Weak convergence of stochastic integrals on Skorokhod space in Skorokhod’s $$J_1$$ and $$M_1$$ topologies

Abstract We provide criteria for Itô integration to behave continuously with respect to Skorokhod’s $$J_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$M_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the $$M_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> setting and unify existing theories in the $$J_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, $$M_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> tightness in fact implies $$J_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak $$M_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$J_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.