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.
625 questions
-4
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Most probable path between two points, Onsager–Machlup functional
Say I have an SDE $$dx_t = b(x_t)dt + dB_t$$ where $B_t$ is a Brownian motion. I would like to compute the most probable path between two points $y_0$ and $y_1$.
I know the Onsager–Machlup functional ...
- 41
0
votes
0
answers
91
views
Is the Matérn field $(\kappa^2 - \Delta)^{\alpha/2} u = W$ the stationary distribution of an infinite-dimensional Ornstein-Uhlenbeck SDE?
In spatial statistics, the Matérn Gaussian field on $\mathbb{R}^d$ is often defined as the (weak) solution to the SPDE
$$
(\kappa^2 - \Delta)^{\alpha/2} u \;=\; W,
$$
where $W$ is Gaussian spatial ...
- 787
0
votes
0
answers
18
views
Solving decoupleable families of FBSDEs
Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs
\begin{align}
Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
- 4,842
1
vote
0
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56
views
Can Atlas-model minimize the distance between two stochastic processes?
Consider a filtered probability space in which there exist two independent Brownian motions $W$ and $B$. For every progressively measurable process $u=(u_t)_{t\ge 0}$ taking values in $[0,1]$, define ...
- 189
1
vote
0
answers
77
views
Generalized Malliavin Chain Rule
Let $ E $ be the Banach space of continuous functions on $[0, T]$, equipped with the supremum norm. Let $ \sigma : E \to \mathbb{R} $ be a function of class $ C^1 $ (Fréchet differentiable). Let $ X = ...
- 696
3
votes
0
answers
48
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General Conditions for NFLVR
Background:
An $\mathbb{R}^d$-valued process $Y_{\cdot}=(Y_t)_{t \ge 0}$ is called a $\sigma$-martingale if there exists some $\mathbb{R}^d$-valued martingale $M_{\cdot}$ and a $\mathcal{F}_{\cdot}^M :...
2
votes
0
answers
78
views
Estimates on SDE with Sobolev drift
I am looking for references about the following SDE:
$$\begin{cases}dX_t = b(t, X_t )dt + \sigma dW_t\\X_0=x \end{cases} ,$$
when th drift $b$ is bounded and Sobolev in the space variable. Namely, I ...
- 911
1
vote
0
answers
457
views
How do we prove 1) existence and uniqueness of the global positive solution and 2) extinction for this stochastic system?
Suppose we have the following system:
\begin{equation}
\begin{aligned}
dS &= \Big[\Lambda -\beta_1 S L_2 -\beta_2 S I - \beta_3 S A - \nu S\Big]dt +\sigma_1 SdB_1(t),\\[2ex]
dL_1 &= \Big[p \...
- 73
2
votes
0
answers
194
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Integration by parts for Functional Ito Calculus
I briefly recall the definition of Functionnal Ito Calculus, developped by Bruno Dupire in https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551. The paths we consider are càdlàg (right ...
- 161
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0
answers
73
views
Substitute deterministic with random data
$
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- 1,163
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0
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59
views
References on SDE with jump
I've some questions related to SDEs with jump. Let $W$ be a real-valued Brownian motion, and $N$ be an independent Poisson random measure on $\mathbb{R}_+\times \mathbb N$ of the intensity measure $\...
- 1,468
2
votes
2
answers
582
views
What does stochastic quantization have to do with quantization?
Let me define the way I understand stochastic quantization. Stochastic quantization to me is a way of constructing infinite dimensional measures with a given "density".
Suppose we are given ...
- 2,345
2
votes
0
answers
56
views
Path-wise uniqueness of SDE driven by Poisson process (or more generally by Lévy process)
Consider the SDE for the right-continuous process $(X_t)$ taking values in $\mathbb R^n$ :
$$dX_t=b(X_{t-})dt+\int_{|z|<1}a\big(X_{t-},z\big)\tilde{N}(dt,dz),$$
where $b:\mathbb{R}^n\longrightarrow ...
- 1,701
1
vote
1
answer
92
views
stochastic integral on the canonical space
Let $C[0,T]$ be the space of all continuous functions on $[0,T]$. Let $X$ be the coordinate mapping process, that is $$X_t(\pi)=\pi_t,\;\forall \pi\in C[0,T].$$ Finally, let $\mathbb{F}$ be the ...
- 111
4
votes
1
answer
232
views
References for stochastic flow of diffeomorphisms on compact manifold
Let $M$ be a smooth compact manifold and let $A_0,A_1, \dots, A_N$ be time dependent vector fields. We consider the stochastic differential equation :
$$
dX_t(x) = \sum_{i=1}^N A_i(t,X_t(x)) \circ dW^...
- 195
3
votes
0
answers
207
views
Simple way to simulate SDE solutions?
To simulate an Ito diffusion,
$$dX_t = f(t,X_t)dt + g(t,X_t)dW_t,$$
you can discretize this in time by using an equidistant partition $(t_0^N,t_1^N,\dots,t_N^N)$ of the time interval $[0,T]$. The ...
- 718
1
vote
0
answers
87
views
Generalization of Fernique theorem for SDE
Let $(\Omega,\mathcal{F},\mathbb{P})$ a probability space and $(W_t)_{t \ge 0}$ a standard Brownian motion, with $\mathbb{F}$ is natural filtration. Let $(X_t)_{t \ge 0}$ a continuous adapted process ...
- 143
1
vote
0
answers
95
views
Regularity of backwards linear parabolic PDEs
In [1] or [2] one can find a regularity theory for (quasi-)linear parabolic PDEs of with Cauchy data, on Euclidean domains. However, in each case the boundary data is "forward in time", ...
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4
votes
1
answer
164
views
Reference for density of dynamic $\Phi_2^4$ SPDE wrt SHE
I am interested in the SPDE
$$\partial_t u=\Delta u-u^3+\xi$$
on the $2$ torus $\Lambda$, started at the invariant measure $\mu$ on $C^{-\alpha}$. It is well known that $\mu$ has a density with ...
- 2,345
0
votes
1
answer
127
views
Why do 1D diffusion exit times scale with the square of interval width?
Migrated from the MSE.
Let $W_t^y$ denote a one-dimensional Brownian motion with initial position $y\in(a,b)$. Furthermore, let
$\tau_{a,b}^y=\inf\{t\geq 0:W_t^y\notin(a,b)\}$ denote the exit time of ...
2
votes
1
answer
154
views
Conditional hitting time distribution for OU process between two fixed barriers
For an the OU process $\mathrm dX_t=-\omega X_t\,\mathrm dt+\omega\mathrm dW_t$ with $X_0=x_0$ and $L<x_0<U$ let
$$
\tau=\inf\{t>0:X_t\notin(L,U)\}
$$
denote the hitting time of either ...
4
votes
0
answers
119
views
Most general form of Yamada-Watanabe's result
Let $E$ be a Banach space. Consider the stochastic differential equation valued in $E$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dW_t,\quad \forall t>0,$$
where $b, a: \mathbb R_+ \times E \to E$ are ...
- 1,701
3
votes
1
answer
132
views
A special SDE with random coefficient
Let $(S_t: t\ge 0)$ be a continuous process defined on some filtered probability space and satisfying
$$dS_t = C_t(1-S_t) dW_t,\quad \forall t>0,$$
where $(C_t: t\ge 0)$ is adapted so that $|C_t| \...
- 1,701
2
votes
0
answers
146
views
Continuity of Law of stopping times with respect to the starting point of Brownian motion
Let $W = C(\mathbb{R}; \mathbb{R})$ equipped with the topology of locally uniform convergence. We use $W$ as a canonical Wiener space and equip it with a family of probability measures $(\mathbb{P}_x)...
- 389
4
votes
1
answer
186
views
Stability of stochastic differential equations
Consider an SDE of the form
$$dX^\mu_t = a(t, X^\mu_t) dt + \sigma(t, X^\mu_t) dB_t$$
with initial condition $X^\mu_0 \sim \mu$, where $\mu$ is some measure on $\mathbb{R}^d$.
I am searching for ...
- 389
0
votes
0
answers
57
views
Conditions on SDE coefficients for well-posedness of Fokker-Planck equation
Consider the following $n$-dimensional Ito-SDE:
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
What are the necessary regularity conditions on $\mu$ and $\sigma$ to ensure that the ...
- 121
1
vote
0
answers
91
views
What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 454
2
votes
1
answer
388
views
Self-adjointness of generator and semigroup of an SDE
$
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- 1,163
1
vote
0
answers
81
views
$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 321
1
vote
0
answers
140
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 321
3
votes
1
answer
145
views
Designing an SDE satisfied by $\frac{B(t)}{1+t}$
Let $B$ be the Brownian motion. I want to find a stochastic differential equation satisfied by the process $$X(t) = \frac{B(t)}{1+t}.$$ I am trying to use Itô's lemma for $f(x,t) = \frac{x}{1+t}$ but ...
- 73
2
votes
0
answers
78
views
Existence and moment estimation for a linear stochastic differential equation (SDE) with random coefficients
Let $W$ be one-demensional Brownian motion, and suppose $X$ satisfies the following SDE
$$
\mathrm{d}X_s=(A_sX_s+B_s)\mathrm{d}s+(C_sX_s+D_s)\mathrm{d}W_s, \quad X_0=x_0\in\mathbb{R}^n,
$$
where $A, C\...
- 135
2
votes
0
answers
143
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 9,980
2
votes
0
answers
111
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,701
2
votes
0
answers
75
views
Approximate the adjoint generator of the discretization of an SDE
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$;
$p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$
$(X_t)_{t\ge0}$ denote ...
- 249
3
votes
0
answers
117
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 ...
2
votes
1
answer
221
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 9,980
4
votes
2
answers
851
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 ...
- 1,017
2
votes
2
answers
159
views
Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
- 9,980
5
votes
1
answer
357
views
Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
- 389
0
votes
0
answers
115
views
When we should integrate on both side over a SDE?
Maybe I am quite stupid, I am quite confused about, when we should use ito formula to solve SDE and when it is appropriate to integrate directly to get the solution?
Specifically, let us consider the ...
- 9
5
votes
0
answers
519
views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 607
2
votes
0
answers
152
views
Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
- 135
5
votes
1
answer
313
views
When does an Itô diffusion give a semigroup on $L^2$
I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$.
I have a time-homogeneous Itô diffusion of the form
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t$...
- 323
2
votes
0
answers
59
views
Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
4
votes
1
answer
383
views
Impulse signal detection
Notation: Here $\mathcal Y_t$ denotes the natural filtration of the process $Y_t$, and $\{\cdot\}$ denotes the fractional part of a real number.
This question concerns detecting the presence (or ...
- 9,980
4
votes
1
answer
176
views
Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...
- 73
6
votes
1
answer
189
views
Simulation of SDEs using Karhunen Loeve expansion
A very common and easy way to simulate the solution of a SDE is to use the Euler-Maruyama method. At each time step the only random part comes from the realization of the increment of the Brownian.
It ...
2
votes
0
answers
47
views
Uniqueness of the solution to switching coefficient SDEs
Consider the following SDE driven by real-valued Brownian motion $W=(W_t)_{t\ge 0}$:
$$dX_t = \left(\sigma {\bf 1}_{\{X_t>1\}} + \sigma' {\bf 1}_{\{0<X_t\le 1\}}\right)dW_t,\quad \forall t>0,$...
- 1,468
4
votes
0
answers
470
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 ...
- 1,017