Custom walker pulldown display current page

WordPress add “current-menu-item” class for the active menu item, you could check if this class is set and the set the option for that item be selected.

class Walker_Nav_Menu_Dropdown extends Walker_Nav_Menu {

    function start_lvl($output, $depth) {    }

    function end_lvl($output, $depth) {    }

    function start_el($output, $item, $depth, $args) {
        // Here is where we create each option.
        $item_output="";

        // add spacing to the title based on the depth
        $item->title = str_repeat(" ", $depth * 4) . $item->title;

        // Get the attributes.. Though we likely don't need them for this...
        $attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
        $attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
        $attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
        $attributes .= ! empty( $item->url )        ? ' value="'   . esc_attr( $item->url        ) .'"' : '';

        $classes = empty( $item->classes ) ? array() : (array) $item->classes; 
        if ( in_array( 'current-menu-item', $classes ) ) {
             $attributes .= ' selected';
        }

        // Add the html
        $item_output .= '<option'. $attributes .'>';
        $item_output .= apply_filters( 'the_title_attribute', $item->title );

        // Add this new item to the output string.
        $output .= $item_output;

    }

    function end_el($output, $item, $depth) {
        // Close the item.
        $output .= "</option>\n";
    }