There is no such thing built in by default in activiti. What you could do, is add a task-listener to your tasks which gets triggered on event "assignement". In this listener, you can set the current time in a task-local variable (delegateTask.setVariableLocal()).
When you task is completed (perhaps in another tasklistener) you have start-time, time that task was claimed and end time. Alternatively, if you wet your history level high enough, the task-local variables (and task) will be available in the history, so you can fetch the times for completed tasks.