Existence and uniqueness of global solutions to the stochastic heat equation with superlinear drift on an unbounded spatial domain

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition [Formula: see text] along with additional restrictions. For example, consider the forcing [Formula: see text]. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

SOME ASPECTS OF SOLUTIONS OF SPACE-TIME FRACTIONAL STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH OSGOOD CONDITION

In this paper we discuss the following problem with additive noise, \[\begin{cases} \frac{\partial^{\beta} u(t,x) }{\partial t}=-(-\triangle)^{\frac{\alpha}{2}} u(t,x)+b(u(t,x))+\sigma\dot{W}(t,x),~~t>0, \\u(0,x)=u_{0}(x),\end{cases},\] where $\alpha \in(0,2) $ and $ \beta \in (0,1)$, the fractional time derivative is in the sense of Caputo, $-(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian, $\sigma$ is a positive parameter, $\dot{W}$ is a space-time white noise, $u_0(x)$ is assumed to be non-negative, continuous and bounded. We study first the equation on $[0,\,1]$ with homogeneous Drichlet boundary condition and show that the solution of the equation blows up in finite time if and only if $b$ satisfies the Osgood condition, \[ \int_{c}^{\infty} \frac{ds}{b(s)} <\infty \] for some constant $c>0$. We then consider the same equation on the whole line and show that the above Osgood condition is satisfied whenever the solution of the equation blows up.

$${{\varvec{L}}}^{{\varvec{p}}}$$-Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

We present a unified approach to $$L^p$$ L p -solutions ($$p > 1$$ p > 1 ) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.