How to display posts by current user in a drop down

Use 'author' => get_current_user_id() as an argument for get_posts(). This will restrict the found posts to these of the user which is currently logged in.

The best reference for the available parameters is still WP_Query::parse_query() in wp-includes/query.php. It is not exactly an atomic function, but you can find parameters not documented elsewhere – like this one.

Tow notes abut your code:

  1. Your onchange handler looks a little bit strange. It doesn’t do anything, does it?
  2. Do not use get_the_title() for attribute values. The post title can contain markup.
    Use the_title_attribute( array ( 'echo' => FALSE ) ); instead.