Showing papers in "Stochastics and Stochastics Reports in 2003"
TL;DR: In this paper, a stochastic calculus for fractional Brownian motion with Hurst parameter H > 1/2 was developed using the techniques of the Malliavin calculus, and the authors established estimates in L p, maximal inequalities and a continuity criterion for the Stochastic integral.
Abstract: We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H > 1/2 using the techniques of the Malliavin calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Ito's formula for integral processes.
238 citations
TL;DR: In this paper, an optimal stopping time for a Brownian path is determined such that the path is as close as possible to its unknown ultimate maximum over a finite time interval, measured by a q-mean or by a probability distance.
Abstract: At time 0 start to observe a Brownian path. Based upon the information, which is continuously updated through the observation of the path, a stopping time is determined such that the path is as close as possible to its unknown ultimate maximum over a finite time interval. The closeness is measured by a q-mean or by a probability distance. This can be formulated as an optimal stopping problem. The method of proof relies upon a representation of a conditional expectation of the gain process and the principle of smooth fit (at a single point).
50 citations
TL;DR: In this article, the authors considered increasing self-similar Markov processes (X t, t \qeq 0 ) on ]0,∞[.0] with probability 1.
Abstract: We consider increasing self-similar Markov processes (X t , t \qeq 0 ) on ]0,∞[. By using the Lamperti's bijection between self-similar Markov processes and Levy processes, we determine the functions f for which there exists a constant c∈R+\{0} such that lim inf t→∞ X t /f(t)=c with probability 1. The determination of such functions depends on the subordinator ξ associated to X through the distribution of the Levy exponential functional and the Laplace exponent of ξ. We provide an analogous result for the self-similar Markov process associated to the opposite of a subordinator.
49 citations
TL;DR: In this article, the authors consider the diffusion scaling limit of the vicious walker model and derive two types of nonintersecting Brownian motions, in which the non-intersectings condition is imposed in a finite time interval (0,T] for the first type and in an infinite time interval(0, ∞ ) for the second type, respectively.
Abstract: We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval (0,T] for the first type and in an infinite time interval {\rm (0, \infty )} for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.
39 citations
TL;DR: In this paper, the authors examined the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure and showed that for convex payoffs, the option price is increasing in the jump risk parameter.
Abstract: In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.
32 citations
TL;DR: In this article, the problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered.
Abstract: The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J 1 and so the methods of approximation based on the J 1-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R+, R d ) introduced recently by Jakubowski. The S-topology is weaker than J 1 but stronger than the Meyer-Zheng topology and shares many useful properties with J 1.
31 citations
TL;DR: In this paper, the authors consider an infinite assets model driven by marked point process and Wiener process and show that the space of measure-valued strategies is not complete with respect to the semimartingale topology.
Abstract: Motivated by the theory of bond markets, we consider an infinite assets model driven by marked point process and Wiener process. The self-financed wealth processes are defined by using measure-valued strategies. Going further on the works of Bjork et al. [“Bond market structure in the presence of marked point processes”, Mathematical Finance, 7 (1997a) pp. 211–239; “Towards a general theory of bond markets”, Finance and Stochastics, 1 (1997b) pp. 141–174] who focus on the existence of martingale measures and market completeness questions, we study here the incompleteness case. Our main result is a predictable decomposition theorem for supermartingales in this infinite assets model context. The concept of approximate wealth processes is introduced, and we show in an example that the space of measure-valued strategies is not complete with respect to the semimartingale topology. As in the case of stock markets, one can then derive a dual representation of the super-replication cost and study the problem of u...
21 citations
TL;DR: By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening sample path large deviation principles for fractional Brownian motion to this topology, this paper obtained large deviation decay rates for steady-state tail probabilities of certain queueing systems in multidimensional heavy traffic models driven by fractional brownian motions.
Abstract: By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.
21 citations
TL;DR: In this article, an irreversible capacity expansion problem where the industry demand is described by a double exponential jump diffusion process is studied, and the goal is to maximize the discounted overall profit, net the cost of investing.
Abstract: This paper studies an irreversible capacity expansion problem where the industry demand is described by a double exponential jump diffusion process. Formally, we consider the following optimization problem over all adapted, non-decreasing process X = (X_{t}). Here we understand X t as the cumulative capital investment up to time t (it is non-decreasing since the investment is assumed to be irreversible). D = (D_{t}) is the exogenous industry demand process, which is modeled by a double exponential jump diffusion. The profit flow is assumed to have rate D \hspace{0.167em} \cdot \hspace{0.167em} H(X) for some concave function H, while k denotes the cost of unit capital. The goal is choosing a capital investment strategy X so as to maximize the discounted overall profit, net the cost of investing. We explicitly solve the problem when H is assumed to be the Cobb–Douglas production function, that is, H(x) = x^{ \mgreek{a} } for some \mgreek{a} \in (0,1). Throughout the paper, a slightly different (but essentia...
20 citations
TL;DR: In this paper, a class of linear parabolic stochastic boundary value problems of Wick-type is studied, and a numerical method based on a Galerkin type of approximation is discussed.
Abstract: A class of linear parabolic stochastic boundary value problems of Wick-type is studied. The equations are understood in a weak sense on a suitable stochastic distribution space, and existence and uniqueness results are provided. The paper continues to discuss a numerical method for this type of problem, based on a Galerkin type of approximation. Estimates showing linear convergence in time and space are derived, and rate of convergence results for the stochastic dimension are reported.
20 citations
TL;DR: In this paper, it was shown that multiplication with an indicator function can increase the corresponding norm of the fractional Clark-Ocone derivative with respect to a fractional Brownian motion with Hurst parameter 1/2 < H < 1.
Abstract: Integration with respect to a fractional Brownian motion with Hurst parameter 1/2 < H < 1 is related to the inner product: In this paper we provide an example, which shows that multiplication with an indicator function can increase the corresponding norm. We discuss the significance of this result for the quasi-conditional expectation and the fractional Clark-Ocone derivative introduced in Hu and Oksendal [“Fractional White Noise Calculus and Applications to Finance”, IDAQPRT, 6 (2003) 1–32]. Finally, we prove a new version of the fractional Clark-Ocone formula.
TL;DR: In this article, general results concerning the asymptotic behavior of the Robbins-Monro type stochastic differential equations are presented, and the rate of convergence of the solution Z = (Z{t}){t \qeq 0} as t \rightarrow \infty is established.
Abstract: General results concerning the asymptotic behaviour of the solution of the Robbins–Monro type stochastic differential equations are presented. In particular, the rate of convergence of the solution Z = (Z_{t})_{t \qeq 0} as t \rightarrow \infty is established. Moreover, it is shown that Z admits an asymptotic expansion which enables one to obtain the asymptotic distribution of the randomly normed solution from a martingale limit theorem.
TL;DR: In this paper, the existence of generalized solutions of the Navier-Stokes equation with additive noise was studied and path-wise uniqueness was proven for this kind of generalized solution.
Abstract: A bidimensional stochastic Navier–Stokes equation with an additive noise is considered. Known results on existence of generalized solutions are reviewed, in order to state an existence result in the most complete form. For this kind of generalized solutions, pathwise uniqueness is proven.
TL;DR: In this paper, the authors considered stochastic partial differential equations driven by space-time white noise in high dimensions and proved that the law of the solution admits a density with respect to Lebesgue measure.
Abstract: In this paper, we consider stochastic partial differential equations driven by space-time white noise in high dimensions. We prove, under reasonable conditions, that the law of the solution admits a density with respect to Lebesgue measure. The stability of the equation, as the higher order differential operator tends to zero, is also studied in the paper.
TL;DR: In this article, the authors apply the WKB method to obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients.
Abstract: The paper is devoted to the study of stochastic heat equations driven by Levy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.
TL;DR: In this paper, the authors prove the existence of the Cramer type asymptotics for the probabilities of large deviations from an averaged motion, which implies the central limit theorem.
Abstract: This paper is devoted to the averaging principle for stochastic systems with slow and intermixing fast motions. Here we (i) prove the existence of the Cramer type asymptotics for the probabilities of large deviations from an averaged motion, which implies the central limit theorem, and (ii) develop an analytic procedure for computation of this asymptotics. We use general apparatus of superregular perturbations of fiber ergodic semigroups to investigate two systems in the same way. The first of them is a cascade in which slow motions are determined by a vector field depending both on slow and fast variables, and fast motions compose a Markov chain depending on the slow variable. The second is a process defined by a system of two stochastic differential equations.
TL;DR: In this paper, a semigroup of conditional expectation operators, generated by a stochastic system with slow and intermixing fast motions, in which the slow motions have a speed of order e.
Abstract: In this paper, we investigate a semigroup of conditional expectation operators A_{ \epsilon }^{t}, generated by a stochastic system with slow and intermixing fast motions, in which the slow motions have a speed of order e. This semigroup, considered as perturbation of A_{0}^{t}, possess a number of unexpected properties; the most important of them is superregularity. First we study these properties. Then we construct an asymptotic expansion for the family A_{ \epsilon }^{t/ \epsilon }e^{ \mgreek{x} F/ \epsilon } by powers of ξ, e, where ξ is a small complex parameter and F is a function of the slow variable. We reveal a new non-trivial phenomenon: each coefficient of the last expansion appears as a sum of four terms of different types. This expansion gives a powerful tool for proving some probability limit theorems for the slow motion behavior over the time periods of order e -1.
TL;DR: In this article, the maximal inequalities c p, n log 1 n +1 (1+ J ) p h A X * A p h C p, n log n + 1 n+1 ( 1+ J ǫ +1) p (0 < p < n +2) hold for all stopping times for nonnegative continuous functions.
Abstract: Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , both are continuous on R and ( x )>0 if x p 0. Denote X * =sup 0 h t h | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n 2 ( x ) h | w ( x )| h K 2 | x | n 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J ) p h A X * A p h C p , n log 1 n +1 (1+ J ) p (0< p < n +1) hold for all stopping times .
TL;DR: In this paper, the problem of ergodic control of reflecting diffusion in a compact domain is analyzed under the condition of partial degeneracy, i.e., when its transition kernel after some time is absolutely continuous with respect to the Lebesgue measure on a part of the state space.
Abstract: The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect to the Lebesgue measure on a part of the state space. Existence of a value function and a "martingale dynamic programming principle" are established by mapping the problem to a discrete time control problem. Implications for existence of optimal controls are derived.
TL;DR: In this article, a new characterization of the Snell envelope of a given process as the unique solution of the stochastic variational inequality (SVI) is given, which leads to several a priori estimates for Snell envelopes and their components.
Abstract: We give a new characterization of the Snell envelope of a given process as the unique solution of the stochastic variational inequality (SVI) in this article. This approach leads to several a priori estimates for the Snell envelopes and their components. The valuation for American Contingent Claims (ACC) in general financial market model is considered as an application. The robustness of the optimal portfolio/consumption processes with respect to the payoff function is established.
TL;DR: In this article, it was shown that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as n \rightarrow \infty.
Abstract: The circular ensembles of Dyson satisfy isoperimetric inequalities and concentration of measure phenomena for large particle numbers analogous to the isoperimetric inequality for surface measure on the sphere in Euclidean space of high dimension. This leads to a geometrical proof of a result of Johansson [Bull. Sci. Math. (2) 112, (1988), 257–304] that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as n \rightarrow \infty .
TL;DR: In this paper, a scale invariant diffusion whose state space is a closed cone in R d, minus the vertex is considered, and the diffusion process is either recurrent, transient to ∞ or transient to the vertex of the cone.
Abstract: Consider a scale invariant diffusion whose state space is a closed cone in R d , minus the vertex. Then the process is either recurrent, transient to ∞ or transient to the vertex of the cone. In the latter case, the diffusion has finite lifetime (a.s.) and converges to the vertex at the lifetime. The Martin boundary consists of two points, and the corresponding minimal harmonic functions are of the form 1 and |x| α ψ(x/|x|).