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Existence and uniqueness for the solutions of non-autonomous stochastic differential algebraic equations with locally Lipschitz coefficients

Oana Silvia Serea, Antoine Tambue, Guy Tsafack

Abstract

In this paper, we study the well-posedness and regularity of non-autonomous stochastic differential algebraic equations (SDAEs) with nonlinear, locally Lipschitz and monotone (2) coefficients of the form (1). The main difficulty is the fact that the operator A(.) is non-autonomous, i.~e. depends on t and the matrix $A(t)$ is singular for all $t\in \left[0,T\right]$. Our interest is in SDAE of index-1. This means that in order to solve the problem, we can transform the initial SDAEs into an ordinary stochastic differential equation with algebraic constraints. Under appropriate hypothesizes, the main result establishes the existence and uniqueness of the solution in $\mathcal{M}^p(\left[0, T\right], \mathbb{R}^n)$, $p\geq 2$, $p\in \mathbb{N}$. Several strong estimations and regularity results are also provided. Note that, in this paper, we use various techniques such as Itô's lemma, Burkholder-Davis-Gundy inequality, and Young inequality.

Existence and uniqueness for the solutions of non-autonomous stochastic differential algebraic equations with locally Lipschitz coefficients

Abstract

In this paper, we study the well-posedness and regularity of non-autonomous stochastic differential algebraic equations (SDAEs) with nonlinear, locally Lipschitz and monotone (2) coefficients of the form (1). The main difficulty is the fact that the operator A(.) is non-autonomous, i.~e. depends on t and the matrix is singular for all . Our interest is in SDAE of index-1. This means that in order to solve the problem, we can transform the initial SDAEs into an ordinary stochastic differential equation with algebraic constraints. Under appropriate hypothesizes, the main result establishes the existence and uniqueness of the solution in , , . Several strong estimations and regularity results are also provided. Note that, in this paper, we use various techniques such as Itô's lemma, Burkholder-Davis-Gundy inequality, and Young inequality.
Paper Structure (13 sections, 10 theorems, 79 equations)

This paper contains 13 sections, 10 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Notations and assumptions
  3. Some classical results on stochastic differential equations and singular matrices
  4. Stochastic differential equations
  5. Some results on singular matrices
  6. Non-autonomous stochastic differential-algebraic equations: Main results
  7. Existence and uniqueness results for non-autonomous SDAEs
  8. More results on the regularity of the solution for the non-autonomous SDAEs
  9. Example
  10. Conclusion and Perspectives
  11. Appendix
  12. Proof of Proposition \ref{['prop2']}
  13. Declarations

Key Result

Proposition 1

mukam2015 Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a complete probability space, $\{W(t)\}_{t\in \left[0,T \right]}$ a one-dimensional Brownian motion and $K: \left[0,T \right] \times\mathbb{R}\to \mathbb{R}$ such that $K$ is once differentiable with respect to the first variable $t$ and twice dif Equivalently we have the differential form

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: Itô lemma
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 4
  • Proposition 2
  • ...and 10 more