Boundary value problems with displacement for one mixed hyperbolic equation of the second order

The paper studies two nonlocal problems with a displacement for the conjugation of two equations of second-order hyperbolic type, with a wave equation in one part of the domain and a degenerate hyperbolic equation of the first kind in the other part. As a non-local boundary condition in the considered problems, a linear system of FDEs is specified with variable coefficients involving the first-order derivative and derivatives of fractional (in the sense of Riemann-Liouville) orders of the desired function on one of the characteristics and on the line of type changing. Using the integral equation method, the first problem is equivalently reduced to a question of the solvability for the Volterra integral equation of the second kind with a weak singularity; and a question of the solvability for the second problem is equivalently reduced to a question of the solvability for the Fredholm integral equation of the second kind with a weak singularity. For the first problem, we prove the uniform convergence of the resolvent kernel for the resulting Volterra integral equation of the second kind and we prove that its solution belongs to the required class. As to the second problem, sufficient conditions are found for the given functions that ensure the existence of a unique solution to the Fredholm integral equation of the second kind with a weak singularity of the required class. In some particular cases, the solutions are written out explicitly

Affine Volterra processes

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance

Non-linear Rough Heat Equations

This article is devoted to define and solve an evolution equation of the form $dy_t=\Delta y_t dt+ dX_t(y_t)$, where $\Delta$ stands for the Laplace operator on a space of the form $L^p(\mathbb{R}^n)$, and $X$ is a finite dimensional noisy nonlinearity whose typical form is given by $X_t(\varphi)=\sum_{i=1}^N x^{i}_t f_i(\varphi)$, where each $x=(x^{(1)},...,x^{(N)})$ is a $\gamma$-H\"older function generating a rough path and each $f_i$ is a smooth enough function defined on $L^p(\mathbb{R}^n)$. The generalization of the usual rough path theory allowing to cope with such kind of systems is carefully constructed.Comment: 36 page