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This paper studies a stochastic control problem with regime switching in a fairly general abstract setting. Such problems may arise from production planning management. We perform a full mathematical analysis of this stochastic control problem via the HJB equation and verification. The connection of the optimal controls and subgame perfect controls is discussed, and it is shown that the optimal controls solve the generalized HJB equation as well. In a special case we provide a closed form solution.

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition betw...

We investigate a one-period portfolio optimization problem of a cumulative prospect theory (CPT) investor with multiple risky assets and one risk-free asset. The returns of the multiple risky asset...

In this article we demonstrate the practical application of standard partial equilibrium, in backward stochastic differential equation (BSDE) framework, to pricing longevity bonds which are essential for functioning of the life market. The market for mortality linked instruments or so-called life market, is experiencing steady progress in its evolution (see Blake et al. (2018)), however, it is regretfully far from reaching its full potential estimated to be of tens of trillions of dollars (see Michaelson and Mulholland (2014)). Highly illiquid and, compared to equity markets, with a relatively very low number of transactions, the life market is comprised of series of negotiated, high monetary value, over-the-counter transactions between few agents that have different risk preferences. To accommodate these realities we demonstrate an application of a partial equilibrium framework for pricing longevity bonds. We do this under the assumption of stochastic mortality intensity that affects the income of economic agents who trade in risky financial security and longevity bond to maximize their monetary utilities. Thus, by rooting ourselves in a foundational economic principle, as a practical contribution, we find the endogenous equilibrium bond price which is numerically computed. In a realistic setting of two agents in a transaction, numerical experiments confirm the expected intuition of price dependence on model parameters. As a theoretical contribution, we prove that our longevity bond completes the market.

This paper deals with the following elliptic equation \begin{equation*} -2\sigma^{2}\Delta z+\left\| \nabla z\right\| ^{2}+4\alpha z=4\left\| x\right\| ^{2}\text{ for }x\in \mathbb{R}^{N}\text{, (}% N\geq 1\text{),} \end{equation*}% where $\alpha >0,$ $\sigma>0$ are some real parameters. The solution method is based on the sub- and super-solutions approach. The case $N>1$ seemed not considered before. This equation models a stochastic production planning problem.

This paper performs the numerical analysis and the computation of a Spread option in a market with imperfect liquidity. The number of shares traded in the stock market has a direct impact on the stock’s price. Thus, we consider a full-feedback model in which price impact is fully incorporated into the model. The price of a Spread option is characterize by a nonlinear partial differential equation. This is reduced to linear equations by asymptotic expansions. The Peaceman-Rachford scheme as an alternating direction implicit method is employed to solve the linear equations numerically. We discuss the stability and the convergence of the numerical scheme. Illustrative examples are included to demonstrate the validity and applicability of the presented method. Finally we provide a numerical analysis of the illiquidity effect in replicating an European Spread option; compared to the Black-Scholes case, a trader generally buys more stock to replicate this option.

We consider the problem of portfolio optimization with a correlation constraint. The framework is the multiperiod stochastic financial market setting with one tradable stock, stochastic income and a non-tradable index. The correlation constraint is imposed on the portfolio and the non-tradable index at some benchmark time horizon. The goal is to maximize portofolio's expected exponential utility subject to the correlation constraint. Two types of optimal portfolio strategies are considered: the subgame perfect and the precommitment ones. We find analytical expressions for the constrained subgame perfect (CSGP) and the constrained precommitment (CPC) portfolio strategies. Both these portfolio strategies yield significantly lower risk when compared to the unconstrained setting, at the cost of a small utility loss. The performance of the CSGP and CPC portfolio strategies is similar., 19 pages

In this article, inspired by Shi et al., we investigate the optimal portfolio selection with one risk-free asset and one risky asset in a multiple period setting under the cumulative prospect theory (CPT) risk criterion. Compared with their study, our novelty is that we consider a stochastic benchmark and portfolio constraints. By performing a numerical analysis, we test the sensitivity of the optimal CPT investment strategies to different model parameters.