Let E be a metric space, ( X t ) t ≥ 0 be an...
Let be a metric space, be an -valued right-continuous process and be locally bounded and Borel measurable. Is this enough to ensure that
for all ? The question is clearly trivial, when and are continuous.
Answer & Explanation
Example: , if , and if . This (non-random) path is right continuous, but with (certainly locally bounded) the integral diverges.
Fix: Strengthen the hypothesis on to "right continuous with left limits". Suppose that by "locally bounded" you mean that is bounded on each metric ball . Fix and . The real-valued function is then right continuous with left limits for . As such, it is bounded. Therefore there is a constant such that for all . The integral therefore converges.