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Hi All,
does anybody know how to invert a two-sided Laplace transform (numerically)? The one-sided Laplace transform ($F$) of the function ($f$) is given by
($$F(s)=\int_0^{+\infty} \exp(-s\cdot t)\cdot f(t) dt$$),
which can be easily inverted numerically using the inversion formula of (Dubner and Abate, 1968, Numerical Inversion of Laplace Transforms by Relating Them to the Finite Fourier Cosine Transform) given by:
($$f(t) =\frac{\exp(A/2)}{2} \cdot \left[ \frac{1}{2} \cdot F \large( \frac{A}{2 \cdot t} \right) + \sum_{k=1}^{+\infty} (-1)^k \cdot \Re \left\{ F\large( \frac{A+ 2 \cdot \pi \cdot k \cdot \sqrt{-1}}{2 \cdot t} \right) \right\} \right] $$),
where ($A$) is a positive constant and ($\Re$) denotes the real part. This method is extended by (Petrella, 2004, An Extension of the Euler Laplace Transform Inversion Algorithm with Applications in Option Pricing) for functions on the entire real line if those satisfy specific conditions.
The two-sided Laplace transform ($\hat{F}$) of the function ($f$) is given by
($$\hat{F}(s)=\int_{-\infty}^{+\infty} \exp(-s\cdot t)\cdot f(t) dt$$),
which is used, for instance, within the option pricing model of (Kou, Petrella, Wang, 2005, Pring Path-Dependent Options with Jump Risk via Laplace Transforms).
(Kou, Petrella, Wang 2005) say that they apply the extention of (Petrella, 2004) but the extention still refers to the one-sided Laplace transform since the inversion formula remains unchanged except that the discretization error is now on both sides of the real axis. Furthermore (Petrella, 2004) unfortunately does not mention how his extended version can be applied for inverting two-sided Laplace transforms.
Does anyone know the answer? Thanxx for helping me.
does anybody know how to invert a two-sided Laplace transform (numerically)? The one-sided Laplace transform ($F$) of the function ($f$) is given by
($$F(s)=\int_0^{+\infty} \exp(-s\cdot t)\cdot f(t) dt$$),
which can be easily inverted numerically using the inversion formula of (Dubner and Abate, 1968, Numerical Inversion of Laplace Transforms by Relating Them to the Finite Fourier Cosine Transform) given by:
($$f(t) =\frac{\exp(A/2)}{2} \cdot \left[ \frac{1}{2} \cdot F \large( \frac{A}{2 \cdot t} \right) + \sum_{k=1}^{+\infty} (-1)^k \cdot \Re \left\{ F\large( \frac{A+ 2 \cdot \pi \cdot k \cdot \sqrt{-1}}{2 \cdot t} \right) \right\} \right] $$),
where ($A$) is a positive constant and ($\Re$) denotes the real part. This method is extended by (Petrella, 2004, An Extension of the Euler Laplace Transform Inversion Algorithm with Applications in Option Pricing) for functions on the entire real line if those satisfy specific conditions.
The two-sided Laplace transform ($\hat{F}$) of the function ($f$) is given by
($$\hat{F}(s)=\int_{-\infty}^{+\infty} \exp(-s\cdot t)\cdot f(t) dt$$),
which is used, for instance, within the option pricing model of (Kou, Petrella, Wang, 2005, Pring Path-Dependent Options with Jump Risk via Laplace Transforms).
(Kou, Petrella, Wang 2005) say that they apply the extention of (Petrella, 2004) but the extention still refers to the one-sided Laplace transform since the inversion formula remains unchanged except that the discretization error is now on both sides of the real axis. Furthermore (Petrella, 2004) unfortunately does not mention how his extended version can be applied for inverting two-sided Laplace transforms.
Does anyone know the answer? Thanxx for helping me.