Which is correct?

Do you appreciate my writing to the company’s stockholders?

Do you appreciate me writing to the company’s stockholders?

#1 is correct.

And another one:

Do you mind my smoking?

Do you mind me smoking?

#1 is correct.

Let’s turn it around a little to make it easier. We’ll substitute correspondence for writing.

Do you appreciate my correspondence to the stockholders?

Do you appreciate me correspondence to the stockholders?

It’s pretty obvious that #1 is correct, isn’t it? Correspondence is a noun (a thing), and my is the personal pronoun describing that noun.

It’s not obvious when using the word writing, because -ing words can be nouns or verbs (action words). Therefore, they can easily cause us to trip.

The best way to avoid tripping is to substitute a noun for the -ing word and see if it works. If it does, use my before the -ing word.

My boys hate my yelling.

(My boys hate my uproar.)

(. . . which, of course, they do!)

I hope this helps.

Robin