Introduction to Partial Differential Equations
Exercise Sheet 5
Ross Ogilvie 30th September,
2024
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In economics, the Black-Scholes equation is a PDE that describes the price of an (European-style) option under some assumptions about the risk and expected return, as a function of time and current stock price . The equation is
where
and
are constants representing the interest rate and the stock volatility respectively. Describe the
order of this equation, and whether it is elliptic, parabolic, and/or hyperbolic.
(3 points)
Consider the unit circle . In this question we will evaluate the integral
in two different ways, demonstrating that it does not depend on the choice of parametrisation.
Therefore in order to compute the submanifold integration () it is enough to use parameterisation that cover all but finitely many points of .
Using Definitions 2.4 and 2.7, prove Lemma 2.9: The following properties hold for and .
Linearity: . (1 point)
Order Preserving: if on then . (1 point)
Triangle Inequality: . (1 point)
Transformation: If is a euclidean motion (translation, reflection, rotation) and is a scaling factor then . (3 points)
Consider the plane , a disc and the function .
does not depend on the radius . (2 points)