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9.6 Suppose W₁ is a standard one-dimensional Brownian motion. Suppose 1 and Zt satisfies the Bessel equation Zo = dZt - a Zt dt + dWt. Here a is a real number and we only consider t < T = min{s : Z, = 0}. (a) Find a nonconstant differentiable function such that Mt (ZAT) = is a martingale. (Hint: use Itô's formula to find a differential equation that o should satisfy and then solve the equation.) = (b) If 0 < € < 1 < a and S Zt = € or Zt = a, find P{Zs = €}. (c) Find the probability that there exists some time t with Zt = e. For which values of a is this probability equal to one? (d) For which values of a does the process reach the origin in finite time? S(e, a) denotes the first time t such that

9.6 Suppose W₁ is a standard one-dimensional Brownian motion. Suppose 1 and Zt satisfies the Bessel equation Zo = dZt - a Zt dt + dWt. Here a is a real number and we only consider t < T = min{s : Z, = 0}. (a) Find a nonconstant differentiable function such that Mt (ZAT) = is a martingale. (Hint: use Itô's formula to find a differential equation that o should satisfy and then solve the equation.) = (b) If 0 < € < 1 < a and S Zt = € or Zt = a, find P{Zs = €}. (c) Find the probability that there exists some time t with Zt = e. For which values of a is this probability equal to one? (d) For which values of a does the process reach the origin in finite time? S(e, a) denotes the first time t such that

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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9.6 Suppose W₁ is a standard one-dimensional Brownian motion. Suppose Zo = 1 and Zt satisfies the Bessel equation dZt = a Zt dt + dWt. Here a is a real number and we only consider t < T = min{s: Z, = 0}. (a) Find a nonconstant differentiable function & such that Mt (Zt^T) is a martingale. (Hint: use Itô's formula to find a differential equation that o should satisfy and then solve the equation.) (b) If 0 < € < 1 < a and S Zt = € or Z₁ = a, find P{Zs = e}. Zt (c) Find the probability that there exists some time t with Zt = €. For which values of a is this probability equal to one? (d) For which values of a does the process reach the origin in finite time? = S(e, a) denotes the first time t such that
Transcribed Image Text:9.6 Suppose W₁ is a standard one-dimensional Brownian motion. Suppose Zo = 1 and Zt satisfies the Bessel equation dZt = a Zt dt + dWt. Here a is a real number and we only consider t < T = min{s: Z, = 0}. (a) Find a nonconstant differentiable function & such that Mt (Zt^T) is a martingale. (Hint: use Itô's formula to find a differential equation that o should satisfy and then solve the equation.) (b) If 0 < € < 1 < a and S Zt = € or Z₁ = a, find P{Zs = e}. Zt (c) Find the probability that there exists some time t with Zt = €. For which values of a is this probability equal to one? (d) For which values of a does the process reach the origin in finite time? = S(e, a) denotes the first time t such that
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