Lijun Bo

On the Conditional Default Probability in a Regulated Market: A Structural Approach

In this article, we consider a regulated market and explore the default events. By using a so-called reflected Ornstein-Uhlenbeck process with two-sided barriers to formulate the price dynamics, we derive the expression on the conditional default probability. In the cases of single observation and multiple observations, the conditional default probabilities are explicitly expressed in terms of the inverse Laplace transforms. Finally, we present a numerical simulation associated with the conditional default probability.

AN OPTIMAL PORTFOLIO PROBLEM IN A DEFAULTABLE MARKET

We consider a portfolio optimization problem in a defaultable market. The investor can dynamically choose a consumption rate and allocate his/her wealth among three financial securities: a defaultable perpetual bond, a default-free risky asset, and a money market account. Both the default risk premium and the default intensity of the defaultable bond are assumed to rely on some stochastic factor which is described by a diffusion process. The goal is to maximize the infinite-horizon expected discounted log utility of consumption. We apply the dynamic programming principle to deduce a Hamilton-Jacobi-Bellman equation. Then an optimal Markov control policy and the optimal value function is explicitly presented in a verification theorem. Finally, a numerical analysis is presented for illustration.

STOCHASTIC CAHN–HILLIARD PARTIAL DIFFERENTIAL EQUATIONS WITH LÉVY SPACETIME WHITE NOISES

In this paper, we consider a class of stochastic Cahn–Hilliard partial differential equations driven by Levy spacetime white noises with Neumann boundary conditions. By a dedicate construction we prove that a (unique) local solution exists for the SPDE under some mild assumptions on the coefficients.

Some Integral Functionals of Reflected Sdes and Their Applications in Finance

In this paper, we consider a class of reflected stochastic differential equations (abbr. SDEs) and we are particularly interested in some integral functionals of the solutions to the equations. We explicitly derive the Laplace transforms of those integral functionals, which are subsequently applied for the financial arguments. Here we consider a regulated market, in which the price dynamics is driven by a reflected SDE. We will calculate the conditional default probability under such price dynamics, and meanwhile we also give the pricing on some digital options. Finally, for practical purpose, an illustration for the numerical inversion of the Laplace transforms is presented in the Appendix.

On a Class of Stochastic Anderson Models with Fractional Noises

Abstract In this article, we are concerned with a class of one-dimensional fourth order stochastic Anderson models with double-parameter fractional noises with Hurst parameter . The unique solution is constructed for the model in some appropriate Hilbert space. On the other hand, we shall estimate the Lyapunov exponent of the solution and study its regularity.

On the first passage times of reflected O-U processes with two-sided barriers

In this article, we give the Laplace transform of the first passage times of reflected Ornstein-Uhlenbeck process with two-sided barriers.

STOCHASTIC CAHN–HILLIARD EQUATION WITH FRACTIONAL NOISE

We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument.

Systemic Risk in Interbanking Networks

We develop a mean field model of interbanking borrowing and lending activities. Each bank borrows from or lends to other counterparties at an idiosyncratic rate and is exposed to sudden shocks affe...

Optimal Investment in Credit Derivatives Portfolio Under Contagion Risk

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamiltonian-Jacobi-Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoullis type ODEs. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.

ON A NONLOCAL STOCHASTIC KURAMOTO-SIVASHINSKY EQUATION WITH JUMPS

In this paper, we study a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures and show the existence and uniqueness of the weak solution to the equation. Furthermore, we prove that an invariant measure of the equation indeed exists under some appropriate assumptions.