Numerical Analysis
Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 9.4, Problem 1E

Use Ito’s formula to show that the solutions of the SDE initial value problems a { d y = B t d t + t d B t y ( 0 ) = c b { d y = 2 B t d B t y ( 0 ) = c are (a) y ( t ) = t B t + c (b) y ( t ) = B 2 t t + c .

a.

Expert Solution
Check Mark
To determine

To show: The SDE dy=Btdt+tdBt initial value problem has a solution y(t)=tBt+c equation with specified boundary y(0)=c condition.

Explanation of Solution

Given information:

The initial conditions that are given are,

  Ito's Formula y(t)=f(t,Bt)=tBt+cdy=Btdt+tdBty(0)=cbrownian motion Bt

Calculation:

As it’s known that by apply Ito's Formula to given equation shown below.

  y(t)=f(t,Bt)=tBt+c .

Note that  f(t,x)=tx+c The partial derivatives are

  ft=x,fx=t, and 2fx2=0.

Therefore,using Ito's Formula gives, the following step below.

   dy= f t (t, B t )dt+ f x (t, B t )d B t + 1 2 2 f x 2 (t, B t ) d B t  d B t = B t  dt+t d B t as required.

By using initial conditions:

  y(0)=0Bt+c=c .

b.

Expert Solution
Check Mark
To determine

To show: The SDE dy=2BtdBt initial value problem has a solution y(t)=tBt+c equation with specified boundary y(0)=c condition.

Explanation of Solution

Given information:

The initial conditions that are given are,

  Ito's Formula y(t)=f(t,Bt)=tBt+cdy=2BtdBty(0)=cbrownian motion Bt

Calculation:

As it’s known that by apply Ito's Formula

  Ito's Formula (brownian motion Bt)y(t)=f(t,Bt)=tBt+cdy=2BtdBty(0)=c

and the partial derivatives of  f(t,x)=x2t+c are ft=1,fx=2x, and 2fx2=2. Ito's Formula gives 

  dy=1dt+2Bt dBt+122 dBt dBt=dt+2Bt dBt+dt=2Bt dBt  as required.

By using initial conditions.

  y(0)=020+c=c

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Chapter 9 Solutions

Numerical Analysis

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