Fluctuations in statistical mechanics are usually modeled by adding a **stochastic** term to the de- terministic **differential** equation. By doing this one obtains what is called **stochastic** **differential** **equations** (SDEs), and the term **stochastic** called noise [1]. Then, a **SDE** is a **differential** equation in which one or more of the terms is a **stochastic** process, and resulting in a solution which is itself a **stochastic** process. Every unwanted signal that adds to the information called noise. Noise in dynamical system is usually considered a nuisance. Noise has the most important role in the **SDE** [2].

follows) to be the **stochastic** processes and also because our control must be partial information adapted, this problem is not of Markovian type and hence cannot be solved by dynamic programming even if the mean term were not present. We instead investigate the maximum principle, and will derive an explicit form for the adjoint process. The approach we employ is Malliavin calculus which enables us to express the duality involved via the Malliavin derivative. Our paper is related to the recent paper [6] and [7]. In [6], they consider a mean-field type **stochastic** control problem where the dynamics is governed by a controlled forward **SDE** with jumps and the information available to the controller is possibly less than the overall information. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed. [7] presents various versions of the maximum principle for optimal control (not mean-field type) of forward-backward **stochastic** **differential** **equations** with jumps and a Malliavin calculus approach which allow us to handle non-Markovian system. The motivation of [7] is risk minimization via g -expectation.

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Diffusions in the generalized sense were studied by several authors begin- ning with, at least in our knowledge [20]; later on, many authors considered special cases of **stochastic** **differential** **equations** with generalized coefficients, it is difficult to quote them all: in particular, we refer to the case when b is a measure, [4, 7, 18, 22]. [4] has even considered the case when b is a not nec- essarily locally finite signed measure and the process is a possibly exploding semimartingale. In all these cases solutions were semimartingales. In fact, [8] considered special cases of non-semimartingales solving **stochastic** differ- ential **equations** with generalized drift; those cases include examples coming from Bessel processes.

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**equations** are used as in the study of queues, insurance risks, dams and more re- cently in mathematical finance. On the other hand, some recent research in auto- matic control such as Boukas and Liu (2002) and Ji and Chizeck (1990) have been devoted to **stochastic** **differential** **equations** with Markovian jumps. As a popular and important topic, the stability property of **stochastic** **differential** **equations** has always lain at the center of our understanding concerning **stochastic** models de- scribed by these **equations**. Dong and Xu (2007) proved the global existence and uniqueness of the strong, weak and mild solutions and the existence of invariant measure for one-dimensional Burgers equation in [0, 1] with a random perturba- tion of the body forces in the form of Poisson and Brownian motion. Later the uniqueness of invariant measure is given in Dong (2008). R¨ ockner and Zhang (2007) established the existence and uniqueness for solutions of **stochastic** evolu- tion **equations** driven both by Brownian motion and by Poisson point processes via successive approximations. In addition, a large deviation principle is obtained for **stochastic** evolution **equations** driven by additive L´ evy noise. Svishchuk and Kazmerchuk (2002) made a first attempt to study the pth-moment exponential stability of solutions of linear Itˆ o **stochastic** delay **differential** **equations** asso- ciated with Poisson jumps and Markovian switching, which was motivated by some practical applications in mathematical finance. Quite recently, Luo and Liu (2008) considered a strong solutions approximation approach for mild solutions of **stochastic** functional **differential** **equations** with Markovian switching driven by L´ evy martingales in Hilbert space. In addition, the sufficient conditions for the moment exponential stability and almost sure exponential stability of **equations** have been established by the Razumikhin-Lyapunov type function methods and comparison principles.

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In mathematical neuroscience, **stochastic** **differential** **equations** (**SDE**) have been uti- lized to model **stochastic** phenomena that range in scale from molecular transport in neurons, to neuronal firing, to networks of coupled neurons, to cognitive phenom- ena such as decision making [1]. Generally these SDEs are impossible to solve in closed form and must be tackled approximately using methods that include eigen- function decompositions, WKB expansions, and variational methods in the Langevin or Fokker–Planck formalisms [2–4]. Often knowing what method to use is not obvi- ous and their application can be unwieldy, especially in higher dimensions. Here we demonstrate how methods adapted from statistical field theory can provide a unifying framework to produce perturbative approximations to the moments of SDEs [5–13]. Any **stochastic** and even deterministic system can be expressed in terms of a path integral for which asymptotic methods can be systematically applied. Often of inter- est are the moments of x(t ) or the probability density function p(x, t ). Path integral methods provide a convenient tool to compute quantities such as moments and tran- sition probabilities perturbatively. They also make renormalization group methods

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**Stochastic** **differential** **equations** (SDEs) driven by Brownian motion W and an addi- tional deterministic path η of low regularity (so called “mixed SDEs”) have been well-studied. In (Guerra and Nualart 2008), the well-posedness of such SDEs is established if η has finite q-variation with q ∈ [ 1 , 2 ) . 1 The integral with respect to the latter is handled via fractional calculus. Independently, in (Diehl 2012) the same problem is studied using Young integration for the integral with respect to η . Interestingly, both approaches need to establish (unique) existence of solutions via the Yamada–Watanabe theorem. A direct proof using a contraction argument is not obvious to implement.

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Abstract -- Quantum **stochastic** **differential** **equations** (QSDEs) of systems that exhibit discontinuity are introduced with the Kurzweil **equations** associated with this class of **equations**. The formulations are simple extensions of the methods applied by Schwabik [10] to ODEs to this present noncommutative quantum setting. Here the solutions of a QSDE are discontinuous functions of bounded variation that is they have the same properties as the Kurzweil **equations** associated with QSDEs introduced in [1].

For simplicity’s sake, reference is made to a scalar **SDE** in which the excitation is a Gaussian white noise process. Writing down its solution, integrals involving the Brownian motion appear: this process is the parent of the Gaussian white noise as the latter is the derivative in the sense of the mathematical distributions of the former. If the Gaussian white noise acts externally (or additively), these integrals have a unique value. Unfortunately, in the case of a parametric (multip- licative) Gaussian white noise excitation the integral has infinite values depend- ing on the position of the point in the discretization interval. Itô chose the infe- rior point of the interval [8], while Stratonovich chose the midpoint [9]. In the so-called kinetic interpretation the superior point is chosen [10] [11]

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We now compare the accuracy and efficiency of the bridging methods discussed in the previous sections, by using them to make proposals inside a Metropolis–Hastings independence sampler. We consider three examples: a simple birth–death model in which the ODEs governing the LNA are tractable, a Lotka–Volterra system in which the use of numerical solvers are required, and a model of aphid growth inspired by real data taken from Matis et al. (2008). Generating discrete- time realisations from the **SDE** model of aphid growth is particularly challenging due to nonlinear dynamics, and an observation regime in which only one component is observed and is subject to additive Gaussian noise.

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More recently, in [21, 22, 23] we examine mathematically the link of SDEs of mean-reversion type arising from modeling collateralized debt obligations (CDOs) and credit risk to Burgers equation in the spirit of [8, 9] and we ex- plore certain computational advantages for the associated Burgers equation. Remark 1.3 The **SDE** (1.6) considered in [9] is a special case of our **SDE** (1.1) with

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In this paper we have studied the Euler and Milstein schemes which are obtained from the truncated Ito-Taylor expansion already proposed in [7]. Then we implemented these schemes to a nonlinear **stochastic** diﬀerential equation for comparing the EM and Milstein methods to each other while illustrating eﬃciency. Moreover, we calculated estimation values for Euler-Maruyama and Milstein methods so as to analyze similarities between the exact solution and numerical approximations. Then we investigated approximations for 2 9 , 2 10 , 2 11 , 2 12 and 2 13 discretization in the interval [0, 1] with 10,000 diﬀerent sample

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Various aspects of the study of deterministic (Hamiltonian) and **stochastic** evo- lution of configurations γ ∈ Γ( X ) have been discussed by many authors, see e.g. [23, 18, 5, 3, 17] and references given there. It is anticipated that (some of) these results can be combined with the approach proposed in the present paper allowing to build **stochastic** dynamics on the marked configuration space Γ( X , S).

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**Stochastic** di ﬀ erential **equations** (SDEs) are an essential tool to describe the randomness of a dynamic system. For example, physicists use this tool to model the time evolution of particles due to thermal fluctuations (Sobczyk, 2001) and ecologists study two interacting populations such as predator and prey by SDEs (Allen, 2007). In the financial system, many di ﬀ erent SDEs have been developed to model a particular financial product or class of products, e.g. geometric Brownian motion (GBM) by Osborne (1959) for modeling stock prices or stock indices, and for modeling interest rates the Vasicek model (Vasicek, 1977), the CIR model (Cox et al., 1985) and the CKLS model (Chan et al., 1992). Financial institutions make use of SDEs to price their derivatives or measure the risks of their portfolios. For example, when pricing financial products, one needs to specify the form of SDEs driving the appropriate randomness and then estimate the parameters of interest in the equation to generate future scenarios; when measuring risks, one needs to calculate shocks based on these generated scenarios to obtain Value-at-Risk or Expected Shortfall. Therefore, this presumption of functional forms is always considered as a parametric method.

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Abstract. This paper shows that penalized backward **stochastic** **differential** equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both op- timal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of **equations**, namely multidimensional reflected BSDE and reflected BSDE with a constraint on the hedging part, and gives **stochastic** control representations for their corresponding penalized **equations**.

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while the linear growth condition (which can in fact be weakened a bit by allowing additional logarithmic terms) allows us to pass from the existence of a local so- lution to that of a global solution by a Gronwall’s lemma procedure. SDEs which have a global strong solution for each initial condition are said to be complete or weakly complete. A complete **SDE** need not have a continuous modification of the solution as a function of time and the initial data. This marks a departure of the theory of **stochastic** flows from that of deterministic ordinary **differential** **equations**. However there is so far only a pitifully small number of examples of complete **stochastic** **differential** **equations** whose solutions do not admit a continu- ous modification as a function of time and initial data. Not a single such example has coefficients which are locally Lipschitz and of linear growth (in spite of a re- mark in [8] stating the contrary). The basic example is dx t = dW t on R 2 \ { 0 } for a

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To treat this problem, we introduce a new type of **stochastic** **differential** **equations** with Markovian switching (SDEwMSs) with **stochastic** coefficients f, g while the classical f , g are deterministic. In section 2 the existence and uniqueness of the solution to the new equation and the comparison theorem will be given. In section 3 we will get comparison theorem of 1-dimensional classical SDDEwMSs and the approach can also be applied to comparison theorem of 1-dimensional new SDDEwMSs corresponding to the **equations** in Section 2.

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The dynamic nature of time-series of prices often follows **stochastic** and chaos behaviors. Considering random fluctuations in stock prices, using **stochastic** **differential** **equations** can be an efficient workaround for the modeling and prediction of the economic time-series. These models were first introduced to the economy literature by the works of Black & Scholes [19], and Merton [20, 21] which were dedicated to the modeling of stock prices in terms of a geometrical motion **stochastic** **differential** **equations**. In this model, a return on stock or stock price are assumed with the stock prices following a log-normal distribution of fixed fluctuations [22]. Even with non-fixed fluctuations, one can use the average values without any interference with the job flow. Accordingly, if the price fluctuations follow a given time-dependent function, the model recovery principles will remain unchanged with the instantaneous fluctuations considered in terms of the average instantaneous fluctuations [19]. Principally, the future price prediction is highly sensitive to the initial price conditions, so that a given level of error in the present period may generate very significant levels of error in the periods to come.

Together with the result of Bonder and Groisman, this can be seen as an extension of a similar result for **stochastic** **differential** **equations** with additive noise. Indeed in later case, Feller’s test for explosions says that the Osgood condition is necessary and sufficient for blow up of the solution when the noise is a Brownian motion. We will later describe a new method for proving this without appealing to Feller’s test that works for a larger class of processes including the bifractional Brownian motion. This is due to [13] which was also the inspiration for the proof of the above theorem.

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Uncertainty propagation through a dynamic system has enjoyed considerable research at- tention during the past decade due to the wide applications of mathematical models in studying the dynamical behavior of systems. In this paper, we consider uncertainty prop- agation in continuous-time dynamical systems through two types of **differential** **equations**. One is **stochastic** **differential** **equations** (SDEs), a classification reserved for **differential** equa- tions driven by white noise (The value of a white noise at any point is uncorrelated from its value at any other point; that is, its correlation function is some Dirac delta function. Hence, white noise is also referred to as an uncorrelated random field 1 ). The other is random **differential** **equations** (RDEs), a classification reserved for **differential** **equations** driven by other type of random inputs such as colored noise (also referred to as a correlated random field as its value at any point is correlated from its value at any other point) and both colored noise and white noise. Readers can refer to [35, 41, 74] for some interesting discussions on white noise and colored noise as well as their applications in different fields.

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Since Itô introduced his **stochastic** calculus more than 50 years ago, the theory of **stochastic** **differential** **equations** has been developed very quickly. In particular, Lyapunov’s second method has been developed to deal with **stochastic** stability by many authors, and we here only mention Ar- nold [1], Friedman [2], Has’minskii [3], Kolmanovskii and Myshkis [4], Kushner [5], Ladde [6], Lakshmikantham et al. [7], Mao [10-13] and Mohammed [15]. Most of the existing results on **stochastic** stability use a single Lyapunov function. Recently, Mao [14] discussed the as- ymptotic stability of **stochastic** system via multiple Lyapunov functions, but the results obtained in [14] can not be applied to the **stochastic** **differential** delay **equations**. The main aim of this paper is to establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of **stochastic** **differential** delay **equations**. Furthermore, from them follow many useful results on **stochastic** asymptotic stability, which en-