All Questions
26 questions
5
votes
1
answer
1k
views
Correlated Brownian motions across different times and representation with independent processes
This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition.
In our problem we have two correlated Brownian motions $B^1,B^2$ (starting at time $t=0$ ...
- 5,474
4
votes
2
answers
376
views
Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 807
4
votes
1
answer
403
views
When are the transition densities of an SDE symmetric?
We fix $T>0$. Let $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $\sigma:[0, T] \times \mathbb{R}^d \rightarrow \mathcal{M}^\text{sym}_{d \times d}(\mathbb{R})$ be measurable and ...
- 825
4
votes
1
answer
489
views
Regularity for Stochastic heat equation with additive noise in d=2
I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u_{t}=\Delta u+f$ in great detail, especially in dimension 2.
In Hairer's Spde notes , he ...
- 5,474
4
votes
1
answer
610
views
Stochastic differential equation associated with an optimal control problem
We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...
- 355
4
votes
1
answer
262
views
Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
- 128k
4
votes
0
answers
328
views
Convergence to unique stationary distribution for SDEs and Markov processes
I am interested in understanding the behavior of solutions to stochastic differential equations (SDEs) and continuous-time Markov processes with constant coefficients. Specifically, I would like to ...
- 807
4
votes
0
answers
95
views
Reference request on theory about Stochastic Riemann problem
I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit.
On one hand, in the literature it is not hard to find books or papers that deal ...
- 657
4
votes
0
answers
276
views
Exit time of a stochastic process defined by a SDE
Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...
- 205
3
votes
1
answer
209
views
Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?
Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...
- 6,215
3
votes
1
answer
390
views
Reference request for a Riemannian Fokker-Planck equation
The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...
- 187
3
votes
0
answers
74
views
Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
3
votes
0
answers
54
views
Unique weak solution of an SDE for a general initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\varepsilon}
\newcommand{\diff}{\...
- 825
2
votes
1
answer
216
views
Decay estimate of moment of an SDE
We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
- 825
2
votes
1
answer
304
views
When does a solution to SDE have full support?
Suppose an $n$-dimensional process $(X_t)_{0 \leq t \leq 1}$ satisfies an SDE of the form:
$$dX_t = u_t(X_t) \,dt + dB_t, ~~X_0 = 0$$
where $(B_t)_{t\geq 0}$ is a Brownian motion with $B_1 \sim N(0,K)$...
- 716
2
votes
2
answers
255
views
Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$
We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...
- 5,474
1
vote
1
answer
337
views
Bessel process conditioned to stay positive
This question has also been asked on https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive
Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x&...
- 284
1
vote
1
answer
100
views
Reference to log-transition-density of a diffusion process
Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely ...
- 51
1
vote
0
answers
134
views
Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
1
vote
0
answers
34
views
Regime switching stochastic systems references
I'm looking for some good references discussing regime switching stochastic systems (Stochastic systems with markovian jump process) and their solutions.
Given a Continuous-time Markov Chain $\xi$ ...
- 141
1
vote
0
answers
659
views
"Expected Value" of a solution to a differential equation
I'm going to write this question in a very informal way as I'm looking for guidance, rather than a specific answer to a specific problem. So I took a course on stochastic processes and Martingales ...
- 597
1
vote
0
answers
118
views
Full version of Soucaliuc's research announcement "Réflexion entre deux diffusions conjuguées"
Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...
- 43
0
votes
1
answer
379
views
What is the derivative of this integral?
I have asked this question here
https://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...
- 131
0
votes
1
answer
152
views
About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E
We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...
- 2,246
0
votes
1
answer
341
views
Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
