Related papers: Rough differential equations with path-dependent c…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…
Branched rough paths, used to solve ODEs on $\mathbb{R}$, have been generalised in two different directions. In one direction, there are regularity structures aimed at solving SPDEs on $\mathbb{R}$. In the other direction, there are…
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be…
The paper considers a stochastic differential equation of Duffing type with Markov coefficients. The existence of unpredictable solutions is considered. The unpredictability is a property of bounded functions characterized by unbounded…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula…
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of…
We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…
While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
We present a theoretical framework and numerical methods for predicting the large-scale properties of solutions of partial differential equations that are too complex to be properly resolved. We assume that prior statistical information…
We prove a large deviation principle for the slow-fast rough differential equations under the controlled rough path framework. The driver rough paths are lifted from the mixed fractional Brownian motion with Hurst parameter $H\in…
An examples of solutions of nonlinear differential equations associated with developable, ruled and minimal surfaces are constructed.
This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to…
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.
This work concerns a type of path-dependent multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the well-posedness for path-dependent multivalued stochastic differential equations under the Lipschitz…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy…
The solution of rough differential equation, driven by the It\^o signature of a continuous local martingale, exists uniquely a.s. when the vector field is Lip(\beta) for \beta > 1, and coincides a.s. with the It\^o signature of the solution…