Show some menu sub items as dropdown under a menu item

Remove the first closing </li> and wrap the sub-items inside <ul></ul>, then close the parent <li>.

For Example:

function my_custom_menu_item($items, $args)
{
    if(is_user_logged_in() && $args->theme_location == 'primary')
    {
        $user=wp_get_current_user();
        $name=$user->display_name; 
        $items .= '<li><a href="">'.$name.'</a>';
        $items .= '<ul>';
        $items .= '<li><a href="http://wordpress.stackexchange.com/my-profile">My Profile</a></li>';
        $items .= '<li><a href="http://wordpress.stackexchange.com/my-posts">My Posts</a></li>';
        $items .= '<ul>';
        $items .= '</li>';
    }
    return $items;
}
add_filter( 'wp_nav_menu_items', 'my_custom_menu_item', 10, 2);