How to run a background task in a servlet based web application?

Your problem is that you misunderstand the purpose of the servlet. It's intented to act on HTTP requests, nothing more. You want just a background task which runs once on daily basis.

EJB available? Use @Schedule

If your environment happen to support EJB (i.e. a real Java EE server such as WildFly, JBoss, TomEE, Payara, GlassFish, etc), then use @Schedule instead. Here are some examples:

@Singleton
public class BackgroundJobManager {

    @Schedule(hour="0", minute="0", second="0", persistent=false)
    public void someDailyJob() {
        // Do your job here which should run every start of day.
    }

    @Schedule(hour="*/1", minute="0", second="0", persistent=false)
    public void someHourlyJob() {
        // Do your job here which should run every hour of day.
    }

    @Schedule(hour="*", minute="*/15", second="0", persistent=false)
    public void someQuarterlyJob() {
        // Do your job here which should run every 15 minute of hour.
    }

    @Schedule(hour="*", minute="*", second="*/5", persistent=false)
    public void someFiveSecondelyJob() {
        // Do your job here which should run every 5 seconds.
    }

} 

Yes, that's really all. The container will automatically pickup and manage it.

EJB unavailable? Use ScheduledExecutorService

If your environment doesn't support EJB (i.e. you're not using not a real Java EE server, but a barebones servletcontainer such as Tomcat, Jetty, etc), then use ScheduledExecutorService. This can be initiated by a ServletContextListener. Here's a kickoff example:

@WebListener
public class BackgroundJobManager implements ServletContextListener {

    private ScheduledExecutorService scheduler;

    @Override
    public void contextInitialized(ServletContextEvent event) {
        scheduler = Executors.newSingleThreadScheduledExecutor();
        scheduler.scheduleAtFixedRate(new SomeDailyJob(), 0, 1, TimeUnit.DAYS);
        scheduler.scheduleAtFixedRate(new SomeHourlyJob(), 0, 1, TimeUnit.HOURS);
        scheduler.scheduleAtFixedRate(new SomeQuarterlyJob(), 0, 15, TimeUnit.MINUTES);
        scheduler.scheduleAtFixedRate(new SomeFiveSecondelyJob(), 0, 5, TimeUnit.SECONDS);
    }

    @Override
    public void contextDestroyed(ServletContextEvent event) {
        scheduler.shutdownNow();
    }

}

Where the job classes look like this:

public class SomeDailyJob implements Runnable {

    @Override
    public void run() {
        // Do your daily job here.
    }

}

public class SomeHourlyJob implements Runnable {

    @Override
    public void run() {
        // Do your hourly job here.
    }

}

public class SomeQuarterlyJob implements Runnable {

    @Override
    public void run() {
        // Do your quarterly job here.
    }

}

public class SomeFiveSecondelyJob implements Runnable {

    @Override
    public void run() {
        // Do your quarterly job here.
    }

}

Do not ever think about using java.util.Timer/java.lang.Thread in a Java EE / Servlet based environment

Last but not least, never directly use java.util.Timer and/or java.lang.Thread in Java EE. This is recipe for trouble. An elaborate explanation can be found in this JSF-related answer on the same question: Spawning threads in a JSF managed bean for scheduled tasks using a timer.

I would suggest using a library like quartz in order to run the task at regular intervals. What does the servlet really do ? It sends you a report ?