We study an open problem of risk-sensitive portfolio allocation in a regime-switching credit market with default contagion. The state space of the Markovian regime-switching process is assumed to be a countably infinite set. To characterize the value function, we investigate the corresponding recursive infinite-dimensional nonlinear dynamical programming equations (DPEs) based on default states. We propose to work in the following procedure: Applying the theory of monotone dynamical system, we first establish the existence and uniqueness of classical solutions to the recursive DPEs by a truncation argument in the finite state space. The associated optimal feedback strategy is characterized by developing a rigorous verification theorem. Building upon results in the first stage, we construct a sequence of approximating risk sensitive control problems with finite states and prove that the resulting smooth value functions will converge to the classical solution of the original system of DPEs. The construction and approximation of the optimal feedback strategy for the original problem are also thoroughly discussed.AMS 2000 subject classifications: 3E20, 60J20. ∞ j=1 q ij = 0 for i ∈ Z + (i.e., j =i q ij = −q ii for i ∈ Z + ).
This paper studies the optimal dividend for a multi-line insurance group, in which each subsidiary runs a product line and is exposed to some external credit risk. The default contagion is considered such that one default event may increase the default probabilities of all surviving subsidiaries. The total dividend problem for the insurance group is investigated and we find that the optimal dividend strategy is still of the barrier type. Furthermore, we show that the optimal barrier of each subsidiary is modulated by the default state. That is, how many and which subsidiaries have defaulted will determine the dividend threshold of each surviving subsidiary. These conclusions are based on the analysis of the associated recursive system of Hamilton-Jacobi-Bellman variational inequalities (HJBVIs). The existence of the classical solution is established and the verification theorem is proved. In the case of two subsidiaries,
This paper investigates the finite time risk-sensitive portfolio optimization in a regimeswitching credit market with physical and information-induced default contagion. The Markov regime-switching process is assumed to be unobservable, which has countable states that affect default intensities of surviving assets. The stochastic control problem is formulated under partial observations of asset prices and default events. By proving a novel martingale representation theorem based on incomplete and phasing out filtration, we characterize the value function in an equivalent form. This allows us to connect the control problem to a new type of quadratic BSDE with jumps, in which the driver term has non-standard structures and carries the conditional filter as an infinite-dimensional parameter. By proposing some truncation techniques and establishing a uniform a priori estimates, we obtain the existence of a solution to this BSDE using the convergence of solutions associated to some truncated BSDEs. The verification theorem can be concluded with the aid of our BSDE results, which in turn yields the uniqueness of the solution to the BSDE.
We study the relaxed control and Gamma-convergence of a class of mean field stochastic optimization problems arising from training deep residual neural networks. We consider their sample and continuous-time idealization, and establish existence of optimal relaxed solutions to such a class of mean field optimization problems when the training sample is finite. The core of our paper is to show that, when the sample capacity is large, the minimizer of the sampled relaxed optimization problem converges to the minimizer of the limiting relaxed optimization problem. To prove the Gamma-convergence of sampled objective functionals, we establish general convergence properties of empirical measure-valued processes arising from the finite sample controlled model. Then, we connect the limit of the large sampled objective functional to the unique solution of a nonlinear Fokker-Plank-Kolmogorov (FPK) equation in a random environment. We prove the uniqueness of solutions to the FPK equation in the trajectory sense.
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