Explanation Given information: An equation of parabola is given as, y = a x 2 − 1 . Also, a changes from − ∞ to ∞ . Concepts used: 1. ) A standard quadratic equation y = a x 2 + b x + c , a ≠ 0 , represents a parabola and the solution of this quadratic is the point where parabola crosses x − axis. 2. ) If a > 0 then, parabola y = a x 2 + b x + c opens upward. 3. ) If a < 0 then, parabola y = a x 2 + b x + c opens downward. Calculation: An equation of parabola is given as, y = a x 2 − 1 . Solution of this quadratic equation will be given as, x = − b ± b 2 − 4 a c 2 a ⇒ x = − 0 ± 0 2 − 4 a (

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 1.2, Problem 7PQ

To determine

To describe:

The changes of parabolas y=ax2−1 as a changes from −∞ to ∞.

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Chapter 1.2, Problem 6PQ

Chapter 1.2, Problem 8PQ

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The OU process studied in the previous problem is a common model for interest rates. Another common model is the CIR model, which solves the SDE: dX₁ = (a = X₁) dt + σ √X+dWt, - under the condition Xoxo. We cannot solve this SDE explicitly. = (a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler scheme to simulate a trajectory of the CIR process. On a graph, represent both the trajectory of the OU process and the trajectory of the CIR process for the same Brownian path. (b) Repeat the simulation of the CIR process above M times (M large), for a large value of T, and use the result to estimate the long-term expectation and variance of the CIR process. How do they compare to the ones of the OU process? Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000. 1 (c) If you use larger values than above for the parameters, such as the ones in Problem 1, you may encounter errors when implementing the Euler scheme for CIR. Explain why.

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Chapter 1 Solutions

Calculus: Early Transcendentals

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