Showing papers on "Mathematical finance published in 2004"

Copula methods in finance

Abstract: PrefaceList of Common Symbols and Notations1 Derivatives Pricing, Hedging and Risk Management: The State of the Art11 Introduction12 Derivative pricing basics: the binomial model121 Replicating portfolios122 No-arbitrage and the risk-neutral probability measure123 No-arbitrage and the objective probability measure124 Discounting under different probability measures125 Multiple states of the world13 The Black-Scholes model131 Ito's lemma132 Girsanov theorem133 The martingale property134 Digital options14 Interest rate derivatives141 Affine factor models142 Forward martingale measure143 LIBOR market model15 Smile and term structure effects of volatility151 Stochastic volatility models152 Local volatility models153 Implied probability16 Incomplete markets161 Back to utility theory162 Super-hedging strategies17 Credit risk171 Structural models172 Reduced form models173 Implied default probabilities174 Counterparty risk18 Copula methods in finance: a primer181 Joint probabilities, marginal probabilities and copula functions182 Copula functions duality183 Examples of copula functions184 Copula functions and market comovements185 Tail dependence186 Equity-linked products187 Credit-linked products2 Bivariate Copula Functions21 Definition and properties22 Frechet bounds and concordance order23 Sklar's theorem and the probabilistic interpretation of copulas231 Sklar's theorem232 The subcopula in Sklar's theorem233 Modeling consequences234 Sklar's theorem in financial applications: toward a non-Black-Scholes world24 Copulas as dependence functions: basic facts241 Independence242 Comonotonicity243 Monotone transforms and copula invariance244 An application: VaR trade-off25 Survival copula and joint survival function251 An application: default probability with exogenous shocks26 Density and canonical representation27 Bounds for the distribution functions of sum of rvs271 An application: VaR bounds28 Appendix3 Market Comovements and Copula Families31 Measures of association311 Concordance312 Kendall's tau313 Spearman's rhoS314 Linear correlation315 Tail dependence316 Positive quadrant dependency32 Parametric families of bivariate copula321 The bivariate Gaussian copula322 The bivariate Student's t copula323 The Fr-echet family324 Archimedean copulas325 The Marshall-Olkin copula4 Multivariate Copulas41 Definition and basic properties42 Frechet bounds and concordance order: the multidimensional case43 Sklar's theorem and the basic probabilistic interpretation: the multidimensional case431 Modeling consequences44 Survival copula and joint survival function45 Density and canonical representation of a multidimensional copula46 Bounds for distribution functions of sums of n random variables47 Multivariate dependence48 Parametric families of n-dimensional copulas481 The multivariate Gaussian copula482 The multivariate Student's t copula483 The multivariate dispersion copula484 Archimedean copulas5 Estimation and Calibration from Market Data51 Statistical inference for copulas52 Exact maximum likelihood method521 Examples53 IFM method531 Application: estimation of the parametric copula for market data54 CML method541 Application: estimation of the correlation matrix for a Gaussian copula55 Non-parametric estimation551 The empirical copula552 Kernel copula56 Calibration method by using sample dependence measures57 Application58 Evaluation criteria for copulas59 Conditional copula591 Application to an equity portfolio6 Simulation of Market Scenarios61 Monte Carlo application with copulas62 Simulation methods for elliptical copulas63 Conditional sampling631 Clayton n-copula632 Gumbel n-copula633 Frank n-copula64 Marshall and Olkin's method65 Examples of simulations7 Credit Risk Applications71 Credit derivatives72 Overview of some credit derivatives products721 Credit default swap722 Basket default swap723 Other credit derivatives products724 Collateralized debt obligation (CDO)73 Copula approach731 Review of single survival time modeling and calibration732 Multiple survival times: modeling733 Multiple defaults: calibration734 Loss distribution and the pricing of CDOs735 Loss distribution and the pricing of homogeneous basket default swaps74 Application: pricing and risk monitoring a CDO741 Dow Jones EuroStoxx50 CDO742 Application: basket default swap743 Empirical application for the EuroStoxx50 CDO744 EuroStoxx50 pricing and risk monitoring745 Pricing and risk monitoring of the basket default swaps75 Technical appendix751 Derivation of a multivariate Clayton copula density752 Derivation of a 4-variate Frank copula density753 Correlated default times754 Variance-covariance robust estimation755 Interest rates and foreign exchange rates in the analysis8 Option Pricing with Copulas81 Introduction82 Pricing bivariate options in complete markets821 Copula pricing kernels822 Alternative pricing techniques83 Pricing bivariate options in incomplete markets831 Frcicing: super-replication in two dimensions832 Copula pricing kernel84 Pricing vulnerable options841 Vulnerable digital options842 Pricing vulnerable call options843 Pricing vulnerable put options844 Pricing vulnerable options in practice85 Pricing rainbow two-color options851 Call option on the minimum of two assets852 Call option on the maximum of two assets853 Put option on the maximum of two assets854 Put option on the minimum of two assets855 Option to exchange856 Pricing and hedging rainbows with smiles: Everest notes86 Pricing barrier options861 Pricing call barrier options with copulas: the general framework862 Pricing put barrier option: the general framework863 Specifying the trigger event864 Calibrating the dependence structure865 The reflection copula87 Pricing multivariate options: Monte Carlo methods871 Application: basket optionBibliographyIndex

Credit Risk Modeling: Theory and Applications

Abstract: Credit risk is today one of the most intensely studied topics in quantitative finance. This book provides an introduction and overview for readers who seek an up-to-date reference to the central problems of the field and to the tools currently used to analyze them. The book is aimed at researchers and students in finance, at quantitative analysts in banks and other financial institutions, and at regulators interested in the modeling aspects of credit risk. David Lando considers the two broad approaches to credit risk analysis: that based on classical option pricing models on the one hand, and on a direct modeling of the default probability of issuers on the other. He offers insights that can be drawn from each approach and demonstrates that the distinction between the two approaches is not at all clear-cut. The book strikes a fruitful balance between quickly presenting the basic ideas of the models and offering enough detail so readers can derive and implement the models themselves. The discussion of the models and their limitations and five technical appendixes help readers expand and generalize the models themselves or to understand existing generalizations. The book emphasizes models for pricing as well as statistical techniques for estimating their parameters. Applications include rating-based modeling, modeling of dependent defaults, swap- and corporate-yield curve dynamics, credit default swaps, and collateralized debt obligations.

Lévy Processes—From Probability to Finance and Quantum Groups

Abstract: 1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11 T he theory of stochastic processes was one of the most important mathematical developments of the twentieth century. Intuitively, it aims to model the interaction of “chance” with “time”. The tools with which this is made precise were provided by the great Russian mathematician A. N. Kolmogorov in the 1930s. He realized that probability can be rigorously founded on measure theory, and then a stochastic process is a family of random variables (X(t), t ≥ 0) defined on a probability space (Ω,F , P ) and taking values in a measurable space (E,E) . Here Ω is a set (the sample space of possible outcomes), F is a σ-algebra of subsets of Ω (the events), and P is a positive measure of total mass 1 on (Ω,F ) (the probability). E is sometimes called the state space. Each X(t) is a (F ,E) measurable mapping from Ω to E and should be thought of as a random observation made on E made at time t . For many developments, both theoretical and applied, E is Euclidean space Rd (often with d = 1); however, there is also considerable interest in the case where E is an infinite dimensional Hilbert or Banach space, or a finite-dimensional Lie group or manifold. In all of these cases E can be taken to be the Borel σalgebra generated by the open sets. To model probabilities arising within quantum theory, the scheme described above is insufficiently general and must be embedded into a suitable noncommutative structure. Stochastic processes are not only mathematically rich objects. They also have an extensive range of applications in, e.g., physics, engineering, ecology, and economics—indeed, it is difficult to conceive of a quantitative discipline in which they do not feature. There is a limited amount that can be said about the general concept, and much of both theory and applications focusses on the properties of specific classes of process that possess additional structure. Many of these, such as random walks and Markov chains, will be well known to readers. Others, such as semimartingales and measure-valued diffusions, are more esoteric. In this article, I will give an introduction to a class of stochastic processes called Levy processes, in honor of the great French probabilist Paul Levy, who first studied them in the 1930s. Their basic structure was understood during the “heroic age” of probability in the 1930s and 1940s and much of this was due to Paul Levy himself, the Russian mathematician A. N. Khintchine, and to K. Ito in Japan. During the past ten years, there has been a great revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications—particularly to option pricing in mathematical finance. As well as a vast number of research papers, a number of books on the subject have been published ([3], [11], [1], [2], [12]) and there have been annual international conferences devoted to these processes since 1998. Before we begin the main part of the article, it is worth David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is D.Applebaum@sheffield.ac.uk. He is the author of Levy Processes and Stochastic Calculus, Cambridge University Press, 2004, on which part of this article is based.

Liquidity Risk and Arbitrage Pricing Theory

Abstract: Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a security’s price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.

An example of indifference prices under exponential preferences

Abstract: The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivative’s payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy.

The Concepts and Practice of Mathematical Finance

Abstract: Preface 1. Risk 2. Pricing methodologies and arbitrage 3. Trees and option pricing 4. Practicalities 5. The Ito calculus 6. Risk neutrality and martingale measures 7. The practical pricing of a European option 8. Continuous barrier options 9. Multi-look exotic options 10. Static replication 11. Multiple sources of risk 12. Options with early exercise features 13. Interest rate derivatives 14. The pricing of exotic interest rate derivatives 15. Incomplete markets and jump-diffusion processes 16. Stochastic volatility 17. Variance gamma models 18. Smile dynamics and the pricing of exotic options Appendix A. Financial and mathematical jargon Appendix B. Computer projects Appendix C. Elements of probability theory Appendix D. Hints and answers to questions Bibliography Index.

Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain

Abstract: We consider a multi-stock market model where prices satisfy a stochastic differential equation with instantaneous rates of return modeled as a continuous time Markov chain with finitely many states. Partial observation means that only the prices are observable. For the investor’s objective of maximizing the expected utility of the terminal wealth we derive an explicit representation of the optimal trading strategy in terms of the unnormalized filter of the drift process, using HMM filtering results and Malliavin calculus. The optimal strategy can be determined numerically and parameters can be estimated using the EM algorithm. The results are applied to historical prices.

Introduction to the economics and mathematics of financial markets

Abstract: Introduction to the Economics and Mathematics of Financial Markets fills the longstanding need for an accessible yet serious textbook treatment of financial economics. The book provides a rigorous overview of the subject, while its flexible presentation makes it suitable for use with different levels of undergraduate and graduate students. Each chapter presents mathematical models of financial problems at three different degrees of sophistication: single-period, multi-period, and continuous-time. The single-period and multi-period models require only basic calculus and an introductory probability/statistics course, while an advanced undergraduate course in probability is helpful in understanding the continuous-time models. In this way, the material is given complete coverage at different levels; the less advanced student can stop before the more sophisticated mathematics and still be able to grasp the general principles of financial economics. The book is divided into three parts. The first part provides an introduction to basic securities and financial market organization, the concept of interest rates, the main mathematical models, and quantitative ways to measure risks and rewards. The second part treats option pricing and hedging; here and throughout the book, the authors emphasize the Martingale or probabilistic approach. Finally, the third part examines equilibrium models -- a subject often neglected by other texts in financial mathematics, but included here because of the qualitative insight it offers into the behavior of market participants and pricing.

Coherent and convex monetary risk measures for unbounded càdlàg processes

Abstract: Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all cadlag processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted cadlag processes that can be extended to coherent or convex monetary risk measures on the space of all adapted cadlag processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded cadlag processes induced by a so called m-stable set.

On the origin of power-law tails in price fluctuations

Abstract: (2004). On the origin of power-law tails in price fluctuations. Quantitative Finance: Vol. 4, No. 1, pp. 7-11.

The log-normal approximation in financial and other computations

Abstract: Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

Some calculations for Israeli options

Abstract: Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder’s claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a saddle point problem associated with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.

A valuation algorithm for indifference prices in incomplete markets

Abstract: A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserves the conditional distribution of unhedged risks, given the hedgeable ones, from their historical counterparts.

On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals

Abstract: Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solutions exhibit a critical phenomenon: If the capital constraint is below some critical value, then the solution will coincide with a classical solution; above this critical value, the solution is a superposition of a classical solution and a less risky or even risk-free investment. For general risk measures and utility functionals, it is shown that there exists a solution that can be written as a deterministic increasing function of the price density.

Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus

Abstract: The author gives a new numerical computation method of Expectation of Diffusion Processes, which is an improvement of a results in [3].

Pricing derivatives of American and game type in incomplete markets

Abstract: In this paper the neutral valuation approach is applied to American and game options in incomplete markets. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Game contingent claims are derivative contracts that can be terminated by both counterparties at any time before expiration. They generalize American options where this right is limited to the buyer of the claim. It turns out that as in the complete case, the price process of American and game contingent claims corresponds to a Snell envelope or to the value of a Dynkin game, respectively. On the technical level, an important role is played by $\sigma$ -sub- and $\sigma$ -supermartingales. We characterize these processes in terms of semimartingale characteristics.

Computations of Greeks in a market with jumps via the Malliavin calculus

Abstract: Using the Malliavin calculus on Poisson space we compute Greeks in a market driven by a discontinuous process with Poisson jump times and random jump sizes, following a method initiated on the Wiener space in [5]. European options do not satisfy the regularity conditions required in our approach, however we show that Asian options can be considered due to a smoothing effect of the integral over time. Numerical simulations are presented for the Delta and Gamma of Asian options, and confirm the efficiency of this approach over classical finite difference Monte-Carlo approximations of derivatives.

Bessel Processes, the Integral of Geometric Brownian Motion, and Asian Options

Abstract: This paper is motivated by questions about averages of stochastic processeswhich originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with [M. Yor, {\em Adv. Appl. Probab.}, 24 (1992), pp. 509--531], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman--Watson theory of [M. Yor, {\em Z. Wahrsch. Verw.\ Gebiete}, 53 (1980), pp. 71--95]. Consequences of this approach for valuing Asian options proper have been spelled out in [H. Geman and M. Yor, {\em Math. Finance}, 3 (1993), pp. 349--375] whose Laplace transform results were in fact regarded as a significant advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the Hartman--Watson approach itself: this approach is in general applicable without modifications only if it...

Introduction to the Economics and Mathematics of Financial Markets

Abstract: Introduction to the Economics and Mathematics of Financial Markets fills the longstanding need for an accessible yet serious textbook treatment of financial economics. The book provides a rigorous overview of the subject, while its flexible presentation makes it suitable for use with different levels of undergraduate and graduate students. Each chapter presents mathematical models of financial problems at three different degrees of sophistication: single-period, multi-period, and continuous-time. The single-period and multi-period models require only basic calculus and an introductory probability/statistics course, while an advanced undergraduate course in probability is helpful in understanding the continuous-time models. In this way, the material is given complete coverage at different levels; the less advanced student can stop before the more sophisticated mathematics and still be able to grasp the general principles of financial economics. The book is divided into three parts. The first part provides an introduction to basic securities and financial market organization, the concept of interest rates, the main mathematical models, and quantitative ways to measure risks and rewards. The second part treats option pricing and hedging; here and throughout the book, the authors emphasize the Martingale or probabilistic approach. Finally, the third part examines equilibrium models—a subject often neglected by other texts in financial mathematics, but included here because of the qualitative insight it offers into the behavior of market participants and pricing.

A non-Gaussian option pricing model with skew

Abstract: Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, 2, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together with closedform approximate solutions for the fair price of a European call option. In certain limits, the standard Black-Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.

Financial Derivatives: Pricing, Applications, and Mathematics

Abstract: 1. Introduction 2. Preliminary mathematics 3. Principles of financial valuation 4. Interest rate models 5. Mathematics of asset pricing 6. Bibliography.

Optimal portfolios when stock prices follow an exponential Lévy process

Abstract: We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Levy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Levy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Levy process and its stochastic exponential are investigated.

On the martingale property of stochastic exponentials

Abstract: We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a ‘candidate measure’ the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical finance are also given.

Quantitative Finance and Risk Management: A Physicist's Approach

Abstract: Quantitative Finance and Risk Management Topics: Simple and Exotic Derivatives, Market and Credit Risk, Optimized Stressed Correlation Matrices, Fat Tails, Stressed VAR, Model Development, Model Risk and Quality Assurance, Numerical Techniques, Deals and Portfolios, Systems (Buy/Build, Model Integration), Data Issues, Markets, Financial Products, Economic Capital - Case Studies in Corporate Finance and Options - "Life as a Quant": Management Issues, Communication, Sociology, Advice - Risk Lab: The Nuts and Bolts of Risk Management (Interest Rates, Equities, FX) - Research Topic: The Macro-Micro Model Combining Economics/Finance (Multi-Time-Scale, Multifactor, Quasi-Random Trends and Cycles, Mean-Reverting Gaussians, Gap/Jumps) - Feynman Path Integrals, Green Functions and Options

Topics in stochastic processes

Abstract: These notes are based on lectures on stochastic processes given at Scuola Normale Superiore in 1999 and 2000. Some new material was added and only selected, less standard results were presented. We did not include several applications to statistical mechanics and mathematical finance, covered in the lectures, as we hope to write part two of the notes devoted to applications of stochastic processes in modelling. The main themes of the notes are constructions of stochastic processes. We present different approaches to the existence question proposed by Kolmogorov, Wiener, Ito and Prohorov. Special attention is also paid to Levy processes. The lectures are basically self-contained and rely only on elementary measure theory and functional analysis. They might be used for more advanced courses on stochastic processes.

Optimal Control Problem Associated with Jump Processes

Abstract: An optimal portfolio/control problem is considered for a two-dimen\-sional model in finance. A pair consisting of the wealth process and cumulutative consumption process driven by a geometric Levy process is controlled by adapted processes. The value function appears and turns out to be a viscosity solution to some integro-differential equation, by using the Bellman principle.

A short history of stochastic integration and mathematical finance the early years, 1880-1970

Abstract: The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880 [80]. 1 ; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [63] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. We go into a little detail about what happened to Bachelier, since he is now seen by many as the founder of modern Mathematical Finance. Ignorant of the work of Thiele (which was little appreciated in its day) and preceding the work of Einstein, Bachelier attempted to model the market noise of the Paris Bourse. Exploiting the ideas of the Central Limit Theorem, and realizing that market noise should be without memory, he reasoned that increments of stock prices should be independent and normally distributed. He combined his reasoning with the Markov property and semigroups, and connected Brownian motion with the heat equation, using that the Gaussian kernel is the fundamental solution to the heat equation. He was able to define other processes related to Brownian motion, such as the maximum change during a time interval (for one dimensional Brownian motion), by using random walks and letting the time steps go to zero, and by then taking

Mean-variance hedging and stochastic control: beyond the Brownian setting

Abstract: We show for continuous semimartingales in a general filtration how the mean-variance hedging problem can be treated as a linear-quadratic stochastic control problem. The adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process, and we give necessary and sufficient conditions for the solvability of these generalized stochastic Riccati equations. Motivated from mathematical finance, this paper takes a first step toward linear-quadratic stochastic control in more general than Brownian settings.

Representation of Martingales with Jumps and Applications to Mathematical Finance

Abstract: We study representations of martingales with jumps based on the filtration generated by a Levy process. Two types of representation theorem are obtained. The first formula is valid for any martingale and written as the sum of the stochastic integral based on the Brownian motion and that based on the compensated Poisson random measure. See (0.1). The second formula is valid only for a process which is a martingale for any equivalent martingale measure. See (0.2). The latter representation formula is then applied to a problem in mathematical finance. The upper hedging strategy and the lower hedging strategy of a contingent claim is obtained through the representation kernel.