Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle

Proposition A.1.

Let Y be a càdlàg process. Then sup t ∈ [ 0 , T ] ⁡ Y t is a random variable and we have

Definition A.2.

Let ( Y ) t ∈ [ 0 , T ] be an optional process. We say that Y is a strong (optional) supermartingale if the following assertions hold:

1. Y τ is integrable for all τ ∈ 𝒯 0 , T .

2. Y S ≥ E [ Y τ | ℱ S ] a.s., for all S , τ ∈ 𝒯 0 , T such that S ≤ τ a.s.

Theorem A.3 (Gal’chouk–Lenglart).

Let n ∈ Z + . Let X be an n-dimensional optional semimartingale, that is, X = ( X 1 , … , X n ) is an n-dimensional optional process with decomposition

for all k ∈ { 1 , … , n } , where M t k is a (càdlàg) local martingale, A t k is a right-continuous process of finite variation such that A 0 = 0 , and B t k is a left-continuous process of finite variation which is purely discontinuous and such that B 0 - = 0 . Let F be a twice continuously differentiable function on R n . Then, almost surely, for all t ≥ 0 ,

where D k denotes the differentiation operator with respect to the k-th coordinate, and M k ⁢ c denotes the continuous part of M k .

Corollary A.4.

Let Y be a one-dimensional optional semimartingale with decomposition Y t = Y 0 + M t + A t + B t , where M, A and B are as in the above theorem. Let β > 0 . Then, almost surely, for all t in [ 0 , T ] ,

Proof.

For showing the corollary, it suffices to apply the change of variables formula from Theorem A.3 with n = 2 , F ⁢ ( x , y ) = x ⁢ y 2 , X t 1 = e β ⁢ t , and X t 2 = Y t . ∎

Corollary A.5.

Let Y be a one-dimensional optional semimartingale with decomposition Y t = Y 0 + M t + A t + B t , where M, A and B are as in the above theorem. Then, almost surely, for all t in [ 0 , T ] ,

where

and

for all t ∈ [ 0 , T ] .

Proof.

This follows from Theorem A.3 and [22, Corollary 5.5] ∎

Proof of Theorem 3.5.

For all S ∈ 𝒯 0 , T , we define the family Y ¯ ⁢ ( S ) by

We put

By Proposition 3.4, there exists a làdlàg optional process ( Y t ) t ∈ [ 0 , T ] which aggregates the family ( Y ¯ ⁢ ( S ) ) S ∈ 𝒯 0 , T , that is,

Step 1: Let us show that Y ¯ ∈ S 2 , 0 . By using the definition of Y ¯ , (3.19), Jensen’s inequality and the triangular inequality, we get

Thus, we obtain

with

Applying the Cauchy–Schwarz inequality, we get

where c is a positive constant. Now, inequality (A.4) leads to | Y ¯ S | 2 ≤ | E [ X | ℱ S ] | 2 . By taking the essential supremum over S ∈ 𝒯 0 , T , we get

By using [15, Proposition A.3], we get

By using this inequality and Doob’s martingale inequalities, we obtain

where c is a positive constant that changes from line to line. Finally, combining inequalities (A.6) and (A.7), we get

Then Y ¯ S ∈ 𝕊 2 , 0 .

Step 2: The existence of Z, A and C. By the previous step and since

the strong optional supermartingale Y ¯ ¯ is of class ( D ) . Applying Mertens decomposition (see [15, Theorem A.1]) and a result from optimal stopping theory (see [7, Proposition 2.34. p. 131] or [23]), we have

By (A.2), we obtain

where M is a (càdlàg) uniformly integrable martingale such that M 0 = 0 , A is a nondecreasing right-continuous predictable process such that A 0 = 0 , E ⁢ ( A T ) < ∞ and satisfying (3.2), and C is a nondecreasing right-continuous adapted purely discontinuous process such that C 0 - = 0 , E ⁢ ( C T ) < ∞ and satisfying (3.3). By the martingale representation theorem (Lemma 2.3), there exists a unique predictable process Z such that

Moreover, we have Y ¯ T = ξ T a.s. by the definition of Y ¯ . Combining this with equation (A.9), we get the equation (3.1). Also, by the definition of Y ¯ , we have Y ¯ S ≥ ξ S a.s. for all S ∈ 𝒯 0 , T , which, along with Proposition 2.2 (or with [30, Theorem 3.2.]), shows that Y ¯ t ≥ ξ t , 0 ≤ t ≤ T a.s. Finally, to conclude that the process ( Y ¯ , Z , A , C ) is a solution to the reflected BSDE with parameters ( f , ξ ) , it remains to show that

Step 3: Let us prove that A × C ∈ S 2 , 0 × S 2 , 0 . Let us define the process A ¯ ¯ t = A t + C t - where the processes A and C are given by (A.9). By similar arguments to those used in the proof of inequality (A.4), we see that | Y ¯ ¯ S | ≤ E [ X | ℱ S ] with

Then [15, Corollary A.1] ensures the existence of a constant c > 0 such that E ⁢ [ ( A ¯ ¯ T ) 2 ] ≤ c ⁢ E ⁢ [ X 2 ] . By combining this inequality with inequality (A.6), we obtain

where we have again allowed the positive constant c to vary from line to line. We conclude that A ¯ ¯ ∈ L 2 . According to the nondecreasingness of A ¯ ¯ , we have ( A ¯ ¯ τ ) 2 ≤ ( A ¯ ¯ T ) 2 for all τ ∈ 𝒯 0 , T . Thus,

i.e. A ¯ ¯ ∈ 𝕊 2 . Then A ∈ 𝕊 2 , 0 and C ∈ 𝕊 2 , 0 .

Step 4: We now show that Z ∈ H 2 , 0 . We have from step 3 that

where A ¯ ¯ is the process from step 3. Since

Hence,

and consequently Z ∈ ℍ 2 , 0 .

For the uniqueness of the solution, suppose that ( Y , Z , A , C ) is a solution of the reflected BSDE with driver f and obstacle ξ. Then, by Lemma 3.3 (applied with f 1 = f 2 = f ), we obtain Y = Y ¯ in 𝕊 2 , 0 , where Y ¯ is given by (A.1). The uniqueness of A, C and Z follows from the uniqueness of the Mertens decomposition of strong optional supermartingales and from the uniqueness of the martingale representation (Lemma 2.3). ∎