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Here is a question about linear transformation of Levy processes. It is stated in my reference (Cont and Tankov's Financial modelling with jump processes, Theorem 4.1) that a linear transformation of a Levy process is again a Levy process. But then, what would be its Levy exponent? Theorem 4.1 of the Book by Cont and Tankov (googlable) should provide an answer but it is my opinion that the result is at best inaccurate. Indeed equation (4.3) is a sum of a vector with a scalar; this confusion originates in the algebraic manipulation where the authors, after adding and subtracting appropriate quantities, implicitly write the integral of a scalar product as a scalar product against an integral. But this is clearly unfeasible unless the integrand is linear (which is not).

Note that since the proof is based on the derivation of a Levy exponent this also casts (for me) a shadow of doubt on the correctness of the first statement. For example: an independent sum of Levy processes is clearly a Levy process, but if levy processes are closed under linear transformations, then finite sums of any Levy process is a Levy process, regardless of possible dependence. Is it that so? Seems like a rather big statement to me.

Does anyone share my concerns or am I just being silly? Can anyone point out a better reference on results about linear transformation of Levy processes, in particular on the Levy exponent of a transformed Levy process? Thanks.

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up vote 2 down vote accepted

I imagine Cont/Tankov were strongly inspired by Ken-Iti Sato - Levy Processes and Infinitely divisible distributions (1999) as they cite that source repeatedly throughout. In that book, Proposition 11.10 deals with the issue of linear transformations (with a very similar proof to Cont/Tankov), who in turn later (at the end of the chapter) references with regard to linear transformations a paper by Maruyama: Infinitely divisible processes (1970) (to be found online here: http://www.mathnet.ru/links/cfdac58de818016ebdbaa68ff74c7322/tvp1546.pdf)

Although I personally don't find these references more helpful in explaining the notation, maybe they will help you. In any case, they are at least further references on the issue.

My intuition is that the scalar in the sum is to be considered as a vector with all elements containing the scalar. I agree that the notation is not helpful in clarifying this.

As far as dependence structures are concerned - yes, the statement made is a general one, with no limitations to the dependence between elements of the multidimensional Levy process. All dependencies are inherent in the characteristics, specifically matrix $A$.

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