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Math Help - Poisson process question and more...

  1. #1
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    Poisson process question and more...

    Can anyone please help ...



    Let {X}(t)=\int_{0}^{t}\tau{d}{N}(\tau), where {N}(\tau) is a Poisson process with rate \lambda.
    Find {E}{X}(t), and cov\{{X}(t),{X}(\tau)\}.


    Also, let {B}_{s} be Brownian motion and {Y}(t)=\int_{0}^{t}({B}_{s}-{s})ds,
    Find {E}{Y}(t), and cov\{{Y}(t),{Y}(\tau)\}.

    I'm having real trouble with these, so if anyone could give concise details of steps involved with explanations, that would be great.
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  2. #2
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    Quote Originally Posted by symmetry7 View Post
    Can anyone please help ...



    Let {X}(t)=\int_{0}^{t}\tau{d}{N}(\tau), where {N}(\tau) is a Poisson process with rate \lambda.
    Find {E}{X}(t), and cov\{{X}(t),{X}(\tau)\}.


    Also, let {B}_{s} be Brownian motion and {Y}(t)=\int_{0}^{t}({B}_{s}-{s})ds,
    Find {E}{Y}(t), and cov\{{Y}(t),{Y}(\tau)\}.

    I'm having real trouble with these, so if anyone could give concise details of steps involved with explanations, that would be great.
    The second one is a bit simpler, you only have to apply Fubini:
    E[Y_t]=\int_0^t (E[B_s]-s)ds=0 and E[Y_tY_{t'}]=\int_0^t\int_0^{t'} E[(B_s-s)(B_{s'}-s')]ds' ds.
    Expand and use E[B_s B_s']=\min(s,s').


    For the first one, one way is to reduce to the same method as for the first one: by integration by part, X(t)=tN(t)-\int_0^t N(\tau)d\tau = \int_0^t (N(t)-N(\tau))d\tau. Then do like above (but the expectation is nonzero).
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