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By smooth, I mean a density $f$ that lies in the space $C^\infty$, infinitely differentiable.

Are there, in the literature, some known models where the underlying density of the state process is non-smooth?

I have only been able to find one such example: the Variance Gamma model. Would love to hear if people know of more such examples.

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A simple model would be the double exponential model from Kou (2002). It is very similar to the jump-diffusion model from Merton (1976) but instead of modelling the jump size by a normal distribution, Kou employs an asymmetric double exponential distribution (aka Laplace distribution).

The corresponding density is $$f_X(x) = p\zeta e^{-\zeta x}\mathbb{1}_{\{x\geq 0\}}+q\eta e^{\eta x}\mathbb{1}_{\{x<0\}},$$ where $p+q=1$ and $\zeta,\eta>0$ and $p,q\geq0$.

This function is not smooth (at zero) and kind of represents the pasting of two exponential distributions. Like the exponential distribution, the double exponential distribution is memoryless. The model implies a high peaked and heavy tailed distribution. Kou argues (psychologically) that upwards/downwards jumps have different effects on investors and hence he opts for the pasting of two exponential distributions. The model is quite tractable, has an easy characteristic function and allows for closed-form solutions to prices of many derivatives.

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