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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.

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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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21 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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1answer
57 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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52 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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27 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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1answer
182 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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31 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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1answer
51 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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1answer
78 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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21 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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24 views

Application of Gram-Charlier expansion for Swaption pricing with drift extension

I am doing a project for university and I'm stuck at some point. The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension. I already found out how ...
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1answer
151 views

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

I posted this question on mathstack a couple of weeks ago and even with 100 bounty on it Ive not been able to get any feedback. Hence I tought Id try posting it here. https://math.stackexchange.com/...
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63 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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35 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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1answer
86 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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40 views

Approximation of deterministic problem with stochastic problem

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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148 views

Reference request on connection between PDE problems

I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
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70 views

Vanishing viscosity method and random forcing

The vanishing viscosity method consists in viewing problem: $$(A) \hspace{1cm} u_t+g(u)_x = 0,\\[2ex] $$ as the limit of the problem: $$(B) \hspace{1cm} u_t+g(u)_x+\nu \varDelta u = 0,\\[2ex]...
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42 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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1answer
165 views

Conditional stochastic integration

Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g. $$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$ What is the ...
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38 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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54 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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39 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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20 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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86 views

Additive stochastic heat equation and Markov property

For the distribution-valued field called the GFF (Gaussian free field) over a domain $\Omega\subset \mathbb{R}^{2}$: $$ h_{\Omega}=\sum a_{k}f_{k,\Omega}(z),$$ where $a_{k}\sim N(0,1)$ and $f_{k},\...
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1answer
395 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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27 views

Stationary distribution of gradient dynamics

We consider the gradient dynamics $ d X_{t} = d B_{t} - \nabla(U(X_{t}))dt $ in $\mathbb{R}^{d}$. G.Royer in the book "An initiation to logarithmic sobolev inequalities" (p29,30) says that if (1) U ...
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99 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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63 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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49 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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32 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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2answers
335 views

Good papers on stochastic differential equations with applications in finance

I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) ...
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87 views

Exit time of a stochastic process defined by a SDE

Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation \begin{align*} \...
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120 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 ...
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28 views

Generators and Covariance Operators of Diffusions

For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
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1answer
313 views

Is there any reason to use paracontrolled calculus over regularity structures?

Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
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92 views

What are morphisms between regularity structures?

In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here: Is there any way of extending this to morphisms ...
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2answers
321 views

A Stochastic Taylor Expansion/Asymptotics

Question: Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth ...
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0answers
62 views

Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process. Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
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1answer
149 views

Is there an Itō formula for random functions in infinite-dimensions?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
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39 views

Reflected SDE with vanishing diffusion term at the boundary

I have a reflected SDE of the form $dX_{t} = a(X_{t})dt + b(X_{t})dW_{t}+dL_{t}, \quad X_{0}=x_{0}\in (0,\infty) $ where $L$ keeps the process from going below zero. The coefficients $a$ and $b$ ...
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0answers
147 views

Associative law of the stochastic integral in Hilbert spaces

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ ...
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1answer
83 views

Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
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53 views

invariant measure theory for spdes with distributional solutions (Hilbert versus Polish)

SPDEs such as the stochastic heat equation for $d\geq 2$ with space-time white noise and the stochastic quantization equation have distributional solutions and we still try to make sense of their ...
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1answer
143 views

Are Holder Condition and signal to noise ratio (SNR) related?

This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately. This question has evolved from ...
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0answers
91 views

Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
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80 views

“Expanding” around the invariant measure

In the spde literature we have results of the form $$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$ where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $...
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0answers
100 views

Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain

Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where $\xi$ is the space-time white noise $$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...
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2answers
125 views

Is it possible to compare Rough path theory and White noise Theory?

Please accept apology if this question is vague. (Would you please comment rather then downvote, I may be stopped to ask more questions. I will delete my question if required.) It is related to the ...
3
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1answer
105 views

Under what condition we get back path from signatures in rough path theory?

A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...