Mean field game of optimal relative investment with jump risk

In this paper,we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth,in which the interaction occurs by peer competition.In the model with n agents,the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modeled by a multi-dimensional nonlinear Hawkes process.With a continuum of agents,we formulate the mean field game problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions,allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications.Moreover,based on the mean field equilibrium,we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large.The explicit order of the approximation error is also derived.

Optimal portfolio and consumption selection with default risk

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi- Bellman （HJB） equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.

Jump Type Cahn-Hilliard Equations with Fractional Noises

The authors explore a class of jump type Cahn-Hilliard equations with fractional noises. The jump component is described by a (pure jump) Lévy space-time white noise. A fixed point scheme is used to investigate the existence of a unique local mild solution under some appropriate assumptions on coefficients.

Portfolio optimization of credit swap under funding costs

We develop a dynamic optimization framework to assess the impact of funding costs on credit swap investments.A defaultable investor can purchase CDS upfronts,borrow at a rate depending on her credit quality,and invest in the money market account.By viewing the concave drift of the wealth process as a continuous function of admissible strategies,we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and two solutions of a suitably chosen system of first order conditions.Contagion effects between risky investor and reference entity make the optimal strategy coupled with the value function of the control problem.Using the dynamic programming principle,we show that the latter can be recovered as the solution of a nonlinear HJB equation whose coeffcients admit singular growth.By means of a truncation technique relying on the locally Lipschitzcontinuity of the optimal strategy,we establish existence and uniqueness of a global solution to the HJB equation.