Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

Value functions in a regime switching jump diffusion with delay market model

<abstract><p>In this paper, we consider a market model where the risky asset is a jump diffusion whose drift, volatility and jump coefficients are influenced by market regimes and history of the asset itself. Since the trajectory of the risky asset is discontinuous, we modify the delay variable so that it remains defined in this discontinuous setting. Instead of the actual path history of the risky asset, we consider the continuous approximation of its trajectory. With this modification, the delay variable, which is a sliding average of past values of the risky asset, no longer breaks down. We then use the resulting stochastic process in formulating the state variable of a portfolio optimization problem. In this formulation, we obtain the dynamic programming principle and Hamilton Jacobi Bellman equation. We also provide a verification theorem to guarantee the optimal solution of the corresponding stochastic optimization problem. We solve the resulting finite time horizon control problem and show that close form solutions of the stochastic optimization problem exist for the cases of power and logarithmic utility functions. In particular, we show that the HJB equation for the power utility function is a first order linear partial differential equation while that of the logarithmic utility function is a linear ordinary differential equation.</p></abstract>

Some Forms of InvestorRiskTolerance in Investing: Review Theory

Investors in investing are always accompanied by a sense of tolerance for the risk of funds invested in an asset. Each investor has a different form of risk tolerance, depending on the function of the utility. This paper aims to conduct a theoretical study of the forms of investor risk tolerance for several utility functions. This study is carried out by reviewing several utility functions which include: square root utility, cubic fraction utility, quadratic utility, exponential negative utility, and logarithmic utility. Based on the results of the study for each of these utility functions, successively obtained risk tolerance in the form of linear, linear, linear, constant, and linear. Linear risk tolerance illustrates that an investor changes the value of his investment in line with changes in the level of risk faced.

Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations

This work presents a two-player extraction game where the random terminal times follow (different) heavy-tailed distributions which are not necessarily compactly supported. Besides, we delve into the implications of working with logarithmic utility/terminal payoff functions. To this end, we use standard actuarial results and notation, and state a connection between the so-called actuarial equivalence principle, and the feedback controllers found by means of the Dynamic Programming technique. Our conclusions include a conjecture on the form of the optimal premia for insuring the extraction tasks; and a comparison for the intensities of the extraction for each player under different phases of the lifetimes of their respective machineries.

A martingale approach for asset allocation with derivative security and hidden economic risk

Asset allocation with a derivative security is studied in a hidden, Markovian regime-switching, economy using filtering theory and the martingale approach. A generalized delta-hedged ratio and a generalized elasticity of an option are introduced to accommodate the presence of the information state process and the derivative security. Malliavin calculus is applied to derive a solution for a general utility function which includes an exponential utility, a power utility, and a logarithmic utility. A compact solution is obtained for a logarithmic utility. Some economic implications of the solutions are discussed.