Yuan-Chung Sheu

An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model

Abstract Under the assumption that the asset value follows a phase-type jump-diffusion, we show that the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if the downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Lévy measure), we obtain closed-form solutions for the expected discounted penalty. As an application, we work out an example in Leland’s structural model with jumps. For earlier and related results, see Gerber and Landry [Insur. Math. Econ. 22:263–276, 1998], Hilberink and Rogers [Finance Stoch. 6:237–263, 2002], Asmussen et al. [Stoch. Proc. Appl. 109:79–111, 2004], and Kyprianou and Surya [Finance Stoch. 11:131–152, 2007].

Lifetime and compactness of range for super-Brownian motion with a general branching mechanism

Let X be a super-Brownian motion with a general (time-space) homogeneous branching mechanism. We study a relation between lifetime and compactness of range for X. Under a restricted condition on the branching mechanism, we show that the set X survives is the same as that the range of X is unbounded. (For [alpha]-branching super-Brownian motion, 1

On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases

Consider the simple random walk on the n-cycle Zn. For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant α is of the same order as the spectral gap λ. However the exact value of α is not known for n>4. (For n=2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that α is 12. For n=3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that α=12log2<λ2=0.75. For n=4, the fact that α=12 follows from n=2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if n⩾4 is even, then the log-Sobolev constant and the spectral gap satisfy α=λ2. This implies that α=12(1−cos2πn) when n is even and n⩾4.

Disorder Chaos in the Spherical Mean-Field Model

We study the problem of disorder chaos in the spherical mean-field model. It concerns the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra’s replica symmetry breaking scheme, we establish this at the levels of the free energy and the Gibbs measure.

On positive solutions of some nonlinear differential equations -- A probabilistic approach

By using connections between superdiffusions and partial differential equations (established recently by Dynkin, 1991), we study the structure of the set of all positive (bounded or unbounded) solutions for a class of nonlinear elliptic equations. We obtain a complete classification of all bounded solutions. Under more restrictive assumptions, we prove the uniqueness property of unbounded solutions, which was observed earlier by Cheng and Ni (1992).

THE LEAST COST SUPER REPLICATING PORTFOLIO IN THE BOYLE–VORST MODEL WITH TRANSACTION COSTS

Boyle and Vorst work in the framework of the binomial model and derive self-financing strategies perfectly replicating the final payoffs to long and short positions in call and put options, assuming proportional transactions costs on trades in the stock and no transactions costs on trades in the bond. Even when the market is arbitrage-free and a given contingent claim has a unique replicating portfolio, there may exist super replicating portfolios of lower cost. Bensaid et al. gave conditions under which the cost of the replicating portfolio does not exceed the cost of any super replicating portfolio. These results were generalized by Stettner, Rutkowski and Palmer to the case of asymmetric transaction costs.In this paper, we first determine the number of replicating portfolios and then compute the least cost super replicating portfolio for any contingent claim in a one-period binomial model. By using the fundamental theorem of linear programming, we show that there are only finitely many possibilities for a least cost super replicating portfolio for any contingent claim in a two-period binomial model. As an application of our results, we give an example in which we compute the least cost super replicating portfolio for a butterfly spread in a two-period model.

Free boundary problems and perpetual American strangles

We consider the perpetual American strangles in the geometric jump-diffusion models. We assume further that the jump distribution is a mixture of exponential distributions. To solve the corresponding optimal stopping problem for this option, by using the approach in [5], we derive a system of equations that is equivalent to the associated free boundary problem with smooth pasting condition. We verify the existence of the solutions to these equations. Then, in terms of the solutions together with a verification theorem, we solve the optimal stopping problem and hence find the optimal exercise boundaries and the rational price for the perpetual American strangle. In addition we work out an algorithm for computing the optimal exercise boundaries and the rational price of this option.

Pricing Perpetual American Compound Options under a Matrix-Exponential Jump-Diffusion Model

ABSTRACT This paper considers the problem of pricing perpetual American compound options under a matrix-exponential jump-diffusion model. The rational prices of these options are defined as the value functions of the corresponding optimal stopping problems. The general optimal stopping theory and the averaging method for solving the optimal stopping problems are applied to find the value functions and the optimal stopping times and thereby to determine the rational prices and the optimal boundaries of these perpetual American compound options. Explicit formulae for the rational prices and the optimal boundaries are also obtained for hyper-exponential jump-diffusion models.

An ODE Approach for the Expected Discounted Penalty at Ruin in Jump Diffusion Model (Reprint)

Under the assumption that the asset value follows a phase-type jump diffusion, we show the expected discounted penalty satisfies an ODE and obtain a general form ?for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Levy measure), we obtain closed form solutions for the expected discounted penalty. As an application, we work out an example in Leland’s structural model with jumps. For earlier and related results, see Gerber and Landry [Insurance: Mathematics and Economics 22:263–276, 1998], Hilberink and Rogers [Finance Stoch 6:237–263, 2002], Asmussen et al. [Stoch. Proc. and their App. 109:79–111, 2004] and Kyprianou and Surya [Finance Stoch 11:131–152, 2007].

A Generalized Renewal Equation for Perturbed Compound Poisson Processes with Two-Sided Jumps

Abstract In this article, we study the discounted penalty at ruin in a perturbed compound Poisson model with two-sided jumps. We show that it satisfies a renewal equation under suitable conditions and consider an application of this renewal equation to study some perpetual American options. In particular, our renewal equation gives a generalization of the renewal equation in Gerber and Landry [2] where only downward jumps are allowed.