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 coefficients of the form. The main difficulty is the fact that the operator $$A(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is non-autonomous, i. e. depends on t and the matrix A ( t ) is singular for all $$t\in \left[ 0,T\right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . Our interest is in SDAE of index-1. This kind of SDAEs can be transformed into an ordinary stochastic differential equation with algebraic constraints. Under appropriate hypothesizes, the main result establishes the existence and uniqueness of the solution in $$\mathcal {M}^p(\left[ 0, T\right] , \mathbb {R}^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$p\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , $$p\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . 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.