All Questions
250 questions
3
votes
1
answer
546
views
Each diffusion SDE is associated to a *unique* family of transition kernels
I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$.
How can I prove that there exists a unique family of transition ...
- 1,149
2
votes
0
answers
111
views
Bounding from below the distance between SDE started from different initial conditions
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...
- 6,215
1
vote
0
answers
108
views
Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)
Let $X$ be the solution to some stochastic differential equation
$$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$
Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
- 1,399
1
vote
0
answers
237
views
Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
- 11
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
3
votes
0
answers
122
views
Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
- 31
3
votes
1
answer
315
views
Strong blow up limits for SDE
Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications.
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...
- 6,215
1
vote
0
answers
190
views
Eigenvalues/eigenfunctions of a diffusion generator
Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
- 63
5
votes
1
answer
334
views
Does the entropy of a SDE with nondegenerate noise always increase?
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
- 6,215
2
votes
1
answer
416
views
Convergence of the quadratic variation process
Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$
$$
\lim_{n\to\infty}\mathbb{E}(...
- 199
4
votes
0
answers
306
views
A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 11.8k
2
votes
1
answer
392
views
Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
$(\Omega, \mathcal F, \mathbb P)$ be a probability space.
$B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
$X=(X_0^...
- 825
2
votes
0
answers
301
views
Ito lemma for SDEs on a Lie group
I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group.
In doing so i'm trying to define a formula ...
- 293
1
vote
1
answer
107
views
How to obtain this differential relation about moments of a stochastic process?
$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos.
...
- 825
2
votes
0
answers
356
views
KL Divergence between the solution to two SDEs
What is the KL divergence between the laws of solutions to SDEs? That is, let
\begin{align*}
dX^1&=b_1(X^1,t) \, dt+\sigma(X^1,t) \, dB\\
dX^2&=b_2(X^2,t) \, dt+\sigma(X^2,t) \, dB
\end{align*}...
- 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
$$
d X_t = ...
- 657
3
votes
1
answer
390
views
Reference request for a Riemannian Fokker-Planck equation
The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...
- 187
4
votes
1
answer
181
views
Small noise limits with irregular drift
Let $W$ be a standard $d$-dimensional Brownian motion.
Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0,...
- 6,215
0
votes
0
answers
120
views
Predictability of the mild solution of a SPDE
Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
- 573
3
votes
2
answers
554
views
Blow up limits for SDE
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \, , \, X_0 = 0$$
with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous....
- 6,215
1
vote
1
answer
653
views
Expectation of stochastic integral
Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation
$$ \left\langle \int^...
2
votes
0
answers
65
views
Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance
Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...
- 169
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 6,215
3
votes
0
answers
201
views
Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?
In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
- 177
5
votes
1
answer
531
views
Riemannian metric induced by a stochastic differential equation
Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
1
vote
1
answer
604
views
Is there an inverse Lamperti transformation for diffusions?
The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...
2
votes
1
answer
204
views
Comparing diffusion processes in different metrics
I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$.
Is there a way to apply ...
1
vote
0
answers
121
views
Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
2
votes
1
answer
163
views
Does the time of maximum of a diffusion process admit a continuous density?
Let $W$ be a standard one dimensional Brownian motion, and consider the solution $X$ to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $X_0 = 0$ a.s., and where $\mu, \sigma: \mathbb R \...
- 6,215
0
votes
0
answers
75
views
Regularity of solutions to forward-backward stochastic differential equations
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\left\{
\begin{aligned}
d X_t
&= aP_t dt +bdB_t;\\
X_0
&= x_0;\\
d P_t
&=cP_t dt + c^*Z_t dB_t;
\\
P_T
&...
- 54
4
votes
1
answer
343
views
Convergence of a continuous time stochastic gradient descent algorithm
Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions
$$\lim_{x \to -\infty} \nabla f(x) = -\infty, \lim_{x \to \infty} \nabla f(x) = \infty.$$
and let $\...
- 6,215
4
votes
1
answer
509
views
What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$
where $W$ is a standard $d$-dimensional Brownian motion.
I am interested in the case where $\sigma: \mathbb ...
- 6,215
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
1
vote
0
answers
156
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
2
votes
1
answer
549
views
A question related to Girsanov’s theorem
I’ve recently realised there is a subtlety in Girsanov’s theorem that I don’t really understand.
Consider a standard one dimensional Brownian motion $W$, and consider the SDE
$$dZ_t = \mu(t, Z_t) \, ...
- 6,215
2
votes
2
answers
416
views
Short time limits for SDE
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = x_0\;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a ...
- 6,215
2
votes
1
answer
296
views
Large noise limit for SDE with general volatility coefficients
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = 1 \;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a ...
- 6,215
2
votes
1
answer
493
views
Is the solution to this SDE always positive?
Let $W$ be a standard one dimensional Brownian motion, and consider the SDE
$$dX_t = \sigma(X_t) \, dW_t, \, \, \, X_0 = 1 \, \text {a.s.}$$
Assume $\sigma$ is regular enough that the above SDE admits ...
- 6,215
1
vote
1
answer
201
views
A comparison principle for SDE
Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE
$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$
$$dY_t = \mu_Y (t, \...
- 6,215
2
votes
1
answer
179
views
Solution of SDE with time power law singular diffusion
I was wondering if anything could be said at all about the well-psedness of the following time-inhomogeneous singular diffusion SDE:
\begin{align}d X_t&=\sigma(X_t,t ) d W_t , \qquad t\geq 0, ...
- 135
0
votes
0
answers
468
views
The relationship between measurability and weak measurability
For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple
functions, measurability (the ...
- 113
1
vote
2
answers
240
views
Solution to SDE conditional on high maxima of driving Brownian motion
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = X_t \, dW_t \;, \quad X_0 = 1 \;.$$
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
- 6,215
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
0
votes
2
answers
182
views
Distribution of local martingale is absolutly continuous to that of the Brownian motion?
Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
- 227
2
votes
1
answer
240
views
Uniqueness of the solution to some degenerate SDE
Consider the one-dimensional stochastic differential equation:
$$dX_t = {\bf 1}_{\{X_t>0\}}\big(b(t,X_t)dt + a(t,X_t)dW_t\big),\quad \forall t>0,$$
or equivalently
$$dX_t = b(t,X_t)dt + a(t,X_t)...
user478492
0
votes
1
answer
349
views
Probability that a geometric Brownian motion with additional determinstic drift ever hits zero
Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE
$$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$
where $\mu, \sigma, C, k > 0$ are constants, ...
- 6,215
2
votes
0
answers
116
views
Is a Riccati BSDE explicitly solvable?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
- 335
1
vote
1
answer
183
views
Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?
Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
- 11
5
votes
0
answers
400
views
Uniform bound for the occupation time of a diffusion
Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...
- 6,215
1
vote
0
answers
124
views
On the Lipschitz constant of $\Gamma$
Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
- 1,334