Add container to nav_menu sub menu

Is there a way I can wrap a div around wp_nav_menu’s ul.sub-menu

So for example the markup would go from

<ul class="menu">

    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
    <li>
        <a href="https://wordpress.stackexchange.com/">Item 2</a>

        <ul class="sub-menu">

            <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
            <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>

        </ul>
   </li>
    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>

</ul>

to this

<ul class="menu">

    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
    <li>
        <a href="https://wordpress.stackexchange.com/">Item 2</a>

        <div class="sub-menu-wrap">

            <ul class="sub-menu">

                <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
                <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>

            </ul>

        </div>

   </li>

    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>
    <li><a href="https://wordpress.stackexchange.com/">Item 1</a></li>

</ul>

Where “sub-menu-wrap” is the custom div.

Is this possible?

Use a custom walker, extend the methods start_lvl() and end_lvl.

Sample code, not tested:

class WPSE_78121_Sublevel_Walker extends Walker_Nav_Menu
{
    function start_lvl( &$output, $depth = 0, $args = array() ) {
        $indent = str_repeat("\t", $depth);
        $output .= "\n$indent<div class="sub-menu-wrap"><ul class="sub-menu">\n";
    }
    function end_lvl( &$output, $depth = 0, $args = array() ) {
        $indent = str_repeat("\t", $depth);
        $output .= "$indent</ul></div>\n";
    }
}

Usage in your theme:

wp_nav_menu(
    array (
        'theme_location' => 'your-theme-location-EDIT-THIS',
        'walker'         => new WPSE_78121_Sublevel_Walker
    )
);