Conditioning diffusions with respect to partial observations

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Conditioning diffusions with respect to partial observations

Jean-Louis Marchand

To cite this version:

Jean-Louis Marchand. Conditioning diffusions with respect to partial observations. 2011. �hal-

00591374�

Conditioning diffusions with respect to partial observations

Jean-Louis Marchand

IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Abstract

In this paper, we prove a result of equivalence in law between a diffusion conditioned with respect to partial observations and an auxiliary process. By partial observations we mean coordinates (or linear transformation) of the process at a finite collection of deterministic times. Apart from the theoritical interest, this result allows to simulate the conditioned diffusion through Monte Carlo’s method, using the fact that the auxiliary process is easy to simulate.

Keywords: Conditioned diffusion, Partial observations, Simulation

1 Introduction

We are interessed in multidimensional diffusions solutions of stochastic differential equations (SDE’s) generated by a Brownian motion. For an-dimensional diffusion solution on [0, T] of the following

dxt=bt(xt)dt+σt(xt)dwt, x0=u (1) where w is a n-dimensional Brownian motion, it is known (see e.g. [7]) that its conditional law L(x|xT =v) is given by the law of a bridge process (as extension of Brownian bridge)y solution of

dyt=bt(yt)dt+σt(yt)d ˜wt+σt(yt)σt(yt)^∗∇zlogpt,T(z, v)

z=ytdt, y0=u

where ˜wis a Brownian motion andps,t(z, .) is the density ofxtknowingxs=z. But in most cases this density is not explicitely known so that we are not able to simulate it easily. For practical purposes,e.g.

parameter estimation of diffusion processes, simulation of paths corresponding to the conditional law is needed.

In their paper [4], B.Delyon and Y.Hu studied the following equation on [0, T] dyt=bt(yt)dt−yt−v

T −tdt+σt(yt)d ˜wt, y0=u. (2) where ˜wis an-dimensional Brownian motions. Under adequate assumptions, the processyis unique on [0, T], limt→Tyt=v, a.s. and for all positive functionf inC([0, T],Rⁿ) we have

E[f(x)|xT =v] =CE[f(y)R(y)]

whereRis a functional of whole pathyon [0, T]. The quantityR(y) is computable knowing parameters b,σ, T andv. The constantC is unknown, but in practice the conditional law is estimated through

E[f(x)|xT =v]≃ P

if(yⁱ)R(yⁱ) P

iR(yⁱ)

where eachyⁱ is an independant sample of (2). In this case, we call the processya bridge even ifydoes not have the right targeted law. Ifb= 0 andσ=In (identityn-dimensional matrix), the processxis a n-dimensional Brownian motion and processy is an-dimensional Brownian bridge so that C =R= 1.

This theorem applies in the case of more than one observation. The Markov property indeed implies that the conditional law is the tensor product of each bridge.

The aim of this paper is to extend this result to solve this problem with only partial observations.

The previous remark does not apply; indeed we have to treat simultaneously all conditionings . To give an idea, let w= (^w_w¹2) be a 2-dimensional Brownian motion. The law ofw conditioned onw¹_S =uand w²_T =v withS < T is given by that one ofy solution of

dyt= d ˜wt−

y_t¹−u S−t1_t<S

y_t²−v T−t1_t<T

!

dt, y0=u each coordinate is a Brownian bridge.

Let us define our observations. At each deterministic positive observation time of the sequence 0< T1<· · ·< Tk<· · ·< TN =T, we get a partial information given by a linear transformation ofxTk, LkxTk, where Lk is a deterministic matrix in Mmk,n(R) whose mk rows form an orthonormal family.

So that our aim is to be able to describe the conditional lawL(x|(LkxTk =vk)1≤k≤N) where vk is an arbitrary deterministicmk-dimensional vector.

We define processy to be the solution of

dyt=bt(yt)dt+σt(yt)d ˜wt−PN

k=1P_t^k(yt)^y_T^t−_k−t^uk1_{(Tk−εk,Tk)}(t)dt

y0=u (3)

where for all time t, all vectorz and for all 1 ≤k ≤N, the matrixP_t^k(z) is an oblique projection and uk is any vector satisfying Lkuk =vk. The correction term operates only on the interval (Tk−εk, Tk) where Tk−εk < Tk for technical reasons. We will show that with a good choice for those projections (see Equation (6)) we have the following equivalence in law

L(x|(LkxTk =vk)1≤k≤N)∼L(y)

with an explicit density (Theorem 1 below).

In this paper a first part is devoted to the study of general bridges which will provide us the good candidate whose law is absolutely continuous with respect to targeted one. The second one provides the main result. Some properties and proofs are postponed in the appendix to ease the reading.

Notations For the sake of readibility, we choose not to specify arguments when not necessary. For example (1) becomes

dxt=btdt+σtdwt

For allz, the matrix at(z) is defined by

at(z) =σt(z)σt(z)^∗

we suppose that there exists a positive numberρsuch that for all (t, z) ρ⁻¹In< at(z)< ρIn

in the sense of symmetric matrices, whereIn is then- dimensional identity matrix. The functiona⁻¹ is defined by

a⁻¹: [0, T]×Rⁿ → Mn(Rⁿ) (t, x) 7→ at(x)−1

We define the infimum of all theεk

ε0=mink{εk}

2 Bridges and bridges approximations

Bridges

We recall that a bridge is defined as a solution of (3) dyt=bt(yt)dt+σt(yt)d ˜wt−PN

k=1P_t^k(yt)^y_T^t−uk

k−t1_{(Tk−εk,Tk)}(t)dt y0=u0

We assume that the deterministic parametersbandσareC_b^1,2 functions (bounded with bounded deriva- tives). We assume that

(t, z)7→P_t^k(z) is a C_b^1,2function and that for anyz

LkP_t^k(z) =Lk and ker(Lk) = ker(P_t^k(z)) (4) First of all, a lemma to describe the behaviour of processy

Lemma 1. The SDE (3) admits a unique solution on[0, T]in the absolute convergence’s sense meaning that

Z Tk

Tk−εk

kP_t^k(yt)(yt−uk)k

Tk−t dt <+∞

For all k we haveLkyTk =Lkuk almost surely (a.s.). Moreover for Tk−εk < t < Tk, kLk(yt−uk)k ≤ Ck(ω)(Tk−t) log log[(Tk−t)⁻¹+e] a.s., whereCk is a positive random variable.

Proof. Let us remark that for times in [T_k−1, Tk] (withT0= 0) the SDE (3) becomes dyt=btdt−P_t^kyt−uk

Tk−t 1_{(Tk−εk,Tk)}(t)dt+σ(yt)d ˜wt

So that we may reduce the proof to the study of (3) with only one observation time, but we here have to consider random initial conditions. If unicity holds it will lead to the result by concatenation. The proof in the caseN = 1 is given in the appendix with Lemma 6.

Bridges approximations

We now introduce approximations that will be useful in the proof of the main result in next section. Let 0< ε < ε0, we set

dy_t^ε=b^ε_t(y_t^ε)dt+σt(y^ε_t)d ˜wt−X

k

P_t^k(y_t^ε)y^ε_t−uk

Tk−t 1_{(Tk−εk,Tk−ε)}(t)dt, y₀^ε=u0 (5) The only difference with the Bridge Equation (3) is that each correction term is stopped from a distance εfrom the observation time.

Lemma 2. There exists a constant0< κ <1 such that sup

t∈[0,T]

E[ky^ε_t−ytk²]≤Cε^κ

for all0< ε <(ε0∧1)whereCis a positive constant. The numbersC andκdepend onT,N, the(εk)k, the (Ak)k, and the bounds for b andσ.

Proof. Given in the appendix, the proof uses classical techniques and auxiliary processes each defined on [Tk−1, Tk].

3 Result in the case of partial observation

Case where b is bounded

We aim to obtain a Delyon&Hu-type theorem that gives absolute continuity of processxsolution of (1) conditioned on observations (LkxTk =vk)1≤k≤N with respect to a bridge processy solution of (3).

We now consider a peculiar projectionP, for allk andz

P_t^k(z) =at(z)L^∗_k(Lkat(z)L^∗_k)⁻¹Lk (6) We set

A^k_t(z) = (Lkat(z)L^∗_k)⁻¹ and also

βt(z)^k =σt(z)^∗L^∗_kA^k_t(z) and ηk(z) = q

det(A^k_t(z)) Let us remark that

β_t^k(z)^∗β_t^k(z) =A^k_t(z) and Lkσt(z)β_t^k(z) =Imk (7) whereImk is the identitymk−dimensional matrix. Here are both systems we now consider

dxt=bt(xt)dt+σt(xt)dwt, x0=u dyt=bt(yt)dt+σt(yt)d ˜wt−

N

X

k=1

σt(yt)β_t^k(yt)Lkyt−vk

Tk−t 1_{(Tk−εk,Tk)}dt, y0=u (8) The result is the following

Theorem 1. Supposeb,σanda⁻¹ to beC_b^1,2-functions. Then for any bounded continuous function f E[f(x)|(LkxTk =vk)1≤k≤N]

=CEh f(y)

N

Y

k=1

ηk(yTk) exp

−kβ_T^k_k−εk(LkyTk−εk−vk)k² 2εk

+ Z Tk

Tk−εk

−(Lkys−vk)^∗Lkbs(ys)ds Tk−s

−(Lkys−vk)^∗d A^k_t(yt)

(Lkys−vk)

2(Tk−s) − X

1≤i,j≤mk

d

A^k(y.)i,j,(Lky.−vk)i(Lky.−vk)j

s

2(Tk−s)

i (9) whereC is a positive constant.

Proof. This one consists in using approximationsy^εsolutions of (5) of processy solution of (8). Thanks to Girsanov’s theorem, we are able to obtain an equality for all bounded continuous functionf

E[f(x)G^ε(x)] =E[f(y^ε)H^ε(y^ε)]

whereG^ε/H^εis the density given by Girsanov’s theorem. We want to prove that with a good choice for G^εandH^ε, the lefthand member of the last inequality converges to the conditional expectation, and the righthand one converges to what appears in the Theorem 1.

We set for allz∈Rⁿ

h^ε_t(z) =

N

X

k=1

β_t^k(z)vk−Lkz

Tk−t 1_{(Tk−εk,Tk−ε)}(t) Then for all bounded continuous functionf

E[f(y^ε)] =E[f(x) exp{−

Z T 0

h^ε_t(xt)^∗dwt+1

2kh^ε_t(xt)k²dt}]

We are looking for a different expression of the argument of the exponential function. We use Itˆo’s formula forTk−εk< t < Tk−εand use (7) to get

d

kβ^k_t(xt)(Lkxt−vk)k² Tk−t

= 2(Lkxt−vk)^∗A^k_t(xt)Lkdxt

Tk−t +kβ_t^k(xt)(Lkxt−vk)k²

(Tk−t)² dt+ mk

Tk−tdt +(Lkxt−vk)^∗d A^k_t(xt)

(Lkxt−vk)

Tk−t + X

1≤i,j≤mk

d

A^k_i,j(x.),(Lkx.−vk)i(Lkx.−vk)j

t

Tk−t

Thek^th term of (h^ε_t)^∗dwt coming from that one in dxt is now isolated

−2(Lkxt−vk)^∗A^k_tσtdwt

Tk−t −kβ^k_t(Lkxt−vk)k²

(Tk−t)² dt=−d

kβ_t^k(Lkxt−vk)k² Tk−t

+ mk

Tk−tdt +2(Lkxt−vk)^∗A^k_tbtdt

Tk−t +(Lkxt−vk)^∗dA^k_t(Lkxt−vk)

Tk−t + X

1≤i,j≤mk

d

A^k_i,j,(Lkx−vk)i(Lkx−vk)j

t

Tk−t

(10) Since we have

kh^ε_tk²dt=X

k

kβ_t^k(Lkxt−vk)k²

(Tk−t)² 1_{(Tk−εk,Tk−ε)}(t)dt and

(h^ε_t)^∗dwt=−X

k

1_{(Tk−εk,Tk−ε)}(t)(Lkxt−vk)^∗A^k_tσtdwt

Tk−t

we obtain−2(h^ε_t)^∗dwt− kh^ε_tk²dtadding the terms given by (10). Finally, it leads us to a new expression for the density given by Girsanov’s theorem

E[f(y^ε)] =Eh exp

N

X

k=1

−kβ^k_Tk−ε(LkxTk−ε−vk)k²

2ε +kβ^k_Tk−εk(LkxTk−εk−vk)k² 2εk

+ Z Tk−ε

Tk−εk

(Lkxt−vk)^∗A^k_tbtdt

Tk−t + mk

2(Tk−t)dt +(Lkxt−vk)^∗dA^k_t(Lkxt−vk)

2(Tk−t) + X

1≤i,j≤mk

d

A^k_i,j,(Lkx−vk)i(Lkx−vk)j

t

2(Tk−t)

i

In an equivalent way, even if it means changingf

E[f(y^ε)ϕ^ε] =E[f(x)ψ^ε] (11) with

ϕ^ε:=ϕ^ε(y^ε) = Y

k=1^N

ε⁻_k^mk2 η^ε_k(y_T^ε_k−ε) exp

N

X

k=1

−kβ_T^k_k−εk(Lky_T^ε_k−εk−vk)k² 2εk

+ Z Tk−ε

Tk−εk

−(Lky_t^ε−vk)^∗A^k_tbtdt Tk−t

−(Lky^ε_t−vk)^∗dA^k_t(Lky_t^ε−vk)

2(Tk−t) − X

1≤i,j≤mk

d

A^k_i,j,(Lky^ε−vk)i(Lky^ε−vk)j

t

2(Tk−t) (12)

and

ψ^ε=C^ε

N

Y

k=1

η_k^ε(xTk−ε) exp{−kβ^k_Tk−ε(xTk−ε)(LkxTk−ε−vk)k²

2ε } (13)

where for allz∈Rⁿ

η_k^ε(z) =q

det A^k_Tk−ε(z)

and C^ε=Y

k

ε^−mk2 Now using it in the case wheref = 1, we get formally

E[f(y^ε)ϕ^ε]

E[ϕ^ε] = E[f(x)ψ^ε] E[ψ^ε]

the fact that this quantity is finite is given by Proposition 1. The fact that the righthand term converges to the conditional expectation is given by Lemma 9 in the appendix. The proof relies essentially on the use of Aronson’s estimates that provides gaussian bounds for transition probabilities.

The main difficulty of the proof consists in showing almost sure convergence and then uniform one for theϕ^ε. An obvious candidate for the limit is

ϕ=

N

Y

k=1

ε⁻_k^mk2 ηk(yTk) exp

−kβ_T^k_k−εk(yTk−εk)(LkyTk−εk−vk)k² 2εk

+ Z Tk

Tk−εk

−(Lkyt−vk)^∗A^k_tLkbt(yt)ds Tk−t

−(Lkyt−vk)^∗d A^k_t(yt)

(Lkyt−vk)

2(Tk−t) − X

1≤i,j≤mk

d

A^k_i,j(y.),(Lky.−vk)i(Lky.−vk)j

t

2(Tk−t) (14)

Thanks to Lemma 10 given in the appendix, ϕ is well defined. As said before, we want to prove the following

Lemma 3. There exists a decreasing sequence(εi)_i∈^N tending to 0 such that

ilim→∞E[kϕ^εi−ϕk] = 0

Proof. The proof is decomposed into two main parts. First one aims at showing the almost sure conver- gence ofϕ^εi. In second part we prove thatE[ϕ^εi] tends toE[ϕ]. Finally to conclude, we will use Scheff´e’s lemma.

For almost sure convergence, we first use triangular inequality

|ϕ^ε(y^ε)−ϕ(y)| ≤ |ϕ^ε(y^ε)−ϕ^ε(y)|+|ϕ^ε(y)−ϕ(y)|

The second one converges to 0, this is given by Lemma 10. We now treat the term|ϕ^ε(y^ε)−ϕ^ε(y)|. ϕ^ε(y^ε)

ϕ^ε(y) =

N

Y

k=1

η^ε_k(y_T^ε_k−ε) η^ε_k(yTk−ε)expn

−kβ^k_Tk−εk(y_T^ε_k−εk)(Lky^ε_Tk−εk−vk)k²− kβ^k_Tk−εk(yTk−εk)(LkyTk−εk−vk)k² 2εk

+ Z Tk−ε

Tk−εk

−(Lky_t^ε−vk)^∗A^k_t(y^ε_t)Lkbt(y^ε_t)−(Lkyt−vk)^∗A^k_t(yt)Lkbt(yt)

Tk−t dt

−(Lky^ε_t−vk)^∗d A^k_t(y_t^ε)

(Lky_t^ε−vk)−(Lkyt−vk)^∗d A^k_t(yt)

(Lkyt−vk) 2(Tk−t)

−X

i,j

d

A^k_i,j(y_.^ε),(Lky_.^ε−vk)i(Lky_.^ε−vk)j

t−d

A^k_i,j(y.),(Lky.−vk)i(Lky.−vk)j

t

2(Tk−t)

o

We can write it respecting the order above ϕ^ε(y^ε)

ϕ^ε(y)

N otation

=

N

Y

k=1

Ξ^ε_kexp{Υ^ε_k+ Ψ^ε_k+ Θ^ε_k+ Φ^ε_k}

According to Lemma 2 there exists a decreasing sequence (εi)_i∈^Ntending to 0 satisfying for allkthat y_T^εi

k−εi converges almost surely toyTk. From this we obtain the fact that Ξ^ε_kⁱ converges almost surely to 1 and Υ^ε_kⁱ to 0 using regularity ofσ. Then for allk

|Ψ^ε_k| ≤ Z Tk−ε

Tk−εk

Lk(y_t^ε−yt)^∗A^k_t(y_t^ε)Lkbt(y_t^ε) Tk−t

+

(Lkyt−vk)^∗ A^k_t(y_t^ε)Lkbt(y_t^ε)−A^k_t(yt)Lkbt(yt) Tk−t

dt Sinceband σare bounded we use Lemma 1 to get

|Ψ^ε_k| ≤C

Z Tk−ε Tk−εk

ky^ε_t−ytk² Tk−t dt

!¹₂

Z Tk−ε Tk−εk

(1 + log log (Tk−t)⁻¹+e

Tk−t dt

!¹₂

whereC andC^′ are positive random variables. Thanks to Lemma 2, up to an extracted subsequence

ilim→∞

Z Tk−εi

Tk−εk

ky_t^εi−ytk² Tk−t dt= 0

that leads us to convergence for allk of|Ψ^ε_kⁱ|to 0. Now we use Identity (34) Θ^ε_k= kLyt−vk²

T−t pt(yt)dt+kLyt−vk²

T−t qt(yt)dwt+kLyt−vk²

(T−t)² rt(yt)dt

−kLy_t^ε−vk²

T−t pt(y^ε_t)dt−kLy^ε_t−vk²

T−t qt(y_t^ε)dwt−kLy^ε_t−vk²

(T−t)² rt(y^ε_t)dt where p, q, and r are all C_b^1,2 functions. Hence using Lemmas 1 and 2 as above we obtain that limεi→0|Θ^ε_kⁱ| = 0 up to a subsequence. It remains to treat the term Φ^ε_k. Still using Identity (34), Lemmas 1 and 2 we show that limεi→0|Φ^ε_kⁱ|= 0 even if it means extracting once more a subsequence.

We have obtained almost sure convergence ofϕ^εtoϕ. Then we show the convergence of the expectations.

For this we set a preliminary result

Proposition 1. There exist two positive constantsc1 andc2 such that for all 0< ε < ε0

c1≤C^εE[ψ^ε]≤c2

Proof. We give an explicit expression C^εE[ψ^ε] =

Z

q^ε(ζ1, . . . , ζN)

N

Y

k=1

η_k^ε(ζk)^−mk2 exp{−kβ^k_Tk−ε(ζk)(Lkζk−vk)k²

2ε }dζk

whereq^εis the density of (xT1−ε, . . . , xTN−ε). Under theorem’s assumptionsxis a strong Markov process, with positive transition density. Fors, t∈[0, T], we denoteps,t(u, z) the density of x^s,u_t solution of (1) initialized to beuat times. Then thanks to Aronson’s estimates there exist positive constantµ, λ, M and Λ such that the densitypsatisfies fors < t

µ(t−s)⁻ⁿ²e^−λkz−uk

2

t−s ≤ps,t(u, z)≤M(t−s)⁻ⁿ²e^−Λkz−uk

2 t−s

Now usingpwe are able to write

q^ε(ζ1, . . . , ζN) =p0,T1−ε(u, ζ1). . . pTN−1−ε,TN−ε(ζN−1, ζN)

Then we apply Aronson’s estimates and the fact that for alli, j the coordinateA^k_i,j is bounded by two positive constants. We obtain bounds forC^εE[ψ^ε] of the type

λ⁻¹C^ε Z

exp{

N

X

j=1

−λkLkζk−vkk²

2ε −λkζ1−uk² T1−ε −

N

X

k=2

λkζk−ζk−1k² Tk−T_k−1 }

N

Y

k=1

dζk

where λis a positive constant large for the lower bound and small for the upper one. The integral can be interpreted as a gaussian expectation where









 ζ1

ζ2−ζ1

... ζN −ζ_N−1











is a centered gaussian vector with covariance matrix

R^ε= 1 2λ













(T1−ε)In 0 . . . 0

0 (T2−T1)In . .. ...

... . .. . .. 0

0 . . . 0 (TN−TN−1)In













where In is the n-dimensional identity matrix. As a remark, in the sense of symmetric matrices, there exist two positive constantsc1andc2such that

c1IN n< R^ε< c2IN n

whereIN nis theN n-dimensional identity matrix. Thus the gaussian vector





 ζ1

... ζN







admits for covariance matrix

Γ^ε=G⁻¹R^εG^−∗

where

G=



















In 0 . . . 0

−In . .. ... ... 0 . .. ... ... ... ... . .. ... ... 0 0 . . . 0 −In In



















We still keep bounds for the covariance matrix

c1IN n<Γ^ε< c2IN n

Now we can get bounds forC^εE[ψ^ε] with expectations of type λ^mk2 C^εE[exp{−

N

X

k=1

λkLkXk−vkk²

2ε }]

whereXk is an-dimensional gaussian variable. Then we use Lemma 11 given in the appendix to obtain the fact that

R^m1× · · · ×R^mN → R (v1, . . . , vN) 7→ C^εQN

k=1λ^mk2 E[exp{−^λkLkX2ε^k−vkk2}] is a gaussian density of a variable (L1X1+p_ε

λY1, . . . , LNXN+p_ε

λYN) where theYkaremk-dimensional centered normalized gaussian vectors. Moreover the two families (Xk)k and (Yk)k are independant.

Finally using bounds obtained above for Γ^εwe get the fact that for all 0< ε < ε0

c1< C^εE[ψ^ε]< c2

As a first consequence, thanks to identity (11),E[ϕ^ε] is finite so that E^ϕ[ϕ^εε] is a density. We may also use Fatou’s lemma to get

E[ϕ]≤lim inf

ε→0 E[ϕ^ε]≤c2

It takes more work to control lim sup_ε→0E[ϕ^ε].

Let J > 0 be a large number, we introduce for all process (zt)_t∈[0,T] and for all 1 ≤ k ≤ N the stopping timeτ_k^ε

τ_k^ε= inf{tk< t≤Tk−ε : 1

√Tk−texp{−kLkzt−vkk²

2(Tk−t) D} ≤J⁻¹}

= inf{tk< t≤Tk−ε : kLkzt−vkk²

2(Tk−t) ≥D⁻¹log J

√Tk−t

}

where D is a positive constant such thatDId ≤Ak. We know that such a constant exists according to assumptions on the functiona. Thetk are chosen to be real numbers contained in (Tk−1, Tk−ε). As a convention we setτ_k^ε=Tk if the condition is empty. Letτ^εbe the first of theτ_k^ε such that the condition is non-empty

τ^ε= inf

k {τ_k^ε : τ_k^ε< Tk}

we set as conventionτ^ε=T if if for allk,τ_k^ε=Tk. Even if it means changingf in Equation (11) E[f(y^ε)1_τε<Tϕ^ε]

E[ϕ^ε] = E[f(x)1_τε<TC^εψ^ε] E[C^εψ^ε] We recall that

C^εψ^ε=Y

k

ε^−mk2 exp{−kβ_T^k_k−ε(LkxTk−ε−vkk²

2ε }

We now consider to be on set{τ^ε=τ_k^ε} tk

τ^ε

Tk

Tk−1 Tk−ε

We decomposeC^εψ^ε into three parts as a product of three factors F1=Y

j<k

ε^−mj2 exp{−kβ_T^j_j−ε(LjxTj−ε−vj)k²

2ε }

F2=ε^−mk2 exp{−kβ_T^k_k−ε(LkxTk−ε−vk)k²

2ε }

F3=Y

j>k

ε^−mj2 exp{−kβ_T^j_j−ε(LjxTj−ε−vj)k²

2ε }

We are interessed in

E[C^εψ^ε1_τε=τ_k^ε] =E[F1F2F31_τε=τ_k^ε]

We now use Markov’s property to get independance between Past and Future knowing Present (cf [3]

see last chapter about conditional expectations)

E[C^εψ^ε1_τε=τ_k^ε] =E[F1E[F2F31_τε=τ_k^ε|τ_k^ε, xτ_k^ε]] =E[F1E[F21_τε=τ_k^ε|τ_k^ε, xτ_k^ε]E[F3|xτ_k^ε]] (15)

=E[F1E[F21_τ^ε_=τ^ε

k|τ_k^ε, xτ_k^ε]E[F3|xTk−ε]] (16) In order to study the factorE[F21_τε=τ_k^ε|F_τε

k] we introduce θt= 1

√Tk−texp{−kβ_t^k(Lkxt−vk)k² 2(Tk−t) }

Fort∈[Tk−1, Tk−ε], we set zt=Lkxt−vk, pt=kβ_t^k(Lkxt−vk)k. We recall thatβ_t^k=σ_t^∗L^∗_kA^k_t with A^k_t = (β^k_t)^∗β_t^k= (LkatL^∗_k)⁻¹. With respect to these notations, we have

p²_t =z_t^∗A^k_tzt

It is also easy to see that

dzt=Lkbtdt+Lkσtdwt

and then dhzit=LkatL^∗_kdt= (A^k_t)⁻¹dt. We use Itˆo’s formula d(p²_t) = d(z_t^∗A^k_tzt) = 2ztA^k_tdzt+z_t^∗dA^k_tzt+X

i,j

dhA^k_i,j, zizjit+mkdt Then

d p²_t

Tk−t = 2ztA^k_tdzt

Tk−t + p²_tdt

(Tk−t)²+z_t^∗dA^k_tzt

Tk−t + mkdt Tk−t+

P

i,jdh∆i,j, zizjit

Tk−t First using definitions ofz,β^k andA^k we get

z^∗_tA^k_tdzt=z_t^∗A^k_tLkdxt=z_s^∗A^k_tLkσtσ_t⁻¹dxt

=z_t^∗(β_t^k)^∗σ_t⁻¹btdt+z^∗_t(β_t^k)^∗dwt

This leads us to the existence of two bounded adapted processesr⁽¹⁾ and r⁽²⁾ defined on [T_k−1, Tk−ε]

such that

z^∗_tA^k_tdzt=ptr_t⁽¹⁾dt+ptr⁽²⁾_t dwt

In a same way we remark that there exist two bounded adapted processesr⁽³⁾ andr⁽⁴⁾ such that d(LkatL^∗_k) =r⁽³⁾_t dt+r⁽⁴⁾_t dwt

we get even if it means changingr⁽³⁾ andr⁽⁴⁾

z_t^∗dA^k_tzt=z_t^∗ d(LkatL^∗_k)⁻¹

zt=p²_tr⁽³⁾_t dt+p²_tr_t⁽⁴⁾dwt

Finally, we obtain existence of two bounded adapted processesrandr^′ such that d p²_t

T−t =2ptdwt

T−t + p²_tdt (T−t)² +rt

p²_t

T−tdwt+ dt

T−t+r^′_tp²_t+pt

T −t dt From this we deduce that quadratic variation

d p²_t

Tk−t

= 4p²_t+r_t²p⁴_t+ 4rtp³_t (Tk−t)² dt Now we apply Itˆo’s formula to the functionθ always fort∈[Tk−1, Tk−ε]

dθt= θtdt 2(Tk−t)−1

2θtd p²_t

Tk−t

+1 8θtd

p²_t Tk−t

We deduce from the three last equations after simplification of four terms that there exists a martingale M and a bounded adapted processr^′′ both defined on [Tk−1, Tk−ε] such that

dθt= dMt+θtr^′′_t

p²_t+pt

Tk−t + p⁴_t+p³_t (Tk−t)²

dt

For anyη >0, functionsx7→e^−ηz22|z|^mform= 1,2,3,4 are all bounded, then there exists a constant cη such that

(p

Tk−t θt)^η

p²_t+pt

Tk−t + p⁴_t+p³_t (Tk−t)²

≤ cη

√Tk−t

This gives us the existence of a bounded adapted processπ defined on [Tk−1, Tk−ε] that allows us to write

dθt= dMt+θ_t^1−η(Tk−t)^−hαtdt withh=^1+η₂ . We now integrate it fort∈(τ_k^ε, Tk−ε]

θt=θτ_k^ε+Mt−Mτ_k^ε+ Z t

τ_k^ε

πsθ¹_s^−η(Tk−s)^−hds This leads to the following

E[θt1_τ^ε

k<t]≤J⁻¹+ ¯π Z t

tk

E[θs1_τ^ε

k<s]^1−η(Tk−s)^−hds where ¯π= sup_s|πs|. SoE[θ1τ_k^ε<t] is bounded by the solutionuof

dus= ¯αu¹_s^−η(Tk−s)^−hds, utk=J⁻¹ and this equation has an explicit solution

ut= ηα¯

1−h[(Tk−tk)^1−h−(Tk−t)^1−h] +J^{−η 1}_η

≤

ck(Tk−tk)^1−h+J^−η

1 η