Questions tagged [stochastic-differential-equations]
Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.
514
questions
6
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0
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Fokker-Planck equation for SDEs on manifold
Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate ...
- 437
3
votes
0
answers
82
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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 ...
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1
vote
0
answers
73
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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:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
- 11
5
votes
1
answer
150
views
Does the entropy of a SDE with nondegenerate noise always increase?
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
- 2,574
1
vote
1
answer
129
views
Convergence of the quadratic variation process
Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$
$$
\lim_{n\to\infty}\mathbb{E}(...
- 79
2
votes
0
answers
68
views
Local martingale for a (two-dimensional) diffusion
Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
- 1,877
2
votes
1
answer
113
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Existence of solution for a non-linear SDE
Since $\exp(\cdot)$ is locally Lipschitz, the following SDE has a strong solution:
$$
\mathrm{d}X_s=\exp(X_s) \, \mathrm{d}B_s,\quad X_0=1,
$$
where $B$ is a standard Brownian motion. I wonder if the ...
- 103
0
votes
0
answers
46
views
Reference request: Gaussian estimates for SDE with discontinuous diffusion coefficient
Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R_+ \times \mathbb R^d \to \mathcal M_{d \times d}^{\text{sym}} (\mathbb R)$ be bounded measurable where $\sigma$ is ...
- 713
4
votes
0
answers
124
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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 ...
2
votes
0
answers
63
views
Feynman-Kac for PIDEs: to jump or not to jump?
Consider the following Cauchy problem for a $\mathscr{C}^2$ function $F$ characterized by a PIDE:
\begin{align}
\begin{cases}
& F_t(t,x)+\alpha(t,x)F_x(t,x)+\frac{1}{2}\beta^2(t,x)F_{xx}(t,x)
\\
&...
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5
votes
0
answers
44
views
Feynman-Kac statement with no boundedness condition
Theorem 5.3 of Friedman (1975, Volume I) and its version in Theorem 7.6 of Karatzas & Shreve (1991) both establish conditions under which the Feynman-Kac formula holds, namely there is a ...
- 171
1
vote
1
answer
74
views
Phase space Brownian bridge
I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$
s.t. $x_0=0$ and $x_1=0$
where $dw_t$ is a Wiener process.
I am thinking about ...
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6
votes
2
answers
215
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
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0
votes
0
answers
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Langevin dynamics or stochastic gradient flow for grand canonical ensemble
We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...
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2
votes
1
answer
153
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 $B_1 \sim N(0,K)$...
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2
votes
0
answers
47
views
Stochastic differential equations driven by composed Poisson process
Consider the stochastic differential equation as follows:
$$X_t = x + \int_0^t b(X_s)\,ds + \int_0^t a(X_{s-})\,dL_s,\quad \forall t\ge 0,$$
where $L=(L_t)_{t\ge 0}$ denotes some Lévy process. What ...
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votes
1
answer
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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^...
- 713
2
votes
0
answers
139
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 ...
- 243
2
votes
1
answer
106
views
Uniqueness of the solution to stochastic differential equation
Let $W$ be a Brownian motion and consider the SDE
$$dX_t = b(t,X_t) \, dt + a(t,X_t)\,dW_t,\quad \forall t\ge 0. \tag{$\ast$} $$
Assume that $x\mapsto b(t,x), a(t,x)$ are locally Lipschitz in $x$ ...
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1
vote
1
answer
105
views
On a martingale defined via some SDE
Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE)
$$dX_t = C(t)(1-X_t)dW_t,\quad \forall t\ge 0,$$
where $C$ is a continuous and bounded function. Under ...
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4
votes
1
answer
119
views
Finite number of ergodic random Dirac measures
Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map
\begin{align*}
\theta\colon T\times \Omega &\to \Omega\\
(t,...
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1
vote
1
answer
49
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.
...
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votes
0
answers
25
views
How can I obtain a SDE with an advection function that contains the difference in covariates?
Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
- 101
0
votes
0
answers
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views
Can I use a derivative in my SDE's advection function?
Suppose that I have the following SDE:
$$\frac{d\mathbf{x}(t)}{dt}=\mathbf{f}(\mathbf{x}(t)) + \boldsymbol{\eta}(t),$$ where $\boldsymbol{\eta}(t)$ is white noise and $\mathbf{f}(\cdot)$ is an ...
- 101
1
vote
0
answers
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views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 31
0
votes
1
answer
87
views
Existence of linear stochastic differential equation given solution
Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
0
votes
1
answer
107
views
Can we define the divergence of a stochastic process?
Suppose I have a stochastic process $(X_t)_{t\in \mathbb{R}^d}$ with infinitesimal generator $\mathcal{A}$, for example $\mathcal{A}f(X) = -\mu f'(X) + \frac{1}{2}\sigma^2f''(X)+\lambda \int (f(X')-f(...
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votes
0
answers
107
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
73
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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
$$
d X_t = ...
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3
votes
1
answer
124
views
Reference request for a Riemannian Fokker-Planck equation
The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...
- 33
1
vote
0
answers
155
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 11
3
votes
1
answer
136
views
Small noise limits with irregular drift
Let $W$ be a standard $d$-dimensional Brownian motion.
Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0,...
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0
votes
0
answers
91
views
Predictability of the mild solution of a SPDE
Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
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0
votes
0
answers
55
views
Recursive formula for approximate multiple Wiener integrals
Given $m$ $d$-dimensional Brownian motion and a multi-index $(j_1,...,j_l)$ with $j_i \in \{0,1,...,m\}$ we can define the multiple Stratonovich integral
$\int_0^t \circ dW_{s_1}^{j_1}...\int_0^{s_{l-...
- 243
3
votes
2
answers
370
views
Blow up limits for SDE
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \, , \, X_0 = 0$$
with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous....
- 2,574
1
vote
1
answer
172
views
SDE with non-degenerate diffusion visits every point
I am asking an extension of the question here for SDEs of the Ito form.
Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...
- 405
0
votes
1
answer
108
views
Expectation of stochastic integral
Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation
$$ \left\langle \int^...
1
vote
0
answers
47
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},(\...
4
votes
1
answer
269
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 ...
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1
vote
1
answer
138
views
Textbook definition for "path measure" or "probability measure over paths"
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
3
votes
0
answers
130
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 ...
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1
vote
1
answer
90
views
How to rigorously prove that this sequence of stochastic processes converges to a deterministic process?
Assume that for each $n\in\mathbb{N}$, there's a stochastic function $f_n$ of type $\mathbb{R}^{m}\to\Delta\mathbb{R}^{m}$, and for each $x\in\mathbb{R}^{m}$, the distributions $\frac{f_n(x)-x}{\frac{...
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4
votes
1
answer
347
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
166
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
149
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
86
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
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 417
2
votes
1
answer
123
views
Does the time of maximum of a diffusion process admit a continuous density?
Let $W$ be a standard one dimensional Brownian motion, and consider the solution $X$ to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $X_0 = 0$ a.s., and where $\mu, \sigma: \mathbb R \...
- 2,574
0
votes
0
answers
66
views
Regularity of solutions to forward-backward stochastic differential equations
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\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
&...
- 208
0
votes
0
answers
46
views
Solutions to forward-backward stochastic differential equations in special Ansatz
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\left\{
\begin{aligned}
d X_t
&= aP_t dt +bdB_t;\\
X_0
&= x_0;\\
d P_t
&=cP_t dt + c^*Z_t dB_t;
\\
P_T
&...
- 208
4
votes
1
answer
239
views
Convergence of a continuous time stochastic gradient descent algorithm
Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions
$$\lim_{x \to -\infty} \nabla f(x) = -\infty, \lim_{x \to \infty} \nabla f(x) = \infty.$$
and let $\...
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