Showing papers in "Stochastics and Stochastics Reports in 2002"
TL;DR: In this paper, reflected backward stochastic differential equations with right continuous and left limited barrier were considered and the existence and uniqueness of the solution were proved. But the authors did not consider the link between such an equation with stochastically mixed control problems.
Abstract: We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems.
142 citations
TL;DR: In this article, the authors considered a continuous time portfolio optimization problem on an infinite time horizon for a factor model, where the mean returns of individual securities or asset categories are explicitly affected by economic factors.
Abstract: We consider a continuous time portfolio optimization problems on an infinite time horizon for a factor model, recently treated by Bielecki and Pliska ["Risk-sensitive dynamic asset management", Appl. Math. Optim. , 39 (1990) 337-360], where the mean returns of individual securities or asset categories are explicitly affected by economic factors. The factors are assumed to be Gaussian processes. We see new features in constructing optimal strategies for risk-sensitive criteria of the portfolio optimization on an infinite time horizon, which are obtained from the solutions of matrix Riccati equations.
104 citations
TL;DR: In this article, the authors consider optimal control problems for systems described by stochastic differential equations with delay (SDDE) and prove a version of the dynamic programming principle for a general class of such problems.
Abstract: We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.
104 citations
TL;DR: In this article, the authors consider continuous financial markets with a regular trader and an insider who is able to invest into one risky asset, and show that the information drift caused by the insider's knowledge cannot be eliminated by an equivalent change of probability measure.
Abstract: We consider models of time continuous financial markets with a regular trader and an insider who are able to invest into one risky asset. The insider's additional knowledge consists in his ability to stop at a random time which is inaccessible to the regular trader, such as the last passage of a certain level before maturity by some stock price process, or the time at which the stock price reaches its maximum during the trading interval. We show that under very mild assumptions on the coefficients of the diffusion process describing these price processes the information drift caused by the additional knowledge of the insider cannot be eliminated by an equivalent change of probability measure. As a consequence, all our models allow the insider to have free lunches with vanishing risk, or even to exercise arbitrage.
78 citations
TL;DR: It is shown that by using non-equidistant time nets, in contrast to equidistantTime nets, approximation rates can be improved considerably, and rates of convergence are asked for.
Abstract: We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.
63 citations
TL;DR: In this paper, the authors consider a forward-backward system of stochastic evolution equations in a Hilbert space and prove an analogue of the well-known Bismut-Elworthy formula.
Abstract: We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bell...
61 citations
TL;DR: In this paper, the authors present a large deviation principle for partial sums processes indexed by the half line, which is particularly suited to queueing applications and is established in a topology that is finer than the topology of uniform convergence on compacts and in which the queueing map is continuous.
Abstract: In this paper, we present a large deviation principle for partial sums processes indexed by the half line, which is particularly suited to queueing applications. The large deviation principle is established in a topology that is finer than the topology of uniform convergence on compacts and in which the queueing map is continuous. Consequently, a large deviation principle for steady-state queue lengths can be obtained immediately via the contraction principle.
50 citations
TL;DR: For a R d -valued sequence of martingale differences, Stoch et al. as mentioned in this paper obtained a moderate deviation principle for the sequence of partial sums in the space of cadlag functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process and a condition of "Lindeberg" type.
Abstract: For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of cadlag functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les deviations moderees des sommes de variables aleatoires vectorielles independantes de meme loi". Ann. Inst. H. Poincare , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prov...
47 citations
TL;DR: In this article, an infinite horizon investment-consumption model is proposed in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks).
Abstract: We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks...
44 citations
TL;DR: In this paper, a comparison theorem for infinite-dimensional Ito processes is proved, which is a generalization of that of [8] referring to the finite-dimensional case.
Abstract: In the paper we prove a comparison theorem for infinite-dimensional Ito processes, which is a generalization of that of [8], referring to the finite-dimensional case. As an application we obtain necessary and sufficient conditions for the relationship X> Y between function-valued processes. A positivity criterion for such processes is also given.
37 citations
TL;DR: In this article, Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes is used to find the value of the hedging portfolio and its derivatives (the deltas).
Abstract: For evaluating a hedging strategy we have to know at every moment the solution of the Cauchy problem for a corresponding parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these quantities by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. As illustrations we consider a Markovian multi-asset model with an instantaneously riskless saving bond and also some applications to the LIBOR rate model of Brace, Gatarck, Musiela and Jamshidian.
TL;DR: In this article, the stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale.
Abstract: The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Levy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.
TL;DR: In this paper, a planar random motion with constant velocity and three directions forming the angles is considered, such that the random times between consecutive changes of direction perform an alternating renewal process, and the transition densities of the motion are expressed in terms of a suitable modified two-index Bessel function.
Abstract: Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.
TL;DR: In this article, the authors considered optimal control problems where the system is driven by a stochastic differential equation of the Ito type and established necessary conditions for optimality satisfied by an optimal relaxed control.
Abstract: In this paper, we are concerned with optimal control problems where the system is driven by a stochastic differential equation of the Ito type. We study the relaxed model for which an optimal solution exists. This is an extension of the initial control problem, where admissible controls are measure valued processes. Using Ekeland's variational principle and some stability properties of the corresponding state equation and adjoint processes, we establish necessary conditions for optimality satisfied by an optimal relaxed control. This is the first version of the stochastic maximum principle that covers relaxed controls.
TL;DR: In this paper, the Cahn-Hilliard stochastic equation driven by a space-time white noise with a non-linear diffusion coefficient was studied and the local existence of the density without non-degeneracy condition in a case of Holder continuous trajectories was shown.
Abstract: This paper deals with the Cahn-Hilliard stochastic equation driven by a space-time white noise with a non-linear diffusion coefficient. Using new lower estimate of the kernel, we prove the local existence of the density without non-degeneracy condition in a case of Holder continuous trajectories, and we show that the density of any vector is lower bounded by a strictly positive continuous function under a non-degeneracy condition.
TL;DR: In this article, Kroeker et al. showed that the orthogonal functionals constructed in Kroeker, J.P. (1980) for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals.
Abstract: We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker, J.P. (1980) "Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals", Biol. Cybernetics 36 , 243-248, [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube.
TL;DR: In this paper, a Cameron-Martin type formula is derived for the Laplace transform of some integrals of the square of a general continuous Gaussian process, which involves the variance of the filtering error in some auxiliary optimal filtering problem which is used in the proof.
Abstract: A Cameron-Martin type formula is derived for the Laplace transform of some integrals of the square of a general continuous Gaussian process The formula involves in particular the variance of the filtering error in some auxiliary optimal filtering problem which is used in the proof This variance is expressed in terms of the solution of a Riccati-Volterra type integral equation containing the covariance function of the process In various specific cases this equation is solved and then the formula becomes completely explicit
TL;DR: In this paper, the exact distribution of a cyclic planar motion with three directions is explicitly derived in terms of Bessel functions of order three (suitably combined) and the absolutely continuous part of the distribution is proved to satisfy suitable boundary conditions and some of its properties are analyzed.
Abstract: The exact distribution of a cyclic planar motion with three directions is explicitly derived in terms of Bessel functions of order three (suitably combined). The absolutely continuous part of the distribution is proved to satisfy suitable boundary conditions and some of its properties are analyzed. The transformations converting the governing equations of order three is presented and its solutions (used here) derived by applying the Frobenius method.
TL;DR: In this article, the authors studied optimal stopping problems for diffusion processes with discontinuous reward function and gave some results about the regularity of the value function and showed that, under suitable mild conditions on the underlying process, it is lower (respectively: upper) semicontinuous if the reward function is.
Abstract: We study optimal stopping problems for diffusion processes with discontinuous reward function. We give some results about the regularity of the value function and we show that, under suitable mild conditions on the underlying process, it has the same regularity of the reward function, namely, it is lower (respectively: upper) semicontinuous if the reward function is. The proofs for the two cases are quite different, and the upper semicontinuous case requires stronger conditions. Finally, we show that, in the case of lower semicontinuous reward, under suitable conditions the value function is a (discontinuous) viscosity solution of the associated variational inequalities.
TL;DR: In this article, the authors considered optimal stopping problems for Markov processes with a semicontinuous reward function and showed that the value function w = w [ g ] is itself a viscosity solution of the associated variational inequality.
Abstract: We consider optimal stopping problems for Markov processes with a semicontinuous reward function g , and we show that under suitable conditions the value function w = w [ g ] is itself semicontinuous and is a viscosity solution of the associated variational inequality.
TL;DR: In this paper, the authors develop the theory of stochastic distributions with values in a separable Hilbert space, and apply this theory to the investigation of abstract Stochastic evolution equations with additive noise.
Abstract: We develop the theory of stochastic distributions with values in a separable Hilbert space, and apply this theory to the investigation of abstract stochastic evolution equations with additive noise.
TL;DR: In this article, the existence of the solution of a general infinite dimensional backward stochastic differential equation is discussed and generalization of the existence problem by a new approach is presented.
Abstract: The existence of the solution of a general infinite dimensional backward stochastic differential equation is discussed. In our setting, we generalize many works concerning the existence problem (by a new approach).
TL;DR: In this paper, generalized calculus and sdes with non-regular drift with non regular drift are discussed. But they do not consider the non-convexity of non-constant drift.
Abstract: (2002). Generalized calculus and sdes with non regular drift. Stochastics and Stochastic Reports: Vol. 72, No. 1-2, pp. 11-54.
TL;DR: In this article, a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control was studied and the existence and uniqueness of a viscosity solution was shown.
Abstract: We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inequations Variationnelles en Controle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.
TL;DR: In this paper, a sub-threshold signal is transmitted through a channel and may be detected when some noise, with known structure and proportional to some level, is added to the data.
Abstract: A subthreshold signal is transmitted through a channel and may be detected when some noise--with known structure and proportional to some level--is added to the data. There is an optimal noise level, called stochastic resonance that corresponds to the highest Fisher information in the problem of estimation of the signal. As noise we consider an ergodic diffusion process and the asymptotic is considered as time goes to infinity. We propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.
TL;DR: In this paper, limit theorems for the occupation times of one-dimensional symmetric stable processes are studied in the context of finite derivatives of the local time of stable processes.
Abstract: Dans cet article nous etudions des theoremes limites pour les temps locaux des processus stables symetriques en dimension un dans une classe d'espaces de Besov. Ce qui presente une generalisation des resultants de Fitzsimmons Getoor et celui de Rosen pour un processus stable dans l'espace des fonctions continues. Les processus obtenus a la limite sont les derivees fractionnaires et les transformes de Hilbert des temps locaux. This paper studies limit theorems for the occupation times of one-dimensional symmetric stable processes. Our goals is to extend Fitzsimmons and Getoor's results (Fitzsimmons, P.J., Getoor, R.K. "Limit theorems and variation properties for fractional derivatives of the local time of stable process," Ann. Inst. Henri Poincare , 28 (2) 1992, pp. 311-333) and those of Rosen (Rosen, J., "Second Order Limit Laws for the Local Times of Stable Processes", Seminaire de Probabilite XXV. Lecture Notes in Mathematics , Vol. 1485 (1990) pp. 407-425. Springer: New York), established in the unifor...
TL;DR: In this paper, the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions, was proved by using the equivalence result, provided by the Stochastic characteristics method.
Abstract: In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.
TL;DR: In this paper, a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall is considered, and a drift term b is added to the lifetime process.
Abstract: We consider a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall. More precisely we add a drift term b to the lifetime process, which may depends on...
TL;DR: In this paper, it was shown that the paths of the indefinite Skorohod integral with respect to the fractional Brownian motion with Hurst parameter less than 1/2 belong to the Besov space B p, X H, for any p >(1/H ).
Abstract: Using the techniques of the Malliavin calculus and the properties of Gaussian processes, we prove that the paths of the indefinite Skorohod integral with respect to the fractional Brownian motion (fBm) with Hurst parameter less than 1/2 belongs to the Besov space B p , X H , for any p >(1/ H ).
TL;DR: In this paper, the authors prove limit theorems for solutions of backward stochastic differential equations of the form Y t = + Z t T h (s, Y s, Z s ) d s + Z R (L T a (Y ) m L t a ( Y )) x (d a ) m Z t t Z s d W s for 0 h t h T.
Abstract: Given a d -dimensional Wiener process W , with its natural filtration { F t }, a F T -measurable random variable in R , a bounded measure x on R , and an adapted process ( s , y , z ) M h ( s , y , z ), we consider the following BSDE: Y t = + Z t T h ( s , Y s , Z s ) d s + Z R ( L T a ( Y ) m L t a ( Y )) x (d a ) m Z t T Z s d W s for 0 h t h T . Here L t a ( Y ) stands for the local time of Y at level a . For h =0, we establish the existence and the uniqueness of the processes ( Y , Z ), and if h is continuous with linear growth we establish the existence of a solution. We prove limit theorems for solutions of backward stochastic differential equations of the above form. Those limit theorems permit us to deduce that any solution of that equation is the limit, in a strong sense, of a sequence of semi-martingales, which are solutions of ordinary BSDEs of the form Y t = + Z t T f ( Y s ) Z s 2 d s m Z t T Z s d W s . A comparison theorem for BSDEs involving measures is discussed. As an application w...