In this paper we consider a variation of the Merton’s problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.
We consider two models of summation of independent identically distributed random variables with a parameter. The first is motivated by financial applications and the second by contact models for species migration. We characterize the limiting distributions and their bifurcations under different relationships between the parameter and the number of summands. We find that in the phase transition we may get limiting distributions that are quite different from those that come up in standard limit theorems. Our results suggest that these limiting distributions may provide better models, at least for certain aggregation levels. Moreover, we show how the parameter determines at which aggregation levels these models apply.\
This paper introduces matrix-variate t-distributions for which degree of freedom is a multivariate parameter. A relation for a density function is obtained. © 2015 Society for Industrial and Applied Mathematics.
We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem for systems with limitations. We also consider the hierarchical networks of queueing systems
The main result of this paper is a counterpart of the theorem of Monroe [Ann. Probab., 6 (1978), pp. 42--56] for a geometric Brownian motion: A process is equivalent to a time change of a geometric Brownian motion if and only if it is a nonnegative supermartingale. We also provide a link between our main result and Monroe's [Ann. Math. Statist., 43 (1972), pp. 1293--1311]. This is based on the concept of a minimal stopping time, which is characterized in Monroe [Ann. Math. Statist., 43 (1972), pp. 1293--1311] and Cox and Hobson [Probab. Theory Related Fields, 135 (2006), pp. 395--414] in the Brownian case. Finally, we suggest a sufficient condition for minimality (for the processes other than a Brownian motion) complementing the discussion in the aforementioned papers.
We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock price appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein- Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor Y in an interval [0, T0] for fixed T0 > 0, and use sequential estimation. We show that the consumption and investment strategy calculated through this sequential procedure is δ-optimal.
We study the Markov exclusion process for a particle system with a local interaction in the integer strip. This process models the exchange of velocities and particle-hole exchange of the liquid molecules. It is shown that the mean velocity profile corresponds to the behavior which is characteristic for incompressible viscous liquid. We prove the existence of phase transition between laminar and turbulent profiles.
В работе для псевдозолотого сечения и для произвольного параметра Бернулли получена формула для вычисления энтропии инвариантной меры Эрдеша. Формула позволяет производить вычисление энтропии и хаусдорфовой размерности инвариантной меры Эрдеша с высокой точностью.
We consider a model of an insurance company investing its reserve into a risky asset whose price follows a geometric Lévy process. We show that the nonruin probability is a viscosity solution of a second order integro-differential equation and prove a uniqueness theorem for the latter.