Learn Italian: Perché continui ad assillare e stuzzicare tua sorella? - Why do you continue to pester and tease your sister?

You could say, Perché non la smetti di assillare tua sorella? (or non la finisci, non la pianti). Notice that while continuare takes the preposition a + infinitive, these verbs of cessation require di + infinitive.

Why does the sentence use ad assillare instead of a assillare?

The preposition a becomes ad because the following word, assillare, also begins with the letter 'a'. This 'd eufonica' (euphonic d) is added to make the phrase sound smoother and avoid the awkward pause of two identical vowels clashing. In modern Italian, you generally only add the 'd' when the adjacent vowels are the same.

Why isn't there a definite article before tua sorella?

In Italian, you must drop the definite article before a possessive adjective when talking about a singular, unmodified family member. You would only use the article if the noun were plural (le tue sorelle), modified by a suffix or adjective (la tua sorellina), or if the possessive were loro (la loro sorella).

What is the difference in nuance between assillare and stuzzicare?