Is there any filter which can be used in a plugin to process the content of the text widget before it is rendered?

Filter

1. widget_text (for the text)
2. widget_title (for the title)

Example

function add_smiley($content) {
$new_content="";
$new_content.= $content . ':)';
return $new_content;
}   
add_filter('widget_text', 'add_smiley');

Note that this works only for the content so not if you have a widget with only a title.

Reference

1. http://codex.wordpress.org/WordPress_Widgets
2. https://stackoverflow.com/questions/1385954/do-wordpress-widget-or-sidebar-hooks-exist
3. https://stackoverflow.com/search?q=wordpress+widget