Initial value PDE: the Black-Scholes model (part c)

kreedman

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How do I start with part c ?? Should I find the norm? If so, which norm?

(c) Recall the terminal-value PDE (1.27)-(1.28) below:

[quote:8zmerpil]. . . . .\(\displaystyle \L u_t\, +\, \frac{1}{2}\sigma^2 x^2 u_{xx}\, +\, rxu_x\, -\, ru\, =\, 0,\, \mbox{ }\, 0\, <\, x\, <\, \infty,\, \mbox{ }\, t\, \leq\, T\, \, \mbox{ }(1.27)\)

...with the terminal condition:

. . . . .\(\displaystyle \L u(T,\,x)\, =\, max(x\,-\,E,\,0)\,\,\mbox{ }\,(1.28a)\)

...and the boundary conditions:

. . . . .\(\displaystyle \L u(t,\,0)\, =\, 0,\,\,u(t,\,x)\,\sim \, x\, -\, Ee^{-r(T\,-\,t)}\,\,\mbox{as}\,x\, \rightarrow \,\infty\,\,\mbox{ }\,(1.28b)\)
Prove that (1.27)-(1.28) is well-posed.

Recall (1.30):


. . . . .\(\displaystyle \L w_s \,=\, w_{yy},\,\,\mbox{ } -\infty \,< y\, < \, \infty,\,\, s\, \geq \, 0\). . . . .(1.30)

. . . . .\(\displaystyle \L w(0,\,y)\, =\, \mbox{max}\left(e^{\frac{1}{2}(\kappa\, +\, 1)y}\, -\, e^{\frac{1}{2}(\kappa\, +\, 1)y}\, ,\,0\right)\)
Note that the solution of (1.30), and therefore also of (1.27)-(1.28) can be specified exactly in terms of the following integral:

. . . . .\(\displaystyle \L N(z)\, =\, \frac{1}{\sqrt{2\pi}}\,\int_{-\infty}^{\zeta}\,e^{-\frac{\zeta^2}{2}}\,d\zeta\)

However, you don't need this for the purpose of the present exercise.
[/quote:8zmerpil]

The textbook says that a problem is well posed if there are constant K and a such that:

. . . . .|| u(t) || <= K e^at ||u(0)||

Is that what the quesion is pointing to? How should I find the norm?

Thank you!

FreeMathHelp thread for part a

FreeMathHelp thread for part b

screen-shot of entire page
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Edited by stapel -- Reason for edit: Converting fuzzy graphic into clear text.
 
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