# 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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### 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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42 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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22 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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61 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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31 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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41 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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25 views

### A SDE of Mean-Field Type

Let $X$ be a stochastic process with
$$
E[ \sup_{t \in [0,T] } |X_t|^2 ] < \infty.
$$
Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ be bounded. I am looking for a solution of the mean-field SDE
$$...

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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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85 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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69 views

### Solutions to the Bond Pricing Equation

Consider a spot rate of the form:
$dr = (\eta - \gamma r) dt + \sqrt{\alpha r + \beta} dW$
where all parameters are constants.
Lets look for a solution of the form $Z(r; t) = e^{A(t;T) - r B(r; T)...

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36 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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148 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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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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390 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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25 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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84 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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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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21 views

### limit of large time of an ergodic diffusion and differentiability with respect to a parameter : towards Ellis-Gartner theorem

Consider an ergodic diffusion. For instance, we can think of $X_t \in \mathbb{R}^d$ satisfying
$$
d X_t = b(X_t) d t + \sigma d W_t,
$$
here $W$ is a $\mathbb{R}^m$ valued Wiener process, $\sigma$ is ...

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44 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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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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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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113 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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**2**answers

329 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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27 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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81 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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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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61 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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82 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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142 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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52 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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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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79 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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**1**answer

93 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 ...

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90 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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47 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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82 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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110 views

### An SDE version of a Fokker Planck Equation

Assume $\rho$ is probability density defined on $\mathbb{R}^d\times\mathbb{R}^d$. I am interested in the Wasserstein gradient flow of a functional:
\begin{equation*}
\mathcal{E}(\rho)=\iint_{\mathbb{...

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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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121 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 ...

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59 views

### Ornstein-Uhlenbeck type process with thresholding

(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...

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20 views

### Pathwise closeness of solutions of SDE's

Given two SDE's, if the diffusion coefficients are pathwise uniformly close, can we say the same about the solutions to corresponding SDE's?
More precisely, consider
$$
dX^1_t = b(X^1_t) dt + \sigma^...

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301 views

### Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...

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146 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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24 views

### Conditions for a affine term structure

I have an stochastic process $r_t$ (short rate model) with dynamics
$dr(t)=a(t,r(t)) \, dt + b(t,r(t)) \, dW(t)$,
where $W$ is a standard brownian motion.
I want to show:
If there is a function $F(...

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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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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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55 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 $(f_n)_{n\in\mathbb N}$...