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Gordan Zitkovic

Publications and source records attributed to Gordan Zitkovic.

At least 19 recordsLinked to original sources
arXiv2024

Convergence of nonhomogeneous Hawkes processes and Feller random measures

We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L\' evy-type operators of order at most $1$, including fractional derivatives of any order $\alpha>0$ (formally corresponding to possibly negative Hurst parameters).

math.PR↗
arXiv2024

A functional limit theorem for additive functionals

We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to L\'evy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.

math.PR↗
arXiv2023

Representation of Random Variables as Lebesgue Integrals

We study representations of a random variable $ξ$ as an integral of an adapted process with respect to the Lebesgue measure. The existence of such representations in two different regularity classes is characterized in terms of the quadratic variation of (local) martingales closed by $ξ$.

math.PR↗
arXiv2019

A Framework for the Dynamic Programming Principle and Martingale-generated Control Correspondences

We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the "martingale formulation" with particular ease. We give two illustrations; first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton-Jacobi-Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem.

math.OC↗
arXiv2018

An incomplete equilibrium with a stochastic annuity

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimize their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and, along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion, and, in particular, includes mean-reverting dynamics. The equilibrium is characterized by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions.

q-fin.MF↗
arXiv2016

Existence, Characterization and Approximation in the Generalized Monotone-Follower Problem

We revisit the classical monotone-follower problem and consider it in a generalized formulation. Our approach is based on a compactness substitute for nondecreasing processes, the Meyer-Zheng weak convergence, and the maximum principle of Pontryagin. It establishes existence under weak conditions, produces general approximation results and further elucidates the celebrated connection between singular stochastic control and stopping.

math.OC↗
arXiv2014

Facelifting in Utility Maximization

We establish the existence and characterization of a primal and a dual facelift - discontinuity of the value function at the terminal time - for utility-maximization in incomplete semimartingale-driven financial markets. Unlike in the lower- and upper-hedging problems, and somewhat unexpectedly, a facelift turns out to exist in utility-maximization despite strict convexity in the objective function. In addition to discussing our results in their natural, Markovian environment, we also use them to show that the dual optimizer cannot be found in the set of countably-additive (martingale) measures in a wide variety of situations.

q-fin.PM↗
arXiv2014

Dynamic Programming for controlled Markov families: abstractly and over Martingale Measures

We describe an abstract control-theoretic framework in which the validity of the dynamic programming principle can be established in continuous time by a verification of a small number of structural properties. As an application we treat several cases of interest, most notably the lower-hedging and utility-maximization problems of financial mathematics both of which are naturally posed over ``sets of martingale measures''.

math.OC↗
arXiv2012

Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.

q-fin.PM↗
arXiv2010

Convex-compactness and its applications

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization - attainment of infima and a version of the Minimax theorem - are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem.

math.FA↗
arXiv2010

An example of a stochastic equilibrium with incomplete markets

We prove existence and uniqueness of stochastic equilibria in a class of incomplete continuous-time financial environments where the market participants are exponential utility maximizers with heterogeneous risk-aversion coefficients and general Markovian random endowments. The incompleteness featured in our setting - the source of which can be thought of as a credit event or a catastrophe - is genuine in the sense that not only the prices, but also the family of replicable claims itself is determined as a part of the equilibrium. Consequently, equilibrium allocations are not necessarily Pareto optimal and the related representative-agent techniques cannot be used. Instead, we follow a novel route based on new stability results for a class of semilinear partial differential equations related to the Hamilton-Jacobi-Bellman equation for the agents' utility-maximization problems. This approach leads to a reformulation of the problem where the Banach fixed point theorem can be used not only to show existence and uniqueness, but also to provide a simple and efficient numerical procedure for its computation.

q-fin.GN↗
arXiv2010

Stability of the utility maximization problem with random endowment in incomplete markets

We perform a stability analysis for the utility maximization problem in a general semimartingale model where both liquid and illiquid assets (random endowments) are present. Small misspecifications of preferences (as modeled via expected utility), as well as views of the world or the market model (as modeled via subjective probabilities) are considered. Simple sufficient conditions are given for the problem to be well-posed, in the sense the optimal wealth and the marginal utility-based prices are continuous functionals of preferences and probabilistic views.

q-fin.PM↗
arXiv2009

Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability

In an incomplete semimartingale model of a financial market, we consider several risk-averse financial agents who negotiate the price of a bundle of contingent claims. Assuming that the agents' risk preferences are modelled by convex capital requirements, we define and analyze their demand functions and propose a notion of a partial equilibrium price. In addition to sufficient conditions for the existence and uniqueness, we also show that the equilibrium prices are stable with respect to misspecifications of agents' risk preferences.

q-fin.RM↗
arXiv2008

Maturity-independent risk measures

The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used to build maturity-independent counterparts. We construct a large class of maturity-independent risk measures and give representative examples in both continuous- and discrete-time financial models.

q-fin.RM↗
arXiv2008

On Agents' Agreement and Partial-Equilibrium Pricing in Incomplete Markets

We consider two risk-averse financial agents who negotiate the price of an illiquid indivisible contingent claim in an incomplete semimartingale market environment. Under the assumption that the agents are exponential utility maximizers with non-traded random endowments, we provide necessary and sufficient conditions for negotiation to be successful, i.e., for the trade to occur. We also study the asymptotic case where the size of the claim is small compared to the random endowments and we give a full characterization in this case. Finally, we study a partial-equilibrium problem for a bundle of divisible claims and establish existence and uniqueness. A number of technical results on conditional indifference prices are provided.

q-fin.PR↗
arXiv2007

Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies -- those strategies whose wealth process is a supermartingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.

q-fin.PM↗
arXiv2007

Financial equilibria in the semimartingale setting: complete markets and markets with withdrawal constraints

Existence of stochastic financial equilibria giving rise to semimartingale asset prices is established under a general class of assumptions. These equilibria are expressed in real terms and span complete markets or markets with withdrawal constraints.We deal with random endowment density streams which admit jumps and general time-dependent utility functions on which only regularity conditions are imposed. As an integral part of the proof of the main result, we establish a novel characterization of semimartingale functions.

q-fin.PR↗