Stochastic differential equations and diffusion processes

Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative L\'{e}vy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

On the optimal dividend problem for a spectrally negative L\'{e}vy process

We study the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time (e.g., to pay for a liability, to consume, or to pay dividends). The investor is allowed to invest in any or all of a given number of risky stocks, whose prices follow geometric Brownian motion, as well as in a riskless asset which has a constant rate of return. The fact that the withdrawal is continuously enforced, regardless of the wealth level, ensures that there is a region where there is a positive probability of ruin. In the complementary region ruin can be avoided with certainty. Call the former region the danger-zone and the latter region the safe-region. We first consider the problem of maximizing the probability that the safe-region is reached before bankruptcy, which we call the survival problem. While we show, among other results, that an optimal policy does not exist for this problem. we are able to construct explicit ∊-optimal policies, for any ∊ > 0. In the safe-region, where ultimate survival is assured, we turn our attention to growth. Among other results, we find the optimal growth policy for the investor, i.e., the policy which reaches another (higher valued) goal as quickly as possible. Other variants of both the survival problem as well as the growth problem are also discussed. Our results for the latter are intimately related to the theory of Constant Proportions Portfolio Insurance.

Survival And Growth With A Liability: Optimal Portfolio Strategies In Continuous Time

Impact of economic policy uncertainty on exchange rate volatility of China

This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (explosion suppression) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations. This work reveals the impact of various random effects on the underlying systems for almost sure and $p$th-moment stability and provides insight on stability and stabilization of switching jump diffusion systems.

ALMOST SURE AND pTH-MOMENT STABILITY AND STABILIZATION OF REGIME-SWITCHING JUMP DIFFUSION SYSTEMS ∗

This work develops a stochastic differential game model between two insurance companies who adopt the optimal reinsurance strategies to reduce the risk. The surplus is modeled by a regime-switching jump diffusion process. A single payoff function is imposed, and one player devises an optimal strategy to maximize the expected payoff function, whereas the other player is trying to minimize the same quantity. Using dynamic programming principle, the upper and lower values of the game satisfy a coupled system of nonlinear integro-differential Hamilton–Jacobi–Isaacs (HJI) equations. Moreover, the existence of the saddle point for this game problem is verified. Because of the jumps and regime-switching, closed-form solutions are virtually impossible to obtain. Our effort is devoted to designing numerical methods. We use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value functions and optimal controls. Convergence of the approximation algorithms is proved. Examples are presented to illustrate the applicability of the numerical methods.

Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods

http://hub.hku.hk/bitstream/10722/198097/1/Content.pdf

Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections

In this paper, we consider a stochastic differential reinsurance game between two insurance companies with nonlinear (quadratic) risk control processes. We assume that the goal of each insurance company is to maximize the exponential utility of the difference between its terminal surplus and that of its competitor at a fixed terminal time T . First, we give an explicit partition (including nine subsets) of time interval [ 0 , T ] . Further, on every subset, an explicit Nash equilibrium strategy is derived by solving a pair of Hamilton–Jacobi–Bellman equations. Finally, for some special cases, we analyze the impact of time t and quadratic control parameter on the Nash equilibrium strategy and obtain some simple partition of [ 0 , T ] . Based on these results, we apply some numerical analysis of the time t , quadratic control parameter and competition sensitivity parameter on the Nash equilibrium strategy and the value function.

Zhuo Jin

Papers

ALMOST SURE AND pTH-MOMENT STABILITY AND STABILIZATION OF REGIME-SWITCHING JUMP DIFFUSION SYSTEMS ∗

Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods

Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections

A reinsurance game between two insurance companies with nonlinear risk processes

Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation