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# Mathematical Models in Finance and Industry Math Problem (Math Problem Sample)

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Please ensure the writer have the maths knowledge of my problem. I will also upload my lecture notes to help writer sovle the problem.

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Mathematical Models in Finance and Industry

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Question one

qDx,t=-k∂μ∂x(x, t) with the k>0 which is the diffusion coefficient

To make this equation readable we denote qDx,t by μx,t such that

μx,t=-k∂μ∂x(x, t)

Then

ddtabμx,t dx=abμx,t dt

=-(q (b,t)-q(a,t)=-abqA,xx,t+qDxx,tdx

=-abμνxx,t-kμxxx,tdx

Implying that

=>abμxx,t-kμxxx,t+μνxx,tdx……………i

Equation (i) only when all intervals [a, b] and for all t>0.

Integrating equation (i) we get

μxx,t-kμxxx,t+μνxx,t+cμ=0 For all xϵR and for all t>0

Question two

* Ljn=μxj,tn+1-μ(xj,tn)∆t+μxj+1,tn-μ(xj,tn)h

Expansion using Taylor expansion we get

|Ljn|=∆t2μxj,tn+1-μ(xj,tn)∆t+Ch(μxj+1,tn-μxj,tn)h, expansion using Taylor expansion theorem

Simplifying the R.H.S Expressions we get

|Ljn|=∆t2μxj,tn+1-μ(xj,tn)∆t+h2(μxj+1,tn-μxj,tn)h…………………(i)

= ∆tμxj,tn+1-μxj,tn+h (μxj,tn+1-μxj,tn……………ii

, simplifying the two expressions in equation above

=( ∆t+h) (μxj,tn+1-μxj,tn) factoring the ( ∆t+h)

But μxj,tn+1-μxj,tn=CL

Replacing μxj,tn+1-μxj,tn with CL from the equation( i) we get

|Ljn|=CL( ∆t+h) Hence proven.

* To show that ejn=μμj-tn-μj satisfiesejn+1-ejn∆t+aej+1-ejh

We start at n=o by finding out the iterations of ejn

ej1=ej0-a∆tej0-ej-10h, j∈Z

ej2=ej1-a∆tej1-ej-11h, j∈Z

Up to

ejn+1=ejn-a∆tejn-ej-1nh, j∈Z and n>1

The exact solution for this expression

uμj-tn=e0x-at

Hence it satisfies ejn+1-ejn∆t+aej+1-ejh

Using induction on n to prove

supj∈Z| ejn|≤n∆tCL( ∆t+h)

|ejn|≤|(1-a∆th)ejn|+|a∆thμj-1n dropping the sup and expanding the R.H.S expression

...

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