All Questions
405 questions
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 1,334
4
votes
1
answer
190
views
Probability that a drifted Gaussian process does not hit zero
Let $m: \mathbb R_+\to [0,1]$ be continuous and decreasing. Consider
$$X_t=1+bt+\int_0^t\frac{\sigma}{1+m(s)}dW_s,\quad \forall t\ge 0,$$
where $b, \sigma>0$ are given and $(W_t)_{t\ge 0}$ is a ...
user128095
1
vote
1
answer
233
views
Martingale representation of time-changed Brownian motion
Let $(B_t)_{t\geq 0}$ be a standard Brownian motion. Let $\phi: [0,1)\to [0,\infty)$ be defined by $
\phi(t):=t/(1-t)$. Then $(M_t)_{0\le t<1}$ is a continuous Markov martingale with $M_t:=B_{\phi(...
user420828
4
votes
1
answer
181
views
Conditions for the SDE be transitive
This question was previously posted on MSE.
Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...
- 333
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
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
1
vote
1
answer
210
views
First hitting time for non-homogeneous diffusion martingale
This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$
Consider the martingale given as
$$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$
...
- 1,334
5
votes
2
answers
556
views
Conditioning an SDE on the event that the driving noise is small
Let $X$ be the solution to the one dimensional SDE
$dX_t = \mu(t, X_t)dt + \sigma(t, X_t) dW_t$, for $t \in [0, T]$.
with $X_0= x_0$ a.s. for some $x_0 \in \mathbb R$.
Here $W_t$ denotes a standard ...
- 6,195
3
votes
1
answer
952
views
How to get speed measure $m(dx)$, scale function $s$, and killing measure $k(dx)$ of a diffusion from the infinitesimal generator? [closed]
This question comes from P13 and P17 of the book Andrei N.Borodin and Paavo Salminen.
Page P13 defines the speed measure $m(dx)$, the scale function $s$, and the killing measure $k(dx)$.
Case 9 on P17:...
2
votes
1
answer
224
views
Perturbation of volatility term in an SDE
Suppose $X, X^{\varepsilon}$, for $\varepsilon > 0$ are real valued stochastic processes satisfying the following SDE on $[0, T]$:
$dX = \mu(t, X_t) dt + \sigma (t, X_t) dW_t,$
$dX^{\varepsilon} = \...
- 6,195
2
votes
1
answer
309
views
A bound for the occupation time of a diffusion
Let $\sigma: \mathbb R \times \mathbb R \to \mathbb R$ be a Lipschitz continuous function bounded below by some $M > 0$.
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the ...
- 6,195
2
votes
1
answer
591
views
Existence/uniqueness of the solution to some SDE with discontinuous coefficient
Consider a SDE
$$dX_t = b(t,X_t)dt + f\big(a(t,X_t)\big)dW_t,\quad \quad\quad\quad\quad\quad\quad\quad\quad(\ast)$$
where $(W_t)_{t\ge 0}$ is a Brownian motion and
$$f(z):={\bf 1}_{\{z>0\}} +\frac{...
user128095
0
votes
1
answer
897
views
How to understand the transition density of reflected Brownian motion
We can see from the above picture the transition density of reflecting Browninan motion is given by (19). As we know, the first part ($2p(t,x,y)$) is the transition density of a Brownian motion (from $...
- 165
2
votes
0
answers
86
views
existence/uniqueness of the (weak) solution to SDEs with discontinuous volatility
Consider a sequence of parametrized SDEs :
$$X^{a}_t = z + \int_0^t b(a,s,X^a_s)ds+\int_0^t\frac{\sigma(a,s,X^a_s)}{1+{\bf 1}_{\{b(a,s,X^a_s)>0\}}}dW_s,\quad \forall t\ge 0,~~~~~~~~~~~~~~~~~~~~~(\...
- 1,334
3
votes
1
answer
277
views
Question on the martingale representation theorem
Let $(X_t)_{0\le t\le 1}$ be a continuous Markov martingale (with respect to its natural filtration) s.t. $X_0=0$ and $X_1\in\{-1,1\}$. Can we prove the existence of some measurable function $\sigma: [...
- 1,334
1
vote
1
answer
275
views
consequence of "the best coupling" of two SDEs with different diffusion matrices
My question comes form a potion of the long review paper, which is attached below
In the set-up, $\sigma_1$ and $\sigma_2$ are possibly different, constant diffusion matrices. To my knowledge, if we ...
- 730
2
votes
1
answer
599
views
Convergence in law of stopped stochastic processes
Let $X^n$ and $X$ be stochastic processes defined by
$$X^n_t=1+\int_0^tb_n(s)ds+\int_0^t\sigma_n(s)dW_s \quad\mbox{and}\quad X_t=1+\int_0^tb(s)ds+\int_0^t\sigma(s)dW_s,$$
where $b_n, \sigma_n, b, \...
- 1,334
1
vote
0
answers
54
views
Conditions ensuring that conditional law of a process belongs to a given exponential family
Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
- 77
2
votes
1
answer
773
views
On the continuity of map $\Gamma$
Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>...
- 1,334
1
vote
1
answer
107
views
Law of OU process with time-dependent dynamics
Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
- 77
2
votes
1
answer
389
views
A mean field SDE with hitting time
Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.
$$\int_{(0,\infty)}\mu_0(dx)=1,$$
consider the mean field SDE :
$$dX_t = \mathbf{1}_{\...
- 1,334
1
vote
1
answer
337
views
Bessel process conditioned to stay positive
This question has also been asked on https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive
Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x&...
- 284
2
votes
0
answers
108
views
Existence of solutions to some Mckean-Vlasov SDE
Let $\mathcal P(\mathbb R)$ be the space of probability measures and $(W_t)_{t\ge 0}$ be a standard Brownian motion.
For given functions $b, \sigma, \beta: \mathbb R_+\times \mathbb R\times \mathbb R\...
user128095
1
vote
0
answers
76
views
Gronwall type lemma for an Ito process
For all $t\in \mathbb{R}$ let $h_t = \frac{1}{2} + \int_0^t v_s\cdot dB_s$ be an Itô process, where $B_s$ is a standard Brownian of $\mathbb{R}^d$ and $v_t$ an $\mathbb{R}^d$ valued adapted process, ...
- 245
8
votes
2
answers
3k
views
Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
- 1,927
2
votes
1
answer
538
views
Generalized Fokker-Planck equation
Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...
- 773
1
vote
0
answers
78
views
If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?
Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
- 309
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 93
1
vote
0
answers
222
views
Is my quadratic variation derivative bounded?
Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
- 335
0
votes
1
answer
461
views
Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure
I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
- 227
1
vote
1
answer
82
views
Local inverse bound of Cameron Martin and Banach norms
Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
- 133
3
votes
1
answer
202
views
Onsager--Machlup functional as the density across a mesh of discrete points
It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...
1
vote
0
answers
766
views
Derivative of the function of random variable
Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...
- 195
1
vote
0
answers
57
views
Choice of Banach space for stochastic processes
In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used:
$$
\sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p}
$$
and
$$
\mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p]^...
- 379
2
votes
0
answers
173
views
When is the dual infinitesimal generator of a S.D.E self-adjoint and negative definite?
Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be,
self-adjoint i.e $\int f ({\...
- 2,246
1
vote
1
answer
512
views
Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
- 5,405
1
vote
1
answer
293
views
Time-Reversal of BSDE = SDE
Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$:
$$
Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;
$$
...
- 5,405
0
votes
1
answer
195
views
How to get the mean, skewness of an Itō integral?
If $B_t$ denotes a standard Brownian motion, and let $X_t = \int f(s)dB_s$, $f(s)$ is a deterministic integrand. I know $B_t$ is a martingale. Is $X_t$ also a martingale? And how can I get the formula ...
- 21
0
votes
1
answer
152
views
About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E
We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...
- 2,246
2
votes
0
answers
96
views
Invariant measures of Levy S.D.Es
Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...
- 2,246
0
votes
1
answer
340
views
Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$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\...
0
votes
2
answers
313
views
Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...
1
vote
0
answers
94
views
Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
- 167
2
votes
1
answer
352
views
Estimating the hitting time for a SDE solution
Consider a the following OU process in one dimension,
$$dX = -\theta(X -x_0)dt + \sqrt{s}dW $$
Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$.
Then ...
- 2,246
2
votes
0
answers
140
views
Convergence of the probability that hitting times being infinity
Let $X^n=(X^n_t)_{t\ge 0}$ and $X=(X_t)_{t\ge 0}$ be RCLL (right-continuous with left limits) processes such that
$$\lim_{n\to\infty}X^n=X,\quad \quad \mbox{almost surely},$$
where this convergence ...
user128095
1
vote
0
answers
276
views
Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
1
vote
1
answer
259
views
Show an SDE's solution has positive probability to visit every set in the state space
Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...
- 503
3
votes
0
answers
235
views
Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
user128095
1
vote
1
answer
472
views
Can derivatives of 2 stochastic processes be multiplied?
We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is ...
- 2,246
3
votes
1
answer
234
views
Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...
- 6,853