Lie point symmetry analysis of a dependent jump-diffusion risk model with exponential claim sizes

We carry out a Lie point symmetry analysis of the coupled integro-differential system satisfied by the conditional Gerber--Shiu functions of a dependent jump-diffusion risk model. In the model, claim sizes are exponential, the intensity of the next claim arrival depends on whether the current claim exceeds a random threshold, and both the volatility and the premium rate may depend on the surplus. We write the system as the restriction to the half-line of a far-field operator with prescribed exterior data, derive the complete set of determining equations, including the condition imposed by the nonlocal convolution term, and solve them. The resulting group classification holds for every surplus-dependent volatility and premium of the form considered and for every absolutely continuous or constant threshold distribution. A non-trivial symmetry beyond scaling and superposition exists if and only if the volatility and the premium rate are constant, and it is then exactly the translation in the surplus variable. We construct the optimal system of one-dimensional subalgebras. We show that the invariant solutions of the combined translation--scaling generators exist precisely when a $2\times2$ generalized Lundberg determinant vanishes, and we give this determinant for an arbitrary threshold distribution, with explicit forms for the constant and exponential thresholds. We prove existence and sign properties of the adjustment coefficient, derive its sensitivities to all model parameters, and establish Cramér--Lundberg-type asymptotics for the Gerber--Shiu functions, including the limit of the ratio between the two regimes. A numerical study shows that claim-size-driven dependence lowers the adjustment coefficient markedly below that of an independent model with the same mean claim intensity.