Stochastic Differential Equations With Discontinuous Diffusion Coefficients

We study one-dimensional stochastic differential equations of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d upper X Subscript t Baseline equals sigma left-parenthesis upper X Subscript t Baseline right-parenthesis d upper Y Subscript t"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">dX_t = \sigma (X_t)dY_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable Hölder continuous driver such as the fractional Brownian motion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript upper H"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">B^H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H greater-than one half"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">H>\frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which we assume very mild conditions. In particular, we allow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.