Corrigendum to “Stability of solutions of BSDEs with random terminal time”

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ESAIM: PS ESAIM: Probability and Statistics

August 2007, Vol. 11, p. 381–384 www.edpsciences.org/ps

DOI: 10.1051/ps:2007025

CORRIGENDUM TO “STABILITY OF SOLUTIONS OF BSDES WITH RANDOM TERMINAL TIME”

Sandrine Toldo

¹

Abstract. This paper is a corrigendum to paper Toldo,ESAIM, P&S 10(2006) 141–163 where we study the stability of the solutions of Backward Stochastic Diﬀerential Equations (BSDE for short) with an almost surely ﬁnite random terminal time.

Mathematics Subject Classification. 60H10, 60Fxx, 60G40.

Received March 27, 2006. Revised July 17 and December 5, 2006.

1. Introduction

The proofs of Theorems 1.4 and 2.3 of my paper [2] are not correct: there is no reason to have the convergence of the sequence (Y^p, Z^p)_p of the Picard approximations when the terminal time is not bounded.

However, using another method, we can get similar results that are given in Theorems 2.1 and 3.1.

2. Stability of BSDEs when the Brownian motion is approximated by a scaled random walk

Letf : Ω×R⁺×R²→Rbe a Lipschitz function: there existsγ∈R⁺such that, for every (t, y, z),(t, y, z)∈ R⁺ ×R², we have |f(t, y, z)−f(t, y, z)| γ[|y −y|+|z −z|]. We also suppose that f is bounded and that f fills the following property of monotonicity w.r.t. y: there exists µ > 0 such that ∀(t, y, z),(t, y, z)∈ R⁺×R²,(y−y)(f(t, y, z)−f(t, y, z))−µ(y−y)². LetW be a Brownian motion,F its natural filtration, τ a F-stopping time almost surely finite and ξ a bounded F_τ-mesurable random variable. We consider the following stochastic differential equation:

Y_t∧τ =ξ+ _τ

t∧τ

f(s, Y_s, Z_s)ds− _τ

t∧τ

Z_sdW_s, t0.

Keywords and phrases.Backward Stochastic Diﬀerential Equations (BSDE), stability of BSDEs, weak convergence of ﬁltrations, stopping times.

1 IRMAR, antenne de Bretagne de l’ENS Cachan, Campus de Ker Lann, 35170 BRUZ, France;

[email protected]

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.esaim-ps.org or http://dx.doi.org/10.1051/ps:2007025

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382 S. TOLDO

In this section, we approximate the previous equation on the following way. We consider the sequence of scaled random walks (Wⁿ)_n1 deﬁned by W_tⁿ= √¹

n

_[nt]

k=1εⁿ_k, t0 where (εⁿ_k)_k∈N∗ is a sequence of i.i.d. symmetric Bernoulli variables. Let Fⁿ be the natural ﬁltrations of Wⁿ, n 1. We have F_tⁿ = σ(εⁿ_k, k [nt]). Let (τⁿ)_n be a sequence of bounded (Fⁿ)-stopping times (for eachn, we can ﬁndT_n ∈Nsuch that τⁿ T_n) and (ξⁿ) a sequence of (F_τⁿn)-measurable integrable random variables. We consider the processes Yⁿ and Zⁿ that are constant on the intervals [k/n,(k+ 1)/n[ and ]k/n,(k+ 1)/n] respectively and that satisfy the following equation:

Y_tⁿ=ξⁿ+ _τⁿ

t∧τⁿ

f(s, Y_(s∧τⁿ _n₎₋, Z_s∧τⁿ n)dAⁿ_s− _τⁿ

t∧τⁿ

Z_s∧τⁿ ndW_sⁿ whereAⁿ_s = [ns]

n · We also suppose that sup_nξⁿ∞<+∞(it implies that (Yⁿ) is uniformly bounded).

Instead of Theorem 1.4 of [2], we have the following result of convergence:

Theorem 2.1. We suppose that all the assumptions given before are filled and that we have the convergences ξⁿ −→^P ξ,τⁿ−→^P τ andWⁿ−→^P W. Then

sup

t∈R⁺e^−µt|Y_tⁿ−Y_t|+ _+∞

0

e^−2µt|Z_tⁿ−Z_t|²dt −→^P 0,

∀L, sup

t∈[0,L]

_t

0

Z_sⁿdW_sⁿ− _t

0

Z_sdW_s

−→^P 0,

t∈Rsup⁺ _t

0 e^−µsZ_sⁿdW_sⁿ− _t

0 e^−µsZ_sdW_s

−→^P 0.

As in [2], we obtain a corollary under assumption of convergence in law:

Corollary 2.2. We suppose that ∀n,∀k, εⁿ_k =ε_k,ξ=g(W)and ξⁿ =g(Wⁿ)with g bounded continuous. We also suppose that we have the convergence(Wⁿ, τⁿ)−→^L (W, τ). Then

Y_.∧τⁿ n,_.∧τⁿ

0 Z_sⁿdW_sⁿ

n converges in law to

Y_.∧τ,_.∧τ

0 Z_sdW_s

for the Skorokhod topology.

Here, we don’t have the assumptions of uniform integrability for the terminal conditions and the stopping times but we have an assumption of uniform boundedness of the terminal conditions. Concerning the convergences, we have uniform convergence on every compact set and if we want a convergence on R⁺, we have to add an exponential factor.

We can’t prove Theorem 2.1 as it was explained in [2] because when the stopping times are not bounded but only almost surely ﬁnite there is no reason to have the convergence of Picard approximation.

We just give the main steps of the proof of the ﬁrst convergence of Theorem 2.1 (this is the wrong proof in [2]). We denote this convergence by (Yⁿ, Zⁿ)→(Y, Z). We prove successively the following convergences:

– for everyK, (Y^n,K, Z^n,K)→(Y^K, Z^K) whenngoes to +∞;

– (Y^K, Z^K)→(Y, Z) whenK goes to +∞;

– (Y^n,K, Z^n,K)→(Yⁿ, Zⁿ) uniformly w.r.t. nwhenK goes to +∞;

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CORRIGENDUM TO “STABILITY OF SOLUTIONS OF BSDES WITH RANDOM TERMINAL TIME” 383 where (Y^K, Z^K) and (Y^n,K, Z^n,K) are solutions of:

Y_t∧τ∧K^K = ξ1τK+ _τ∧K

t∧τ∧K

f(s, Y_s^K, Z_s^K)ds− _τ∧K

t∧τ∧K

Z_s^KdW_s, ∀t0,

and Y_t^n,K = ξⁿ1τⁿK+ _τⁿ_∧K

t∧τⁿ∧K

f(s, Y_s−^n,K, Z_s^n,K)dAⁿ_s − _τⁿ_∧K

t∧τⁿ∧K

Z_s^n,KdW_sⁿ, ∀t0.

The reader can ﬁnd more details in Chapter 3 Section 2 of [1].

3. Stability of BSDEs when the Brownian motion is approximated by a sequence of martingales

Let W be a Brownian motion, F its natural ﬁltration, τ a F-stopping time almost surely ﬁnite and ξ a bounded random variableF_τ-measurable. We consider the following BSDE:

Y_t∧τ=ξ+ _τ

t∧τ

f(r, Y_r, Z_r)dr− _τ

t∧τ

Z_rdW_r, t0. (1)

We approximate this equation on the following way. Let (Wⁿ)_n be a sequence of càdlàg processes and (Fⁿ)_n the natural filtrations for these processes. We suppose that (Wⁿ) is a sequence of square integrable (Fⁿ)- martingales which converges in probability toW. We don’t suppose thatWⁿhas the predictable representation property. Let (τⁿ)_n be a sequence of (Fⁿ)-stopping times that converges almost surely toτ. Then, we consider the following BSDE:

Y_tⁿ=ξⁿ+ _τⁿ

t∧τⁿ

fⁿ(r, Y_r−ⁿ, Z_rⁿ)dWⁿ_r− _τⁿ

t∧τⁿ

Z_rⁿdW_rⁿ−(N_τⁿn−N_t∧τⁿ n), t0 (2)

where (ξⁿ)_n is a sequence of random variables (F_τⁿn)-measurable, (Nⁿ) is a sequence of (Fⁿ) martingales orthogonal to (W^n,τⁿ).

We put the following assumptions on the martingales and on the terminal conditions:

(H1) (i)∀L,Wⁿ−−→^S^L² W,

(ii)|< Wⁿ >_t−t|a_n where (a_n)↓0.

(H2) (i)ξ^{n L}−−→² ξ,

(ii)ξ_∞+ sup_nE[|ξⁿ|]<∞.

Let us put some assumptions on the generatorf: (Hf ) (i)f isγ-Lipschitz iny andz,

(ii)f is monotone iny in the following way with constantµ >0, (iii)f is bounded,

(iv) for every (y, z),{f(t, y, z)}tis progressively measurable w.r.t. F.

Let us put some assumptions on the generators (fⁿ):

(Hfn) (i) for everyn,fⁿ isγ-Lipschitz iny andz,

(iifor everyn,fⁿ is monotone w.r.t. y with constantµ, (iii) sup_nfⁿ<∞,

(iv) for every (y, z),{fⁿ(t, y, z)}tis progressively measurable w.r.t. (Fⁿ)_n, (v)∀(y, z),{fⁿ(t, y, z)}_thas c`adl`ag trajectories andfⁿ(., y, z)−−→^S^L² f(., y, z),∀L.

The only diﬀerence with [2] is the assumption(H1) that has been replaced by a weaker one.

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384 S. TOLDO

Instead of Theorem 2.3 of [2], we have the following result of convergence:

Theorem 3.1. We suppose that all the assumptions given before are filled. Then,

E

sup

t∈R⁺e^−2µt|Y_tⁿ−Y_t|²+ sup

t∈R⁺e^−2µt|N_tⁿ|²

−−−−−→

n→+∞ 0,

∀L, E sup

t∈[0,L]

_t

0

Z_sⁿdW_sⁿ− _t

0

Z_sdW_s

−−−−−→

n→+∞ 0.

Here again, we can’t argue as in [2] to prove the theorem because there is no reason to have the convergence of Picard approximations. To prove the theorem, we argue using the same steps as in the proof of Theorem 2.1.

The reader can ﬁnd more details in Chapter 3 Section 3 of [1].

References

[1] S. Toldo, Convergence de filtrations ; application à la discrétisation de processus et à la stabilité de temps d’arrêt. Ph.D.

Thesis, Universit´e de Rennes 1 (Novembre 2005).

[2] S. Toldo, Stability of solutions of BSDEs with random terminal time.ESAIM, P&S10(2006) 141–163.