Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.     

Stochastic Nicholson’s blowflies delayed differential equations

In this paper, we consider a class of stochastic Nicholson’s blowflies delayed differential equations. Firstly, we obtain the existence and uniqueness of the global positive solution with nonnegative initial conditions. Then the ultimate boundedness in mean of solution is derived under the same condition. Moreover, we estimate the sample Lyapunov exponent of the solution, which is less than a positive constant. In the end, an example with its numerical simulations is carried out to validate the analytical results.     

Stochastic differential equations with polar-decomposed L関y measures and applications to stochastic optimization

This paper is concerned with the optimal control of jump-type stochastic differential equations associated with polar-decomposed Lévy measures with the feature of explicit construction on the jump term. The concrete construction is then utilized for analysis of two portfolio optimization problems for financial market models driven by stable-like processes.      

Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations

In this paper we study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu, Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton–Jacobi–Bellman equations with integral-differential operators. For this proof, we adapt Peng’s BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure.     

Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework.     

Ax–Schanuel for linear differential equations

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.     

Stochastic continuity equations–A general uniqueness result

We showexistence and uniqueness of L^∞-solutions to stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition. The main issue of uniqueness is proved reeling on stochastic characteristic method.     

Examining students’ variational reasoning in differential equations

In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) =0 in Ω, u=0 on Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u~3,u~2(u-p),u~2(u~2 -p),     

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

Hyers–Ulam stability of hypergeometric differential equations

Aequationes mathematicae - In the present paper by applying the series method we prove the Hyers–Ulam stability of the homogeneous hypergeometric differential equation in a subclass of...     

Hyers-Ulam-Rassias stability of κ-Caputo fractional differential equations

The paper is connected with the existence of solutions and Hyers-Ulam stability for a class of nonlinear fractional differential equations with κ-Caputo fractional derivative in boundary value problems. The existence and uniqueness results are obtained by utilizing the Banach fixed point theorem and Leray-Schauder nonlinear alternative theorem. In addition, two sufficient conditions to guarantee the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of boundary value problems of fractional differential equations are also presented. Finally, theoretical results are illustrated by two numerical examples.     

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of non-linear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the “geometric form” of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations (BSDEs) and (ii) pricing of European and American derivatives via BSDEs. Regarding (i), we extend previous results on BSDEs in a Lévy setting and the connection to semilinear integro-partial differential equations.     

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

     

Second-order differential equations in the Laguerre–Hahn class

Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.