Showing papers in "Stochastic Analysis and Applications in 2005"
TL;DR: In this paper, a general Khasminskii-type test for nonlinear SDDEs is proposed, which covers a wide class of highly nonlinear nonlinear SDEs, including the stochastic delay power logistic model.
Abstract: The classical Khasminskii theorem (see [6]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied.
104 citations
TL;DR: In this paper, a class of backward doubly stochastic differential equations (BDSDEs) is studied and a comparison theorem of these BDSDEs is derived for continuous coefficients.
Abstract: A class of backward doubly stochastic differential equations (BDSDEs for short) are studied. We obtain a comparison theorem of these BDSDEs. As one of its applications, we derive the existence of solutions for BDSDEs with continuous coefficients.
96 citations
TL;DR: In this article, the authors consider a stock with price process defined by a stochastic differential equation driven by a process Y(t) different from Brownian motion, and they show that the process has a good MA(∞)-type representation.
Abstract: This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process Y(t) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y(t) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show that the process Y(t) has a good MA(∞)-type representation. The existence of such simultaneous good AR(∞) and MA(∞) representations enables us to apply a new method for the calculation of relevant conditional expectations, whence to obtain various explicit results for problems such as portfolio optimization. The financial market defined by the above stock price process is complete, and if the coefficients are constant, then the prices of European calls and puts are given by the Black-Scholes formulas as in the Black-Scholes model. Unlike the latter, however, the model allows for differences between the historical and implied volatilities. The model includes a special case in which only two additional parameters are introduced to describe the memory of the market, compared with the Black-Scholes model. Analysis based on real market data shows that this simple model with two additional parameters is more realistic in capturing the memory effect of the market, while retaining the simplicity and usefulness of the Black-Scholes model.
76 citations
TL;DR: In this article, a direct definition of stochastic integrals for deterministic Banach valued functions on separable Banach spaces with respect to compensated Poisson random measures is given.
Abstract: A direct definition of stochastic integrals for deterministic Banach valued functions on separable Banach spaces with respect to compensated Poisson random measures is given. This definition yields...
68 citations
TL;DR: This approach allows us to prove, in some cases, when the signal is nonergodic, the stability of the optimal filter in mean over the observations and the uniform convergence in meanover the observations of a special interacting particle filter to the optimalfilter.
Abstract: In this paper, we propose a new approach to study the stability of the optimal filter w.r.t. its initial condition in introducing a “robust filter, ” which approximates the optimal filter uniformly in time. This approach allows us to prove, in some cases, when the signal is nonergodic, the stability of the optimal filter in mean over the observations and the uniform convergence in mean over the observations of a special interacting particle filter to the optimal filter.
64 citations
TL;DR: In this article, a sufficient maximum principle for the optimal control of systems described by a quasilinear stochastic heat equation was proved for the problem of optimal harvesting.
Abstract: We prove a sufficient maximum principle for the optimal control of systems described by a quasilinear stochastic heat equation. The result is applied to solve a problem of optimal harvesting from a system described by a stochastic reaction-diffusion equation.
60 citations
TL;DR: In this article, an explicit solution to the Merton's portfolio optimization problem with the Schwartz mean-reversion dynamics is presented, and the optimal investment strategy is also given explicitly.
Abstract: We analyze the classical Merton's portfolio optimization prob- lem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman equation is a two-dimensional nonlinear parabolic partial differential equation. We produce an explicit solution to this equa- tion by reducing it to a simpler one-dimensional linear parabolic equation. This reduction is achieved through a Cole-Hopf type transformation, recently introduced in portfolio optimization theory by Zariphopoulou (9). A verifi- cation argument is then used to prove that this solution coincides with the value function of the control problem. The optimal investment strategy is also given explicitly. On the technical side, the main problem we are facing here is the necessity to identify conditions on the parameters of the control problem ensuring uniform integrability of a family of random variables that roughly speaking are the exponentials of squared Wiener integrals. We consider the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process. This process was sug- gested by Schwartz (8) as a reasonable model for the spot price evolution of com- modities like gold or coffee, and energies like oil, gas, and electricity (see also Clewlow and Strickland (2) or Pilipovic (6)), but could also be a model for the price of a bond or the interest-rate dynamics. In this paper we analyze the optimal in- vestment problem for an agent in a financial market where the risky asset follows the price dynamics of Schwartz, and the risk preferences is described by a power utility (also known as HARA utility). This portfolio optimization problem is a variant of the classical Merton problem (3). However, using the Schwartz dynamics rather than the classical geometric Brownian motion to model the risky asset price leads to some interesting problems that needs to be overcome in order to rigorously derive a solution. We shall use the dynamic programming approach to derive an optimal investment strategy and the indirect utility. The main ingredient in the solution approach will be a transformation of Cole-Hopf type which reduces the dimension and removes the nonlinearity of the related Hamilton-Jacobi-Bellman (HJB henceforth) equa- tion, enabling us to find an explicit solution. This transformation was introduced
58 citations
TL;DR: Convergence rates of adaptive algorithms for weak approximations of Itoˆ stochastic differential equations are proved for the Monte Carlo Euler method and both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm.
Abstract: Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil optimal stocha ...
48 citations
TL;DR: In this article, the M|G|1 retrial queue with nonpersistent customers and orbital search is considered, and the structure of the busy period and its analysis in terms of Laplace transform is discussed.
Abstract: The M|G|1 retrial queue with nonpersistent customers and orbital search is considered. If the server is busy at the time of arrival of a primary customer, then with probability 1 − H 1 it leaves the system without service, and with probability H 1 > 0, it enters into an orbit. Similarly, if the server is occupied at the time of arrival of an orbital customer, with probability 1 − H 2, it leaves the system without service, and with probability H 2 > 0, it goes back to the orbit. Immediately after the completion of each service, the server searches for customers in the orbit with probability p > 0, and remains idle with probability 1 − p. Search time is assumed to be negligible. In the case H 2 = 1, the model is analyzed in full detail using the supplementary variable method. The joint distribution of the server state and the orbit length in steady state is studied. The structure of the busy period and its analysis in terms of Laplace transform is discussed. We also provide a direct method of calcu...
42 citations
TL;DR: In this paper, the authors developed a prediction theory for a class of processes with stationary increments, and proved an explicit representation of the innovation processes associated with the stationary increments processes, and applied the representation to obtain a closed-form solution to the problem of expected logarithmic utility maximization for the financial markets with memory.
Abstract: We develop a prediction theory for a class of processes with stationary increments. In particular, we prove a prediction formula for these processes from a finite segment of the past. Using the formula, we prove an explicit representation of the innovation processes associated with the stationary increments processes. We apply the representation to obtain a closed-form solution to the problem of expected logarithmic utility maximization for the financial markets with memory introduced by the first and second authors.
42 citations
TL;DR: In this article, the authors address a dynamic portfolio optimization problem where the expected utility from terminal wealth has to be maximized, where the special feature of this paper is an additional constraint on the portfolio strategy modeling bounded shortfall risks, which are measured by value at risk or expected loss.
Abstract: We address a dynamic portfolio optimization problem where the expected utility from terminal wealth has to be maximized. The special feature of this paper is an additional constraint on the portfolio strategy modeling bounded shortfall risks, which are measured by value at risk or expected loss. Using a continuous-time model of a complete financial market and applying martingale methods, analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Finally, some numerical results are presented.
TL;DR: The Wiener-Itoˆ chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters was studied in this article.
Abstract: We give the Wiener–Itoˆ chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.
TL;DR: In this paper, a Meyer-Tanaka formula involving weighted local time was derived for fractional Brownian motion and geometric fractional brownian motion for the stop-loss-start-gain portfolio in a fractional Black-Scholes market.
Abstract: A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the European call and put option prices into their intrinsic and time values.
TL;DR: In this paper, the existence of mild solutions of a class of non-linear neutral stochastic differential inclusions in Hilbert space is studied by using a new fixed point theorem for a condensing map due to Martelli.
Abstract: In this paper, the existence of mild solutions of a class of non-linear neutral stochastic differential inclusions in Hilbert space is studied. The results are obtained by using a new fixed point theorem for a condensing map due to Martelli. For an application of the result, the neutral stochastic reaction-diffusion inclusion is also discussed.
TL;DR: In this article, a transient solution for the system size in the M/M/1 queueing system with the possibility of catastrophes and server failures is analyzed, and the steady state probabilities of system size are also derived.
Abstract: A transient solution for the system size in the M/M/1 queueing system with the possibility of catastrophes and server failures is analyzed. The steady state probabilities of the system size are also derived. Some important performance measures are discussed. Finally, the reliability and availability of the system are obtained.
TL;DR: The aim is to relate frequently made assumption on the covariance operator Q to assumptions on the correlation function of the noise process, which is the kernel of Q, when the Wiener process is given as a series expansion in terms of eigenfunctions of the Laplacian.
Abstract: In this article, we comment on the well known connection between noise and Q-Wiener processes. In physical applications, noise is usually given as a generalized Gaussian process and all assumptions are formulated in terms of the correlation functional. In contrast, the mathematical theory of stochastic partial differential equations on bounded domains is in many cases formulated with Q-Wiener processes. Our aim is to relate frequently made assumptions on the covariance operator Q to assumptions on the correlation function of the noise process, which is the kernel of Q. One of our main results gives necessary and sufficient conditions when the Wiener process is given as a series expansion in terms of eigenfunctions of the Laplacian.
TL;DR: In this paper, the existence and uniqueness of the solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process were proved.
Abstract: In this paper, by using a penalization as well as a fixed point methods, we prove existence and uniqueness of the solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process.
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 problem of hedging contingent claims in the framework of a two factors jump-diffusion model under initial budget constraint is studied and the authors give explicit formulas for the so called efficient hedging.
Abstract: This paper is devoted to the problem of hedging contingent claims in the framework of a two factors jump-diffusion model under initial budget constraint We give explicit formulas for the so called efficient hedging These results are applied for the pricing of equity linked-life insurance contracts
TL;DR: In this article, the role played by the survival function and the corresponding hazard rate with respect to capture of a point by the so-called crystalline phase were revisited by classical methods of survival analysis.
Abstract: Many real phenomena, including phase changes, such as crystallization processes, tumor growth, forest growth, etc., may be modelled as stochastic birth-and-growth processes, in which crystals develop from points (nuclei) that are born at random both in space and time. In this paper, we revisit these processes by classical methods of survival analysis with specific reference to the role played by the survival function and the corresponding hazard rate with respect to capture of a point by the so called crystalline phase. General expressions for the hazard and survival functions associated with a point are provided. Known results for Poisson type processes follow as particular cases. Further, a link between hazard functions and contact distribution function of stochastic geometry is also obtained.
TL;DR: In this article, the authors established a combinatorial central limit theorem for an array of independent random variables (X ij ), where X ij is a suitable normalization of the strandrad normal distribution.
Abstract: This paper establishes a combinatorial central limit theorem for an array of independent random variables (X ij ), 1 ≤ i, j ≤ n, (n → ∞) with finite third moments Let π = (π(1), π(2), …, π(n)) be a permutation of {1, 2, …, n}, and define W n = ∑ i X iπ(i) Then the authors prove the following uniform central limit property: , where F n is the distribution of is the strandrad normal distribution, and with [Xcirc] ij is a suitable normalization of X ij The proof uses Stein's method and the result generalizes and improves a number of known results
TL;DR: In this article, the authors show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process under few technical assumptions and allowing for the absence of an equivalent martingale measure.
Abstract: Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market prices of risk as markets become approximately complete and consider the limits of traded securities, characterizing explicitly the growth optimal portfolio and investigating arbitrage and diversification in such markets.
TL;DR: In this paper, the authors studied discrete-time Markov decision processes with average expected costs (AEC) and discount-sensitive criteria in Borel state and action spaces, and proposed another set of conditions on the system's primitive data, and under which they proved (1) AEC optimality and strong − − 1-discount optimality are equivalent; (2) a condition equivalent to strong 0discount optimal stationary policies; and (3) the existence of strong n (n−− 1, 0)-discount policy.
Abstract: In this paper we study discrete-time Markov decision processes with average expected costs (AEC) and discount-sensitive criteria in Borel state and action spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions on the system's primitive data, and under which we prove (1) AEC optimality and strong − 1-discount optimality are equivalent; (2) a condition equivalent to strong 0-discount optimal stationary policies; and (3) the existence of strong n (n = −1, 0)-discount optimal stationary policies. Our conditions are weaker than those in the previous literature. In particular, the “stochastic monotonicity condition” in this paper has been first used to study strong n (n = −1, 0)-discount optimality. Moreover, we provide a new approach to prove the existence of strong 0-discount optimal stationary policies. It should be noted that our way is slightly different from those in the previous literature. Finally, we apply our results to an inventory system and a contro...
TL;DR: In this paper, an adaptive numerical scheme that reproduces the explosive behavior was proposed to reproduce the explosion of the solutions of stochastic ODEs under adequate hypotheses, and the time step was adapted according to the size of the computed solution in such a way that the explosion was reproduced.
Abstract: Stochastic ordinary differential equations may have solutions that explode in finite or infinite time. In this article we design an adaptive numerical scheme that reproduces the explosive behavior. The time step is adapted according to the size of the computed solution in such a way that, under adequate hypotheses, the explosion of the solutions is reproduced.
TL;DR: In this article, the authors introduce a concept of detectability for discrete-time infinite Markov jump linear systems that relates the stochastic convergence of the output with the stochiastic convergence in the state of the system.
Abstract: This paper introduces a concept of detectability for discrete-time infinite Markov jump linear systems that relates the stochastic convergence of the output with the stochastic convergence of the state. It is shown that the new concept generalizes a known stochastic detectability concept and, in the finite dimension scenario, it is reduced to the weak detectability concept. It is also shown that the detectability concept proposed here retrieves the well-known property of linear deterministic systems that observability is stricter than detectability.
TL;DR: In this article, the authors studied the queue length of the M X /G/1 queue under D-policy and derived the mean queue length, which is then used to derive the PGF at an arbitrary point of time.
Abstract: We study the queue length of the M X /G/1 queue under D-policy. We derive the queue length PGF at an arbitrary point of time. Then, we derive the mean queue length. As special cases, M/G/1, M X /M/1, and M/M/1 queue under D-policy are investigated. Finally, the effects of employing D-policy are discussed.
TL;DR: In this paper, a unified approach for studying block-structured fluid models is proposed by means of the RG-factorization, where the stochastic environment (or background) is assumed to be a quasi-birth-and death (QBD) process, with either infinitely many levels or finitely many levels.
Abstract: In this paper, a unified approach for studying block-structured fluid models is proposed by means of the RG-factorization. When the stochastic environment (or background) is assumed to be a quasi-birth-and death (QBD) process, with either infinitely many levels or finitely many levels, the Laplace transform for the stationary probability distribution of the buffer content is expressed in terms of the R-measure. At the same time, the Laplace-Stieltjes transforms for both the conditional distribution and the conditional mean of a first passage time in such a fluid queue are derived by the same approach.
TL;DR: In this paper, the authors prove the existence of a weak mild solution to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space where W is a Wiener process, A is a linear operator that generates a C 0-semigroup, F and G are multifunctions with convex compact values satisfying some growth condition.
Abstract: We prove the existence of a weak mild solution to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space where W is a Wiener process, A is a linear operator that generates a C 0-semigroup, F and G are multifunctions with convex compact values satisfying some growth condition, and with respect to the second variable, a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures.
TL;DR: A limiting problem for a stochastic evolution equation is studied in this article, where it is shown that, in the limit, the mild solution to the evolution equation tends to the solution of an ordinary Ito equation.
Abstract: A limiting problem for a stochastic evolution equation is studied in the paper. In the equation, the linear operator is non-positive with a pure point spectrum, the non-linearity is monotone, and the Brownian motion is cylindrical. It is shown that, in the limit, the mild solution to the evolution equation tends to the solution of an ordinary Ito equation.