Igor Cialenco

Illinois Institute of Technology | IIT · Department of Applied Mathematics

We develop an innovative learning framework that incorporate the noise structure to infer the governing equations from observation of trajectory data generated by stochastic dynamics. Our approach can proficiently captures both the noise and the drift terms. Furthermore, it can also accommodate a wide range of noise types, including correlated and...

We study a dynamic stochastic control problem subject to Knightian uncertainty with multi-objective (vector-valued) criteria. Assuming the preferences across expected multi-loss vectors are represented by a given, yet general, preorder, we address the model uncertainty by adopting a robust or minimax perspective, minimizing expected loss across the...

The aim of this work is to study risk measures generated by distortion functions in a dynamic discrete time setup, and to investigate the corresponding dynamic coherent acceptability indices (DCAIs) generated by families of such risk measures. First we show that conditional version of Choquet integrals indeed are dynamic coherent risk measures (DCR...

We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, while the state process and the controls are observed at time t, the actual cost that may depend on the unknown parameter is not known at time t. The cont...

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic processes with arbitrary finitely regular paths. These results are proved by using the notion of \(\Delta \)-power variations, introduced herein, along with the Hölder-Zygmund...

We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, while the state process and the controls are observed at time $t$, the actual cost that may depend on the unknown parameter is not known at time $t$. The...

In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.

We derive consistent and asymptotically normal estimators for the drift and volatility parameters of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space and the bounded domain. We establish the exact spatial regularity of the soluti...

In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of $\Delta$-power variations, introduced herein, along with the H\"older-Zygmund no...

In this paper, we study a class of time-inconsistent terminal Markovian control problems in discrete time subject to model uncertainty. We combine the concept of the sub-game perfect strategies with the adaptive robust stochastic control method to tackle the theoretical aspects of the considered stochastic control problem. Consequently, as an impor...

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of r...

The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spe...

This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced by Altmeyer, Reiss (2020) retains its asymptotic p...

We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual...

The main goal of this paper is to build consistent and asymptotically normal estimators for the drift and volatility parameter of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space R and the bounded domain (0,\pi). First, we establ...

In this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide expl...

In this paper we study a class of time-inconsistent terminal Markovian control problems in discrete time subject to model uncertainty. We combine the concept of the sub-game perfect strategies with the adaptive robust stochastic to tackle the theoretical aspects of the considered stochastic control problem. Consequently, as an important application...

This work contributes to the theory of Wiener-Hopf type factorization for finite Markov chains. This theory originated in the seminal paper [Barlow et al., Wiener-Hopf factorization for matrices, Séminaire de Probabilités de Strasbourg 14 (1980), pp. 324–331], which treated the case of finite time-homogeneous Markov chains. Since then, several work...

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of r...

The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spe...

We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual...

The main goal of this paper is to study the parameter estimation problem, using the Bayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We take the spectral approach by assuming that one path of the first N Fourier modes of the solution is continuously observed over...

This work contributes to the theory of Wiener-Hopf type factorization for finite Markov chains. This theory originated in the seminal paper Barlow et al. (1980), which treated the case of finite time-homogeneous Markov chains. Since then, several works extended the results of Barlow et al. (1980) in many directions. However, all these extensions we...

In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide expli...

The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is the most studied sampling scheme under which the observations are done in the Fourier space over some finite...

The main goal of this paper is to study the parameter estimation problem, using the Bayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We take the spectral approach by assuming that one path of the first $N$ Fourier modes of the solution are continuously observed o...

In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equation (SPDE) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed traj...

We introduce a dynamic model of the default waterfall of derivatives CCPs and propose a risk sensitive method for sizing the initial margin (IM), and the default fund (DF) and its allocation among clearing members. Using a Markovian structure model of joint credit migrations, our evaluation of DF takes into account the joint credit quality of clear...

In this paper we derive the Wiener-Hopf factorization for a finite-state time-inhomogeneous Markov chain. To the best of our knowledge, this study is the first attempt to investigate the Wiener-Hopf factorization for time-inhomogeneous Markov chains. In this work we only deal with a special class of time-inhomogeneous Markovian generators, namely p...

We consider a parameter estimation problem for one dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We establish some general results on derivation of consistent and asymptotically normal estimators based on computation of the $p$-variations of stochastic processes and their smooth perturbations....

In this paper, we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures (also known as acceptability indices). The proposed framework not only integrates existing forms of time consistency, but also provides a comprehensive toolbox for analysis and synthesis of the concept of time co...

In this paper we propose a new methodology for solving an uncertain stochastic Markovian control problem in discrete time. We call the proposed methodology the adaptive robust control. We demonstrate that the uncertain control problem under consideration can be solved in terms of associated adaptive robust Bellman equation. The success of our appro...

The objective of this paper is to provide a comprehensive study of the no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in several respects the nonlinea...

The objective of this paper is to provide a comprehensive study no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in several respects the nonlinear prici...

Assuming that one-step transition kernel of a discrete time, time-homogenous Markov chain model is parameterized by a parameter $\theta\in \boldsymbol \Theta$, we derive a recursive (in time) construction of confidence regions for the unknown parameter of interest, say $\theta^*$. The key step in this construction is derivation of a recursive schem...

In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, with focus on discrete time setup. The two key operational concepts used throughout are the notion of the LM-measure and the notion of the update rule that, we believe, are the key tools for studying the time consistency in a uni...

In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LM-measure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a uni...

We present an arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities in a discrete time setup using theory of dynamic acceptability indices. In the first part of the paper we develop the theory of dynamic subscale invariant performance measures, on a general probability space, and discrete time setup. We...

In this paper we provide a unified and flexible framework for study of the time consistency of risk and performance measures. The proposed framework integrates existing forms of time consistency as well as various connections between them. In our approach the time consistency is studied for a large class of maps that are postulated to satisfy only...

In this paper, we find sufficient conditions on the coefficients and the kernels of a general class of nonselfadjoint integro-differential operators of any order that guarantee the finiteness of the point spectrum of these operators. The results are obtained by the direct investigation of the analyticity of the resolvent function near the essential...

In the recent paper [CX13], we study the simple hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. Assuming that one path of the projected solution is observed continuously over time interval $[0,T]$, we established `the proper asymptotic...

We propose a new class of mappings, called Dynamic Limit Growth Indices, that are designed to measure the long-run performance of a financial portfolio in discrete time setup. We study various important properties for this new class of measures, and in particular, we provide necessary and sufficient condition for a Dynamic Limit Growth Index to be...

We study the simple hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. We assume that the first $N$ Fourier modes of the solution are observed continuously over time interval $[0,T]$. We introduce the notion of asymptotically the most pow...

This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of $L^0$-modules. In the first part of the paper we develop th...

In this paper we discuss the issue of computation of the bilateral credit valuation adjustment (CVA) under rating triggers, and in presence of ratings-linked margin agreements. Specifically, we consider collateralized OTC contracts, that are subject to rating triggers, between two parties -- an investor and a counterparty. Moreover, we model the ma...

We prove a version of First Fundamental Theorem of Asset Pricing under transaction costs for discrete-time markets with dividend-paying securities. Specifically, we show that the no-arbitrage condition under the efficient friction assumption is equivalent to the existence of a risk-neutral measure. We derive dual representations for the superhedgin...

In this paper we present a theoretical framework for determining dynamic ask and bid prices of derivatives using the theory of dynamic coherent acceptability indices in discrete time. We prove a version of the First Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures. We introduce the dynamic ask and bid prices of a deriva...

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic...

We analyze the counterparty risk embedded in CDS contracts, in presence of a bilateral margin agreement. First, we investigate the pricing of collateralized counterparty risk and we derive the bilateral Credit Valuation Adjustment (CVA), unilateral Credit Valuation Adjustment (UCVA) and Debt Valuation Adjustment (DVA). We propose a model for the co...

We consider a parameter estimation problem of determining the viscosity ν of a stochastically perturbed 2D Navier–Stokes system. We derive several different classes of estimators based on the first N Fourier modes of a single sample path observed on a finite time interval. We study the consistency and asymptotic normality of these estimators. Our a...

We examine the in- and out-of-sample behavior of two popular trading systems, Alexander and Double MA filters, for 14 developed-country currencies using daily data with bid-ask spreads. We find significant in-sample returns in the early periods. But out-of-sample returns are lower and only occasionally significant. We show that a currency risk fact...

In this paper we present a theoretical framework for studying coherent acceptability indices in a dynamic setup. We study dynamic coherent acceptability indices and dynamic coherent risk measures, and we establish a duality between them. We derive a representation theorem for dynamic coherent risk measures in terms of so called dynamically consiste...

We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter $H\in(0,1)$. Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general resu...

A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximu...

Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\mathbf{L}_{p}(\mathbb{R}_{+})$ or $\mathbf{L}_{p}(\mathbb{R}) (1\leq p<\infty)$. As an application of general results, new sp...

A parameter estimation problem is considered for a stochastic parabolic equation with multiplicative noise under the assumption that the equation can be reduced to an infinite system of uncoupled diffusion processes. From the point of view of classical statistics, this problem turns out to be singular not only for the original infinite-dimensional...

Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differenti...

This paper analyzes the response of the Term Structure of discount rates to the changes in the Federal Funds Target Rate. It also suggests a method of hedging fixed income portfolio's risk to the unexpected changes in monetary policy. We use two alternative widely used models of term structure of interest rates - the Extended Nelson-Siegel and the...

We examine the in- and out-of-sample behavior of two popular trading systems, Alexander and Double MA filters, for 14 developed-country currencies using daily data with bid-ask spreads. We find significant in-sample returns in the early periods. But out-of-sample returns are lower and only occasionally significant. We show that a currency risk fact...

We discuss the spectrum of the integral Wiener-Hopf operator of the form (Hφ)(x)=∫ ℝ + a(x-y)φ(y)dy+∫ ℝ + b(x,y)φ(y)dy, where a∈L 1 (ℝ), b(·,·) is a measurable function with respect to both variables x,y∈ℝ + . In general, the operator H is supposed to be a nonselfadjoint operator acting on the space L 2 (ℝ + ). Finiteness criteria are established f...

In this paper there are obtained results on the flniteness of the point spectrum of some nonselfadjoint operators. In particular the operators of Wiener- Hopf type acting in arbitrary Hilbert space, l2 and L2(R+) are considered. In the present paper there is examined the problem of flniteness of the point spectrum of some nonselfadjoint operators....

In this note, results on flniteness of point spectrum of operators generated by integro-difierential expression of arbitrary order are announced.

The main purpose of the present paper is to give sufficient conditions for the finiteness of the point spectrum of some nonselfadjoint operators.