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Math Help - Brownian Motion

  1. #1
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    Brownian Motion

    Hi!!! I hope someone can help me with the following exercise...
    n>=1, 0=t_{0}<t_{1}<...<t_{n}, a_{1},a_{2},...,a_{n} ε R. Show that the random variable a_{1}*B(t_{1})+...+a_{n}*B(t_{n}) is normally distributed and find its mean value and variance.
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  2. #2
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    Re: Brownian Motion

    Hey mathman.

    Hint: Use the property of a Brownian motion that if if B_a(t) and B_b(t) are brownian motions over disjoint times then they are both independent.
    Thanks from mathmari
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  3. #3
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    Re: Brownian Motion

    How can I find the mean value E(a_{1}*B(t_{1})+...+a_{n}*B(t_{n})) and the variance Var(a_{1}*B(t_{1})+...+a_{n}*B(t_{n}))? Using the property of a Brownian motion that E(B(t)-B(s))=0 and Var(B(t)-B(s))=t-s, 0<=s<t????
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  4. #4
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    Re: Brownian Motion

    Hint: If two variables are independent then E[XY] = E[X]E[Y]
    Thanks from mathmari
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