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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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172 views

Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
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85 views

Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is ...
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94 views

Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
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149 views

Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution. The SDE is - $ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$ where $b_1$ and $b_2$ are two ...
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197 views

The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
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110 views

probabilistic interpretation of a finite difference scheme

Let me start with some simple background. Consider the heat equation : $ \frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} \quad \mbox{in} \quad \mathbb{R}\times (0,\...
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944 views

Expected value of a stochastic integral expression

I am wondering if the following expression can be processed a bit analytically, $$ E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right], $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
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120 views

Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-...
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57 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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57 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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37 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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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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65 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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43 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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41 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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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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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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30 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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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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126 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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48 views

Has this type of pathwise (S)DE been studied before?

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before. Let $(G,\ast)$ be an abelian $C^1$ Lie group....
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124 views

Moment Estimate

Let $\epsilon > 0$ be a small parameter and consider the following lemma. Lemma. Let $B(t)$ be a bounded, continuous, $R^{n \times n}$-valued function defined on a time interval $[0,T]$ such that ...
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41 views

Martingale covariation operator in infinite-dimensions

Let $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space $U,H$ be separable $\mathbb R$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
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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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53 views

Domain of a Reflected SDE reference

I am currently investigating the domain of the infinitesimal generator of a reflected stochstic differential equation (for a smooth and bounded domain) with Lipshitz coefficients. Namely SDEs of the ...
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57 views

Onsager-Machlup Function of a Killed Diffusion Process

Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
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224 views

Construction of the quadratic variation for Hilbert space valued local martingales

Let $H$ be a separable $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a ...
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66 views

Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Remark: I've asked this question on MSE as well. Let $T>0$ $I:=[0,T]$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
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74 views

Ito's formula for jump diffusions

Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...
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47 views

Stochastic Control with Stochastic Cost-functional

Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also? That is, let $X_t^u$ is the solution to a controlled SDE $$ dX_t = \mu(t,u_t,X_t^u)dt ...
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24 views

Generalized polynomial chaos with 2D independent uniform variables for first order equation

I'm trying to transform the first order differential equation $\dot{x} = k x(t)$ into the corresponding set of stochastic differential equations. I have two independent uniformly distributed ...
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49 views

Matching Numbers in Ito McKean

Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as $e_1 = \lim_{b \...
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30 views

numerical scheme for SDE and empirical estimation of rate of convergence

Consider $\{X_t , t \geq 0 \}$ real valued diffusion satisfying $$ d X_t = b(X_t) d t + \sigma (X_t) d W_t, \quad X_0 = x \in \mathbb{R} $$ where $b, \sigma$ are well-behaved functions and $W$ is a ...
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49 views

Time discretization of the (stochastic) Navier-Stokes equation

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonnempty and open $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$ I've found a thesis where ...
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0answers
166 views

Mean and Variance of SDE

What is the mean and the variance of $y_t$, given the following SDE: $dy_t = -x_t y_t dt + \sigma_1 dW^1_t$ $dx_t = -\sigma_2 y_t dW^2_t$ $W^1$ and $W^2$ are (possibly correlated) Wiener ...
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0answers
137 views

Transforming reaction-diffusion equations to random walk processes

I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
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0answers
143 views

Conditional Expectation of solution to SDE

Suppose $g \in C^{1,2,2}$ and $X_t$ and $Y_t$ solve some SDEs $$ dY_t = a(t,X_t,Y_t)dt +b(t,X_t,Y_t)dW_t $$ and $$ dX_t = c(t,X_t)dt + d(t,X_t)d\tilde{W}_t $$, then Iio's lemma implies $$ g(t,X_t,...
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0answers
24 views

Usually trivial Excursion-type process

How Can i construct a stochastic process $X_t$ which has the property that: $X_t \in [0,1]$ for all $t \in [0,T]$ and $m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]...
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0answers
61 views

Stochastic Inverse

Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying $$ \int_0^Tf(H_t,...
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0answers
77 views

Continuity of solution map to Stratonovich Integral

For paths $x:[0, T] \rightarrow \mathbb{R}^n$, the Stratonovich integral along a one form $\omega$ on $\mathbb{R}^n$ can be defined by $$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\...
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0answers
95 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
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0answers
79 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
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0answers
84 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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188 views

Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
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0answers
106 views

Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes: [1] F. Soucaliuc, Réflexion entre deux diffusions ...
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123 views

Convergence of approximate quadratic variation in $L^p$

For a diffusion $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...
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0answers
370 views

Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is $$ P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
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0answers
71 views

The Stratonovich formulation of the Double Mean Reverting Model

I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice ...
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0answers
87 views

Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...