... Let Xn Xo + k=1 $k, n = 0, 1, 2, . . . denote...

 

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... Let Xn Xo + Σk=1 $k, n = 0, 1, 2, . . . denote a Random Walk process, with Xo = x integer x, and (k), k = 1, 2, . is a sequence of i.i.d. random variables with distribution P(§₁ = −1) = 2/3 = 1 − P(§₁ = 2). Let T time when the Random Walk hits the set [100, ∞). = T[100,∞) be first = (a) Show that T = T{100,101}, the first hitting time of the set (b) Write the difference equation for (x) P(T < ∞|Xo First step analysis and specify the domain where it holds. {100, 101}. = - x) using the = (c) Show that the function G(x) bx+c for any constants b, c is a solution to the difference equation in (b), disregarding the boundary conditions. (d) Show that the function H(x) = a* is a solution to the difference equation in (b) for the appropriate choice of constant a. Find all such solutions. Hint: Factorize the equation by using the fact that a = 1 is a root. (e) Use the fact that the general solution of the difference equation in (b) is of the form c₁G(x) + c₂H (x) for some constants c₁, C2 to find (x). Hint: Deduce that (x) is a constant, and then find it. It might help to show that (x) is monotone non-decreasing. ... Let Xn Xo + Σk=1 $k, n = 0, 1, 2, . . . denote a Random Walk process, with Xo = x integer x, and (k), k = 1, 2, . is a sequence of i.i.d. random variables with distribution P(§₁ = −1) = 2/3 = 1 − P(§₁ = 2). Let T time when the Random Walk hits the set [100, ∞). = T[100,∞) be first = (a) Show that T = T{100,101}, the first hitting time of the set (b) Write the difference equation for (x) P(T < ∞|Xo First step analysis and specify the domain where it holds. {100, 101}. = - x) using the = (c) Show that the function G(x) bx+c for any constants b, c is a solution to the difference equation in (b), disregarding the boundary conditions. (d) Show that the function H(x) = a* is a solution to the difference equation in (b) for the appropriate choice of constant a. Find all such solutions. Hint: Factorize the equation by using the fact that a = 1 is a root. (e) Use the fact that the general solution of the difference equation in (b) is of the form c₁G(x) + c₂H (x) for some constants c₁, C2 to find (x). Hint: Deduce that (x) is a constant, and then find it. It might help to show that (x) is monotone non-decreasing.

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Solution Random Walk process 100 101 No... View the full answer