Hi There is a reference to measurable stochastic processes in the chapter- Brownian Motion and Martingales under the topic of Martingales in continuous time. "A stochastic process Y(t), t>=0 in ADAPTED to the filtration F(t) if Y(t) if F(t)-measurable for all t"... 1. Pls explain this line... 2. What does it mean for a stochastic process to be adapted to information? 3. What is meant by F(t)measurable? Or When is a process called measurable? 4. There is reference to sigma algebra when I googled about measurable stochastic processes. What is sigma algebra? This is what I googled---- " Let Ω be a set of outcomes. We denote by P(Ω) its power set, i.e. the collection of all the subsets of Ω. If the cardinality |Ω| of Ω is finite, then |P(Ω)| = 2|Ω| Definition 1. A σ-algebra on Ω is a collection F of subsets of Ω satisfying: 1. ∅ ∈ F. 2. If A ∈ F then Ac ∈ F. 3. F is stable by countable union: (∀n ≥ 1, An ∈ F) ⇒ ∪ n≥1 An ∈ F. The couple (Ω, F) is called a measurable space. In probabilty theory, an element of F is often called an event of F " Can some one pls explain all this? I shall really appreciate if some one can help me in understanding these concepts Regards Anand Sachdeva
Hi Anand, These concepts can appear daunting at first, but they're only trying to get across some quite simple ideas - it's just the jargon that can be confusing. The info you've googled goes more deeply and precisely into Measure Theory, but we really don't need that level of detail. The following should help to answer your questions: A filtration, F(t), captures all the information known up to and including time t. So if we're conditioning on F(t), this just means we're taking account of all the information we know by time t. For our purposes, "sigma algebra" is just another term for "filtration". A random variable is F(t)-measurable if it's value (or outcome) is known at time t. So you can think of "measurable" as meaning "known". So if we define Y to be the value of the FTSE-100 at time t then it's F(t)-measurable. But if we define Y to be the value of the FTSE-100 at time t+1 then it's not F(t)-measurable. Moving on to a stochastic process, this is a whole series of random variables. We introduce the concept of "adaption" and say that a process is adapted to F(t) if at all times t, the value of the process is F(t)-measurable. This is simply saying we always know the current value of the process, whatever the time is. Is this enough to give you a picture of what's going on?
Thanks Thanks a ton, Mike. You have made life much easier. You have explained things really nicely....Really appreciate!! Thanks pal
