AERONAUTIC
Examples.docx

# Λx 1 t 2 σ u ye λy px y tdy exp 4 1 λt cosh x 1

• 8

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λ(x + 1 t 2 ) Σ u 0 (y)e λy p(x, y, t)dy = √ exp 4 . 0 (1 + λt) cosh( x) 1 + λt Inversion of the Laplace transform gives the fundamental solution 1 . Σ t 4 t cosh( x) exp y t (x, y, t) = . t I α . t I α t 4 t cosh( x) exp y t (x, y, t) = This reduces to the transition density of the process at µ = 0. (See [9] for the derivation of the transition density). This fundamental so- lution is integrable near zero for 0 ≤ µ < 1 . We may calculate by the Feynman-Kac formula E x Σ e λX t µ t ds 0 X s = e λy p(x, y, t)dy but 0 this integral again does not seem to be easy to evaluate analytically. Of course it is possible to evaluate it numerically. We leave it to the reader to show that for X = {X t : t ≥ 0} where dX 1 X coth( X )) dt + 2 X dW , we may solve the equation u t = xu xx + ( 1 + x coth( x))u x µ u to obtain the fundamental solution 1 y t (x, y, t) = 2 x . Σ , t I t 4 t 0 X s 2

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at µ = 0. y = 0 for all µ > 1 . 2 x t 0 X s . Σ 0 X s 2 4 y d p(x, y, t)e λy dy = p(x, y, t) = 2t 2 2 I 2 d + n 1 = e 2 t +4 t 2 λ Γ(β)(1 + t 0 X s x , t I t 4 t sinh( x) exp y t (x, y, t) = where α = 1+16 µ . This reduces to the transition density of the process It is integrable at 16 The range of µ values in these examples for which the fundamental solution is integrable at y = 0 depends on the properties of the process. For µ outside the given ranges we may find other fundamental solutions, and investigate their properties. Some will contain distributional terms. A full study of this is beyond the scope of the current paper. Example 5. Let X = { X t : t 0 } b e a squared Bessel pro- cess, where dX t = ndt + 2 X t dW t , X 0 = x. The joint density of (X , t ds ) arises in the pricing of Asian options and other prob- lems, see [6]. To obtain its Laplace transform we require a funda- mental solution for the PDE u t = 2xu xx + nu x µ u. From Theorem 2.3 the reader may check that the stationary solution u 0 (x) = x d , where d = 1 (2 n + ( n 2) 2 + 8 µ
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• Fall '16
• Dr Salim Zahir
• Complex number, Dirac delta function, Pierre-Simon Laplace

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