Killed process

In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.

For killed processes in computer operating systems, see Process state § Terminated.

Definition

Let X : T × Ω → S be a stochastic process defined for "times" t in some ordered index set T, on a probability space (Ω, Σ, P), and taking values in a measurable space S. Let ζ : Ω → T be a random time, referred to as the killing time. Then the killed process Y associated to X is defined by

and Yt is left undefined for t ≥ ζ. Alternatively, one may set Yt = c for t ≥ ζ, where c is a "coffin state" not in S.

gollark: Even gets around the evil Global Interpreter Lock for maximum performance.
gollark: Enterprise-level, fully deterministic execution of random code. Totally secure!
gollark: ```pythonimport itertoolsimport multiprocessing as mchars = [chr(x) for x in range(32, 126)]chars.extend(["\t", "\n"])def generate_for_length(length): return map(lambda chars: "".join(chars),itertools.combinations(chars, length))def generate(): for i in itertools.count(): for s in generate_for_length(i): yield sif __name__ == "__main__": def start(func): proc = m.Process(target=func) proc.start() io_queue = m.Queue(maxsize=128) def printer(): while True: print(io_queue.get()) code_queue = m.Queue(maxsize=128) def generator(): for x in generate(): code_queue.put(x) def executor(): while True: code = code_queue.get() result = None try: result = repr(exec(code)) except Exception as e: result = repr(e) if result != None: io_queue.put(code + ": " + result) start(generator) for _ in range(4): start(executor) printer()```
gollark: Which do you think is slower: the execution or generation of strings?
gollark: ```pythonimport itertoolschars = [chr(x) for x in range(32, 126)]chars.extend(["\t", "\n"])def generate_for_length(length): return map(lambda chars: "".join(chars),itertools.combinations(chars, length))def generate(): for i in itertools.count(): for s in generate_for_length(i): yield sfor x in generate(): try: print(exec(x)) except Exception as e: print(f"{e} from {x}")```

See also

References
  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 8.2)
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