Adding a class (arrows) to main menu links that have children?

Question

I'm wondering if it is possible to add different classes to second/third/fourth/etc-level items that have children in Appearance > Menus tree?

That's how I call the menu:

 <?php $menu_args = array(
    'container'       => '', 
    'menu_class'      => '', 
    'menu_id'         => 'my-menu',
    'walker'          => new Description_Walker);

    wp_nav_menu($menu_args );   
 ?>

I know every link owns different ID like <li id="menu-item-3230">, but I want the solution to be dynamic so I won't have to edit code every time I change something. I guess the easiest way will be to attach additional class to these items but how?