# 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

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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 ...

3
votes

0
answers

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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 ...

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 \...

5
votes

1
answer

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### 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 ...

1
vote

1
answer

129
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### 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}(...

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 ...

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 ...

0
votes

0
answers

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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 ...

4
votes

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answers

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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

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### 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)
\\
&...

5
votes

0
answers

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### 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 ...

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 ...

6
votes

2
answers

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### 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 $...

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 ...

2
votes

1
answer

153
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### 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 $B_1 \sim N(0,K)$...

2
votes

0
answers

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### 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 ...

2
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^...

2
votes

0
answers

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### 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 ...

2
votes

1
answer

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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 $x$ ...

1
vote

1
answer

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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 ...

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,...

1
vote

1
answer

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### 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.
...

0
votes

0
answers

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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 ...

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 ...

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 ...

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(...

2
votes

0
answers

107
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### 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*}...

1
vote

0
answers

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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 = ...

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:...

1
vote

0
answers

155
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### 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,~...

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,...

0
votes

0
answers

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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+...

0
votes

0
answers

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### 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-...

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....

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}...

0
votes

1
answer

108
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### 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

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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 ...

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

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### 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 ...

1
vote

1
answer

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### 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{...

4
votes

1
answer

347
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### 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(\...

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 \...

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
&...

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
&...

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 $\...