Explanation Given: The given equation is ∂ 2 u ∂ x 2 − R C ∂ u ∂ t − R G u = 0 , x > 0 , t > 0 and the boundary conditions are u ( 0 , t ) = 0 , lim x → ∞ ∂ u ∂ x = 0 and u ( x , 0 ) = u 0 . Calculation: The given equation is, ∂ 2 u ∂ x 2 − R C ∂ u ∂ t − R G u = 0 (1) Take Laplace transform of both sides of the above equation, L { ∂ 2 u ∂ x 2 } − R C [ L { ∂ u ∂ t } ] − R G [ L { u } ] = 0 d 2 U d x 2 − R C [ s U ( x , s ) − u ( x , 0 ) ] − R G 1 s = 0 Substitute u ( x , 0 ) = u 0 in the above equation, d 2 U d x 2 − R C [ s U ( x , s ) − u 0 ] − R G 1 s = 0 The particular solution of the above equation is, U ( x , s ) = c 1 e − R C s + R G x + c 2 e R C s + R G x + C u 0 C s + G (2) At the given boundary conditions u ( 0 , t ) = 0 and lim x → ∞ ∂ u ∂ x = 0 , lim x → ∞ d U d x = 0 and

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.2, Problem 25E

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To solve: The potential $u(x,t)$ in the line.

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Chapter 14.2, Problem 24E

Chapter 14.2, Problem 26E

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For each month of the year, Taylor collected the average high temperatures in Jackson, Mississippi. He used the data to create the histogram shown. Which set of data did he use to create the histogram? A 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92\ 55,\ 60,\ 64,\ 72,\ 73,\ 75,\ 77,\ 81,\ 83,\ 91,\ 91,\ 92 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92 B 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91\ 55,\ 57,\ 60,\ 65,\ 70,\ 71,\ 78,\ 79,\ 85,\ 86,\ 88,\ 91 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91 C 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93\ 55,\ 60,\ 63,\ 64,\ 65,\ 71,\ 83,\ 87,\ 88,\ 88,\ 89,\ 93 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93 D 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91\ 55,\ 58,\ 60,\ 66,\ 68,\ 75,\ 77,\ 82,\ 86,\ 89,\ 91,\ 91 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91

In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

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Chapter 14 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

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The potential u ( x , t ) in the line.

Solution Summary: The author explains the solution of potential in the line: (x,t)=u_0e-Gt

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To solve: The potential $u(x,t)$ in the line.

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Chapter 14 Solutions

The potential u ( x , t ) in the line.

Solution Summary: The author explains the solution of potential in the line: (x,t)=u_0e-Gt

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To solve: The potential u(x,t)<m:math><m:mrow><m:mi>u</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>t</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> in the line.

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Chapter 14 Solutions

To solve: The potential $u(x,t)$ in the line.