Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

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Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

Abass Sagna, Christophe Profeta

To cite this version:

Abass Sagna, Christophe Profeta. Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method. 2012. �hal-00753887�

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

CHRISTOPHE PROFETA ^∗ ABASS SAGNA^†

Abstract

The model consists of a signal processX which is a general Brownian diffusion process and an observation processY, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the processY is observed from time0tos >0at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed processX crosses a fixed barrier after a given timet > s. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.

1 Introduction

We consider in this work a nonlinear filtering model where the signal processXand the observation processY evolve following the stochastic differential equations:

(dX_t =b(X_t, t)dt+σ(X_t, t)dW_t, X₀ =x₀,

dY_t=h(Y_t, X_t, t)dt+ν(Y_t, t)dW_t+δ(Y_t, t)dfW_t, Y₀ =y₀. (1) In these equations, W and fW are two independent standard real valued Brownian motions. We suppose that the functions b, σ, h, ν, and δ are Lipschitz and that, for every (x, t) ∈]0,+∞)², δ(x, t)>0,ν(x, t)>0andσ(x, t)>0.

Letabe a real number such that0<a< x₀and let

τa^X = inf{u≥0, X_u ≤a}

be the first hitting time of the barrieraby the signal process. As usually, we consider thatinf∅= +∞. Our aim is to estimate the distribution of the conditional hitting time

P τa^X > t| Fs^Y

(2)

fort≥s >0and where(Ft^Y)_t≥0 is the filtration generated by the observation processY: Fs^Y =σ(Yu, u≤s).

More generally, we shall denote by(Ft^Z)_t≥0the filtration generated by the processZ.

Such a problem arises for example in credit risk when modeling a credit event in a structural model as the first hitting time of a barrieraby the firm value processX. Investors are supposed to

∗Fédération de Mathématiques d’Evry, Laboratoire d’Analyse et de Probabilités, 23 Boulevard de France, 91037 Evry, e-mail:christophe.profeta@univ-evry.fr

†Fédération de Mathématiques d’Evry, Laboratoire d’Analyse et de Probabilités, 23 Boulevard de France, 91037 Evry,

& ENSIIE, e-mail:abass.sagna@ensiie.fr. Both author’s researches are supported by an AMaMeF exchange grant and the “Chaire Risque de Crédit” of the French Banking Federation.

have no access to the true value of the firm but only to the observation processY, which is correlated to the value of the firm (see e.g. [3, 4]). We will typically suppose that we observe the processY at regular discrete timest₀ = 0 < t₁ < . . . < t_m = sover the time interval [0, s]and intend for estimating the quantity

P τa^X > t|Y_t0, . . . , Y_tm

for every t ≥ s. Note that, if t < s, then, applying the Markov property to the diffusion Y, the computations boil down to the cases=t. In [2], the quantity

P

u∈[s,t]inf X_u>a|Y_t0, . . . , Y_tm

has been estimated by a hybrid Monte Carlo-Optimal quantization method in the case where the observation process dynamics is given by:

dYt=Yt

h(Xt)dt+ν(t)dWt+δ(t)dWft

, Y0 =y0,

whereνandδare deterministic functions. However, the approach used in the previous work does not apply to our framework because we want to compute the conditional distribution of a function of the whole trajectory of the signal process from0totgiven the observations from0tos, withs≤t.

Example 1. A particular case of model (1) one may consider is the following “Black-Scholes” case:

(dX_t=X_t(µdt+σdW_t), X₀=x₀,

dY_t=Y_t(rdt+νdW_t+δdWf_t), Y₀=y₀, (3) so that

dYt

Y_t = ν σ

dXt

X_t + r−µν σ

dt+δdfW_t, (4) or

Y_t= y₀

(x₀)^ν/σX_t^ν/σexp

δfW_t+

r−ν²+δ² 2 −µν

σ + νσ 2

t

. (5)

Observe that settingr =µandσ=ν yields dY_t

Y_t = dX_t

X_t +δdfWt,

meaning that the return onY is the return onXaffected by a noise (see e.g. [3]).

Of course, in this case, we may compute theoretically the expression (2) by noticing that:

τa^X = inf{u≥0, X_u ≤a}

= inf

u≥0, δfW_u−δ²

2u≥ln(Y_u) + ln x₀

y₀

−ln(a)

and that, conditionally toFs^Y, the process

fW_u, u≤s

has the same law as σ

√σ²+δ²B_u+ δ σ²+δ²

ln(Y_u) + ln x₀

y₀

+σ²+δ²

2 u

, u≤s

whereBis a standard Brownian motion independent from the processY. This follows from the fact that, sinceW andfW are two independent Brownian motion, so are

B= σW +δWf

√σ²+δ² and Be = δW −σWf

√σ²+δ² ,

and from the relations :

B_t= 1

√σ²+δ²

ln(Y_t) + ln x₀

y₀

+σ²+δ² 2 t

and Wf_t= δB_t−σBe_t

√σ²+δ² .

Therefore, in this particular setting, the problem boils down to the computation of the first passage time of a Brownian motion to a curved boundary. We refer to [3] for some similar, and far more general, considerations.

Example 2. Another particularly simple case is given when both the signal and the observation pro- cess evolve following the Ornstein-Uhlenbeck dynamics:

(dX_t=λ(θ−X_t)dt+σdW_t, X₀ =x₀,

dYt=λ(θ−Yt)dt+σdWt+δdfWt, Y0 =y0. (6) In this case, we have :

X_t=Y_t−(y₀−x₀)e^−λt−δe^−λt Z t

0

e^λudfW_u, so that

τa^X = inf

u≥0, Yu−a−(y0−x0)e^−λu≤δe^−λu Z u

0

e^λvdfWv

and, conditionally toFs^Y, the process

e^−λu Z _u

0

e^λvdfW_v, u≤s

has the same law as σ

√σ²+δ²U_u+ δe^λu σ²+δ²

Y_u−y₀e^−λu−θ(1−e^−λu)

, u≤s

whereU is an Ornstein-Uhlenbeck process with parametersλand 0 started from 0, and independent fromY. This follows from Knight’s representation theorem combined with the same ideas as above.

The rest of the paper is organized as follows: in Section 2, we state and prove the main theorems, i.e. we give an approximation of the expectation (2) when X is replaced by its continuous Euler schemeX, see especially Subsection 2.3. Then, in Section 3, we introduce some numerical tools in¯ order to compute the quantities involved. We finally conclude the paper by a few simulations.

2 Estimation of the conditional survival probability

2.1 Preliminaries results

To deal with the computation of the conditional hitting time (2), we introduce the first hitting time from times:

τ^(s) = inf{u≥s, X_u≤a}. Define furthermore the filtration(Gt)_t≥0 by, for everyt≥0,

Gt=σ{(W_u,fW_u), u≤t}. (7) We have the following result.

Lemma 2.1. For everyt≥s,

P τa^X > t| Fs^Y

= Eh 11_{τX

a >s}F(s, t, X_s)| Fs^Y

i (8)

with

F(s, t, X_s) =P

u∈[s,t]inf X_u >a|X_s

. Proof. Ift≥s, one has

{τa^X > t}={τa^X > s} ∩ {τ^(s)> t}, so that

P τa^X > t| Fs^Y

= P {τa^X > s} ∩ {τ^(s)> t}| Fs^Y

= E

h E

h 11_{τ^X

a >s}11_{τ(s)>t}|Gs

i|Fs^Y

i ,

where theσ-algebra Gsis defined in (7). Hence, using the fact thatWf is independent fromW and the Markov property of the processX:

P τa^X > t| Fs^Y

= Eh 11_{τX

a >s}Eh

11_{τ(s)>t}|Gs

i|Fs^Y

i

= E

h 11_{τ^X

a >s}E h

11_{τ(s)>t}|Fs^W

i|Fs^Y

i

= Eh 11_{τX

a >s}P τ^(s)> t|Fs^X

|Fs^Y

i

= E

h 11_{τ^X

a >s}P inf

u∈[s,t]Xu >a|Xs

|Fs^Y

i . This shows assertion (8).

Note that in general, there is no closed-form expression for computingP inf

u∈[s,t]X_u>a|X_s , except in some very special cases, such as Brownian motion or Bessel processes... We shall therefore need an approximation of this expression, which is the purpose of the next subsection.

2.2 The continuous Euler scheme

Let us denote by X¯ the continuous Euler scheme process associated to the signal process X and defined by

X¯_s= ¯X_s+b( ¯X_s, s)(s−s) +σ( ¯X_s, s)(W_s−W_s), X¯₀=x₀,

withs=t_kifs∈[t_k, t_k+1). To estimate the distribution of the conditional hitting time given in (2), consider that we observe the processY at discrete and regularly spaced times: t₀, t₁, . . . , t_m, with 0 = t₀ < · · · < t_m = s < t_m+1 < · · · < t_n = t. In our general model (see Equation (1)), the discrete time observation processes{X¯_tk, k = 0, . . . , n}and{Y_tk, k= 0, . . . , m}are obtained from Euler scheme as:

(X¯t_k+1 = ¯Xtk+b( ¯Xtk, t_k)∆_k+σ( ¯Xtk, t_k)(Wt_k+1−Wtk)

Y_tk+1 =Y_tk+h(Y_tk,X¯_tk, t_k)∆_k+ν(Y_tk, t_k)(W_tk+1−W_tk) +δ(Y_tk, t_k)(fW_tk+1−fW_tk) (9) wherek = 0, . . . , m−1for the observation process, kgoing tilln−1for the signal process and

∆_k=t_k+1−t_k. Note that ift > s, the number of discretization steps over[0, s]may differ from the number of discretization steps over[s, t]so that we choose





t_k= ^ks_m fork= 0, . . . , m t_k=s+^k(t−s)_n fork=m+ 1, . . . , n.

Supposing that we have observed the trajectory(Y_t0, . . . , Y_tm)of the observation processY we aim to estimate

P(τa^X > t_n|Y_t0, . . . , Y_tm) by

P τa^X¯ > t_n|Y_t0, . . . , Y_tm

, (10)

where

τa^X¯ = inf{u≥0,X¯_u≤a}.

To this end, we will use a useful result called the regular Brownian bridge method. This result recalled below allows us to compute the distribution of the minimum (or the maximum) of the continuous Euler schemeX¯ of the processXover the time interval[0, t], given its values at discrete and regular time observation points0 =t₀< t₁<· · ·< t_n=t(see, e.g. [6, 7]).

Lemma 2.2. Let(X_t)_t≥0 be a diffusion process with dynamics given by X_t=x+

Z t 0

b(X_u, u)du+ Z t

0

σ(X_u, u)dW_u

and let( ¯X_t)_t≥0be its associated continuous Euler process. Then, the following equality in law holds:

L

u∈[0,t]min

X¯_u|X¯_tk =x_k, k= 0,· · ·, n

=L

k=0,···,n−1min

H_∆^xk,xk+1

kσ²(xk,tk)

−1

(Λ_k)

, (11) where (Λ_k)_{k=0,···,n−1} are i.i.d. random variables uniformly distributed over the unit interval and H_∆^xk,xk+1

kσ²(xk,tk)

−1

is the inverse function of the conditional cumulative functionH_∆^xk,xk+1

kσ²(xk,tk), defined by

H_∆^xk,xk+1

kσ²(xk,tk)(u) :=

(exp

−_∆kσ2(x²k,tk)(u−x_k)(u−x_k+1)

ifu≤min(x_k, x_k+1)

1 otherwise.

(12) In the following, we shall replace the expressionσ(x_k, t_k)in the expression ofHbyσ_kand rather write:H_∆^xk,xk+1

kσ²_k .

Short proof. Observe first that, conditionally to{X¯_tk = x_k; k = 0,· · · , n}, the random variables

u∈[tmink,tk+1]

X¯u; k= 0, . . . , n−1

are mutually independent, thanks to the independent increments property of Brownian motion. Then, it suffices to notice that computing the law of min

u∈[tk,t_k+1]

X¯_u conditionally to{X¯_tk =x_k,X¯_tk+1 =x_k+1}amounts to computing the law of the hitting times of the bridge of a Brownian motion with drift. This law is well-known to be independent from the drift (i.e.

from the functionbhere) and to be given by an expression such as (12).

In the rest of the paper, the function 1 −H_∆^xk,xk+1

kσ²_k (a) will be often used. We shall denote it by G^x_∆^k,xk+1

kσ²_k (a), so that G^x_∆^k,xk+1

kσ²_k (a) =

1−exp

−2(x_k−a)(x_k+1−a)

∆_kσ²(x_k, t_k)

11_{xk≥a;x_k+1≥a}. (13) Now, in our set-up, we shall apply the Brownian bridge method to obtain the following lemma, which is taken from [2]:

Lemma 2.3. We have P

u∈[tinfm,tn]

X¯_u >a( ¯X_tk)_k=0,...,m

=

n−1Y

k=m

G^{X¯tk,X¯tk+1}

∆kσ²_k (a). (14) Proof. It follows from Lemma 2.2 that

P

u∈[tinfm,tn]

X¯_u > a( ¯X_tk)_k=0,...,m

=P min

k=m,···,n−1

H^{X¯tk,X¯tk+1}

∆kσ²_k

−1

(Λ_k)>a

! . Since the functionH^zk,zk+1

∆kσ²_k (·)is non-decreasing and theΛ_k’s are i.i.d. uniformly distributed random variables, this reduces to:

P

u∈[tminm,tn]

X¯_u > a( ¯X_tk)_k=0,...,m

=

n−1Y

k=m

P

Λ_k≥H_∆^{X¯tk,X¯tk+1}

kσ²_k (a)

=

n−1Y

k=m

1−H^{X¯tk,X¯tk+1}

∆kσ_k² (a)

. This completes the proof.

2.3 Main theorem

We now state and prove the main theorems of this section. We first show that the conditional hit- ting time of the continuous Euler processX¯ given the discrete path observationsY_t0, . . . , Y_tm may be written as an expectation of an explicit functional of the discrete pathX¯_t0, . . . ,X¯_tm of the signal given the observationsYt0, . . . , Ytm. Therefore, we reduce the initial problem to the characterization of the conditional distribution ofX¯_t0, . . . ,X¯_tm givenY_t0, . . . , Y_tm.

Theorem 2.4. SetX¯^m= ( ¯X_t0, . . . ,X¯_tm)andY^m = (Y_t0, . . . , Y_tm). We have:

P τa^X¯ > t_n|Y_t0, . . . , Y_tm

= Ψ(Y_t0, . . . , Y_tm), (15) where fory^m:= (y₀, . . . , y_m),

Ψ(y^m) =E

Φ ¯X^mY^m=y^m

. (16)

The functionΦis defined for everyz^m= (z0, . . . , zm)by Φ(z^m) = ¯F(t_m, t_n, z_m)

m−1Y

k=0

G^z_∆^k,zk+1

kσ²_k (a), with

F¯(t_m, t_n, x) =E

"_n−1 Y

k=m

G^X_∆^¯tk,X¯tk+1

kσ_k² (a)X¯_tm =x

# . Proof. We may show similarly to Equation (8) in Lemma 2.1 that

P τa^X¯ > t_nY_t0, . . . , Y_tm

=E h

11_{τX^¯

a >tm}F¯(t_m, t_n,X¯_tm)|Y_t0, . . . , Y_tmi , with

F¯(t_m, t_n,X¯_tm) =P

u∈[tinfm,tn]

X¯_u>aX¯_tm

.

It follows from Lemma 2.3 that F(t¯ _m, t_n,X¯_tm) =E

P

u∈[tminm,tn]

X¯_u >a( ¯X_tk)_k=0,...,m X¯_tm

=E

"_n−1 Y

k=m

G^X_∆^¯tk,X¯tk+1

kσ²_k (a)X¯_tm

# . On the other hand, combining a successive conditioning rule with the independence betweenW and Wfgives :

P τa^X¯ > t_n|Y_t0, . . . , Y_tm

= E h

E h11_{τX^¯

a >tm}F(t¯ _m, t_n,X¯_tm)|(W_tk,Wf_tk)_k=0,...,mi

|Y^mi

= E h

E h

11_{τX^¯

a >tm}F(t¯ _n, t_m,X¯_tm)|(W_tk)_k=0,...,mi

|Y^mi

= Eh

F¯(t_m, t_n,X¯_tm)P

τa^X¯ > t_m|(W_tk)_k=0,...,m

|Y^mi

= E

hF¯(t_m, t_n,X¯_tm)P

τa^X¯ > t_m|( ¯X_tk)_k=0,...,m

|Y^mi . Now, using Lemma 2.3 once again, we have

P

τa^X¯ > t_m|( ¯X_tk)_k=0,...,m

=

m−1Y

k=0

G^X_∆^¯tk,X¯tk+1

kσ_k² (a) so that

P τa^X¯ > t_n|Y_t0, . . . , Y_tm

=E

Φ( ¯X^m)|Y_t0, . . . , Y_tm where for everyx^m:= (x₀, . . . , x_m),

Φ(x^m) = ¯F(t_m, t_n, x_m)

m−1Y

k=0

G^x_∆^k,xk+1

kσ²_k (a).

This completes the proof.

It remains now to characterize the law ofX¯^m|Y^m = y^mappearing in (16). This is the purpose of the following theorem, in which we shall write the conditional survival probability in a usual form with respect to standard nonlinear filtering problems.

We set from now onσ_k = σ(x_k, t_k), b_k = b(x_k, t_k), δ_k = δ(y_k, t_k), h_k = h(y_k, x_k, t_k), andν_k = ν(y_k, t_k). We next give the main result of the paper.

Theorem 2.5. We have:

P τa^X¯ > t_n|Y_t0, . . . , Y_tm

= Ψ(Y_t0, . . . , Y_tm), (17) where fory^m:= (y₀, . . . , y_m),

Ψ(y^m) = EF¯(t_m, t_n,X¯_tm)K^mL^m_y

E[L^m_y ] , (18) with

K^m=

m−1Y

k=0

G^{X¯tk,X¯tk+1}

∆kσ_k² (a) and L^m_y =

m−1Y

k=0

g_k( ¯X_tk, y_k; ¯X_tk+1, y_k+1).

The functiong_kis defined by

g_k(x_k, y_k;x_k+1, y_k+1) = 1

(2π∆_k)^3/2σ²_kδ_kexp − ν_k² 2δ²_k∆_k

x_k+1−m¹_k

σ_k −y_k+1−m²_k ν_k

2! (19) withm¹_k:=x_k+b_k∆_k and m²_k :=y_k+h_k∆_k.

Proof. It follows from Theorem 2.4 that P τa^X¯ > tn|Yt0, . . . , Ytm

=E

Φ( ¯X^m)|Yt0, . . . , Ytm

where for everyx^m:= (x0, . . . , xm),

Φ(x^m) = ¯F(t_m, t_n, x_m)

m−1Y

k=0

G^xk,xk+1

∆kσ²_k (a).

It remains to characterize the conditional distribution ofX¯^m givenY^m. Recall that the dynamics of the processes( ¯X_tk)and(Y_tk)are given fork= 0, . . . , mby

(X¯t_k+1 = ¯Xtk +b( ¯Xtk, t_k)∆_k+σ( ¯Xtk, t_k)(Wt_k+1−Wtk)

Y_tk+1 =Y_tk +h(Y_tk,X¯_tk, t_k)∆_k+ν(Y_tk, t_k)(W_tk+1−W_tk) +δ(Y_tk, t_k)(fW_tk+1−Wf_tk) (20) LetPxk,yk( ¯X_tk+1 ∈dx_k+1, Y_tk+1 ∈dy_k+1)denote the density function of the couple( ¯X_tk+1, Y_tk+1) given( ¯X_tk, Y_tk) = (x_k, y_k). Using Bayes formula and the Markov property of the process( ¯X_tk, Y_tk)_k we get:

Px0,y0( ¯X_t1 ∈dx₁, . . . ,X¯_tm ∈dx_m|Y_t1 ∈dy₁, . . . , Y_tm ∈dy_m)

= Px0,y0 ( ¯X_t1 ∈dx₁, Y_t1 ∈dy₁);. . .; ( ¯X_tm ∈dx_m, Y_tm ∈dy_m) Px0,y0(Y_t1 ∈dy₁, . . . , Y_tm∈dy_m)

= Px0,y0 X¯_t1 ∈dx₁, Y_t1 ∈dy₁)× · · · ×Pxm−1,ym−1( ¯X_tm ∈dx_m, Y_tm ∈dy_m) Px0,y0(Y_t1 ∈dy₁, . . . , Y_tm ∈dy_m) . For everyx^m = (x₀, . . . , x_m)andy^m = (y₀, . . . , y_m), set :

N_k(x^m;y^m) :=Px0,y0 X¯_t1 ∈dx₁, Y_t1 ∈dy₁)× · · · ×Pxm−1,ym−1( ¯X_tm ∈dx_m, Y_tm ∈dy_m).

With this notation, the denominator reads:

Px0,y0(Yt1 ∈dy1, . . . , Ytm ∈dym) = Z

N_k(x^m;y^m)dx1. . . dxm.

On the other hand, the random vector ( ¯X_tk+1, Y_tk+1)( ¯X_tk, Y_tk) = (x_k, y_k)has a Gaussian distribu- tion with meanm_kand covariance matrixΣ_kgiven for everyk= 0, . . . , m−1by

m_k =

x_k+b_k∆_k y_k+h_k∆_k

, Σ_k= ∆_k

σ²_k σ_kν_k σ_kν_k ν_k²+δ²_k

. (21)

Then, for everyk= 0, . . . , m−1, the densityf_k(x_k, y_k;x_k+1, y_k+1)of( ¯X_tk+1, Y_tk+1)( ¯X_tk, Y_tk) = (x_k, y_k)reads

f_k(x_k, y_k;x_k+1, y_k+1) = 1

2πσ_kδ_k∆_kexp −ν_k²+δ_k² 2δ_k²∆_k

n(x_k+1−m¹_k)²

σ²_k +(y_k+1−m²_k)² ν_k²+δ²_k

− 2ν_k

σ_k(ν_k²+δ_k²)(x_k+1−m¹_k)(y_k+1−m²_k)o! , (22) wherem¹_k:=x_k+b_k∆_k and m²_k :=y_k+h_k∆_k. As a consequence,

P

τa^X¯ > t_n|(Y_t0, . . . , Y_tm) =y^m

= E

Φ( ¯X^m)|(Y_t0, . . . , Y_tm) =y^m

=

R Φ(x^m)N_k(x^m;y^m)dx₁. . . dx_m

R N_k(x^m;y^m)dx₁. . . dx_m . (23)

Now, we know that the random variableX¯_tk+1|X¯_tk =x_khas a Gaussian distribution with meanm¹_k and varianceσ²_k∆_k. Its densityP_k(x_k, x_k+1)dx_k+1is therefore given by :

P_k(x_k, x_k+1)dx_k+1= 1

√2π∆_kσ_kexp

−(x_k+1−m¹_k)² 2σ²_k∆_k

dx_k+1.

Then, by definition ofN_k, we may write:

Z

N_k(x^m;y^m)dx₁. . . dx_m

= Z ^m−1Y

k=0

f_k(x_k, y_k;x_k+1, y_k+1)dx_k+1

= Z m−1Y

k=0

f_k(x_k, y_k;x_k+1, y_k+1)

P_k(x_k, x_k+1) P_k(x_k, x_k+1)dx_k+1

=

Z ^m−1Y

k=0

f_k(x_k, y_k;x_k+1, y_k+1) P_k(x_k, x_k+1)

!_m−1 Y

k=0

P_k(x_k, x_k+1)dx_k+1

=

Z m−1Y

k=0

g_k(x_k, y_k;x_k+1, y_k+1)

!

Px0,y0( ¯X_t1 ∈dx₁, . . . ,X¯_tm ∈dx_m) =E[L^m_y ],

where g_k is defined by (19) and the next-to-last equality follows from the Markov property of the process( ¯X_tk, k= 0, . . . , m). Now, looking at the numerator of (23), similar computations lead to:

Z

Φ(x^m)N_k(x^m;y^m)dx₁. . . dx_m

= Z

F¯(t_m, t_n, x_m)

m−1Y

k=0

G^x_∆^k,xk+1

kσ²_k (a)f_k(x_k, y_k;x_k+1, y_k+1)dx_k+1,

= Z

F¯(t_m, t_n, x_m)

m−1Y

k=0

G^x_∆^k,xk+1

kσ²_k (a)

!

×

m−1Y

k=0

g_k(x_k, y_k;x_k+1, y_k+1)

!

Px0,y0( ¯Xt1 ∈dx1, . . . ,X¯tm ∈dxm)

=EF¯(t_m, t_n,X¯_tm)K^mL^m_y . Finally, going back to (23), we obtain

P

τa^X¯ > t_n|(Y_t0, . . . , Y_tm) =y^m

= EF¯(tm, tn,X¯tm)K^mL^m_y E[L^m_y ]

which is the announced result.

Remark 2.1. Remark that more generally we have for every real valued bounded functionf, E[f( ¯X_tm)|Y_t0 =y₀, . . . , Y_tm =y_m] = E

f( ¯X_tm)L^m_y

E[L^m_y ] . (24) In this case we can estimate the right hand side quantity of (24) using recursive algorithms. In our setting, we can adapt these algorithms by noting that the expression (18) may be read in the similar form of (24) as:

P

τa^X¯ > t_n|(Y_t0, . . . , Y_tm) =y^m

= EF¯(t_m, t_n,X¯_tm)L^′_y^m E[L^m_y ]

where

L^′_y^m =

m−1Y

k=0

g_k( ¯X_tk, y_k; ¯X_tk+1, y_k+1)G^X_∆^¯tk,X¯tk+1

kσ²_k (a) and where the involved functions are defined in Theorem 2.5.

Considering the model given in Example 1, one may apply the result of Theorem 2.5 using the continuous Euler process X¯ of the signal. However since in this framework the solutions of the stochastic differential equations are explicit, we refrain from using the Euler scheme to avoid adding additional error. In the next result we deduce a similar representation of the conditional survival probability using the explicit solutions of the stochastic differential equations of the model (3).

Corollary 2.6. Consider that the signal process X and the observation processY evolve following the stochastic differential equations given in (3):

(dX_t=X_t(µdt+σdW_t), X₀=x₀,

dY_t=Y_t(rdt+νdW_t+δdWf_t), Y₀=y₀. (25) Then,

Ψ(y^m) = E

F(t_m, t_n, X_tm)K^mL^m_y

E[L^m_y ] , (26) with

K^m=

m−1Y

k=0

G^X_∆^tk,Xtk+1

kσ_k² (a) and L^m_y =

m−1Y

k=0

g_k(Xtk, y_k;Xt_k+1, y_k+1).

The functiong_kis defined by g_k(x_k, y_k;x_k+1, y_k+1) = 1

(2π∆_k)^3/2π_kexp − ν² 2δ²∆_k

logx_k+1−m¹_k

σ −logy_k+1−m²_k ν

2! (27) with

π_k:=σ²δx²_k+1y_k+1, m¹_k:= logx_k+ (µ−1

2σ²)∆_k and m²_k:= logy_k+ (r−1 2ν²−1

2δ²)∆_k. Proof. It follows from Itô formula that for everys≤t,

(X_t=X_sexp (µ−¹₂σ²)(t−s) +σ(W_t−W_s) Y_t=Y_sexp

(r−¹₂ν²− ¹₂δ²)(t−s) +ν(W_t−W_s) +δ(fW_t−Wf_s)

. (28)

Then, for every k = 0, . . . , m−1 the random vector (X_tk+1, Y_tk+1)|(X_tk, Y_tk) = (x_k, y_k) has a bivariate lognormal distribution with meanm_kand covariance matrixΣgiven by

m_k=

logx_k+ (µ−¹₂σ²)∆_k logy_k+ (r−¹₂ν²−¹₂δ²)∆_k

, Σ = ∆_k

σ² σν σν ν²+δ²

. Hence, its density reads (settingµ_k=σδx_k+1y_k+1)

f_k(x_k, y_k;x_k+1, y_k+1) = 1

2π∆_kµ_kexp −ν²+δ² 2δ²∆_k

n(logx_k+1−m¹_k)² σ²

− 2ν

σ(ν²+δ²)(logx_k+1−m¹_k)(logy_k+1−m²_k) +(logy_k+1−m²_k)²

ν²+δ²

o! ,

wherem¹_k= logx_k+ (µ−¹₂σ²)∆_kandm²_k:= logy_k+ (r−¹₂ν²−¹₂δ²)∆_k. Furthermore, for every k= 0, . . . , m−1, the random variableX_tk+1|X_tk =x_khas a lognormal distribution with meanm¹_k and varianceσ²so that its density distributionP_k(x_k, x_k+1)is given by

P_k(x_k, x_k+1) = 1

√2π∆_kσx_k+1 exp

−(logx_k+1−m¹_k)² 2σ²∆_k

.

We then conclude the proof by using the same arguments than those of the proof of Theorem 2.5.

There exist several methods to estimate the above representation of the conditional survival prob- ability. These methods involve, amount others, Monte Carlo simulations and optimal quantization methods. Owing to the numerical performance of the optimal quantization method due for example to its fast performability as soon as the optimal grids are obtained, we will use optimal quantization methods to estimate the conditional survival probability.

The use of optimal quantization methods to estimate the filter supposes to have numerical access to optimal (or stationary) quantizers of the marginals of the signal process. One may use the opti- mal vector quantization method (as done in the seminal work [11] and used in [2]) to estimate these marginals. This method requires the use of some algorithms, like stochastic algorithms or Lloyd’s algorithm, to obtain numerically the optimal (or stationary) quantizers of the marginals of the sig- nal process. Given these optimal quantizers, the filter estimation is obtained quite instantaneously.

However, the step of search of stationary quantizers is very time consuming.

We propose here an alternative method, the (quadratic) marginal functional quantization method (introduce in [14] to price barrier options), to quantize the marginals of the signal process. The marginal functional quantization method consists first in considering the ordinary differential equa- tion (ODE) resulting to the substitution of the Brownian motion appearing in the dynamics of the signal process by a quadratic quantization of the Brownian motion. Then, by constructing some

“good” marginal quantization of the signal process based on the solution of the previous ODE’s, we will show how to estimate the nonlinear filter. Since this procedure is based on the quantization of the Brownian motion and skips the use of algorithms to perform the stationary quantizers, the computa- tion of the marginal quantizers is quite instantaneous. This reduce drastically the time computation of the procedure with respect to the vector quantization method since we skip the step of the use of al- gorithms search of marginal quantizers by using instead, the marginals of the functional quantization of the signal process.

In the rest of the paper we deal with the estimations methods of the conditional survival probabil- ity.

3 Numerical tools

Our aim in this section is to derive a way to compute numerically the conditional survival probability using Equation (18). To this end, we have to estimate three quantities: the quantity F(t¯ _m, t_n,X¯_tm), the expectationE[ ¯F(tm, tn,X¯tm)K^mL^m_y ]as soon asF¯(tm, tn,X¯tm)is estimated, and finally, the ex- pectationE[L^m_y ]. Both expectations will be estimated using marginal functional quantization method.

The probabilityF¯(t_m, t_n,X¯_tm)will be estimated by Monte Carlo simulations.

Before dealing with the estimation tools, we shall recall first some basic results about both optimal vector quantization and functional quantization. Then, we will show how to construct the marginal functional quantization process which will be used to estimate the quantities of interest.

3.1 Overview on optimal quantization methods

The optimal vector quantization on a gridΓ = {x₁,· · · , x_N}(which will be called a quantizer) of anR^d-valued random vectorXdefined on a probability space(Ω,A,P)with finiter-th moment and

probability distributionPX consists in finding the best approximation ofXby a Borel function ofX taking at mostN values. This turns out to find the solution of the following minimization problem:

eN,r(X) = inf{kX−Xb^Γkr,Γ⊂R^d,|Γ| ≤N}, (29) where kXkr := (E[|X|^r])^1/r and whereXb^Γ := P_N

i=1x_i11_{X∈Ci(Γ)} is the quantization of X (we will writeXb^N instead ofXb^Γ) on the gridΓand(C_i(Γ))_i=1,...,N corresponds to a Voronoi tessellation ofR^d(with respect to a norm| · |onR^d), that is, a Borel partition ofR^dsatisfying for everyi,

C_i(Γ)⊂ {x∈R^d:|x−x_i|= min

j=1,...,N|x−x_j|}.

We know that for every N ≥ 1, the infimum in (29) is reached at one grid Γ^⋆ at least, called a L^r-optimalN-quantizer. It is also known that if card(supp(PX)) ≥ N then|Γ| = N (see e.g. [8]

or[10]). Moreover, theL^r-mean quantization errore_N,r(X)decreases to zero at an N^−1/d-rate as the sizeN of the gridΓgoes to infinity. This convergence rate has been investigated in [1] and [15]

for absolutely continuous probability measures under the quadratic norm onR^d, and studied in great details in [8] under an arbitrary norm onR^dfor absolutely continuous measures and some singular measures.

From the numerical integration viewpoint, finding an optimal quantization grid Γ^⋆ may be a challenging task. In practice (we will only consider the quadratic case, i.e. when r = 2) we are sometimes led to find some “good” quantizations Proj_Γ(X) (with Γ = {x₁, . . . , x_N}) which are close to X in distribution, so that for every Borel function F : R^d 7→ R, we can approximate E[F(X)]by

E h

F Xb^Ni

= XN i=1

F(x_i)p_i, (30)

wherep_i =P Proj_Γ(X) =x_i

.Amount “good” quantizations ofX we have stationary quantizers.

A gridΓinducing the quantizationXb^N ofXis said stationary if

∀i6=j, xi 6=xj and P(X∈ ∪i∂Ci(Γ)) = 0 (31) and

Eh

X|Xb^Ni

=Xb^N.

The stationary quantizers search is based on zero search recursive procedures like Newton algorithm in the one dimensional framework and some algorithms as Lloyd’s I algorithms (see e.g. [5]), the Competitive Learning Vector Quantization (CLVQ) algorithm (see [5]) or stochastic algorithms (see [12]) in the multidimensional framework. Note that optimal quantizers estimates of the multivariate Gaussian random vector are available in the websitewww.quantize.math-fi.com.

We next recall some error bounds induced from approximating E[F(X)] by (30). Let Γ be a stationary quantizer andF be a Borel function onR^d.

(i) IfF is convex then

E

F(Xb^N)

≤E[F(X)].

(ii) Lipschitz functions:

– IfF is Lipschitz continuous then (this error bound doesn’t require the quantizerΓto be stationary) E[F(X)]−E[F(Xb^N)]≤[F]LipkX−Xb^Nk2,

where

[F]_Lip:= sup

x6=y

|F(x)−F(y)|

|x−y| .