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 formulate a general Bayesian disorder detection problem, which generalizes models considered in the literature. We study properties of basic statistics, which allow us to reduce problems of quickest detection of disorder moments to optimal stopping problems. Using general results, we consider in detail a disorder problem for Brownian motion on a finite time segment.

We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not assume the knowledge of eigenfunctions of the correspondinge covariance operator. This makes it possible to generalize many previous results in this area. We also obtain ultimate results connected with exact small deviation of Brownian excursion and Brownian meander as well as for Bessel processes and their local times.

The asymptotic behavior of probability of large massive excursions above a high level is evaluated for Gaussian isotropic twice differentiable Gaussian fields. A massive excursion is the excursion with base diameter exceeding a fixed positive number. For proofs, we introduce and study a vector Gaussian process with components that are independent copies of the initial field, and consider the point process of exits of trajectories of this process from an appropriate infinitely expanding set. By studying the asymptotic behavior of the first and second moments of distribution of this point process, the desired asymptotic behavior is obtained. Moreover, general results on the logarithmic (rough) asymptotic behavior of the large massive excursion probabilities are put forward

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.\

In this paper we study the asymptotic distributions, under appropriate normalization, of the sum $S_t = \sum_{i=1}^{N_t} e^{t X_i}$, the maximum $M_t = \max_{i\in\{1,2,\dots,N_t\}} e^{tX_i}$, and the $l_t$ norm $R_t=S_t^{1/t}$, when $N_t\to\infty$ as $t\to\infty$ and $X_1,X_2,\dots$ are independent and identically distributed random variables in the maximum domain of attraction of the reverse-Weibull distribution.

We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that ${{E}} |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < \infty$ for some $\delta > 0$. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-1/2} X$ and Wigner's semicircle law is of order $(nv)^{-1}$, where $v$ is the distance in the complex plane to the real line. Furthermore, we outline applications such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.

The asymptotic behavior of probability of large massive excursions above a high level is evaluated for Gaussian isotropic twice differentiable Gaussian fields. A massive excursion is the excursion with base diameter exceeding a fixed positive number. For proofs, we introduce and study a vector Gaussian process with components that are independent copies of the initial field, and consider the point process of exits of trajectories of this process from an appropriate infinitely expanding set. By studying the asymptotic behavior of the first and second moments of distribution of this point process, the desired asymptotic behavior is obtained. Moreover, general results on the logarithmic (rough) asymptotic behavior of the large massive excursion probabilities are put forward.

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

This paper contains detailed exposition of the results presented in the short communication [M. V. Zhitlukhin and A. A. Muravlev, Russian Math. Surveys, 66 (2011), pp. 1012–1013]. We consider Chernoff’s problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed. We obtain an integral equation which characterizes the optimal decision rule and find its solution numerically.

We obtain upper functions that serve as almost sure asymptotic upper bounds for a displacement process given by an integrated time-varying Ornstein--Uhlenbeck process. The form of upper functions depends on the characteristics (the stability rate and the diffusion coefficient) of a stochastic linear differential equation. We introduce the notion of anomalous diffusion related to behavior of upper functions and compare the results of diffusion classification (normal diffusion, subdiffusion, and superdiffusion) with those obtained on the basis of mean square displacements.

We consider the problem of optimal estimation of the value of a vector parameter \thetavector=(\theta_0,...,\theta_n)^⊤ of the drift term in a fractional Brownian motion represented by the finite sum i=0^nii(t) over known functions \varphi_i(t), \alli. For the value of parameter \thetavector, we obtain a maximum likelihood estimate as well as Bayesian estimates for normal and uniform a priori distributions.

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.

Let (X1; Y1),..., (XN; YN) be independent identically distributed random vectors. We show, that if the statistics LX = a1*X1+... + aN*XN and LY = b1*Y1 + ... + bN*YN are epsilon - independent, then under some conditions X = (X1,..., XN), Y = (Y1,..., YN) are "epsilon in the power alpha"-independent for some alpha> 0.

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.

We consider a basic stochastic particle system consisting of *N* identical particles with isotropic *k*-particle synchronization, ${k\ge 2}$. In the limit when both the number of particles *N* and the time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on *N* and increases to infinity as $N\rightarrow\infty$.