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Problem 3. A two-dimensional system. Let (W)to be a one-dimensional Brownian motion. We consider two processes (X+)+20 and (Y)to solutions of the system of stochastic differential equations = with Xo 1, Yo = 0. dXt = Xx₁ dt - Y₁dWt, dyt = -Y dt +X+dWt, 1 (a) We denote Rt and t the radius and the angle in polar coordinates for the point (X, Y) in the two-dimensional plane. Use the multidimensional Itô Lemma to find the differential equations satisfied by Rt and Ot. (b) Solve the differential equations provided in (a) and give an intuitive description of the solution of the system. (c) Write an expression for the solution (X, Y) to the system.

Problem 3. A two-dimensional system. Let (W)to be a one-dimensional Brownian motion. We consider two processes (X+)+20 and (Y)to solutions of the system of stochastic differential equations = with Xo 1, Yo = 0. dXt = Xx₁ dt - Y₁dWt, dyt = -Y dt +X+dWt, 1 (a) We denote Rt and t the radius and the angle in polar coordinates for the point (X, Y) in the two-dimensional plane. Use the multidimensional Itô Lemma to find the differential equations satisfied by Rt and Ot. (b) Solve the differential equations provided in (a) and give an intuitive description of the solution of the system. (c) Write an expression for the solution (X, Y) to the system.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Problem 3. A two-dimensional system. Let (W)to be a one-dimensional Brownian motion. We consider two processes (X+)+20 and (Y)to solutions of the system of stochastic differential equations = with Xo 1, Yo = 0. dXt = Xx₁ dt - Y₁dWt, dyt = -Y dt +X+dWt, 1 (a) We denote Rt and t the radius and the angle in polar coordinates for the point (X, Y) in the two-dimensional plane. Use the multidimensional Itô Lemma to find the differential equations satisfied by Rt and Ot. (b) Solve the differential equations provided in (a) and give an intuitive description of the solution of the system. (c) Write an expression for the solution (X, Y) to the system.
Transcribed Image Text:Problem 3. A two-dimensional system. Let (W)to be a one-dimensional Brownian motion. We consider two processes (X+)+20 and (Y)to solutions of the system of stochastic differential equations = with Xo 1, Yo = 0. dXt = Xx₁ dt - Y₁dWt, dyt = -Y dt +X+dWt, 1 (a) We denote Rt and t the radius and the angle in polar coordinates for the point (X, Y) in the two-dimensional plane. Use the multidimensional Itô Lemma to find the differential equations satisfied by Rt and Ot. (b) Solve the differential equations provided in (a) and give an intuitive description of the solution of the system. (c) Write an expression for the solution (X, Y) to the system.
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