To show that Euler’s Method with time step h = ( b − a ) / N for N steps yields the N t h left-endpoint approximation to y ( b ) = ∫ a b f ( u ) d u

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Answer 1

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Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

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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 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

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Answer 3

Question

Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

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Answer 4

Question

Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

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Answer 5

Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

Answer 6

Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

Answer 7

Chapter 9.3, Problem 23E

To determine

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$

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 show that Euler’s Method with time step h = ( b − a ) / N for N steps yields the N t h left-endpoint approximation to y ( b ) = ∫ a b f ( u ) d u

To show that Euler’s Method with time step h=(b−a)/N for N steps yields the Nth left-endpoint approximation to y(b)=∫abf(u)du

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Chapter 9 Solutions

To show that Euler’s Method with time step $h = (b - a) / N$ for $N$ steps yields the $N t h$ left-endpoint approximation to $y (b) = \int_{a}^{b} f (u) d u$