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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10 views

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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47 views

Solve the SDE $d X_{t}=X_{t} B_{t} d t-X_{t} d B_{t}$

I am not sure about my attempt so far as I hit a road block: let $Y_{t}=X_{t} B_{t}$ then $d Y_{t}= B_{t}dX_{t}+X_{t}d B_{t}+ d B_{t} d X_{t}$ Substituting $dX_{t}$ we get: $dY_{t}=B_{t}\left(X_{t} B_{...
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63 views

Find the solution $f(x, t)$ to the partial differential equation for $0 \leq t \leq 1$

Find the solution $f(x, t)$ to the partial differential equation for $0 \leq t \leq 1$ $$ \frac{x^{2}}{2} \frac{\partial^{2}}{\partial x^{2}} f(x, t)+\frac{\partial}{\partial t} f(x, t)=f(x, t), \quad ...
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40 views

Hilbert Schmidt norm

Suppose $Q \in L(U)$ is non negative operator with $Tr Q < \infty$ where $(U, \langle,\rangle)$ is Hilbert space. Suppose that $e_k, k=0...$ is an orthonormal base of $U$ consisting of eigenvectors ...
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43 views

Solution of SPDE can be expressed by expectation of solution to the SDE?

I am confused by the idea that when the data is stochastic, the solution u($\omega$ ,x) of the SPDE can be expressed as the conditional expectation of some functional solution to the SDE Take Darcy ...
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28 views

Affine Structure PDE resolution for interest rates

I would like to now how to solve the PDE of the affine structure under Vasicek.I am delineating the steps: First let's posit the OU process under a Risk Neutral Measure such as : \begin{align*} \...
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46 views

Properties of Stratonovich and Ito integrals

I'm struggling to understand if I have this idea of an orthogonal transform (to simplify a set of SDE's in Ito form) understood. Suppose I have the orthogonal transform \begin{align} \begin{bmatrix} ...
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43 views

Divergence when calculating moments with a space-time Gaussian white noise

Consider a variable $M(x,t)$ driven by space-time Gaussian white noise: $$ \partial_t M(x,t) = -k M + \xi(x,t).$$ The noise has mean $\langle \xi(x,t) \rangle = 0$ and correlation function $\langle \...
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68 views

Most approachable text / intro on Faynman-Kac Theorem?

I am looking for an advice / a recommendation on a PDF or a text-book that would be suitable for tackling the Feynman-Kac Theorem for the first time (think of a Graduate probability student with keen ...
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1answer
206 views

Derivative of a Stochastic Integral with respect to Limit & with respect to Integrator

I have recently come across an attempt to differentiate the following function with respect to $t$ and with respect to $W_t$: $$F(W_t, t):=\int_{h=0}^{h=t}W_hdW_h$$ Is it possible (i.e. is it "...
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78 views

Space-time white noise grows at infinity?

I am reading some introduction texts on SPDE's and I often find the phrase "noise grows at infinity" (for instance in Gubinelli & Hofmanova's paper on $\Phi^4$ https://arxiv.org/abs/1804....
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59 views

Ito formula to asian option

I was reading "2001The Integral of Geometric Brownian Motion" am getting confused whether $A_t^{(\mu)}=\int_0^t e^{2 \mu \tau + 2 B_\tau} d\tau$ is a function of $t$ only or a function of $t$...
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1answer
60 views

Stochastic integration by parts for random point processes

I'm trying to understand this proof of the following specifing integration by parts. Introduction Let $\Omega=Point_{\mathbb{R}}$ the set of point distributions in $\mathbb{R}^3$ (i.e an element $w \...
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26 views

Gaussian Measure of a submanifold is 0

Let $\mu$ be a non-degenerate Gaussian measure on a seperable Hilbert space $\mathcal{H}$. Let $A \subset \mathcal{H}$ be a hyperplane. I would like to show that $\phi(A)$ has measure $0$ for a smooth ...
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38 views

Proof of unique existence of Stochastic Wave Equation (d=1)

I'm trying to prove the uniqueness and existence of the solution for the following Stochastic Wave Equation in one-spatial dimension as follows: $ \frac{d^{2}u}{dt^{2}} - \frac{d^{2}u}{dt^{2}} = \...
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1answer
141 views

Poisson equation with stochastic source

In a physical set-up, one can consider an electrostatic problem where the charge density at each point in space is a random variable, and try to find the electric potential or electric field. To be ...
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1answer
40 views

Reference request : existence and uniqueness of solution to a certain class of SPDE

Is there any papers/reference for the existence and uniqueness of the following type of Stochastic Partial Differential Equations (perhaps a much larger class of SPDES containing the following) $$ dv(...
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100 views

$\operatorname{cov}\left(\int_0^tf(s,T_1)\,dW(s,T_1),\int_0^tf(s,T_2)\,dW(s,T_1)\right)=\text{?}$

Define a random field $W$ so that: $W(t,T)$ is a standard Brownian motion for every $T$. $dW(t,T_1) \, dW(t,T_2)=c(t,T_1,T_2)\,dt$ $W(t,\cdot)$ is a a continuous function for every $t$. Suppose $f$ ...
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22 views

Hamilton-Jacobi-Bellman with expected-time cost function

I have a question about the Hamilton-Jacobi-Bellman equations when one takes the expected cost to go as equalling the time taken to reach a destination. Typically the boundary condition for HJB is ...
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1answer
196 views

Infinitesimal generator of the Brownian motion on a circle

As explained here, the infinitesimal generator of a Brownian motion is $\frac{1}{2}\Delta$. Can someone please provide (for a non-stochastic-student) the proof of finding the infinitesimal generator ...
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66 views

If $X:\Omega\times[0,\infty)\times E\to E$ is a stochatic flow, is $\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]$ a semigroup?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space and $X:\Omega\times[0,\infty)\times E\to E$ be a stochastic flow, i.e. $X$ is $(\mathcal A\...
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33 views

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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1answer
58 views

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{equation} \begin{cases} X_t = X_0 + B_t -\...
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162 views

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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46 views

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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53 views

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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61 views

Bounding Solution of Nonlinear Stochastic Heat Equation with Coloured Noise

My question concerns solutions to the nonlinear stochastic heat equation: $$\begin{align}u_{t}&=u_{xx}+u^{\gamma}\dot{F}, && \gamma\geq 1,&& t>0& 0\leq x\leq J,\\ u(t,0)&...
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1answer
134 views

Book recommendation: Numerical method for SPDE (stochastic partial differential equation)

I saw many classical numerical books for stochastic differential equations (SDE) and partial differential equations (PDE). But I rarely found any related to SPDE. Why? I have enough foundation in SDE ...
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30 views

Numerically integrating the same SPDE on different dimensions results in different outcomes

Let us consider a general SPDE of the form $$\partial_t h = F(h, \partial_x h, (\partial_x h)^2, \partial_x^2 h,..) + \eta,$$ where $\eta$ is a normal random variable in space and time with $$<\...
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1answer
22 views

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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1answer
56 views

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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74 views

Difference between formal tensor series and vector space of polynomial?

The question is related to the link https://www.sciencedirect.com/science/article/pii/S0022123619302460#br0110 where in page 12 they define the vector space of formal tensor series over $\mathbb{R}^{d+...
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1answer
48 views

Solving $Y_t$ = $e^{-\alpha t}X_t$ using Ito's lemma

G'day I am trying to solve $Y_t$ = $e^{-\alpha t}X_t$, where $dX_t = \alpha X_tdt + \sigma_tdW_t$ using Ito's lemma. Defining f(t, X) = $e^{-\alpha t}X$. $\frac{df(t,X)}{dt}$ = $-\alpha e^{-\alpha t}...
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1answer
112 views

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 \begin{equation} dX(t) = \mu(t,X(t)) dt + \sigma(t,X(...
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1answer
74 views

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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1answer
47 views

A simple question regarding the derivation of the Black-Scholes formula

i am taking a derivatives class, and what is of course obligatory is to derive the Black Scholes formula. I am simply stuck or puzzled by one simple thing in the derivation: How do they get from $$...
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1answer
108 views

Separability of Reproducing Kernel Hilbert Space of a positive definite covariance function.

I am reading the book Random Fields and Geometry by R. Adler and J. Taylor. In chapter 3 they introduce the concept of reproducing kernel Hilbert space (RKHS). I will summarize the definition in what ...
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3answers
114 views

Bond Option Hedging

(My question) Please show me how to solve from (2) to (4) with computation processes. These are too difficult to solve. Thank you for your help in advance. (Cross-link) I have posted the same ...
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2answers
96 views

Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price

(My Question) Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M. $$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
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91 views

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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2answers
136 views

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 ...
3
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0answers
162 views

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 ...
3
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2answers
343 views

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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1answer
221 views

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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0answers
96 views

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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88 views

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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1answer
91 views

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 ...
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77 views

Bounds for the Green function of heat equation

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 ...
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1answer
271 views

Change of variables in stochastic PDE

I have the following stochastic partial differential equation (SPDE): $d v = -\mu \frac{\partial v}{\partial x} dt + \frac{1}{2} \frac{\partial^2 v}{\partial x^2} dt - \sqrt{\rho} \frac{\partial v}{\...
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1answer
53 views

Finding relationship between solutions of two different first order linear PDEs (using Feynman-Kac)

The first partial differential equation, using the shorthand $\mathcal{L}u(t,x) = \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t)$, is: ...