MathFinance 345/Stat390 Homework 5 Due November 7galton.uchicago.edu/~lalley/Courses/390/HW5.pdf ·...
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MathFinance 345/Stat390 Homework 5 Due November 7 1. One–Sided Stable-1/2 Law. For each a> 0, let τ a be the first–passage time to the level a by the Brownian path W (t). Let T 1 ,T 2 ,... be a sequence of independent, identically distributed random variables each having the same distribution as τ 1 . (a) Show that ET 1 = ∞. (b) Show that τ n has the same distribution as ∑ n j =1 T j . (c) Show that T 1 has the same distribution as n -2 ∑ n j =1 T j . Hint for (b): Use the strong Markov property repeatedly. 2. First Passage to a Tilted Line. Define τ = min{t> 0: W (t)= a - bt} where a, b > 0 are positive constants. Find the Laplace transform and/or the probability density function of τ . 3. Nondifferentiability of Brownian Paths. Prove that, with probability one, W (t) is not differentiable at t = 0. Hint: If W (t) were differentiable at t = 0 then for all sufficiently small ε> 0 the graph of {W (t)} 0≤t≤ε would have to lie between the lines w = -t/ε and w = t/ε. But the probability that even the endpoint (ε, W (ε)) lies between these two lines is only P {|W (ε)| <ε} = 1 √ 2π √ ε - √ ε e -x 2 /2 dx. Note: It is a theorem of Dvoretsky, Erd¨ os, and Kakutani that, with probability one, W (t) is not differentiable anywhere. This is considerably more difficult to prove than nondifferentiability at 0. 4. Verify that the Gauss kernel p t (x, y) solves the heat equation. 5. * Embedded Simple Random Walk. Define stopping times τ n inductively as follows: let τ 0 = 0, and let τ 1 = min{t : |W (t)| =1}; τ n+1 = min{t : |W (t + τ n ) - W (τ n )| =1} (a) Use the strong Markov property to show that the sequence {W (τ n )} n≥0 is a simple random walk. (b) Use the Optional Sampling Formula to show that Eτ 1 = 1. (c) Use the SLLN to conclude that τ n /n → 1. (d) Use (a), (c), path continuity, and the scaling property of Brownian motion to give a new proof of the DeMoivre–Laplace Central Limit theorem. 1
Transcript of MathFinance 345/Stat390 Homework 5 Due November 7galton.uchicago.edu/~lalley/Courses/390/HW5.pdf ·...
MathFinance 345/Stat390Homework 5Due November 7
1. One–Sided Stable-1/2 Law. For each a > 0, let τa be the first–passage time to thelevel a by the Brownian path W (t). Let T1, T2, . . . be a sequence of independent, identicallydistributed random variables each having the same distribution as τ1.
(a) Show that ET1 = ∞.(b) Show that τn has the same distribution as
∑nj=1 Tj.
(c) Show that T1 has the same distribution as n−2∑n
j=1 Tj.
Hint for (b): Use the strong Markov property repeatedly.
2. First Passage to a Tilted Line. Define τ = min{t > 0 : W (t) = a−bt} where a, b > 0are positive constants. Find the Laplace transform and/or the probability density functionof τ .
3. Nondifferentiability of Brownian Paths.Prove that, with probability one, W (t) is not differentiable at t = 0.
Hint: If W (t) were differentiable at t = 0 then for all sufficiently small ε > 0 the graph of{W (t)}0≤t≤ε would have to lie between the lines w = −t/ε and w = t/ε. But the probabilitythat even the endpoint (ε, W (ε)) lies between these two lines is only
P{|W (ε)| < ε} =1√2π
∫ √ε
−√
ε
e−x2/2 dx.
Note: It is a theorem of Dvoretsky, Erdos, and Kakutani that, with probabilityone, W (t) is not differentiable anywhere. This is considerably more difficult to prove thannondifferentiability at 0.
4. Verify that the Gauss kernel pt(x, y) solves the heat equation.
5.∗ Embedded Simple Random Walk. Define stopping times τn inductively as follows:let τ0 = 0, and let
τ1 = min{t : |W (t)| = 1};τn+1 = min{t : |W (t + τn)−W (τn)| = 1}
(a) Use the strong Markov property to show that the sequence {W (τn)}n≥0 is a simplerandom walk.(b) Use the Optional Sampling Formula to show that Eτ1 = 1.(c) Use the SLLN to conclude that τn/n → 1.(d) Use (a), (c), path continuity, and the scaling property of Brownian motion to give a newproof of the DeMoivre–Laplace Central Limit theorem.
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