On Stochastic Differential Equation and Modified Black-Scholes Option Pricing Model

General population growth models with Allee effects in a random environment

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions.

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at $\varepsilon \to 0$ were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al . ( Basak et al . 1999 J. Math. Anal. Appl. 202 , 604–622) and Yuan et al . ( Yuan et al . 2003 Syst. Control Lett. 50 , 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa ( Ichikawa 1982 J. Math. Anal. Appl. 90 , 12–44) and Luo & Liu ( Luo & Liu 2008 Stoch. Proc. Appl. 118 , 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

We present a novel approach to the mean square exponential stability of stochastic delay differential equations. Consequently, some new explicit criteria for the mean square exponential stability of general nonlinear stochastic delay differential equations are derived. Both delay-independent and delay-dependent stability criteria are given, and illustrative examples are also provided.

Fractional Brownian Motions, Fractional Noises and Applications

Previous article Next article A Simple Proof of the Existence of a Solution of Itô’s Equation with Monotone CoefficientsN. V. KrylovN. V. Krylovhttps://doi.org/10.1137/1135082PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. V. Krylov and , B. L. Rozovskii, Stochastic evolution equationsCurrent problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71–147, 256 81m:60116 Google Scholar[2] I. Gyongy and , N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1–21 82j:60104 CrossrefGoogle Scholar[3] N. V. Krylov, Extremal properties of solutions of stochastic equations, Theory Probab. Appl., 29 (1984), 205–217 10.1137/1129033 0557.60042 LinkGoogle Scholar[4] L. A. Alyushina, Euler polygonal lines for Itô equations with monotone coefficients, Theory Probab. Appl., 32 (1987), 340–346 10.1137/1132046 0663.60049 LinkGoogle Scholar[5] Gisirō Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2), 4 (1955), 48–90 17,166f 0053.40901 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics | 1 Apr 2022 Cross Ref Approximation of BSDEs with super-linearly growing generators by Euler’s polygonal line method: A simple proof of the existenceSystems & Control Letters, Vol. 153 | 1 Jul 2021 Cross Ref Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General CaseJournal of Mathematical Fluid Mechanics, Vol. 22, No. 4 | 4 September 2020 Cross Ref A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditionsInternational Journal of Computer Mathematics, Vol. 5 | 9 March 2020 Cross Ref Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility ModelsAndrei Cozma and Christoph ReisingerSIAM Journal on Numerical Analysis, Vol. 56, No. 6 | 11 December 2018AbstractPDF (1095 KB)Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded DomainsJournal of Mathematical Fluid Mechanics, Vol. 18, No. 1 | 22 October 2015 Cross Ref An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz CoefficientsJean-François Chassagneux, Antoine Jacquier, and Ivo MihaylovSIAM Journal on Financial Mathematics, Vol. 7, No. 1 | 14 December 2016AbstractPDF (602 KB)A New Type of Stability Theorem for Stochastic Systems With Application to Stochastic StabilizationIEEE Transactions on Automatic Control, Vol. 61, No. 1 | 1 Jan 2016 Cross Ref Loss of regularity for Kolmogorov equationsThe Annals of Probability, Vol. 43, No. 2 | 1 Mar 2015 Cross Ref Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz ConditionStochastic Analysis and Applications, Vol. 32, No. 2 | 28 February 2014 Cross Ref A Note on Euler Approximations for Stochastic Differential Equations with DelayApplied Mathematics & Optimization, Vol. 68, No. 3 | 24 August 2013 Cross Ref Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equationsThe Annals of Applied Probability, Vol. 23, No. 5 | 1 Oct 2013 Cross Ref Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficientsActa Mathematica Sinica, English Series, Vol. 29, No. 8 | 15 July 2013 Cross Ref Stochastic Itô inclusion with upper separated multifunctionsJournal of Mathematical Analysis and Applications, Vol. 400, No. 2 | 1 Apr 2013 Cross Ref Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficientsThe Annals of Applied Probability, Vol. 22, No. 4 | 1 Aug 2012 Cross Ref Stochastic 3D tamed Navier–Stokes equations: Existence, uniqueness and small time large deviation principlesJournal of Differential Equations, Vol. 252, No. 1 | 1 Jan 2012 Cross Ref On the Strong Solution for the 3D Stochastic Leray-Alpha ModelBoundary Value Problems, Vol. 2010, No. 1 | 1 Jan 2010 Cross Ref ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTSStochastics and Dynamics, Vol. 09, No. 04 | 21 November 2011 Cross Ref Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicityProbability Theory and Related Fields, Vol. 145, No. 1-2 | 19 July 2008 Cross Ref Large deviations in the Langevin dynamics of a random field Ising modelStochastic Processes and their Applications, Vol. 105, No. 2 | 1 Jun 2003 Cross Ref Viable solutions of set-valued stochastic equationOptimization, Vol. 48, No. 2 | 1 Jan 2000 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications411-623 History Submitted:05 September 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135082Article page range:pp. 583-587ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Optimal Control of Stochastic Systems

Under a non-Lipschitz condition, the averaging principle for the general stochastic differential delay equations (SDDEs) is established. The solutions of convergence in mean square and convergence in probability between standard SDDEs and the corresponding averaged SDDEs are considered.MSC:34K50, 34C29.

By a novel approach, we present several new criteria for the mean square exponential stability of general non-linear stochastic delay differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order $1$ of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.

This paper investigates the existence and uniqueness of mild solutions to the general nonlinear stochastic impulsive differential equations. By using Schaefer′s fixed theorem and stochastic analysis technique, we propose sufficient conditions on existence and uniqueness of solution for stochastic differential equations with impulses. An example is also discussed to illustrate the effectiveness of the obtained results.