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Characteristic Functionfunction of Quadratic Variationquadratic variation of Compoundcompound Poisson Processprocess

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Characteristic Function of Quadratic Variation of Compound Poisson Process

If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process?

If not possible in general, are there specific examples of compound Poisson processes for which this function is known in closed form?