Research - preprints
(2016)
A class of globally solvable Markovian quadratic BSDE systems and applications
submitted for publication
[arXiv]
[bib]
[abstract]
Abstract: We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds.
@unpublished{XinZit16,
author = {Xing, Hao and {\v{Z}}itkovi{\'c}, Gordan}, year = {2016}, note = {submitted for publication}, title = {A class of globally solvable {M}arkovian quadratic BSDE systems and applications}, }
(2015)
Incomplete stochastic equilibria with exponential utilities close to Pareto optimality
submitted for publication
[arXiv]
[bib]
[abstract]
Abstract: We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete markets model. An assumption of `smallness' type - imposed through the new notion of 'closeness to Pareto optimality' - is shown to be sufficient for existence and uniqueness. Central role in our analysis is played by a fully-coupled nonlinear system of quadratic BSDEs.
@unpublished{KarXinZit15,
author = {Kardaras, Constantinos and Xing, Hao and {\v{Z}}itkovi{\'c}, Gordan}, year = {2015}, note = {submitted for publication}, title = {Incomplete stochastic equilibria with exponential utilities close to {P}areto optimality}, }
(2014)
An expansion in the model space in the context of utility maximization
submitted for publication
[arXiv]
[bib]
[abstract]
Abstract: In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale, an explicit first-order expansion formula for the power investor's value function - seen as a function of the underlying market price of risk process - is provided and its second-order error is quantified. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.
@unpublished{LarMosZit14,
author = {Larsen, Kasper and Mostovyi, Oleksii and {\v{Z}}itkovi{\'c}, Gordan}, title = {An expansion in the model space in the context of utility maximization}, note = {submitted for publication}, year = {2014}, } Research - published/accepted
(2017)
Existence, characterization and approximation in the generalized monotone follower problem
to appear in SIAM Journal of Control and Optimization
[arXiv]
[bib]
[abstract]
Abstract: We revisit the classical monotone-follower problem and consider it in a generalized formulation. Our approach, based on a compactness substitute for nondecreasing processes, the Meyer-Zheng weak convergence, and the maximum principle of Pontryagin, establishes existence under minimal conditions, produces general approximation results and further elucidates the celebrated connection between optimal stochastic control and stopping.
@unpublished{LiZit15,
author = {Li, Jiexian and {\v{Z}}itkovi{\'c}, Gordan}, title = {Existence, characterization and approximation in the generalized monotone follower problem}, year = {2017}, note = {to appear in SIAM Journal of Control and Optimization}, }
(2016)
Facelifting in utility maximization
Finance and Stochastics
vol. 20
no. 1
pp. 99-121
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{LarSonZit16,
author = {Larsen, Kasper and Soner, {H}. {M.}and {\v{Z}}itkovi\'{c}, Gordan}, title = {Facelifting in utility maximization}, journal = {Finance and Stochastics}, doi = {10.1007/s00780-015-0274-y}, volume = {20}, year = {2016}, number = {1}, pages = {99-121}, year = {2016}, }
(2014)
Dynamic programming for controlled Markov families: abstractly and over martingale measures
SIAM Journal of Control and Optimization
vol. 52
no. 3
pp. 1597-1621
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We describe an abstract control-theoretic setting 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''
@article{Zit14,
author = {Gordan {\v{Z}}itkovi\'{c}}, title = {Dynamic programming for controlled Markov families: abstractly and over martingale measures}, journal = {SIAM Journal of Control and Optimization}, volume = {52}, number = {3}, pages = {1597-1621}, doi = {10.1137/130926481}, year = {2014}, }
(2013)
Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs
SIAM Journal of Control and Optimization
vol. 51
no. 6
pp. 4414-4449
[arXiv]
[bib]
[abstract]
Abstract: 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.
@article{ChoSirZit13,
author = {Jinhyuk Choi and Mihai Sirbu and Gordan {\v{Z}}itkovi\'{c}}, title = {Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs}, journal = {SIAM Journal of Control and Optimization}, volume = {51}, number = {6}, pages = {4414-4449}, year = {2013}, }
(2013)
Utility maximization under convex portfolio constraints
Annals of Applied Probability
vol. 23
no. 2
pp. 665-692
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We consider a utility-maximization problem in a general semimartingale financial market, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin, i.e., no risky investment at all may be inadmissible. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present.
Our main result establishes existence of the optimal trading strategies in such markets under no smoothness requirements on the utility function, and relates them to the corresponding dual objects. Moreover, we show that the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.
@article{LarZit12,
author = {Kasper Larsen and Gordan {\v{Z}}itkovi\'{c}}, title = {Utility maximization under convex portfolio constraints}, year = {2013}, journal = {Annals of Applied Probability}, volume = {23}, number = {2}, pages = {665-692}, doi = {10.1214/12-AAP850}, }
(2013)
Forward-convex convergence of sequences of nonnegative random variables
Proceeding of the American Mathematical Society
vol. 141
pp. 919-929
[arXiv]
[doi]
[bib]
[abstract]
Abstract: For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance.
@article{KarZit10a,
author = {Constantinos Kardaras and Gordan {\v{Z}}itkovi\'{c}}, title = {Forward-convex convergence of sequences of nonnegative random variables}, journal = {Proceeding of the American Mathematical Society}, volume = {141}, year = {2013}, pages = {919-929}, doi = {10.1090/S0002-9939-2012-11373-5}, }
(2012)
An example of a stochastic equilibrium with incomplete markets
Finance and Stochastics
vol. 16
no. 2
pp. 177-206
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{Zit12,
author = {Gordan {\v{Z}}itkovi{\'{c}}}, title = {An example of a stochastic equilibrium with incomplete markets}, journal = {Finance and Stochastics}, year = {2012}, volume = {16}, number = {2}, pages = {177-206}, doi = {10.1007/s00780-011-0161-0}, }
(2011)
Stability of the utility maximization problem with random endowment in incomplete markets
Mathematical Finance
vol. 21
no. 2
pp. 313-333
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{KarZit11,
author = {Constantinos Kardaras and Gordan {\v{Z}}itkovi{\'{c}}}, title = {Stability of the utility maximization problem with random endowment in incomplete markets}, journal = {Mathematical Finance}, year = {2011}, volume = {21}, number = {2}, pages = {313-333}, doi = {10.1111/j.1467-9965.2010.00433.x}, }
(2010)
On Agents' Agreement and Partial-Equilibrium Pricing in Incomplete Markets
Mathematical Finance
vol. 20
no. 3
pp. 411-446
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{AntZit10,
author = {Michail Anthropelos and Gordan {\v{Z}}itkovi{\'{c}}}, title = {On Agents' Agreement and Partial-Equilibrium Pricing in Incomplete Markets}, journal = {Mathematical Finance}, year = {2010}, volume = {20}, pages = {411-446}, number = {3}, doi = {10.1111/j.1467-9965.2010.00405.x}, }
(2010)
Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability
Annals of Finance
vol. 6
no. 1
pp. 107-135
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{AntZit10a,
author = {Michail Anthropelos and Gordan {\v{Z}}itkovi{\'{c}}}, title = {Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability}, journal = {Annals of Finance}, year = {2010}, volume = {6}, pages = {107-135}, number = {1}, doi = {10.1007/s10436-009-0134-x}, }
(2010)
Maturity-Independent Risk Measures
SIAM Journal on Financial Mathematics
vol. 1
pp. 266-288
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{ZarZit10,
author = {Thaleia Zariphopoulou and Gordan {\v{Z}}itkovi\'{c}}, title = {Maturity-Independent Risk Measures}, journal = {SIAM Journal on Financial Mathematics}, year = {2010}, volume = {1}, pages = {266-288}, doi = {10.1137/080739732}, }
(2010)
Utility theory: historical perspectives
Encyclopedia of Quantitative Finance
John Wiley and Sons Inc.
New York
[doi]
[bib]
[abstract]
Abstract: We present a short survey of the history of utility theory. Starting from the seminal idea of Daniel Bernoulli, we proceed to discuss the work of Herman Gossen and its importance for the “Marginalist revolution”. The next milestone is the advent of axiomatic approaches, starting with the work of Franz Alt, John von Neumann, and Oskar Morgenstern. Various critiques of the expected-utility model are discussed. On the conceptual side, we mention the “Subjectivist revolution” lead by Leonard Savage, and, on the empirical side, we discuss the paradoxes of Maurice Allais and Daniel Ellsberg and their partial resolution in the work by Daniel Kahneman and Amos Tversky. Finally, we touch upon the concept of risk aversion and the place of utility theory in modern finance.
@incollection{Zit09a,
author = {Gordan {\v{Z}}itkovi{\' c}}, title = {Utility theory: historical perspectives}, booktitle = {Encyclopedia of Quantitative Finance}, publisher = {John Wiley and Sons Inc.}, year = {2010}, address = {New York}, doi = {10.1002/9780470061602.eqf01011}, }
(2009)
A dual characterization of self-generation and exponential forward performances
Ann. Appl. Probab.
vol. 19
no. 6
pp. 2176-2210
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We propose a mathematical framework for the study of a family of random fields-called forward performances-which arise as numerical representation of certain rational preference relations in mathematical finance. Their spatial structure corresponds to that of utility functions, while the temporal one reflects a Nisio-type semigroup property, referred to as self-generation. In the setting of semimartingale financial markets, we provide a dual formulation of self-generation in addition to the original one, and show equivalence between the two, thus giving a dual characterization of forward performances. Then we focus on random fields with an exponential structure and provide necessary and sufficient conditions for self-generation in that case. Finally, we illustrate our methods in financial markets driven by Itô-processes, where we obtain an explicit parametrization of all exponential forward performances.
@article{Zit09b,
author = {Gordan {\v{Z}}itkovi\'{c}}, title = {A dual characterization of self-generation and exponential forward performances}, journal = {Ann. Appl. Probab.}, year = {2009}, volume = {19}, pages = {2176-2210}, number = {6}, doi = {10.1214/09-AAP607}, }
(2009)
Convex-compactness and its applications
Mathematics and Financial Economics
vol. 3
no. 1
pp. 1-12
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{Zit09,
author = {Gordan {\v{Z}}itkovi{\'{c}}}, title = {Convex-compactness and its applications}, journal = {Mathematics and Financial Economics}, year = {2009}, volume = {3}, pages = {1-12}, number = {1}, doi = {10.1007/s11579-010-0024-z}, }
(2009)
Hydrophilicity Matching : a Prerequisite for the Formation of Protein-Protein Complexes
PLoS One
vol. 5
no. 6
pp. e11169
[doi]
[pdf]
[bib]
[abstract]
Abstract: A binding event between two proteins typically consists of a diffusional search of binding partners for one another, followed by a specific recognition of the compatible binding sites resulting in the formation of the complex. However, it is unclear how binding partners find each other in the context of the crowded, constantly fluctuating, and interaction-rich cellular environment. Here we examine the non-specific component of protein-protein interactions, which refers to all physicochemical properties of the binding partners that are independent of the exact details of their binding sites, but which can affect their localization or diffusional search for one another. We show that, for a large set of high-resolution experimental 3D structures of binary, transient protein complexes taken from the DOCKGROUND database, the binding partners display a surprising, statistically significant similarity in terms of their total hydration free energies normalized by a size-dependent variable. We propose that colocalization of binding partners, even within individual cellular compartments such as the cytoplasm, may be influenced by their relative hydrophilicity in response to intra- compartmental hydrophilic gradients.
@article{HleZitZag09,
author = {Mario Hlevnjak and Gordan {\v{Z}}itkovi{\' c}and Bojan {\v{Z}}agrovi{\' c}}, title = {Hydrophilicity Matching : a Prerequisite for the Formation of Protein-Protein Complexes}, journal = {PLoS One}, year = {2009}, volume = {5}, pages = {e11169}, number = {6}, doi = {10.1371/journal.pone.0011169}, }
(2009)
Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing
Mathematical Finance
vol. 19
no. 1
pp. 129-159
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{OweZit09,
author = {Mark Owen and Gordan {\v{Z}}itkovi{\' c}}, title = {Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing}, journal = {Mathematical Finance}, year = {2009}, volume = {19}, pages = {129-159}, number = {1}, doi = {10.1111/j.1467-9965.2008.00360.x}, }
(2009)
Maximizing the growth rate under risk constraints
Mathematical Finance
vol. 19
no. 3
pp. 423-455
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, Itô-process models of financial markets with random ergodic coefficients. Including value-at-risk (VaR), tail-value-at-risk (TVaR), and limited expected loss (LEL), these constraints can be both wealth-dependent(relative) and wealth-independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state-dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk-constrained wealth-growth optimizer locally behaves like a CRRA-investor, with the relative risk-aversion coefficient depending on the current values of the market coefficients.
@article{PirZit09,
author = {Traian A. Pirvu and Gordan {\v{Z}}itkovi\'{c}}, title = {Maximizing the growth rate under risk constraints}, journal = {Mathematical Finance}, year = {2009}, volume = {19}, pages = {423-455}, number = {3}, doi = {10.1111/j.1467-9965.2009.00378.x}, }
(2008)
On the semimartingale property via bounded logarithmic utility
Annals of Finance
vol. 4
no. 2
pp. 255-268
[arXiv]
[doi]
[bib]
[abstract]
Abstract: This paper provides a new version of the condition of Di Nunno et al. (2003), Ankirchner and Imkeller (2005) and Biagini and Øksendal (2005) ensuring the semimartingale property for a large class of continuous stochastic processes. Unlike our predecessors, we base our modeling framework on the concept of portfolio proportions which yields a short self-contained proof of the main theorem, as well as a counterexample, showing that analogues of our results do not hold in the discontinuous setting.
@article{LarZit08,
author = {Kasper Larsen and Gordan {\v{Z}}itkovi{\' c}}, title = {On the semimartingale property via bounded logarithmic utility}, journal = {Annals of Finance}, year = {2008}, volume = {4}, pages = {255-268}, number = {2}, doi = {10.1007/s10436-006-0067-6}, }
(2008)
On maturity-independent risk measures
Proceedings of 47th IEEE Conference on Decision and Control
[doi]
[bib]
[abstract]
Abstract: This paper deals with the problem one faces when the maturity (horizon, expiration date, etc.) associated with a particular risky position is not fixed. We take the view that the mechanism used to measure the risk content of a certain random variable should not depend on any a priori choice of the measurement horizon. The case in complete financial markets is given as an example.
@inproceedings{ZarZit08,
author = {Thaleia Zariphopoulou and Gordan {\v{Z}}itkovi{\' c}}, title = {On maturity-independent risk measures}, booktitle = {Proceedings of 47th IEEE Conference on Decision and Control}, year = {2008}, pages = {5596-6501}, doi = {10.1109/CDC.2008.4739244}, }
(2007)
Stability of utility-maximization in incomplete markets
Stochastic Processes and their Applications
vol. 117
no. 11
pp. 1642-1662
[arXiv]
[doi]
[bib]
[abstract]
Abstract: The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent's optimal behavior derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem where prices are modeled by continuous semimartingales is established as an offshoot of the proof of our central theorem.
@article{LarZit07,
author = {Kasper Larsen and Gordan {\v{Z}}itkovi{\' c}}, title = {Stability of utility-maximization in incomplete markets}, journal = {Stochastic Processes and their Applications}, year = {2007}, volume = {117}, pages = {1642-1662}, number = {11}, doi = {10.1016/j.spa.2006.10.012}, }
(2006)
Financial equilibria in the semimartingale setting: complete markets and markets with withdrawal constraints
Finance and Stochastics
vol. 10
no. 1
pp. 99-119
[arXiv]
[doi]
[bib]
[abstract]
Abstract: 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.
@article{Zit06,
author = {Gordan {\v{Z}}itkovi{\' c}}, title = {Financial equilibria in the semimartingale setting: complete markets and markets with withdrawal constraints}, journal = {Finance and Stochastics}, year = {2006}, volume = {10}, pages = {99-119}, number = {1}, doi = {10.1007/s00780-005-0175-6}, }
(2005)
Utility maximization with a stochastic clock and an unbounded random endowment
Annals of Applied Probability
vol. 15
no. 1B
pp. 748-777
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility maximization problems including the classical ones of terminal wealth or consumption, as well as the problems depending on a random time-horizon or multiple consumption instances. As an example we treat explicitly the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein-Uhlenbeck process acts as a stochastic clock.
@article{Zit05,
author = {Gordan {\v{Z}}itkovi{\'c}}, title = {Utility maximization with a stochastic clock and an unbounded random endowment}, journal = {Annals of Applied Probability}, year = {2005}, volume = {15}, pages = {748-777}, number = {1B}, doi = {10.1214/105051604000000738}, }
(2003)
Optimal consumption from investment and random endowment in incomplete semimartingale markets
Annals of Probability
vol. 31
no. 4
pp. 1821-1858
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption/terminal wealth problems, in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of L^1 to its topological bidual (L^{\infty})^*, a space of finitely-additive measures. As an application, we treat the case of a constrained Itô-process market-model.
@article{KarZit03,
author = {Ioannis Karatzas and Gordan {\v{Z}}itkovi{\' c}}, title = {Optimal consumption from investment and random endowment in incomplete semimartingale markets}, journal = {Annals of Probability}, year = {2003}, volume = {31}, pages = {1821-1858}, number = {4}, doi = {10.1214/aop/1068646367}, }
(2002)
A filtered version of the bipolar theorem of Brannath and Schachermayer
Journal of Theoretical Probability
vol. 15
no. 1
pp. 41-61
[arXiv]
[doi]
[bib]
[abstract]
Abstract: We extend the Bipolar Theorem of Brannath and Schachermayer (1999) to the space of nonnegative cadlag supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer (1999) and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.
@article{Zit02,
author = {Gordan {\v{Z}}itkovi{\'c}}, title = {A filtered version of the bipolar theorem of {B}rannath and {S}chachermayer}, journal = {Journal of Theoretical Probability}, year = {2002}, volume = {15}, pages = {41-61}, number = {1}, doi = {10.1023/A:1013885121598}, } This material is based upon work supported by the National Science Foundation under Grant No. DMS-0706947 (August 2010 - present) and Grant No. DMS-0955614 (August 2007 - June 2010). Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF). |
Lecture Notes
All (other) lecture notes are
[here].
Current TeachingFall 2016: Intro to Mathematical Statistics (M378K) and Theory of Probability (M385C)Slides for both classes are [here]. You probably want the "handout" version. |