# 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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### 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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55 views

### Solution of stochastic ODE stationary?

Consider the following ODE:
$$\frac{ d \gamma(x,t;\tau)}{d \tau} = R(\gamma(x,t;\tau)) ; \qquad \gamma(x,t,t)=x. $$
$R$ is smooth enough, bounded away from zero and a stationary process. Is there a ...

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56 views

### Computing the Malliavin Derivative

Let $X_t$ be a continuous local-martingale modeling the stock price given by
$$
X_t = \int_0^t \sigma_t(T,K)dW_t
,
$$
and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...

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40 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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320 views

### Tanaka-Meyer formula

I have a simple question about Tanaka-Meyer formula, I am having difficulty applying it. Let $X$ be a continous martingale vanishing at zero. From Tanaka-Meyer formula it holds $$d|X_t| = sgn(X_t)dX_t+...

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117 views

### Optimal control of SDEs

I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...

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227 views

### Estimate for the composition of two Hilbert-Schmidt operators

Let
$U$, $H$, $\tilde H$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
$Q$ be a self-adjoint and nonnegative nuclear linear operator on $U$
$\Psi$ be a Hilbert-Schmidt operator from$^1$...

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**1**answer

311 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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160 views

### Stopping time property

Hi, I am reading a textbook about SDE, and am very confused about the transition
$$X_T 1_{T\lt t} + E\{X_T 1_{T\geq t} | F_{t\wedge T}\}$$
$$= X_T 1_{T\lt t} + E\{X_T | F_t\} 1_{T\geq t}$$
I ...

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247 views

### Almost sure stability of a scalar, nonautonomous, nonlinear SDE

I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...

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**1**answer

197 views

### Walker whose Velocity is a Brownian Bridge

Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...

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152 views

### Walk with randomised boosts

The classical random walk can be described as the evolution of the position $X_t$ of a walker for integers $t \geqslant 0$, where $X_0 = 0$ and $X_t = X_{t-1} + V_t$ for $t \geqslant 1$, where the "...

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272 views

### How to define (and solve) the diffusion equation with a sticky boundary at the origin?

For the diffusion equation $\frac{\partial} {\partial t} P_t(x)=D \frac{\partial^2} {\partial x^2} P_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x} P_t(...

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192 views

### Infinite-time, Path-Dependent Expected Value of an Orstein-Uhlenbeck process

I am dealing with an Orstein-Uhlenbeck process $X_t$ with its stochastic differential equation being
$$dX_t=(\mu-X_t)dt+\sigma dW_t.$$
I want to show
$$\mathbb{E}\left[\frac{|X_\infty|}{\int_{0}^{\...

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63 views

### Generalisation of Lyapunov time to stochastic dynamical systems

Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...

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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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92 views

### What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?

Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...

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**1**answer

190 views

### Asymptotic form of pdf of Escape Time of arithmetic fBm

I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...

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124 views

### Time Interval of Existence of an SDE solution with Locally Lipschitz Drift

Consider the stochastic ODE $$
dX = F(X)dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "almost ...

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51 views

### Convergence of empirical measure in case of proliferation

I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:
I don't understand why he calls $u(t,x)$ a time dependent ...

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182 views

### Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE
$$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$
with $W_t$ a Wiener process, then the associated ...

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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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99 views

### Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...

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444 views

### For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$

Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...

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74 views

### Stochastic stability of “open” continuous-time stochastic systems: reference request

I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...

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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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99 views

### Generator of spacetime Markov Process is Parabolic?

Suppose one considers some non-autonomous SDE thereby the Markov transition function is not homogeneous. In order to "recover" some homogeneity, one can consider the "spacetime" or "lifted" ...

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98 views

### Is the stochastic integral invariant under equivalent change of probability?

Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\...

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334 views

### Solution of multivariate Geometric Brownian Motion?

It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve
\begin{equation} \begin{cases}
dX=X \, dW_1+...

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150 views

### Regularity for Stochastic heat equation with additive noise in d=2

I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u_{t}=\Delta u+f$ in great detail, especially in dimension 2.
In Hairer's Spde notes , he ...

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46 views

### Assertion of Local Martingale

I am currently reading a proof of the Feynman-Kac representation theorem. The main step in the proof is to consider an "interpolation martingale" which has the form $$M_s := \varphi(t-s, x+B_s)\exp \...

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62 views

### Uniqueness of a stationary measure for a simple ODE

How to prove uniqueness of the stationary measure using coupling approach to the following process: $u_{n+1}=S(1)u_n+\eta_{n+1}(\omega)$, where
a) $S(1)y_0:=y(1)$, $y(t)$ solves the following Cauchy ...

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52 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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122 views

### stochastic recurrence relation “convergence”

Consider a recurrence relation $x_n = f(x_{n-1})$. Suppose $f(x_n)<0$ for some "large enough" $n$. Now consider a "stochastic variant" $x_n = f(x_{n-1})+y_n$ where $y_n$ is a sequence of random ...

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201 views

### Uniqueness of a SDE with non-negativity constraint

I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...

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**1**answer

243 views

### Why would one work with Kushner-FKK equation over Zakai equation?

In stochastic filtering you are interested in a process called the optimal filter $\pi_t$ which is a probability measure(d stochastic process). You can consider the unnormalized version $V_t$.
The ...

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84 views

### Ito formula between manifolds

I have seen many Ito formulae giving dynamics for $f(X_t)$ where $f:M\to \mathbb{R}$ is a smooth function from a manifold $M$ and $X_t$ is a (continous) manifold-valued semi-martingale.
My question ...

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**1**answer

318 views

### Existence of normal number except random numbers

For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...

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54 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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**2**answers

359 views

### Kolmogorov continuity theorem and Holder norm

The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...

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223 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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**1**answer

196 views

### Uniqueness of a SDE with positivity constraint

We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...

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26 views

### Finding an SDE given constraints over the final distribution

I want to find the SDE
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t
$$
such that, if the initial condition is $x_{t=0}=x_0$, we have the following constraints:
$\mathbb{E} X_t = \alpha x_0^{\beta}$
$\mathbb{V} ...

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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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**2**answers

176 views

### Large deviation bound for O-U process

Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...

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**1**answer

50 views

### Filtering Mixed Discrete and Continous

Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...

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**1**answer

147 views

### Generalisation of Strassen's (Kellerer's) Theorem

Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first movements, i.e.
$$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$
$\mu$...

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73 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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46 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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**1**answer

102 views

### Sequence of diffusions

Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?