Malliavin Regularity of Non-Markovian Quadratic BSDEs and Their Numerical Schemes

Round 1

Reviewer 1 Report

The paper is interesting and the results, although very technical, seem reasonable.  There are some minor typos

p3, line 52 "axillary" spelling

p4, line 70 "endowed with"

p24, line 393: capitalize "annals of applied probability"

 

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

In this manuscript, authors investigate Malliavin regularity for a class of quadratic backward stochastic differential with terminal data as a function of a forward diffusion. They prove existence and uniqueness of the solution in L^q, q≥2 and  establish the L^p-  Hölder continuity of the solution. Results concerning the Malliavin regularity of the solutions are obtained. The results are illustrated by  three examples. Authors consider  explicit and implicit numerical schemes and inverstigate the order of convergence.

Remarks:

    1)      Some sentences are uncompleted. For example in the Abstract “By using the connection between the QBSDE under study and some backward stochastic 3 differential equations (BSDEs) with globally Lipschitz coefficients”;

 2)      Quadratic backward stochastic differential equations are well studied in the literature, including numerically.   The introduction is incomplete and the references/literature in the manuscript are very scarce.

 

 

Author Response

Kindly see the attached file for answers to your valuable comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

Review:

Malliavin Regularity of non-Markovian Quadratic BSDEs and their Numerical Schemes

by Mhamed Eddahbi et al.

General Comments:

The paper focus on a class of quadratic backward stochastic differential equations. Theoretically, using the analogy between the quadratic backward stochastic differential equations under study and some backward stochastic differential equations, the results of existence and uniqueness are proved. Theoretical findings are illustrated by three simple examples.

The paper uses the well-known results of the backward stochastic differential equation model and examines a few cases in the form of a quadratic backward stochastic differential equation model.

In general, it may be recommended for a publication in Axioms.

Specific Comments:

1.      Lines 15-18: Formalize the incoming functions, processes and the solution in more detail.

2.      Lines 51-52, page 2: The partition of π in the discretization scheme is not formalized. Why is equality ζ=ζ^π not fair?

3.      Line 72: The integral on the right-hand side of the equation defines a new random Gaussian process, so it makes sense to denote it with a different letter.  

Author Response

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Reviewer 4 Report

I found the paper interesting in both theoretical and practical aspects. However, it would be much more  interesting to give applications to real life problems such as problems in finance and/or economics. 

Author Response

Kindly see the attached file for answers to your valuable comments.

Author Response File: Author Response.pdf