Showing papers on "Semimartingale published in 2005"
TL;DR: In this article, the authors studied the dynamics of the exponential utility indifference value process for a contingent claim B in a semimartingale model with a general continuous filtration.
Abstract: We study the dynamics of the exponential utility indifference value process C(B; α) for a contingent claim B in a semimartingale model with a general continuous filtration. We prove that C(B; α) is (the first component of) the unique solution of a backward stochastic differential equation with a quadratic generator and obtain BMO estimates for the components of this solution. This allows us to prove several new results about Ct(B; α). We obtain continuity in B and local Lipschitz-continuity in the risk aversion α, uniformly in t, and we extend earlier results on the asymptotic behavior as α↘0 or α↗∞ to our general setting. Moreover, we also prove convergence of the corresponding hedging strategies.
228 citations
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TL;DR: This work derives the dynamics of the LIBOR rate process in a semimartingale as well as a Lévy Heath-Jarrow-Morton setting and introduces an explicit formula to price caps and floors which uses bilateral Laplace transforms.
Abstract: Models driven by Levy processes are attractive because of their greater flexibility compared to classical diffusion models. First we derive the dynamics of the LIBOR rate process in a semimartingale as well as a Levy Heath-Jarrow-Morton setting. Then we introduce a Levy LIBOR market model. In order to guarantee positive rates, the LIBOR rate process is constructed as an ordinary exponential. Via backward induction we get that the rates are martingales under the corresponding forward measures. An explicit formula to price caps and floors which uses bilateral Laplace transforms is derived.
125 citations
TL;DR: In this article, the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market was studied and it was shown that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utilitybased price is that the solution to the dual problem defines an equivalent local martingale measure.
Abstract: We study the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility-based price is that the solution to the dual problem defines an equivalent local martingale measure.
103 citations
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TL;DR: In this paper, the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market was studied and it was shown that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utilitybased price is that the solution to the dual problem defines an equivalent local martingale measure.
Abstract: We study the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility-based price is that the solution to the dual problem defines an equivalent local martingale measure.
95 citations
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TL;DR: In this article, the authors consider a stochastic delay differential equation driven by a general Levy process and show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space.
Abstract: We consider a stochastic delay differential equation driven by a general Levy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.
55 citations
TL;DR: In this article, the authors show that whenever an agent's expected utility is flnite, S is a semimartingale with a Doob-Meyer decomposition featuring a martingale part and an information drift.
Abstract: We consider flnancial markets with two kinds of small traders: regular traders who perceive the (continuous) asset price process S through its natural flltration, and insiders who possess some information advantage which makes the flltrations through which they experience the evolution of the market richer. We discuss the link between (NFLVR), the semimartingale property of S viewed from the agent’s perspective, and bounded expected utility. We show that whenever an agent’s expected utility is flnite, S is a semimartingale with a Doob-Meyer decomposition featuring a martingale part and an information drift. The expected utility gain of an insider with respect to a regular trader is calculated in a completely general setting. In particular, for the logarithmic utility function, utility gain is a function of the relative information drift alone, regardless of whether the market admits arbitrage.
52 citations
TL;DR: It turns out that in case the mean reversion level and the correlation coefficient are nonzero, an investor who can use trading strategies adapted to the Brownian filtration may achieve a higher expected exponential utility from terminal wealth than an investorwho can only observe the price process.
Abstract: We outline a martingale duality method for determining the minimal entropy martingale measure in a general continuous semimartingale model, and provide the relevant verification results. This method is illustrated by a detailed case study of the Stein and Stein stochastic volatility model driven by two correlated Brownian motions. It turns out that in case the mean reversion level and the correlation coefficient are nonzero, an investor who can use trading strategies adapted to the Brownian filtration may achieve a higher expected exponential utility from terminal wealth than an investor who can only observe the price process.
41 citations
TL;DR: In this paper, the evolution of barycenters of masses transported by stochastic flows is studied in smooth affine manifolds with a certain convexity structure, and it is shown that the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann bary center of the limiting measure.
Abstract: We investigate the evolution of barycenters of masses transported by stochastic flows. The state spaces under consideration are smooth affine manifolds with certain convexity structure. Under suitable conditions on the flow and on the initial measure, the barycenter {Zt} is shown to be a semimartingale and is described by a stochastic differential equation. For the hyperbolic space the barycenter of two independent Brownian particles is a martingale and its conditional law converges to that of a Brownian motion on the limiting geodesic. On the other hand for a large family of discrete measures on suitable Cartan–Hadamard manifolds, the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann barycenter of the limiting measure.
39 citations
TL;DR: In this paper, the authors studied the problem of superreplication and utility maximization from terminal wealth in a semimartingale model with countably many assets, and proved the existence of an optimizer in a suitable class of generalized strategies with the property that maximal expected utility is the limit of maximal expected utilities in finite-dimensional submarkets.
36 citations
TL;DR: Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied and delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained.
Abstract: Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.
36 citations
01 Jan 2005
TL;DR: In this article, the authors give an adequate definition of the stochastic integral up to infinity, where H = (Ht)_{t\ge0} is a predictable process and X = (X_t){t͡ge 0} is semimartingale.
Abstract: The first goal of this paper is to give an adequate definition of the stochastic integral
$$\int_0^\infty {{H_s}} {\text{d}}{X_s},(*)$$
where \(H = (H_t)_{t\ge0}\) is a predictable process and \(X = (X_t)_{t\ge0}\) is a semimartingale. We consider two different definitions of (*): as a stochastic integral up to infinity and as an improper stochastic integral.
22 Jul 2005
TL;DR: The first part of this thesis deals with impacts of changes in the information structure on the appearance of a stochastic process and analyzes the impact of information on utility.
Abstract: Stochastic Analysis provides methods to describe random numerical processes. The descriptions depend strongly on the underlying information structure, which is represented in terms of filtrations. The first part of this thesis deals with impacts of changes in the information structure on the appearance of a stochastic process. More precisely, it analyses the consequences of a filtration enlargement on the semimartingale decomposition of the process. From the martingale part a drift has to be subtracted in order to obtain a martingale in the enlarged filtration. Methods are given how one can compute and analyze this correcting drift. The second and third part discuss the role of information in financial utility calculus: In the framework of the general semimartingale model of financial markets the link between information and utility is analyzed. The second part is of a qualitative nature: It deals with implications of the assumption that the maximal expected utility of an investor is bounded. It is shown that finite utility implies some structure properties of the price process viewed from the intrinsic perspective: At first it follows that the price is a semimartingale. Moreover, one can show for continuous processes that the bounded variation part in the semimartingale decomposition is nicely controlled by the martingale part and does not explode. Thus the second part justifies these widespread assumptions. The third part is of a quantitative nature: It analyzes the impact of information on utility. From an extrinsic point of view traders with different knowledge are compared. In particular, it is shown how additional information increases utility. If the preferences of the investor are described by the logarithmic utility function, then one can calculate the utility increment by means of the so-called information drift. Furthermore, the utility increment coincides with the mutual information between the additional knowledge and the original knowledge, ‘mutual information’ being defined in the sense of information theory. As a consequence the link between two different concepts of ‘information’ is established.
TL;DR: In this article, the authors consider the problem of finding optimal consumption strategies in an incomplete semimartingale market model under model uncertainty and prove the existence of a saddle point and give a characterization of an optimal consumption strategy.
Abstract: In this paper we consider the problem of finding optimal consumption strategies in an incomplete semimartingale market model under model uncertainty. The quality of a consumption strategy is measured by not only one probability measure but as common in risk theory by a class of scenario measures. We formulate a dual version of the optimization problem and prove the existence of a saddle point and give a characterization of an optimal consumption strategy in terms of solutions of the dual problem. This generalizes results of Karatzas and Zitkovic (2003) for the optimal consumption problem under a fixed probability measure.
TL;DR: In this article, the convergence of the p-optimal martingale measures to the minimal entropy measure is established in an incomplete financial market where asset prices are continuous semimartingales.
Abstract: In an incomplete financial market where asset prices are continuous semimartingales, we establish the convergence of the p-optimal martingale measures to the minimal entropy martingale measure as p tends to 1. The result is achieved exploiting the theory of BMO-martingales and semimartingale backward equations.
TL;DR: In this article, the segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem, and the segment calculus is embedded in the theory of anticipating calculus.
Abstract: For a given stochastic process X, its segment Xt at time t represents the \slice" of each path of X over a xed time-interval [t r; t], where r is the length of the \memory" of the process. Segment processes are important in the study of stochastic systems with memory (stochastic functional dieren tial equations, SFDEs). The main objective of this paper is to study non-linear transforms of segment processes. Towards this end, we construct a stochastic integral with respect to the Brownian segment process. The dicult y in this construction is the fact that the stochastic integrator is innite dimensional and is not a (semi)martingale. We overcome this dicult y by employing Malliavin (anticipating) calculus techniques. The segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem. We then prove It^ o’s formula for the segment of a continuous Skorohod-type process and embed the segment calculus in the theory of anticipating calculus. Applications of the It^ o formula include the weak innitesimal generator for the solution segment of a stochastic system with memory, the associated Feynman-Kac formula and the Black-Scholes PDE for stock dynamics with memory.
TL;DR: In this paper, the authors provide a characterisation of mean-variance hedging strategies in a general semimartingale market and introduce a new probability measure P* which turns the dynamic asset allocation problem into a myopic one.
Abstract: We provide a new characterisation of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure P* which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to P* coincides with the variance-optimal martingale measure relative to the original probability measure P.
TL;DR: Theorem 2.4 as discussed by the authors is a characterization of a weaker notion of no-arbitrage in terms of the existence of supermartingale densities, which is equivalent to the absence of immediate arbitrage opportunities.
Abstract: The present paper deals with the characterization of no-arbitrage properties
of a continuous semimartingale. The first main result, Theorem
\refMainTheoremCharNA, extends the no-arbitrage criterion by Levental and
Skorohod [Ann. Appl.
Probab. 5 (1995) 906-925] from diffusion processes to arbitrary continuous
semimartingales. The second main result, Theorem 2.4, is a characterization of
a weaker notion of no-arbitrage in terms of the existence of supermartingale
densities. The pertaining weaker notion of no-arbitrage is equivalent to the
absence of immediate arbitrage opportunities, a concept introduced by Delbaen
and Schachermayer [Ann. Appl. Probab. 5 (1995) 926-945]. Both results are
stated in terms of conditions for any semimartingales starting at arbitrary
stopping times \sigma. The necessity parts of both results are known for the
stopping time \sigma=0 from Delbaen and Schachermayer [Ann. Appl. Probab. 5
(1995) 926-945]. The contribution of the present paper is the proofs of the
corresponding sufficiency parts.
01 Jan 2005
TL;DR: In this paper, the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale financial model is studied, and necessary and sufficient conditions are obtained in terms of the triplet of predictable characteristics of the asset price process.
Abstract: The Numeraire Portfolio and Arbitrage in Semimartingale Models of Financial Markets Konstantinos Kardaras We study the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale financial model. The numeraire portfolio generates a wealth process which makes the relative wealth processes of all other portfolios with respect to it supermartingales. Necessary and sufficient conditions for the existence of the numeraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of an arbitrage-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required, in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger (NFLVR) condition lacks.
TL;DR: In this article, the titular question is investigated for fairly general semimartingale investment and asset price processes, and it is shown that the two are equivalent if the jump part of the price process converges.
Abstract: The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.
TL;DR: Theorem 2.4 as discussed by the authors is a characterization of a weaker notion of no-arbitrage in terms of the existence of supermartingale densities, which is equivalent to the absence of immediate arbitrage opportunities.
Abstract: The present paper deals with the characterization of no-arbitrage properties of a continuous semimartingale. The first main result, Theorem \refMainTheoremCharNA, extends the no-arbitrage criterion by Levental and Skorohod [Ann. Appl.
Probab. 5 (1995) 906-925] from diffusion processes to arbitrary continuous semimartingales. The second main result, Theorem 2.4, is a characterization of a weaker notion of no-arbitrage in terms of the existence of supermartingale densities. The pertaining weaker notion of no-arbitrage is equivalent to the absence of immediate arbitrage opportunities, a concept introduced by Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926-945]. Both results are stated in terms of conditions for any semimartingales starting at arbitrary stopping times \sigma. The necessity parts of both results are known for the stopping time \sigma=0 from Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926-945]. The contribution of the present paper is the proofs of the corresponding sufficiency parts.
TL;DR: In this paper, the stationary distribution for reflected diffusions with jumps in the positive orthant was studied and necessary and sufficient conditions for the existence of a product-form distribution for diffusion with oblique boundary reflections and jumps were provided.
Abstract: In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R n that satisfies certain finiteness conditions, we characterize the Fokker– Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Levy processes. We provide explicit results for some models of interest.
TL;DR: In this paper, the authors focus on properties of the variance-optimal martingale measure for discontinuous semimartingales, and they give sufficient conditions for the variance optimal measure to be a probability measure and for the density process to satisfy the reverse Holder inequality, respectively.
TL;DR: In this paper, the forward rates of interest rates are assumed to be semimartingales, and conditions on their components under which the discounted bond prices are martingales.
Abstract: Assuming that the forward rates f u t are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes f u t = u 0 f v t dv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.
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TL;DR: In this article, the robustness of probability limits and central limit theory for realised multipower variation when adding finite activity and infinite activity jump processes to an underlying Brownian semimartingale was studied.
Abstract: In this paper we provide a systematic study of the robustness of probability limits and central limit theory for realised multipower variation when we add finite activity and infinite activity jump processes to an underlying Brownian semimartingale.
TL;DR: In this paper, the authors consider the solution to a stochastic heat equation and show that it has infinite quadratic variation and is not a semimartingale.
Abstract: We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Ito calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of $t$, converge weakly to Brownian motion.
TL;DR: In this paper, a unified approach to obtaining rates of convergence for the maximum likelihood estimator (MLE) in Brownian semimartingale models of the form dX t ¼ � n,Ł t dt þ � n t dWt, t < Tn is presented.
Abstract: In this paper we present a unified approach to obtaining rates of convergence for the maximum likelihood estimator (MLE) in Brownian semimartingale models of the form dX t ¼ � n,Ł t dt þ � n t dWt, t < Tn: We show that the rate of the MLE is determined by (an appropriate version of) the entropy of the parameter space with respect to the random metric hn, defined by h 2 n(Ł, l) ¼ ð Tn 0 � n,Ł s � � n,l s � n �� 2 ds: Several known results for the rates in certain popular sub-models of the Brownian semimartingale model are shown to be special cases in our general framework.
TL;DR: In this paper, a mean-variance hedging for the discontinuous semimartingale case is obtained under some assumptions related to the variance-optimal martingale measure.
Abstract: Mean-variance hedging for the discontinuous semimartingale case is obtained under some assumptions related to the variance-optimal martingale measure. In the present paper, two remarks on it are discussed. One is an extension of Hou–Karatzas' duality approach from the continuous case to discontinuous. Another is to prove that there is the consistency with the case where the mean-variance trade-off process is continuous and deterministic. In particular, one-dimensional jump diffusion models are discussed as simple examples.
01 Jun 2005
TL;DR: In this article, the existence of a semimartingale of which one-dimensional marginal distributions are given by the solution of the Fokker-Planck equation with the $p$-th integrable drift vector was shown.
Abstract: We show the existence of a semimartingale of which one-dimensional marginal distributions are given by
the solution of the Fokker-Planck equation with the $p$-th integrable drift vector ($p>1$).
TL;DR: In this paper, the evolution of barycenters of masses transported by stochastic flows is studied in smooth affine manifolds with a certain convexity structure, and it is shown that under suitable conditions on the flow and on the initial measure, the barycenter {Z_t} is shown to be a semimartingale and is described by a Stochastic differential equation.
Abstract: We investigate the evolution of barycenters of masses transported by stochastic flows. The state spaces under consideration are smooth affine manifolds with certain convexity structure. Under suitable conditions on the flow and on the initial measure, the barycenter {Z_t} is shown to be a semimartingale and is described by a stochastic differential equation. For the hyperbolic space the barycenter of two independent Brownian particles is a martingale and its conditional law converges to that of a Brownian motion on the limiting geodesic. On the other hand for a large family of discrete measures on suitable Cartan-Hadamard manifolds, the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann barycenter of the limiting measure.
01 Jan 2005
TL;DR: In this paper, Korolyuk et al. presented Poisson approximation results for additive functionals switched by Markov and semi-Markov processes, which are obtained via semimartingale representations of additive functions and the convergence of generators for Markov processes.
Abstract: We present Poisson,approximation results for additive functionals switched by Markov and semi-Markov processes. The weak convergence results are obtained via semimartingale representations of additive functionals and the convergence of generators for Markov processes and of compensative operator of the extended Markov renewal processes. This is a review paper of our previous results given in [Korolyuk, 2002; Korolyuk, 2002A].