Problem 1. X. Notice first that Y₁ = X₁ = %. Then, we use the function x) e'x in Ito Lemma. We have (a) Let Y, f(t, af (1)-2(x)- è', and (2.3)-0 Thus, Its formula leads to Y₁-f(t, x) - Yo+ + ( (s. X.) ds+ x.)dx+ 2% (X) (X)* - 3+ eX 2+ -x+ + Lex eX, da + Loca edW, - x+ ac*ds + acdW -x+(e' -1)+ ae'dW.. We conclude that X="-20" + a(1-e") + ac Lea e'dW (1) (b) Since the Itó integral has zero expectation, we obtain from (1) that E[X₂] = ±â˜º + a(1 − e¨¹). Then, using the definition of variance and Its isometry, we have Var(X)- E[{X; - E[X;])²] -E -21 edW = £ [p*e* ([^<^av.)"] (2) -04-27 ds -- 1 (c) Taking the limit as too in (2) and (3), we obtain and lim EX-a 1-+ - (3) (d) Yes. The random variable X, is normally distributed. The first two terms in (1) are not random. The only random term is Leaw which is normally distributed since the integrand is deterministic. Indeed, you can think of this term as the limit Lea WWW), A-0 for suitable partitions (t. The sum above is a linear combina tion of independent Gaussian random variables and, thus, is normally distributed. Altogether, the distribution of X, is given by N( 2 The purpose of this problem is to simulate the trajectory of an Ornstein-Uhlenbeck (OU) process. The OU process solves the SDE dX = (a Xt) dt +σ dWt, - under the condition Xoxo. The exact solution was computed in HW#6, Prob.1. = (a) Use the software / language of your choice to simulate a trajectory of Brownian motion over an interval [0,T] with N discretization points. (b) Use the Brownian trajectory simulated and use the Euler scheme to simulate one trajectory of the OU process. Represent your result on a graph. (c) Explain how the numerical scheme could be improved if you needed better precision. (You do not need to do it, just briefly explain.) (d) Repeat the simulation above M times (M large), for a large value of T, and use the result to estimate (as in Monte-Carlo simulations) the long-term expectation and variance of the OU process. Compare with the theoretical result seen in HW#6, Prob.1. Numerical application: T = 10, N = 500, a = 1, x0 = 5, σ = 1, M = 1000.

Problem 1. X. Notice first that Y₁ = X₁ = %. Then, we use the function x) e'x in Ito Lemma. We have (a) Let Y, f(t, af (1)-2(x)- è', and (2.3)-0 Thus, Its formula leads to Y₁-f(t, x) - Yo+ + ( (s. X.) ds+ x.)dx+ 2% (X) (X)* - 3+ eX 2+ -x+ + Lex eX, da + Loca edW, - x+ acds + acdW -x+(e' -1)+ ae'dW.. We conclude that X="-20" + a(1-e") + ac Lea e'dW (1) (b) Since the Itó integral has zero expectation, we obtain from (1) that E[X₂] = ±â˜º + a(1 − e¨¹). Then, using the definition of variance and Its isometry, we have Var(X)- E[{X; - E[X;])²] -E -21 edW = £ [pe* ([^<^av.)"] (2) -04-27 ds -- 1 (c) Taking the limit as too in (2) and (3), we obtain and lim EX-a 1-+ - (3) (d) Yes. The random variable X, is normally distributed. The first two terms in (1) are not random. The only random term is Leaw which is normally distributed since the integrand is deterministic. Indeed, you can think of this term as the limit Lea WWW), A-0 for suitable partitions (t. The sum above is a linear combina tion of independent Gaussian random variables and, thus, is normally distributed. Altogether, the distribution of X, is given by N( 2 The purpose of this problem is to simulate the trajectory of an Ornstein-Uhlenbeck (OU) process. The OU process solves the SDE dX = (a Xt) dt +σ dWt, - under the condition Xoxo. The exact solution was computed in HW#6, Prob.1. = (a) Use the software / language of your choice to simulate a trajectory of Brownian motion over an interval [0,T] with N discretization points. (b) Use the Brownian trajectory simulated and use the Euler scheme to simulate one trajectory of the OU process. Represent your result on a graph. (c) Explain how the numerical scheme could be improved if you needed better precision. (You do not need to do it, just briefly explain.) (d) Repeat the simulation above M times (M large), for a large value of T, and use the result to estimate (as in Monte-Carlo simulations) the long-term expectation and variance of the OU process. Compare with the theoretical result seen in HW#6, Prob.1. Numerical application: T = 10, N = 500, a = 1, x0 = 5, σ = 1, M = 1000.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 77E

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