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Register a new userWell-posedness of path-dependent semilinear parabolic master equations
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Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in games and control theory. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent master equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in stochastic control theory and games. We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupires vertical derivative, and applying stochastic forward-backward system argument. Moreover, we consider a general non-smooth case with a functional mollifying method.
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We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to Metivier and Pellaumail.
In this paper, to cope with the shortage of sufficient theoretical support resulted from the fast-growing quantitative financial modeling, we investigate two classes of generalized stochastic volatility models, establish their well-posedness of strong solutions, and conduct the stability analysis with respect to small perturbations. In the first class, a multidimensional path-dependent process is driven by another multidimensional path-dependent process. The second class is a generalized one-dimensional stochastic volatility model with Holder continuous coefficients. What greatly differentiates those two classes of models is that both the process and its correlated driving process have their own subdifferential operators, whose one special case is the general reflection operators for multi-sided barriers. Hence, the models investigated fully cover various newly explored variants of stochastic volatility models whose well-posedness is unknown, and naturally serve as the rigorous mathematical foundation for new stochastic volatility model development in terms of multi-dimension, path-dependence, and multi-sided barrier reflection.
We study the Hardy-Henon parabolic equations on $mathbb{R}^{N}$ ($N=2, 3$) under the effect of an additive fractional Brownian noise with Hurst parameter $H>maxleft(1/2, N/4right).$ We show local existence and uniqueness of a mid $L^{q}$-solution under suitable assumptions on $q$.
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to $-infty$ at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, under the further assumption that the initial datum is in $L^2_{mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and we are able to establish uniqueness of purely (nonnegative) distributional solutions, which to our knowledge was not known before even in the Euclidean space. The required curvature bound is in fact sharp, since on model manifolds it turns out to be equivalent to stochastic completeness, and it was shown in [13] that uniqueness for the FDE fails even in the class of bounded solutions on manifolds that are not stochastically complete. Qualitatively this amounts to asking that the curvature diverges at most quadratically at infinity. A crucial ingredient of the uniqueness result is the proof of nonexistence of distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.
In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n leq 5$. We show this equation, with power $2le ple 1+4/(n-1)$, is (strongly) ill-posed in $H^{s}$ with $s = (n+5)/4$ in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure of nonlinear terms in terms of the regularity. As a byproduct, we give an alternative proof of the failure of the local in time endpoint scale-invariant $L_{t}^{4/(n-1)}L_{x}^{infty}$ Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.
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