# 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.

319
questions

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20 views

### Equivalence of forward and backward laws on $C([0,T], \mathbb R^n)$ for hypoelliptic diffusions

Consider a time-homogeneous diffusion process on $\mathbb R^n$
$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ and suppose that it satisfies Hormander's Lie bracket condition. Suppose that it is stationary at a ...

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11 views

### Some uniform property w.r.t the approximation of the SDE solution

I have a slightly general question that might be related to the Euler Approximation of SDE solution, could someone provide some thoughts on this or share some source could potentially helpful?
...

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21 views

### Moments of Logistic SDE's solution

On this article starting from equation $(30)$ it's presented a derivation of the first moment for the solution the logistic SDE:
$$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$...

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63 views

### BSDE without volatility

Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE
$$
dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t
$$
with terminal condition ...

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**1**answer

125 views

### Can we show that this transition semigroup preserves a certain Wasserstein space?

Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...

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**1**answer

182 views

### Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...

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25 views

### What is the Friedlin Wentzell rate function for $X(t)=\int_0^t f(B(s))ds+B(t)$?

Let $X(t)=\int_0^t f(B(s))ds+B(t)$ satisfy the SDE $dX(t)=\mu(X(t))dt+dB(t)$. Consider the process that solves $dX^\varepsilon(t)=\mu(X^\varepsilon(t))dt+\sqrt\varepsilon dB(t)$. We know that the ...

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54 views

### Itō formula for the solution of a SPDE in the distributional sense

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...

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36 views

### Ito's Lemma (CVF) on product of Poisson processes

I have the following stochastic differential equation:
$da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$,
with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by:
$dq(t)...

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28 views

### Estimate for continuous Ito semimartingale

I'm reading an article in which one defines a continuous Ito semimartingale of the form
$$\hat{V}_{i \Delta_n}' - \bar{V}_{i \Delta_n} = \frac{2}{k_n \Delta_n} \sum_{j=1}^{k_n} \int_{(i+j-1)\Delta_n}^{...

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134 views

### What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?

What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...

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112 views

### What is the drift for a convex combination of Girsanov measures?

Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a ...

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47 views

### Differentiable dependence on the initial condition of the solution of a SDE

Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...

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22 views

### If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...

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64 views

### Large deviations estimate for arbitrary continuous function

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...

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45 views

### Probability that a stochastic flow is near $0$

Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...

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122 views

### Intuition behind Gubinelli derivative

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject.
On page 14 of "A Course on Rough Paths
With an ...

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**1**answer

85 views

### Test for OU-Process

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?
So far, everything I've seen is hand-...

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**1**answer

90 views

### Diffeomorphism for mapping one SDE into another

Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...

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151 views

### How to make sense of recursively defined SPDE solutions, like in Hairer's “Solving the KPZ equation” paper?

In Martin Hairer's 2013 paper "Solving the KPZ equation", the process $X_\epsilon^\bullet$ is defined as the stationary solution to
$$
\partial_t X_\epsilon^{\bullet} = \partial_x^2 X_\epsilon^{\...

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24 views

### Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\
X_0=x_0 \\
dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\
Y_0=y_0
\end{...

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22 views

### Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors.
It's a well ...

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139 views

### Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$

We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...

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210 views

### On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...

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24 views

### SDE conditional expectation

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

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103 views

### Absolute value of a diffusion

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X_0 = Y_0 = 0$
\...

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69 views

### Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...

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10 views

### About the role of total variation measure on boundary reflected stochastic processes

I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that ...

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29 views

### Expected value and variance and ode

I wonder if it is possible to compute statistics of a stochastic differential equation. I begin with a simple question about linear stochastic differential equation. Let $\dot{w} = z$, and $z$ a ...

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45 views

### conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

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28 views

### Hint/Clue on the solution and properties of a particular SDE

I hope you don't mind me asking for clues/hints/help on this particular SDE:
$dX_{t}=(\mu + \gamma(Y(t)))X_{t}dt + \sigma X_{t}dW_{t}$
where $\mu , \sigma \in \mathbb{R}$, $W_{t}$ is a Brownian ...

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**1**answer

60 views

### Stochastic invariant subset

Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...

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119 views

### Diverging solution to a SDE

I'm considering the SDE, with $B$ the brownian motion and $\beta$ a scalar (it can be negative)
$$ X_t = x_0 + \int_0^t (\beta + X_s^2) ds + B_t $$
and I would like to show that $X_t$ almost surely ...

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42 views

### How to find the PDE for the following transition density

Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...

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29 views

### Associating “weak” solutions of stochastic differential equation on manifolds with real valued weak solutions

Let's say we are working on a (real) differentiable manifold $M$. For smooth vector fields $A_0,A_1,...,A_r$ on $M$ we define stochastic differential equations as $$dX_t = A_{\alpha}(X_t) \circ dB^{\...

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53 views

### Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...

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144 views

### “Expected Value” of a solution to a differential equation

I'm going to write this question in a very informal way as I'm looking for guidance, rather than a specific answer to a specific problem. So I took a course on stochastic processes and Martingales ...

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59 views

### McKean–Vlasov diffusion (strongly) Markov?

Consider the McKean–Vlasov diffusion $$dX(t) = b(X(t),\mu(t))dt + \sigma(X(t),\mu(t))dW(t),$$ in which $\mu(t)= \mathcal L_{X(t)}$ is the law of $X(t)$. Assume $b,\sigma$ are bounded and Lipschitz so ...

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26 views

### Reference for an infinite system of SDEs

Consider system of the following form,
\begin{align*}
\mathrm{d} X_k(t) = \big(AX(t)\big)_k\mathrm{d}t + B_k(X_k(t))\mathrm{d}t+\mathrm{d}W_k(t),\quad k\in\mathbb{Z},
\end{align*}
where $A$ is matrix, ...

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78 views

### Rough paths theory for Non-Markovian processes

I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems.
I would appreciate any example or also any other theory (...

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**1**answer

77 views

### What's the role of commutation relations in stochastic mechanics?

In a stochastic context, we can understand a term like
$$ \int_0^T \frac{d q(t)}{dt} dq $$
either as the (Ito) limit
$$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$
or the (Anti-...

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48 views

### Scaling Property Ito diffusion processes

It is well-known that a Brownian motion $W$ has the following scaling property
$$
c^{-1/2}W_{ct} \qquad \mbox{ for any $c,t>0$}.
$$
In particular this means that the increments of the processes $W$ ...

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**1**answer

91 views

### Convergence rate estimates of Monte-Carlo first-passage time estimates

Setup
Let $X_t$ be a $d$-dimensional diffusion process solving the Ito-stochastic differential equation
$$
X_t = x+ \int_0^t f(X_t,u_t)dt + \int_0^t \sigma dW_t,
$$
where $x \in \mathbb{R}^d$, $u_t$ ...

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votes

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510 views

### Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit.
In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...

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40 views

### Large Deviations Principle for First Exit time of a Diffusion Process

Let $b:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be a smooth Lipschitz function, $x \in \mathbb{R}^d$, $\sigma >0$, and consider the solution to the SDE $X_t^x$ defined by
$$
dX_t^x = b(X_t^x)dt + \...

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45 views

### Numerical evaluation of KL divergence for SDE

Consider the SDE
$$
dX_t = v(X_t)dt + dW_t
$$
where $W_t$ is a standard Brownian motion. Girsanov's theorem tells us that the Radon-Nikodym derivative of the measure $\mathbb{P}_v$ of $X_t$ with ...

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55 views

### Undergrad thesis topic involving stochastic differential equations

I am considering a topic for my undergraduate thesis. I want to study the estimation for the parameters in the SDEs. My instructor wants me to think of a generic and unique procedure to solve the ...

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58 views

### Different solution to system of nonlinear second order ODEs

Given the system of the following two ODEs
$$0=\frac{1}{2} \sigma^{2} \Phi_i''\left(x\right)+\left(\mu-\tilde{K} \Phi_1'(x)-\tilde{K} \Phi_2'(x)\right)\ \Phi_i'\left(x\right)-\delta \Phi_{i}\left(x\...

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151 views

### Proving that $dX_t=a_tX_td\tilde{B}_t$ is a martingale

Fix $T>0. $Consider the probability space $(\Omega,F,Q)$ and a Brownian motion $\{B_t\}_{t\leq T}$ and filtration $\{F\}_t$ generated by the Brownian paths. Suppose $a_t,\gamma_t$ are random ...

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**1**answer

68 views

### Hölder continuity for discrete time process

Let $(X_n)_{n\in\mathbb N}$ be a discrete time stochastic process taking values in a Banach space $E.$ Suppose there exist constants $C,\alpha,\beta>0$ such that $\mathbb E\|X_n-X_m\|^\alpha\leq C|...