BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient

Definition A.1.

Let τ∈T[0,T]. An optional process (ξt)t≤T is said to be right upper-semicontinuous (r.u.s.c.) along stopping times at the stopping time τ if for all nonincreasing sequence of stopping times (τn)n≥0 such that τn↘τ a.s., we have ξτ≥lim supn→+∞⁡ξτn a.s. The process (ξt)t≤T is said to be r.u.s.c. along stopping times if it is r.u.s.c. along stopping times at each τ∈T[0,T].

Remark A.2.

If the process (ξt)t≤T has left limits, (ξt)t≤T is left-continuous along stopping times if and only if for each predictable stopping time τ∈T[0,T], we have ξτ-=ξτ a.s.

Definition A.3.

Let (Yt)t≤T be an optional process. We say that Y is a strong (optional) supermartingale if

1. Yτ is integrable for all τ∈𝒯[0,T],

2. Yν≥𝔼⁡[Yτ∣ℱν] a.s. for all ν,τ∈𝒯[0,T] such that ν≤τ a.s.

Theorem A.4 (Mertens decomposition).

Let Y~ be a strong optional supermartingale of class (D). There exists a unique uniformly integrable martingale (càdlàg) M, a unique nondecreasing right-continuous predictable process K with K0=0 and E⁡[KT]<+∞ and a unique nondecreasing right-continuous adapted purely discontinuous process C with C0-=0 and E⁡[CT]<+∞, such that

Theorem A.5 (Dellacherie–Meyer).

Let K be a nondecreasing predictable process. Let U be the potential of the process K, i.e., U:=E[KT∣Ft]-Kt for all t≤T. We assume that there exists a positive FT-measurable random variable X such that |Uν|≤E⁡[X∣Fν] a.s. for all ν∈T[0,T]. Then

where c is a positive constant.

Corollary A.6.

Let Y be a strong optional supermartingale of class (D) such that for all ν∈T[0,T], we have |Yν|≤E⁡[X∣Fν] a.s., where X is a nonnegative FT-measurable random variable. Let K~ be the Mertens process associated with Y. There exists a positive constant c such that

Proof.

Let ν∈𝒯[0,T]. From the Mertens decomposition, we have Yν=Mν-K~ν a.s. and YT=MT-K~T a.s. Taking the conditional expectation in the second equation gives 𝔼⁡[YT∣ℱν]=𝔼⁡[MT∣ℱν]-𝔼⁡[K~T∣ℱν] a.s. By subtracting this equation from the first, we obtain Yν-𝔼⁡[YT∣ℱν]=Mν-𝔼⁡[MT∣ℱν]+𝔼⁡[K~T∣ℱν]-K~ν a.s. Since M is a martingale, we have Yν-𝔼⁡[YT∣ℱν]=𝔼⁡[K~T∣ℱν]-K~τ a.s. Thus

By applying Theorem A.5, we obtain the desired result. ∎

Proposition A.7.

Let X and Y be two optional processes such that Xν≥Yν a.s. for all ν∈T[0,T]. Then X≥Y, up to an evanescent set.

Theorem A.8 (Gal’chouk–Lenglart formula).

Let n∈N. Let Y be an n-dimensional optional semimartingale with decomposition Yk=Y0k+Mk+Vk+Wk for all k=1,…,n, where Mk is a (càdlàg) local martingale, Vk is a right-continuous process of finite variation such that V0k=0 and Wk is a left-continuous process of finite variation which is purely discontinuous and such that W0k=0. Let F be a twice continuously differentiable function on Rn. Then, almost surely, for all t≥0,

where Dk denotes the differentiation operator with respect to the k-th coordinate and Mk,c denotes the continuous part of Mk.

Corollary A.9.

Let Y be a one-dimensional optional semimartingale with decomposition Y=Y0+M+V+W, where M, V and W are as in the above theorem. Let X be a continuous process of finite variation. Then, almost surely, for all t≥0,

where ∂Y is the partial derivative operator with respect to Y.

Theorem A.10.

Let Y be a one-dimensional optional semimartingale. Then, almost surely, for all t≥0,

where (Lt)t≤T is a local time (nondecreasing continuous process). Moreover, Y+ is an optional semimartingale.

Theorem A.11.

If Y be an optional semimartingale, then ∑0≤s<tYs+⁢1{Ys-≤0}+Ys-⁢1{Ys->0} is finite a.s.

Proposition A.12.

𝔅2⁢(β,a) is the Banach space of the processes endowed with the norm