Related papers: Chaos for generalized Black-Scholes equations
In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c…
In this article, we prove null-controllability results for the heat equation associated tofractional Baouendi-Grushin operators $$\partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u= \mathbb{1}_\Omega h$$ where $V$ is a potential that…
A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black--Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option…
Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional…
In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…
We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial…
We investigate static and spherically symmetric black hole (BH) solutions in shift-symmetric quadratic-order degenerate higher-order scalar-tensor (DHOST) theories. We allow a nonconstant kinetic term $X=g^{\mu\nu}…
In this work, we tested the thermal behavior of a class of regular black hole solutions defined as generalized black-bounce spacetimes. We introduce several novel configurations governed by different mass functions and geometric…
We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and…
In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be…
Option contracts can be valued by using the Black-Scholes equation, a partial differential equation with initial conditions. An exact solution for European style options is known. The computation time and the error need to be minimized…
This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence…
We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})=…
We study a method of reducing space dimension in multi-dimensional Black-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the…
We consider non-linear time-fractional stochastic heat type equation $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$,…
In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: \begin{equation*} \left\{\begin{array}{l} \partial_t u-\Delta u= \rho^2 u^2…
We revisit the problem of perturbations of Schwarzschild-AdS$_4$ black holes by using a combination of the Martel-Poisson formalism for perturbations of four-dimensional spherically symmetric spacetimes and the Kodama-Ishibashi formalism.…
Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \] where…
In this paper, we consider the following nonlocal parabolic equation \begin{equation*} u_{t}-\Delta u=\left( \int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right) |u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty),…
We analyze several spherically symmetric exterior vacuum solutions allowed by the Einstein-Aether (EA) theory with a non static aether and study the thermodynamics of their Killing and universal horizons. We show that there are five classes…