# Questions tagged [stochastic-pde]

Stochastic partial differential equations are partial differential equations with a random driving force. Please do not use this tag just because there are stochastic processes and differential equations in your question. Consider if [tag: SDE] is a better choice. This tag is only to be used for PDEs driven by noise.

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### what would be the solution of this Fokker-Planck equation?

Define $A:=\left( \begin{matrix}5/8 & -3/8 \\ -3/8 & 5/8\end{matrix}\right)$ and $\mu =(1,1)$. Define $V(x):= \frac{1}{2} (x-\mu)^T A^{-1} (x-\mu)$. What would be the solution $\rho(x,t)$ for ...
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### What type of PDE is this and what conditions determine the existence of a solution?

So I have a system of PDE's essentially consisting of equations of the form $$\sum_{i=1}^{n}f_{i}(\mathbf{x}(t), t)\frac{\partial u(\mathbf{x}(t),t)}{\partial x_i}=c$$ where $c$ is any nonzero ...
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### Itô Integration by parts on a distribution for defaulted banks

The question is based on the McKean-Vlasov problem formulated in the paper "A McKean-Vlasov equation with positive feedback and blow-ups", namely: \begin{cases} X_t = X_0 + B_t -\...
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### Feller condition for CIR process positivity

I am trying to understand Feller’s square root condition which provides the existence of the positive solution of the Cox-Ingersoll-Ross which introduced by Gikhman, Ilya please find link to the ...
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### Derivative of a stochastic process

I'm reading a engeneering book (Turbulent Flow - Pope) and it isn't very rigorous. Let $X(t)$ be a casual variable, for every fixed $t\in \mathbb{R}$. How can we define the derivative with respect ...
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### Gaussian measure push-forward through linear partial differential operator

Setup Let $\Omega$ be a bounded domain with smooth boundary and $\zeta \in C^\infty_c(\Omega)$ be distributed according to a Gaussian measure such that $\zeta \sim \mathcal{N}(0,\mathcal{C}_\zeta)$ ...
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### Is it possible to define unitary operator-valued stochastic process and related stochastic differential equations?

There are many references on stochastic differential equations valued in Hilbert or Banach space which are driven by diffusion processes also on Hilbert space. Now, I am wondering if there are any ...
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### A stochastic cannibalistic snail problem

Here's a somewhat stupid question, in sum about adding stochasticity to a linear ODE (see the end). Background I was thinking about a silly problem of 4 cannibal snails in a box: Each snail ...
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### Feynman-Kac And Infinite Time Horizon

(1) is there literature about a Feynman-Kac formula on an infinite time horizon? (2) This means: ... Let $X$ be the process satisfying the SDE dX(t) = \mu(t,X(t)) dt + \sigma(t,X(...
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### uniqueness of solution and continuity for differential equations

What is disturbing me is how is it possible that if the solution of a differential solution is unique then, it's continuous ? Do you have any idea how, this, is possible ? Apparently, it has some ...
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### The Ho-Lee Model (1986)'s Bond Call Option Pricing

(My Question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (the details in this ...
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### The Riccatti equation for The Cox-Ingerson-Ross Model.

(My Question) I went through the calculations halfway, but I cannot find out how to calculate the following Riccatti equation. Please tell me how to calculate this The Riccatti equation with its ...
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### The Ho-Lee Model (1986)

(My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (Thank you for your ...
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### The Cox-Ingersoll-Ross Model (1985)

Please show me how to solve (2) with computation processes.(1) was the initial question which I solved. I show the answers (1) below. Consider the equation \begin{eqnarray} dr_t=(\alpha - \beta r_t ) ...
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### The Exponential Vasicek Model (1978)

Please show me how to solve (3) with computation processes. (1) and (2) were initial questions which I solved. I show the answers (1) and (2) below. (1) Solve the S.D.E. \begin{eqnarray} dS_t=\alpha ...
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### The probability distribution of “derivative” of a random variable

Let me set the stage; Consider a stochastic PDE, which has to following form $$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$ where $H$ is a deterministic function, and $\chi(x,t)$ is a random variable. ...
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### Multidimensional Gaussian white noise definition

In one of the papers I came across the following: Let ${W(dt, du): t, u ≥ 0}$ denote a Gaussian white noise with density measure $dtdu$ on $(0,\infty)^2$. Now, if I understand this correctly, $W$ ...
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### Resources on numerically solving a stochastic Navier Stokes Equation

I know these types of questions don't have definitive answers, but I need some assistance. I would like to do some numerical experiments (preferably in python, but any language is fine) for my thesis ...
It is well known that the Green Function of the 1-D Heat equation is given by the density of the normal distribution, i.e. $$G(t,x,y)=\frac{1}{\sqrt{4 \pi t}} e^{\frac{-(x-y)^2}{4t}}.$$ Now, I was ...