Hello, I have 2 questions regarding the numerical inversion of Laplace transforms. Apologies if I seem somewhat pedantic ...(1) The inversion formula (Bromwich integral) requires all singularities of the transform to lie to the left of the imaginary axis over which the integration is performed. However in pretty mcuh every paper I have read, no author ever bothers to check where these singularities are when they do the numerical inversion.Why is this? (Admittedly, I imagine it might be difficult to identify where the singularities are but shouldn't people at least flag this issue in their papers?) Moreover, I imagine this might be an issue if you're using laplace-based prices to calibrate a model as some extreme parameter values (which could appear in the optimization process) could create / shift singularities to the right of the axis over which you are integrating when doing the numerical inversion.(2) Numerical inversion of the Laplace transform, f(s) say, requires the transform for complex s. However some authors just compute the transform for real s. Are they implicitly invoking an analytic continuation argument here to obtain the transform for complex s? If so, why don't they state this? And don't some conditions need to be checked before invoking analytic continuation ?

Yes, you should check for singularities and check that analytic continuation makes sense.As a practical matter, with F(s) = int exp(-s t) f(t) dt, in finance f(t) is usually bounded or growing at worst like exp(r t), where r is some interest rate. If bounded, there are no singularities of F(s) in Re s > 0. If f(t) is growing like a money account, there are no singularities in Re s > r.Even when these things are true, authors should mention them for their readers.regards,

Thanks Alan,And do you have a view on the 2nd question I raised via analytic continuation being invoked to obtain the transform for complex s?

Perhaps I already answered it, although maybe not clearly.For example, once you check that f(t) is bounded, then F(s), defined as I wrote it before, is an analytic function of complex s in the entire half-plane Re s > 0. This also means that you can start at s = u0, where u0 > 0 is real and perform an analytic continuationto anywhere in Re s > 0 as described by the procedure in complex variable books. The final result will be the same function F(s) as defined before by the integral, as the analytic continuation is unqiue.