To apply both Euler’s Method and the Euler Midpoint Method with h = 0.1 to estimate y ( 1.5 ) where y ( t ) satisfies d y d t = y with y ( 0 ) = 1 . Find y ( t ) exactly and compute the errors in these two approximations.

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Answer 1

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Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

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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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Answer 2

Question

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

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Answer 3

Question

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

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Answer 4

Question

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

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Answer 5

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

Answer 6

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

Answer 7

Chapter 9.3, Problem 24E

To determine

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 9.1 - Prob. 1PQ Ch. 9.1 - Prob. 2PQ Ch. 9.1 - Prob. 3PQ Ch. 9.1 - Prob. 4PQ Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E

To apply both Euler’s Method and the Euler Midpoint Method with h = 0.1 to estimate y ( 1.5 ) where y ( t ) satisfies d y d t = y with y ( 0 ) = 1 . Find y ( t ) exactly and compute the errors in these two approximations.

To apply both Euler’s Method and the Euler Midpoint Method with h=0.1 to estimate y(1.5) where y(t) satisfies dydt=y with y(0)=1. Find y(t) exactly and compute the errors in these two approximations.

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Chapter 9 Solutions

To apply both Euler’s Method and the Euler Midpoint Method with $h = 0.1$ to estimate $y (1.5)$ where $y (t)$ satisfies $\frac{d y}{d t} = y$ with $y (0) = 1$ . Find $y (t)$ exactly and compute the errors in these two approximations.