I am proud to announce that I have found the solution thanks to @cjbj good advice. I paste the code here. I know it will help others.
In addition, I used the function preg_quote to protect the path to the image.
protected_path = preg_quote($filename);
This code works great.
// $filename should be the path to a file in the upload directory.
$filename="/path/to/uploads/2013/03/filename.jpg";
// The ID of the post this attachment is for.
$parent_post_id = 37;
// Check the type of file. We'll use this as the 'post_mime_type'.
$filetype = wp_check_filetype( basename( $filename ), null );
// Get the path to the upload directory.
$wp_upload_dir = wp_upload_dir();
// Prepare an array of post data for the attachment.
$attachment = array(
'guid' => $wp_upload_dir['url'] . "https://wordpress.stackexchange.com/" . basename( $filename ),
'post_mime_type' => $filetype['type'],
'post_title' => preg_replace( '/\.[^.]+$/', '', basename( $filename ) ),
'post_content' => '',
'post_status' => 'inherit'
);
// Insert the attachment.
$attach_id = wp_insert_attachment( $attachment, $filename, $parent_post_id );
// Make sure that this file is included, as wp_generate_attachment_metadata() depends on it.
require_once( ABSPATH . 'wp-admin/includes/image.php' );
// Generate the metadata for the attachment, and update the database record.
$attach_data = wp_generate_attachment_metadata( $attach_id, $filename );
wp_update_attachment_metadata( $attach_id, $attach_data );
set_post_thumbnail( $parent_post_id, $attach_id );