Summary
LetX andZ be ℝd -valued solutions of the stochastic differential inequalities dX t ≤a(t,X t )dt+σ(t,X t )dW t andb(t, Z t )dt+σ(t, Z t )dW t ≤dZ t , respectively, with a fixed ℝm-valued Wiener processW. In this paper we give conditions ona, b and σ under which the relationX 0≤Z 0 of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like ‘maximal/minimal solution’ of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such ‘solutions’ as well as some Gronwall-type estimates.
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Assing, S., Manthey, R. The behavior of solutions of stochastic differential inequalities. Probab. Th. Rel. Fields 103, 493–514 (1995). https://doi.org/10.1007/BF01246336
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DOI: https://doi.org/10.1007/BF01246336