# 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.

269
questions

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### If a probability measure is stationary in both forward time and reverse time, does this imply that the measure is incompressible?

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact metric space. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable ...

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19 views

### Derivation of a differential equation from a SDE

Suppose there is a non-homogeneous Markov process with state space $\mathbb{R}_{+}$
driven by this McKean-Vlasov-tipe SDE:
$$ dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$
where $...

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81 views

### Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...

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**1**answer

238 views

### Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...

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104 views

### The probability distribution of “derivative” of a random variable

Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...

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**2**answers

93 views

### Backward Stochastic Differential Equation

Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and
$$
dX_t=f_tdt+B_tdW_t
$$
where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...

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62 views

### Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field"
Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$
Consider the ...

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**1**answer

220 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

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79 views

### Sufficient conditions for taking limits in stochastic partial differential problems

Let's say we have a Cauchy problem:
$$
(1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t)=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$
$$(2) \hspace{0.5cm} u(x,0)=u_0(x),
$$
where $x \in ...

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50 views

### Lebesgue Integral in SDE

In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form
$$
\int_0^t f(s,X(s))ds.
$$
If we ...

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64 views

### Defining weak solutions to infinitely many SDEs on the same probability space

Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...

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25 views

### Example of a “very noisy” SDE on a compact manifold with zero maximal Lyapunov exponent

Setting:
Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure.
Write $T_{\neq 0}M \subset TM$ for the non-...

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107 views

### How to judge the solution process of an SDE to lie on the sphere?

Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...

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59 views

### Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation

On all you geniusses out there: this is a tough one.
Preliminaries and Rigorous Technical Framework
Let $T \in (0, \infty)$ be fixed.
Let $d \in \mathbb{N}_{\geq 1}$ be fixed.
Let $$(\Omega, \...

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**1**answer

138 views

### Simulation of Itô integral processes where integrand depends on terminal (Volterra process)

I need to simulate a process of the form
$$X_t=\int_0^t f(s,t)\mathop{dW_s}$$
where $f$ is deterministic and the integral is an Itô integral. I know I can simply take finite Itô sums of discrete ...

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60 views

### Smoothness of expectation

Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...

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93 views

### Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation
\begin{gather}
\frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\
u(0,x) = \...

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74 views

### Conditioning on an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...

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60 views

### Can there be a explicit expression of g as defined in the link

This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...

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39 views

### Characterization of Time-homogeneous flows for conditional expectation

Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...

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**1**answer

140 views

### Why control a continuous approximation of stochastic gradient descent instead of just the SGD?

In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...

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97 views

### Domain of the Generator of a Bessel process

Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...

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187 views

### Divergence form degenerate pde and Feynman Kac

Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...

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65 views

### 2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...

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36 views

### Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

Let
$\varrho\in C^1(\mathbb R)$ with $\varrho>0$
$\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$
$\mu$ denote the measure with density $\varrho$ with respect to $\lambda$
$b:=2^{-...

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40 views

### Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

Let
$h\in C^2(\mathbb R)$
$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)...

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77 views

### Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...

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60 views

### How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?

Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...

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78 views

### How is the Cauchy-Schwarz inequality used in the proof of Lyapunov's criterion in the book “Analysis and Geometry of Markov Diffusion Operators”

Let $(E,\mu,\Gamma)$ be a full Markov triple (see definition below), $J\in\mathcal A$ with $J\ge1$ and $g\in\mathcal A_0$. In the proof of Theorem 4.6.2 of the book "Analysis and Geometry of Markov ...

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**1**answer

206 views

### Existence of a Lyapunov function for a log-concave measure

Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...

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95 views

### Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...

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28 views

### Bayesian parameter estimation

I am generally not that knowledgeable for math, so if my question is too broad or inaccurate, please let me know.
I am currently reading a paragraph of one paper (https://www.fil.ion.ucl.ac.uk/spm/...

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**1**answer

75 views

### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

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**1**answer

90 views

### Is there solution to a backward stochastic differential equation with $yz$ in the generator?

Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...

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41 views

### Reference: Stochastic Optimal Control with cost functional

There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by
$$
J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right],
$$
function $g$ and ...

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48 views

### Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as
$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$
where $X_0,X_1,...,X_k$ are ...

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71 views

### Approximation of deterministic problems in the PDEs with the stochastic ones

A lot of problems in PDE theory are solved in the following way:
The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and ...

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**1**answer

110 views

### Reference from the article “Random Ordinary Differential Equations”, by J.L. Strand

In the article Random Ordinary Differential Equations, Journal of differential equations 7, 538-553 (1970), by J.L. Strand, reference number 6 refers to his PhD thesis: Stochastic Ordinary ...

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45 views

### Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$

Consider the linear discrete-time stochastic systems:
\begin{equation}
x_{k+1} = Ax_k + v_k,
\end{equation}
with time-instants $k \in \mathbb{N}$, state $x_k \in \mathbb{R}^n$, stochastic process $v_k ...

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338 views

### Is an SDE really equal to an integral equation, or is it rather “its integral” that is?

Ive been told and been reading in some textbooks on SDE's that an SDE really is an integral equation. In other words, that
$ dX= \beta dt + \sigma dW$ $\,$ "really means" $\,$ $X_{t}= X_{0} +\int_{0}...

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57 views

### Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...

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43 views

### Floquet stochastic process

Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...

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22 views

### Singular direction of a particle system

Consider a system of n-sdes in $\mathbb{R}$ ( the formula is not important).
The corresponding particle system $X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$ lives in $\mathbb{R}^{n}$ and assume that ...

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**1**answer

414 views

### English translation of “Les aspects probabilistes du contrôle stochastique”

I am looking for an English translation of "Les aspects probabilistes du contrôle stochastique" written by Nicole El Karoui, or knowledge whether it exists.
Other references with similar content on ...

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69 views

### Stationary distribution of overdamped Langevin dynamics

Consider the over damped Langevin dynamics: $d X_{t} = d B_{t} - \nabla U(X_{t}) dt $ on $\mathbb{R}^{d}$ where $B_t$ is a standard Brownian motion. On pages 29 and 30 of the following book
Royer,...

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71 views

### Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...

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58 views

### Why does the correct scaled dimension for SPDEs count time as two dimensions?

In this video, Felix Otto says that the correct way to count dimensions for parabolic equations is $2+\text{number of space dimensions}$. He said nothing about this. In the accompanying notes it is ...

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106 views

### Is there a distinct Ito-Sasaki version of Riemannian stochastic development?

Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...

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59 views

### Reference request on theory about Stochastic Riemann problem

I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit.
On one hand, in the literature it is not hard to find books or papers that deal ...

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175 views

### Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...