) = Φ

_G

( − x) ≤ 1 x √

2π exp − x

²

2 , ∀ x > 0. (3)

In order to have a Lévy process with non zero jump part, let us introduce

X

t

= mt + W

t

+

Nt

∑

i=1

Y

i

, X

_t^∗

= sup

s≤t

X

t

,

where N is a Poisson process with constant positive intensity λ , (Y

_i

, i ∈

N^∗

) is a sequence of

independent and identically distributed random variables with the distribution function F

Y

and

the sequence of jump times of N is denoted by (T

_i

), i ≥ 1 . Let θ be the shift operator and (U

t

; t ≥ 0) be the

R²

−value process dened by

U

_t

= (X

_t^∗

, X

_t

), t ≥ 0. (4)

The aim is to prove the theorem:

Theorem 2.1. (i) For all t > 0, the law of the pair (X

_t^∗

, X

t

) has a density with respect to the Lebesgue measure, denoted by p(b, a; t).

(ii) For all t > 0, a ∈

R

the map h 7→ p(a + h, a; t) has a limit when h goes to 0 denoted by p(a+, a; t).

(iii) Let be φ :

R²

→

R

a C

_b³

− bounded function with a support in { (b, a), b > 0, b ≥ a } such that there exists δ > 1 satisfying ∫

R

| ∂

₁

φ(a, a) |

^δ

da < ∞ . For any t > 0,

E

(φ(U

t

)) = φ(0, 0) +

∫

_t

0

E

[

m∂

2

φ(U

s

) + 1

2 ∂

₂₂²

φ(U

s

) ]

ds (5)

+

∫

_t

0

1 2

E

[

∂

₁

φ(X

_s

, X

_s

) p(X

_s

+, X

_s

; s g(X

_s

; s)

] ds + λ

∫

_t

0

E

(∫

R

[φ(U

_s

(y)) − φ(U

_s

)] F

_Y

(dy) )

ds.

where g(.; s) is the density of the random variable X

s

and

U

s

= (X

_s^∗

, X

s

), U

s

(y) = (max(X

_s^∗

, X

s

+ y), X

s

+ y), s ≥ 0. (6) In the next sections, details of the proof of Theorem 2.1 are given.

3 Existence of the density of the law of (X

_t^∗

, X

_t

) and its properties

We note that

X

_t^∗

= max { ( sup

u∈[Ti,inf(Ti+1,t)[

X

u

, i = 0, ..., N

t

), X

t

}

and use the joint density of ( ˜ X

_t^∗

, X ˜

t

) given by (2) to show that the pair (X

_t^∗

, X

t

) law has a density which is right continuous on the diagonal, see Proposition 3.1 below which actually is the proof of Theorem 2.1 (i) and (ii).

Proposition 3.1. (i) For all t > 0, the law of the random vector (X

_t^∗

, X

_t

) admits a density with respect to the Lebesgue measure given by

p(b, a, t) =

E

(

_N

∑

t

k=0

˜ p (

b − X

T_k

, a − X

T_k

− Y

_k+11_{Tk+1≤t}

− (X

t

− X

T_k+1∧t

), t ∧ T

_k+1

− T

_k

)

1∆_k,t

(b, a) )

where p ˜ is given by (2) and

∆

_k,t

= {

(b, a) | b > max (

X

_T^∗

k

, a + [X

_t∧Tk+1

− sup

u∈[T_k+1,t]

X

_u

]1

_{Tk+1<t}

)}

. (7)

(ii) Moreover, for all a ∈

R, t >

0 the map h 7→ p(a +h, a; t) has a limit when h goes to 0 denoted by p(a+, a; t) and

p(a+, a; t) =

E

(

_N

∑

t

k=0

˜ p (

(a − X

_Tk

)

⁺

, a − X

_Tk

− Y

_k+11_{Tk+1≤t}

− (X

t

− X

_Tk+1∧t

), t ∧ T

_k+1

− T

_k

)

1_a∈Dk,t

)

where D

_k,t

:= { a : a ≥ max (

X

_T^∗

k

, a + [X

_t∧Tk+1

− sup

_u∈[Tk+1,t]

X

_u

]1

_{Tk+1<t}

) } The proof of (i) relies on the following lemma:

Lemma 3.2. Almost surely, for all t, X

_t^∗

= max

(

X

_Tk

+ sup

u∈[Tk,Tk+1∧t]

( X ˜

_u

− X ˜

_Tk

)

, k = 0, ..., N

_t

)

. (8)

Moreover, almost surely, for all t , there exists a unique k denoted as N

_t^∗

such that

X

_t^∗

= X

_Tk

+ sup

u∈[Tk,Tk+1∧t]

( X ˜

_u

− X ˜

_Tk

)

. (9)

Proof. Let t be xed.

(a) Note that

X

_t^∗

= max {

max (

X

_Tk

+ sup

u∈[Tk,Tk+1∧t[

(X

u

− X

_Tk

) , k = 0, ..., N

t

) , X

t

}

. (10)

For k ∈

N

, for all u ∈ [T

_k

, T

_k+1

[, X

_u

− X

_Tk

= ˜ X

_u

− X ˜

_Tk

where X ˜ is the continuous process dened in (1), thus for k ≤ N

t

,

sup

u∈[T_k,T_k+1∧t[

(X

_u

− X

_Tk

) = sup

u∈[T_k,T_k+1∧t]

( X ˜

_u

− X ˜

_Tk

) (11)

and max

(

X

_TNt

+ sup

u∈[T_Nt,T_Nt+1∧t[

(

X

_u

− X

_TNt

)

, X

_t

)

= X

_TNt

+ sup

u∈[T_Nt,T_Nt+1∧t]

( X ˜

_u

− X ˜

_TNt

)

. (12) Plugging identities (11) and (12) in equality (10) yields (8).

(b) Let two integers i < j then, X

Tj

+ sup

u∈[Tj,Tj+1∧t]

( X ˜

u

− X ˜

Tj

)

= X

Ti

+

( X ˜

Ti+1

− X ˜

Ti

)

+ Y

i+1

+ (

X

Tj

− X

Ti+1

) + sup

u∈[Tj,Tj+1∧t]

( X ˜

u

− X ˜

Tj

)

and

X

_Tj

+ sup

u∈[Tj,Tj+1∧t]

( X ˜

_u

− X ˜

_Tj

) − X

_Ti

− sup

u∈[Ti,Ti+1∧t]

( X ˜

_u

− X ˜

_Ti

)

= {

Y

_i+1

+ (X

_Tj

− X

_Ti+1

) + sup

u∈[Tj,Tj+1∧t]

( ˜ X

_u

− X ˜

_Tj

) }

+ {

( ˜ X

_Ti+1

− X ˜

_Ti

) − sup

u∈[Ti,Ti+1∧t]

( X ˜

_u

− X ˜

_Ti

)}

.

The two following random vectors are independent:

(

sup

u∈[Ti,Ti+1∧t]

( X ˜

u

− X ˜

_Ti

)

; ˜ X

_Ti+1

− X ˜

_Ti

)

, Y

i+1

+ (

X

_Tj

− X

_Ti+1

)

+ sup

u∈[Tj,Tj+1∧t]

( X ˜

u

− X ˜

_Tj

)

and the law of the vector (

sup

_{u∈[Ti,Ti+1∧t]}

( X ˜

_u

− X ˜

_Ti

)

; ˜ X

_Ti+1

− X ˜

_Ti

) admits a density with respect to the Lebesgue measure, hence the law of the random variable

sup

u∈[Ti,Ti+1∧t]

( X ˜

_u

− X ˜

_Ti

)

+ ˜ X

_Ti+1

− X ˜

_Ti

has a density with respect to the Lebesgue measure and is independent of Y

i+1

+ (

X

_Tj

− X

_Ti+1

)

+ sup

u∈[Tj,Tj+1∧t]

( X ˜

u

− X ˜

_Tj

)

. Therefore, X

Tj

+ sup

_{u∈[Tj,Tj+1∧t]}

( X ˜

u

− X ˜

Tj

) − X

Ti

− sup

_{u∈[Ti,Ti+1∧t]}

( X ˜

u

− X ˜

Ti

) is the sum of two independent random variables, one having a density, then also has a density. So for all t , almost surely, whenever i ̸ = j

X

_Tj

+ sup

u∈[Tj,Tj+1∧t]

( X ˜

_u

− X ˜

_Tj

) ̸ = X

_Ti

+ sup

u∈[Ti,Ti+1∧t]

( X ˜

_u

− X ˜

_Ti

)

(c) Above, we can exchange ∀ t > 0 and almost surely, since the processes (N

t

, t ≥ 0) and ((

max (

X

_Tk

+ sup

_{u∈[Tk,Tk+1∧t]}

( X ˜

_u

− X ˜

_Tk

))

, k ≤ N

_t

)

, t ≥ 0

) are right continuous.

Proof. of Proposition 3.1 (i): According to Lemma 3.2, let N

_t^∗

denoting the index k where the maximum below is reached,

X

_t^∗

= max (

X

_Tk

+ sup

u∈[T_k,T_k+1∧t]

( X ˜

_u

− X ˜

_Tk

)

, k = 0, ..., N

_t

)

.

The fact N

_t^∗

= k is equivalent to: the supremum is reached on the interval [T

_k

, T

_k+1

∧ t] , actually meaning X

_t^∗

= sup

_[T

k,T_k+1∧t]

X

_u

and remark that sup

_[T

k,T_k+1∧t]

X

_u

≥ X

_T^∗

k

∨ sup

_[T

k+1∧t,t]

X

_u

. On the interval [T

_k

, T

_k+1

∧ t] , X

_u

= X

_Tk

+ ˜ X

_u

− X ˜

_Tk

. Thus the following inequalities are equivalent to N

_t^∗

= k:

(a) X

_Tk

+ sup

[Tk,Tk+1∧t]

( ˜ X

_u

− X ˜

_Tk

) ≥ X

_T^∗

k

, (b) X

_Tk

+ sup

[Tk,Tk+1∧t]

( ˜ X

_u

− X ˜

_Tk

) ≥ sup

[Tk+1∧t,t]

X

_u

= [X

_Tk+1∧t

+ sup

[Tk+1,t]

(X

_u

− X

_Tk+1∧t

)]1

_{Tk+1<t}

+ X

_t1_{Tk+1≥t}

. Using X

_Tk+1

= X

_Tk

+ ˜ X

_Tk+1

− X ˜

_Tk

+ Y

_k+1

, (b) is equivalent to

sup

[Tk,Tk+1∧t]

( ˜ X

u

− X ˜

T_k

) ≥ [ ˜ X

T_k+1

− X ˜

T_k

+Y

_k+1

+ sup

[Tk+1,t]

(X

u

− X

T_k+1

)]1

_{Tk+1<t}

)+( ˜ X

t

− X ˜

T_k

)1

_{Tk+1≥t}

. As a conclusion we get { N

_t^∗

= k } =

{ sup

[T_k,T_k+1∧t]

( ˜Xu−X˜T_k)≥X_T^∗

k−XT_k}∩{ sup

[T_k,T_k+1∧t]

( ˜Xu−X˜T_k)≥X˜t∧T_k+1−X˜T_k+[Yk+1+ sup

[T_k+1,t]

(Xu−XT_k+1)]1_{Tk+1≤t})}.

Thus

{ N

_t^∗

= k } = {(

sup

u∈[Tk,Tk+1∧t]

( ˜ X

_u

− X ˜

_Tk

), X ˜

_t∧Tk+1

− X ˜

_Tk

)

∈ ∆ ¯

_k,t

}

where

∆ ¯

k,t

= {

(b, a) : | b > max (

X

_T^∗_k

− X

T_k

, a + [Y

k+1

+ sup

u∈[Tk+1,t]

(X

u

− X

T_k+1

)]1

_{Tk+1≤t}

)}

. (13)

Moreover on { k ≤ N

_t

} so on { N

_t^∗

= k } ⊂ { k ≤ N

_t

}

X

t

= X

Tk

+ ( ˜ X

t∧Tk+1

− X ˜

Tk

) + Y

k+11_{t≥Tk+1}

+ (X

t

− X

t∧Tk+1

). (14) Let Φ be a bounded Borel function, hence

E[Φ(Xt^∗, Xt)] =E [_N

∑t k=0

Φ(Xt^∗, Xt)1_{N_t^∗=k}

]

=

E [_N

∑t k=0

1_{N_t^∗=k}Φ (

XT_k+ sup

u∈[Tk,T_k+1∧t]( ˜Xu−X˜T_k), XT_k+ ( ˜X_t∧Tk+1−X˜T_k) +Yk+11_{t≥T_k+1}+ (Xt−X_t∧Tk+1) )]

.

The four following random vectors are independent:

(X

_Tk

, X

_T^∗_k

), Y

_k+1

, (

X

_t

− X

_t∧Tk+1

, sup

u∈[Tk+1∧t,t[

(X

_u

− X

_Tk+1∧t

) )

, (

sup

u∈[Tk,Tk+1∧t]

X ˜

_u

− X ˜

_Tk

, X ˜

_t∧Tk+1

− X ˜

_Tk

)

and conditionally to σ (

F

Tk

, Y

_k+1

, (X

_u

− X

_Tk+1

, u ≥ T

_k+1

∧ t), T

_k

, T

_k+1

)

, the law of the ran-

dom vector (

sup

u∈[T_k,T_k+1∧t]

X ˜

u

− X ˜

Tk

, X ˜

t∧Tk+1

− X ˜

Tk

)

has a density with respect to the Lebesgue measure given by p(b, a, T ˜

_k+1

∧ t − T

_k

) where p ˜ is dened by (2). We obtain that

E

(Φ(X

_t^∗

, X

_t

)) =

∫

E

[

_N

∑

t

k=0

Φ(X

_Tk

+ b, X

_Tk

+ a + Y

_k+11_{t≥Tk+1}

+ (X

_t

− X

_Tk+1∧t

))˜ p(b, a, T

_k+1

∧ t − T

_k

)1

∆¯_k,t

(b, a) ]

dadb.

The change of variable formula v = b + X

Tk

and u = X

Tk

+ a + Y

k+11_{t≥Tk+1}

+ (X

t

− X

Tk+1∧t

) concludes the proof.

Proof of Proposition 3.1 (ii): Let a ∈

R

, t > 0, the map h 7→

Nt

∑

k=0

˜ p (

a + h − X

_Tk

, a − X

_Tk

− Y

_k+11_{Tk+1≤t}

− (X

_t

− X

_Tk+1∧t

), t ∧ T

_k+1

− T

_k

)

1_∆k,t

(b, a) has a right limit when h goes to 0 since both functions h 7→ p(a ˜ + h, a; t) and h →

1∆_k,t

(a + h, a) admit a limit when h decreases to 0 . According to Proposition 6.2 in Appendix the family (

_N

∑

t

k=0

˜ p (

a + h − X

_Tk

, a − X

_Tk

− Y

_k+11_{Tk+1≤t}

− (X

_t

− X

_Tk+1∧t

), t ∧ T

_k+1

− T

_k

)

1_∆k,t

(b, a) )

h∈[0,1]

is uniformly integrable.

Then, we can exchange the limit and the expectation and h 7→ p(a + h, a; t) has a limit when h decreases to 0 and

p(a+, a; t) =

E

(

_N

∑

t

k=0

˜ p (

(a − X

_Tk

)

₊

, a − X

_Tk

− Y

_k+11_{Tk+1≤t}

− (X

_t

− X

_Tk+1∧t

), t ∧ T

_k+1

− T

_k

)

1_a∈Dk,t

)

where D

_k,t

:= { a : a ≥ max (

X

_T^∗

k

, a + [X

_t∧Tk+1

− sup

_u∈[Tk+1,t]

X

_u

]1

_{Tk+1<t}

) }

As a corollary the law of X

_t^∗

is deduced:

Corollary 3.3. For any t > 0, the law of the random variable X

_t^∗

has a density p

^∗

(., t) given by p

^∗

(b, t) = 2E

(

_N

∑

t

k=0

2e

^2(b−XTk)m

H

m

[ (C

t,k

)

⁺1Tk+1<t

+ (b − X

Tk

) + m(t ∧ T

k+1

− T

k

), t ∧ T

k+1

− T

k

]

1_{b>X∗ Tk}

)

(15) where H

_m

: (x, t) →

^√_2πt¹

exp

[ −

^x_2t²

]

− mΦ

_G

( −

^√x_t

) and C

_t,k

= (Y

_k+1

+ sup

_u∈[T

k+1,t]

(X

_u

− X

_Tk+1

))1

_{Tk+1≤t}

.

Proof. Let q ˜ be the function such that p(b, a, .) = ˜ ˜ q(b, a, .)1

_b>a∨0

where

˜

p(b, a; t) = 2(2b − a)

√ 2πt

³

exp [

− (2b − a)

²

2t + ma − m

²

t 2 ]

1_{max(0,a)<b}

. Remark that

(2b − a)

²

− 2mta + m

²

t

²

= [a − (2b + mt)]

²

− 4bmt, (16) thus we obtain

˜

q(b, a, t) = 2e

^2bm

√ 2πt

( 2b + mt − a

t exp

[

− [a − (2b + mt)]

²

2t

]

−m exp [

− [a − (2b + mt)]

²

2t

]) . Hence, for any A,

0 ≤

∫

_A

−∞

p(b, a, t)da ˜ = 2e

^2bm1b>0

H

m

(x, t), x = b ∧ A − 2b − mt. (17) Let k be xed and P

_k^∗

(b, t) be given by

P

_k^∗

(b, t) :=

∫

R

p ˜ (

b − X

Tk

, a − X

Tk

− Y

k+11Tk+1≤t

− (X

t

− X

Tk+1∧t

), t ∧ T

k+1

− T

k

)

1∆k,t

(b, a)da then the density of X

_t^∗

is given by

p

^∗

(b, t) =

E

(

_N

∑

t

k=0

P

_k^∗

(b, t) )

.

With the change of variables u = a − X

_Tk

− Y

_k+11_Tk+1≤t

− (X

_t

− X

_t∧Tk+1

), it follows P

_k^∗

(b, t) :=

∫

R

p ˜ (b − X

_Tk

, u, t ∧ T

_k+1

− T

_k

)

1_∆k,t

(b, u + X

_Tk

+ Y

_k+11_Tk+1≤t

+ (X

t

− X

_t∧Tk+1

))du.

According to the denition of ∆

k,t

(7)

1_∆k,t

(b, u + X

_Tk

+ Y

_k+11_{Tk+1≤t}

+ (X

t

− X

_t∧Tk+1

)) =

1_{b>X∗

Tk}1_{{b>u+XTk+[Yk+1+supu∈[}

Tk+1,t](Xu−Xt)]1_{Tk+1≤t}}

On the event T

k+1

≤ t (id est k < N

t

) P

_k^∗

(b, t) =

∫

R

q(b ˜ − X

_Tk

, u, T

_k+1

− T

_k

)1

_b>X∗

Tk1_{]−∞,min(b−X}

Tk,b−X_Tk−Ct,k[

(u)du since C

_t,k

= (Y

_k+1

+ sup

_u∈[Tk+1,t]

(X

_u

− X

_t

))1

_{Tk+1≤t}

.

And on the event T

_k+1

> t P

_k^∗

(b, t) =

∫

R

q(b ˜ − X

_Tk

, u, t − T

_k

)1

_b>X∗

Tk1_{]−∞,b−X}

Tk[

(u)du.

(a) On the event T

k+1

≤ t applying (17) to A = b − X

T_k

− C

t,k

, with T

k+1

− T

k

and b − X

T_k

instead of t and b, (b − X

_Tk

) ∧ A = b − X

_Tk

− (C

_t,k

)

⁺

(since 0 ∧ ( − x) = − x

⁺

) so on this event P

_k^∗

(b, t) = 2e

^2m(b−XTk)

H

_m

[

(C

_t,k

)

⁺

+ (b − X

_Tk

) + m(T

_k+1

− T

_k

), T

_k+1

− T

_k

] .

(b) On the event T

_k+1

> t , applying (17) to A = b − X

_Tk

and taking t − T

_k

and b − X

_Tk

instead of t and b, so on this event

P

_k^∗

(b, t) = 2e

^2(b−XTk)m

H

m

[ − (b − X

T_k

) − m(t − T

_k

), t − T

_k

] . To summarize both cases

P

_k^∗

(b, t) = 2e

^2(b−XTk)m

H

m

[ (C

t,k

)

⁺1_{Tk+1≤t}

+ (b − X

Tk

) + m(t ∧ T

k+1

− T

k

), t ∧ T

k+1

− T

k

]

and the proof is achieved.

4 Proof of Theorem 2.1 (iii)

To prove the end of this theorem, we proceed as follows: we will compute lim

h→0

h

⁻¹

A(t, h) = a(t) where

A(t, h) :=

E

[φ(U

_t+h

) − φ(U

_t

)] . (18) After that, we will use [8] 11.82 p. 368: If f is a function such that f

^′

is nite everywhere and integrable, then for all a ≤ b, f (b) − f(a) = ∫

_b

a

f

^′

(s)ds. The study of a(t) := lim

_h→0

h

⁻¹

A(t, h) could prove that for all t > t

₀

> 0,

E

(φ(U

t

)) =

∫

_t

t0

a(s)ds +

E

(

φ(X

_t^∗₀

, X

t0

) )

, ∀ t > t

0

.