"मेरे दादा आज दौड़ रहे हैं ।"
Translation:My grandfather is running today.
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Because you are using respect to address/speak about your grandfather, the entire sentence is in the oblique case (hence the मेरे at the beginning of the sentence). रहा है would grammatically make sense if the sentence started with मेरा but because your grandparents are older/wiser than you, in Hindi they are always referred to using the oblique case (using words phrases आप हैं instead of तुम हो).