Stochastic transforms for jump diffusion processes combined with related backward stochastic differential equations

Abstract This study considers three stochastic transforms for jump diffusion processes: the stochastic Laplace transform, stochastic Fourier transform, and stochastic wavelet transform. First, we introduce the stochastic Laplace transform for processes adapted by Brownian filtration as a solution for complex number valued backward stochastic differential equations (BSDEs). This transform can be explained well by the Girsanov theorem, which also allows us to define the stochastic Laplace transform for jump diffusion processes directly. Based on this perspective, we give natural definitions of the stochastic Fourier transform and stochastic wavelet transform. The advantages of these stochastic transforms are all related to the uniqueness of the processes. Compared with the classical transforms, the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes, while they also agree with the corresponding parameters in the classical transforms, which can represent the frequency property of the processes. In addition, these three stochastic transforms can also be regarded as the solutions of related BSDEs with jumps.