arXiv subjects
Publications and source records attributed to Nikolai Dokuchaev.
We study linear backward stochastic partial differential equations of parabolic type with special boundary condition that connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results for the solutions are obtained.
We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.
The possibility of statistical evaluation of the market completeness and incompleteness is investigated for continuous time diffusion stock market models. It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into a incomplete one. The paper shows that market incompleteness is also non-robust: small deviations can convert an incomplete model into a complete one. More precisely, it is shown that, for any incomplete market from a wide class of models, there exists a complete market model with arbitrarily close paths of the stock prices and the market parameters. This leads to a counterintuitive conclusion that the incomplete markets are indistinguishable from the complete markets in the terms of the market statistics.
We study an optimal control problem related to swing option pricing in a general non-Markovian setting in continuous time. As a main result we show that the value process solves a first-order non-linear backward stochastic partial differential equation. Based on this result we can characterize the set of optimal controls and derive a dual minimization problem.
The paper studies estimation of parameters of diffusion market models from historical data. The standard definition of implied volatility for these models presents its value as an implicit function of several parameters, including the risk-free interest rate. In reality, the risk free interest rate is unknown and need to be forecasted, because the option price depends on its future curve. Therefore, the standard implied volatility is {\it conditional}: it depends on the future values of the risk free rate. We study two implied parameters: the implied volatility and the implied average cumulative risk free interest rate. They can be found unconditionally from a system of two equations. We found that very simple models with random volatilities (for instance, with two point distributions) generate various volatility smiles and skews with this approach.
By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential equation. We found that the solution of this replication problem exists and is not unique. This leads to a new optimal control problem: find a replicating process that is minimal in an integral norm. We found an explicit solution of this problem. Possible applications to portfolio selection problems and to bond pricing models are suggested.
Backward stochastic partial differential equations in bounded and unbounded domains are studied. Existence and regularity results are obtained. Duality relationship with forward SPDEs are established. Representation of functionals of Ito processes in bounded domains is discussed.
We study causal dynamic approximation of non-bandlimited processes by band-limited processes such that a part of the historical path of the underlying process is approximated in $L_2$-norm by the trace of a band-limited process. This allows to cover the case of irregular non-smooth processes. We show that this problem has an unique optimal solution. The approximating band-limited process is obtained in time domain in a form of sinc series. To accommodate the current flow of observations, the coefficients of this series and the related band-limited process have to be changed dynamically. This can be interpreted as a causal and linear filter that is not time invariant.
We consider a problem of replication of random vectors by ordinary integrals in the setting when a underlying random variable is generated by a Wiener process. The goal is to find an optimal adapted process such that its cumulative integral at a fixed terminal time matches this variable. The optimal process has to be minimal in an integral norm.
Pathwise predictability and predictors for discrete time processes are studied in deterministic setting. It is suggested to approximate convolution sums over future times by convolution sums over past time. It is shown that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. In addition, a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.
The paper introduces a limit version of multiple stopping options such that the holder selects dynamically a weight function that control the distribution of the payments (benefits) over time. In applications for commodities and energy trading, a control process can represent the quantity that can be purchased by a fixed price at current time. In another example, the control represents the weight of the integral in a modification of the Asian option. The pricing for these options requires to solve a stochastic control problem. Some existence results and pricing rules are obtained via modifications of parabolic Bellman equations.
In this paper we consider the problem of minimizing a quadratic functional for a discrete-time linear stochastic system with multiplicative noise, on a standard probability space, in infinite time horizon. We show that the necessary and sufficient conditions for the existence of the optimal control can be formulated as matrix inequalities in frequency domain. Furthermore, we show that if the optimal control exists, then certain Lyapunov equations must have a solution. The optimal control is obtained by solving a deterministic linear-quadratic optimal control problem whose functional depends on the solution to the Lyapunov equations. Moreover, we show that under certain conditions, solvability of the Lyapunov equations is guaranteed. We also show that, if the frequency inequalities are strict, then the solution is unique up to equivalence.
Smoothing causal linear time-invariant filters are studied for continuous time processes. The paper suggests a family of causal filters with almost exponential damping of the energy on the higher frequencies. These filters are sub-ideal meaning that a faster decay of the frequency response would lead to the loss of causality.
Parabolic equations with homogeneous Dirichlet conditions on the boundary are studied in a setting where the solutions are required to have a prescribed change of the profile in fixed time, instead of a Cauchy condition. It is shown that this problem is well-posed in L_2-setting. Existence and regularity results are established, as well as an analog of the maximum principle.
The following optimization problem is considered. For a linear vector Ito equation. it is required to find an optimal deterministic control vector which minimizes a quadratic the functional. A necessary and sufficient condition for the existence of a optimal control are formulated in the form of frequency inequalities. It is shown that an optimal control can be found by solving a certain linear-quadratic deterministic optimization problem.
The paper studies solutions of ODEs killed on the domain boundary and their first exit times from the domain. Some regularity is obtained in the form of estimates of L_2-norm for functionals on solutions and the first exit times. Some regularity properties for the density of solutions killed on the boundary is also studied for the case of random initial conditions.
The paper studies integral functionals with non-smooth functions from L_2 defined on solutions of ODEs. Some regularity is obtained in the form of estimates of L_2-norm for these functionals. This result is used for regularization of optimal control problems for ODEs with non-smooth functionals: it is suggested to replace the initial vector by a random vector with density. Necessary conditions of optimality and sufficient conditions of existence of optimal control are obtained.
This short note suggests a heuristic method for detecting the dependence of random time series that can be used in the case when this dependence is relatively weak and such that the traditional methods are not effective. The method requires to compare some special functionals on the sample characteristic functions with the same functionals computed for the benchmark time series with a known degree of correlation. Some experiments for financial time series are presented.