Related papers: Kolmogorov equations for stochastic Volterra proce…
Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to compute numerically expected values and…
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional…
We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, existence and uniqueness of a measure…
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a…
We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak H\"ormander condition). We construct an approximate boundary potential which captures the effect…
In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven by an…
In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the…
We consider the following stochastic partial differential equation, \begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=\psi, \end{align*} associated with a stochastic flow $\{X(t,x)\}$, for $t \geq 0$, $x \in \mathbb{R}^d$, as in…
Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV)…
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov…
Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical…
Path-dependence is a defining feature of many real-world systems, with applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra equations (SVEs), such dependence is encoded…
In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$…
We study a class of McKean-Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and…
We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of stochastic control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes that…
In this paper, we are concerned with stochastic Volterra equations with singular kernels and H\"older continuous coefficients. We first establish the well-posedness of these equations by utilising the Yamada-Watanabe approach. Then, we aim…
In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered…
This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $\rho_t-\Delta\beta(\rho)+{\rm div}(Db(\rho)\rho)=0$ in…
In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and…
We obtain Calder{\'o}n-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous coefficients. We then derive the well-posedness of the martingale problem associated to related degenerate operators, and therefore…