Explanation Given: f , g , h are continuous on [ a , b ] and c is a real number. Calculation: f , g , h are continuous on [ a , b ] and c is a real number. An inner product of a function continuous real valued function is, ( f , g ) = ∫ a b f ( x ) g ( x ) d x … (1) Substitute f for g in Equation (1). ( f , f ) = ∫ a b f ( x ) f</ To determine (b) ( f , g ) = ( g , f ) To determine (c) ( f + h , g ) = ( f , g ) + ( g , h ) To determine (d) ( c f , g ) = c ( f , g )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Question

Chapter 11.3, Problem 12E

To determine

(a)

(f,f)≥0 and equals zero only when f=0.

To determine

(b)

(f,g)=(g,f)

To determine

(c)

(f+h,g)=(f,g)+(g,h)

To determine

(d)

(cf,g)=c(f,g)

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Chapter 11.3, Problem 11E

Chapter 11.3, Problem 14E

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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 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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(a) ( f , f ) ≥ 0 and equals zero only when f = 0 .

Solution Summary: The author explains that (f,f)ge 0 and equals zero only when

Videos

(a)

(f,f)≥0 and equals zero only when f=0.

(b)

(f,g)=(g,f)

(c)

(f+h,g)=(f,g)+(g,h)

(d)

(cf,g)=c(f,g)

Want to see the full answer?

Chapter 11 Solutions

(a) ( f , f ) ≥ 0 and equals zero only when f = 0 .

Solution Summary: The author explains that (f,f)ge 0 and equals zero only when

Videos

(a) (f,f)≥0 and equals zero only when f=0.

(b) (f,g)=(g,f)

(c) (f+h,g)=(f,g)+(g,h)

(d) (cf,g)=c(f,g)

Want to see the full answer?

Chapter 11 Solutions

(a)

(f,f)≥0 and equals zero only when f=0.

(b)

(f,g)=(g,f)

(c)

(f+h,g)=(f,g)+(g,h)

(d)

(cf,g)=c(f,g)