How Prove a process is Markov?

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  • #1
vale
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This is my first time here, so... Hi everybody!

I've very little time to figure out the following problem ... and I am wandering if some of you can give me any help or just suggest me any good reading material...

The question is how you can prove a process [tex] P_t[/tex], given the dynamics, is Markov.
In short my process is on alternate intervals, a mean reverting brownian bridge [tex]dP_t = \frac{\alpha}{G-t}(Q-P_t)dt + \sigma dW_t [/tex], and a mean reverting proportional volatility process : [tex]dP_t = K(\theta -P_t)dt + \nu dW_t [/tex]. The length of the intervals and their occurence is determined by an exogenous bootstrap procedure, which I believe, doesn't give any problems, being a resampling procedure with replacement, it doesn't generate any dependence with the past history...

How should I procede on your opinion? Any hints ?

Thank you very much in advance,
Vale
 

Answers and Replies

  • #2
EnumaElish
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Could this be of any help? When you say prove, do you mean empirically or mathematically?
 
  • #3
vale
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Thank you for the reference and the reply!

Actually I meant a mathematical proof...
I think I should show somehow the transition probabilities are independent from the past realizations... but I don't Know how to retrieve them from the dynamic... :uhh:

Many thanks...
Vale
 
  • #4
EnumaElish
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I guess I'd argue W(t) is independent of the past. Then equate Eq. (1) to Eq. (2) and solve for P(t). It'll be a function of t, W(t) and some constants. Since W is independent of the past, so's P(t).

{P.S. Oh, whoops! You said "on alternating intervals." Does that mean the two Eq's do not hold simultaneously?}

{P.P.S. In that case:

P(t+1) = P(t) + dP(t) = P(t) + a(dt) P(t) + b dW(t) = [a(dt)+1] P(t) + b dW(t).

Et+1[P(t+1)|P(t),P(t-1)...,P(0)] = (a+1) Et+1[P(t)|P(t),P(t-1)...,P(0)] + b Et+1[dW(t)|P(t),P(t-1)...,P(0)] = (a+1) P(t) + b Et+1[dW(t)]. QED

The last step is based on two premises: (i) E[X|X,Y,Z,...] = X, and (ii) dW(t) is independent of past history so E[dW(t)|P(t),P(t-1)...,P(0)] = E[dW(t)].}
 
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