Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Optimal control over evolution stochastic systems and its application to stochastic models of financial mathematics

We consider problems of optimal stabilization of controlled evolution stochastic systems in semi-Markov media and their application to financial stochastic models. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 687–698, May, 1998.     

On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension

Existence, uniqueness and regularity of the trajectories of mild solutions of one-dimensional nonlinear stochastic fractional partial differential equations of order $\alpha >1$ containing derivatives of entire order and perturbed by space–time white noise are studied. The fractional derivative operator is defined by means of a generalized Riesz–Feller potential.     

Simplified existence for solutions to stochastic differential equations

Nonstandard methods are used to give a simple construction of a solution to SDEs of the form , where are required only to be measurable, with, bounded. By working with an internal Brownian motion the proof avoids the complicated lifting and approximation arguments needed in previous existence proofs.     

Moment equations and Hermite expansion for nonlinear stochastic differential equations with application to stock price models

Exact moment equations for nonlinear Itô processes are derived. Taylor expansion of the drift and diffusion coefficients around the first conditional moment gives a hierarchy of coupled moment equations which can be closed by truncation or a Gaussian assumption. The state transition density is expanded into a Hermite orthogonal series with leading Gaussian term and the Fourier coefficients are expressed in terms of the moments. The resulting approximate likelihood is maximized by using a quasi Newton algorithm with BFGS secant updates. A simulation study for the CEV stock price model compares the several approximate likelihood estimators with the Euler approximation and the exact ML estimator (Feller, in Ann Math 54: 173–182, 1951).     

Optimal stopping and stochastic control differential games for jump diffusions

We study stochastic differential games of jump diffusions driven by Brownian motions and compensated Poisson random measures, where one of the players can choose the stochastic control and the other player can decide when to stop the system. We prove a verification theorem for such games in terms of a Hamilton–Jacobi–Bellman variational inequality. The results are applied to study some specific examples, including optimal resource extraction in a worst-case scenario, and risk minimizing optimal portfolio and stopping.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

An approach to deducing approximate solutions to the Cauchy problem for nonlinear differential equations

A new technique for the construction of numerical methods based on continued fractions is proposed. A characteristic feature of these algorithms is the fact that for certain values of the parameters it is possible to obtain both novel and traditional (explicit and implicit) numerical methods for the solution of the Cauchy problem for ordinary differential equations. Two-sided formulas are proposed by means of which it is possible to obtain on each integration step not only upper and lower approximations to the exact solution, but also information concerning the magnitude of the leading term of the error without the need for additional calculations of the right-hand side of the initial differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1695–1701, December, 1992.     

On solutions to Ito stochastic differential equations

In this paper we obtain general conditions under which stochastic differential equations possess a strong solution representable in an explicit form as a functional of the Wiener process. Particular interest bears the problem of determining conditions that guarantee non-explosion of the solution. The necessary as well as sufficient condition is derived.     

Weak solutions to gamma-driven stochastic differential equations

We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

Markov solutions of stochastic differential equations

Probability Theory and Related Fields -     

Numerical solutions to time-fractional stochastic partial differential equations

Numerical Algorithms - This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the...     

Smooth densities for solutions to stochastic differential equations with jumps

We consider a solution _xt$x_t$ to a generic Markovian jump diffusion and show that for any _t0>0$t_0>0$ the law of _xt0$x_{t_0}$ has a C^∞$C^{\infty }$ density with respect to the Lebesgue measure under a uniform version of the Hörmander conditions. Unlike previous results in the area the result covers a class of infinite activity jump processes. The result is accomplished using carefully crafted refinements to the classical arguments used in proving the smoothness of density via Malliavin calculus. In particular, we provide a proof that the semimartingale inequality of J. Norris persists for discontinuous semimartingales when the jumps are small.     

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.