Questions tagged [stochastic-differential-equations]
Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).
695
questions with no upvoted or accepted answers
11
votes
0
answers
505
views
Discretization formula for system of differential equations. "Solution to one of these is the initial condition of the other". In which sense?
Consider the following stochastic differential equation
\begin{equation}
dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1}
\end{equation}
where $A$, $B$ and $C$ are parameters ...
9
votes
0
answers
422
views
Forward vs backward formulation in Feynman-Kac
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a nice filtered probability space with an $m$-dimensional standard Brownian motion $W$. Fix a time horizon $T>0$. Let $\mu \colon [0,T] \times \mathbb{R}^d \...
- 1,192
8
votes
0
answers
215
views
Reference request for a Riemannian Fokker-Planck equation
I am looking for any reference which states, and proves, a Fokker-Planck equation for Riemannian manifolds.
In particular, if $\mathrm{d}X_t=\mu(X_t)~\mathrm{d}t + \sigma(X_t)~\mathrm{d}B_t$ is a ...
- 450
8
votes
0
answers
452
views
Asymptotic behaviour of integral. How should I proceed?
Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$...
7
votes
0
answers
142
views
Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$
I am trying to analyse the exit time $T_1:=\inf\{t:X_t\notin[\alpha,2]\}$ and hitting time $T_2:=\inf\{t:X_t=0\}$, where $\alpha<1$ is a constant, and $X_t$ follows the Bessel process defined by ...
- 2,158
7
votes
0
answers
230
views
Solving root stochastic differential equation
I'm concerned with the following SDE:
$$d Y_t= v \,dt + \sqrt {|Y_t|} \,d W_t$$
with $Y_0=-a$, $v>0$ being a constant, $a>0$ and $W_t$ as standard Brownian Motion. Do you have hints how to solve ...
- 71
7
votes
0
answers
191
views
Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.
Let$^1$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$U$ be a separable Hilbert space
$Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace
$(W_t)_{t\ge0}...
- 13.5k
7
votes
0
answers
700
views
proving equalities in stochastic calculus
I am struggling with this question:
FIRST PART (almost done, but stuck somewhere):
Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by
the formula
\begin{equation}
...
- 2,920
6
votes
0
answers
44
views
Intuition for expression of most likely trajectory of an SDE
Consider a stochastic differential equation evolving on $\mathbb R$
\begin{equation}
dx_t = f(x_t)dt + c dw_t ,\quad x_0 = y \in \mathbb R
\end{equation}
where $f: \mathbb R \to \mathbb R, c \in \...
- 1,014
6
votes
0
answers
87
views
Existence and uniqueness of the solution to $dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos X_{t}dt+\cos X_{t}dW_{t}$
Let $\chi$ denotes an indicator variable and $W$ a Wiener process. What can we say about the solution of the following SDE:$$\begin{cases}
dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos ...
- 988
6
votes
0
answers
275
views
Langevin equation and convergence to stationary solutions. Free energy. SDE. FPE.
Let $f\geq 0$ be Lipschtiz. The overdamped Langevin equation
\begin{equation}\label{eq overdamped Langevin SDE}
dX=-\nabla f(X)dt+\sqrt{2} dW_t
\end{equation}
with Kolmogorov forward equation
\...
6
votes
0
answers
477
views
Ito Derivative of White Noise
We know that white noise $w_{t}$ is given by the time derivative of Brownian motion $\beta_{t}$, ie that:
$$ w_{t} = \frac{d \beta_{t}}{dt} $$
Now I want to define a new process, called blue noise $...
- 173
5
votes
0
answers
164
views
If a diffusion is Gaussian, what does it imply to its drift and volatility?
Let $(Y_t)$ be a stochastic process solution to the SDE
$$dY_t= \lambda(Y_t,t) dt + \sigma(Y_t,t) dB_t. $$
If we know that $(Y_t)$ is a Gaussian process, what does it inform us on the drift $\lambda$ ...
- 2,260
5
votes
0
answers
82
views
How are Markov Kernels Related to SDEs
(Disclaimer: I've been working with SDEs for some years now but have not worked with general Markov processes before... so I'm trying to reconcile some ideas with this post.)
I recently read the ...
- 71
5
votes
0
answers
110
views
Estimate to arrive at $\mathbb{E}[(|X^{i}_t|+\frac{1}{N}\sum_{j=1}^N|X^j_t|)^q\mid X_0^{i}=x]\leq C x^q$
Setup
We have a system of $N$ diffusion processes described by
$$X_t^{i}=X_0^{i}+\int_0^t \mu(s,X_s)ds+\int_0^t\sigma(s,X_s)dW^{i}_s,$$
with the Brownian motions $W^{i}$ as well as the drift $\mu$ and ...
- 464
5
votes
0
answers
165
views
Convergence of a stochastic integral to a normal random variable
Let $X_t$ be the Ornstein-Uhlenbeck process defined by:
$$
X_t = X_0 \, e^{-t} + \int_0^t e^{-(t-s)} dW_s.
$$
Is it possible to show using elementary tools,
in particular without using the central ...
- 1,930
5
votes
0
answers
404
views
How can I numerically compute a stochastic integral?
I am trying simulate a to solve a 2-dimensional stochastic process and $Y_t^1$ is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution.
I ...
- 3,517
5
votes
0
answers
189
views
Are SDE's really "differential"?
An SDE of the form
$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$
is really short-hand notation for an equation involving Ito integrals:
$$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,...
- 12.2k
5
votes
0
answers
624
views
Brownian motion on sphere proof?
proving the brownian motion on the sphere equation
the
stratonovich form differential equation
$$\partial X=n(X)\times \partial B$$
the equation in ito's form becomes
$$dX=n(X)\times dB+H(X)n(X)...
- 1,946
5
votes
0
answers
758
views
costruction of brownian motion on sphere?
i am trying to construct a brownian motion on the sphere using the method given in Price and williams paper.$\partial$ represents the SDE of stratonovich type which is converted to ito form in last ...
- 1,946
5
votes
0
answers
764
views
conversion from stratonovich SDE to Ito's form?
conversion of stratonovich SDE to Ito SDE (Where $\partial$ is differential in the stratonovich form and $d$ is in ito's form):
$$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$.
...
- 405
5
votes
0
answers
2k
views
How to solve system of stochastic differential equations?
I have the following two SDEs
$$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$
$$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$
where $W$ is the standard Brownian motion/Wiener process. This isn't homework, I'm ...
- 123
5
votes
0
answers
267
views
Question about a Bessel process
Are there any explicit path-wise solutions for a 3 dimensional Bessel process?
E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$
where $W_t$ is a standard Wiener process.
- 1,074
4
votes
0
answers
31
views
Why does Brownian motion cluster around singularities of the potential?
Suppose I give you the potential plotted on the left (with toroidal boundaries). On the right, I've plotted the associated Gibbs measure, which is how I'd naively expect a Brownian particle to spend ...
- 169
4
votes
1
answer
102
views
Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces
Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
- 1,743
4
votes
0
answers
60
views
Deriving PDE from stochastic representation formula
This is a question from Exercise 5.10 of Arbitrage Theory in Continuous Time (2009) by Tomas Bjork (rest in peace). The problem states that,
Consider the following boundary value problem in the domain ...
- 41
4
votes
0
answers
92
views
Diffusion process and jump process
I am reading diffusion process from a textbook and noticed that the author claims that condition (1) implies the stochastic process $X_{t}$ cannot have instantaneous jumps. So I wonder does "...
- 1,098
4
votes
0
answers
130
views
Radon-Nikodym derivative of pushforward measures and Girsanov theorem
Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
- 611
4
votes
0
answers
136
views
Conditions for the SDE be transitive.
Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\mathbb R^3 \times \mathbb S^2$ (where $\...
- 3,459
4
votes
0
answers
81
views
Efficient numerical solution of brownian motion SDE with non-smooth diffusivity
I'm trying to numerically solve the 1-dimensional stochastic differential equation for brownian motion (Fokker-Planck) with rapidly varying diffusivity $K$,
$$ \textrm{(Ito)}\quad dX_t = \frac{\...
- 163
4
votes
0
answers
194
views
Ito's Lemma for Markov Chain
I'm sorry if this question may have been asked before but given the dearth of references, I doubt it.
Now, here's my problem. I have this continuous function $f(t,w,x)$ which is dependent on time t, ...
4
votes
0
answers
76
views
Solving/Rewriting SDEs in Non-Matrix Lie Groups
I'm working on trying to solve a state estimation problem in a non-matrix Lie group. I have found some good resources for state estimation in certain matrix Lie groups. For instance, in this paper ...
- 12.5k
4
votes
0
answers
153
views
Independence of solution to SDE and initial condition
Let $(\Omega,F,(F_t)_{t\geq 0},P)$ be a filtered probability space with the standard condition.
Let $W_t$ be the $F_t$-Wiener process, and
let $(X_t)_{t\geq 0}\subset {\mathbb{R}}$ be the (strong) ...
- 314
4
votes
0
answers
199
views
Stochastic Gradient Descent Converges Not to a Maximum Point
Let $n\in\mathbb N$, $B$ a $n$-dimensional Brownian motion, $\sigma_t$ a positive (deterministic) caglad (LCRL) function in $\mathcal L^2([0,\infty[)$ and $f\in\mathcal C^2(\mathbb R^n\to\mathbb R)$ ...
- 2,326
4
votes
0
answers
275
views
Deriving Stochastic Differential Equations from Autocorrelation
Suppose I know my stationary stochastic process has the following autocorrelation function:
$$ R(\tau) = \sigma^{2} e^{-\alpha |\tau|} \cos(\omega \tau) $$
How can I derive the stochastic ...
- 173
4
votes
0
answers
424
views
Exact (!) relation between Martingale Problem, SDEs and Markov processes
I am currently trying to understand the big picture/connections of Martingale Problem, Fokker-Planck-equations (although, until now, I have mostly kept these out of my considerations), SDEs and Markov ...
- 731
4
votes
0
answers
121
views
What is the solution to SDEs of the form $dx=f(x)(dt+\rho dW_t)$?
I've been thinking about the class of SDEs which have the form:
$$dx(t)=f(x)(dt+\rho dW_t)$$
Clearly when $\rho=0$ this is just an ODE:
$$\frac{dx}{dt}=f(x) $$
My question is, how does the solution to ...
- 113
4
votes
1
answer
240
views
Uniqueness of the parameters of an Ito process, given initial and terminal conditions
Sufficient conditions for the existence and uniqueness of a weak solution of a stochastic differential equation $$dX_t = \mu(X_t)dt + \sigma dZ_t$$ are known. Loosely speaking, Lipschitz conditions ...
- 330
4
votes
0
answers
199
views
Derivation of continuous Euler-Maruyama approximation
I am currently going through the Euler-Maruyama approximation and have question regarding the linear interpolation of it.
For the stochastic differential equation $$dX_t=a(t,X_t)dt+\sigma(t,X_t)dW_t \...
- 83
4
votes
1
answer
902
views
Proving an SDE has a unique strong solution
I have the stochastic differential equation
$$dX_t = \ln(1+ X_t^2) \, dt + X_t \, dB_t$$
In this equation, $X_0 = x$, and $x \in\mathbb R$.
How can we show that this equation has a unique strong ...
- 43
4
votes
0
answers
358
views
Expected hitting time of a stochastic differential equation with jumps (neuroscience example)
The basic model I'm working with is a neuron that receives input from other neurons which cause instantaneous spikes in the voltage. In a nutshell, I have a differential equation that describes the ...
- 3,556
4
votes
0
answers
118
views
Methods of SDE calibration
There is somewhere summary of methods that can be used to estimate parameters of SDE?
I currently using MLE and regression due to linear dependence between samples. I searching for something ...
- 41
4
votes
1
answer
82
views
If $S_{t}$ satisfies $dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}$, will $S_{t}$ be a martingale?
If the process $S=S_{t}$ satisfies the SDE: $$dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}, \ S_{0}=1.$$ will $S_{t}$ be a martingale? It seems reasonable to say so because $S_{t}$ is clearly ...
- 895
3
votes
0
answers
73
views
Is it possible to find the distribution of this nonlinear SDE?
Consider the nonlinear SDE for $(X_t)_{t\geq 0}$
\begin{equation}
\mathop{dX_t}=X_t\left(\mu\mathop{dt}+\sqrt{v_0+\omega\left(\phi(t)-\ln X_t\right)^2}\mathop{dW_t}\right),
\end{equation}
where $\phi:...
- 145
3
votes
0
answers
35
views
Calculating expectation of product of two Cox-Ingersoll-Ross (CIR) processes
I would like to calculate expectation $\mathbb{E}\left[\nu_t\tilde{\nu}_t\right]$ of product of two Cox-Ingersoll-Ross processes $\nu_t$ and $\tilde{\nu}_t$. They are described by the following SDEs:
$...
- 350
3
votes
0
answers
48
views
Question on regular function from Kolmogorov equation
I am reading about a topic on Kolmogorov equation and diffusion process. However, I am confused by the last using so called "regular function". I wonder what does this "regular function&...
- 1,098
3
votes
0
answers
66
views
For a Brownian motion on a Riemannian manifold, is the log of the transition probability proportional to the squared geodesic distance?
I am trying to gather some intuition on the connection on diffusion processes and Riemannian geometry, with only very limited knowledge of the latter.
First, let us consider a Brownian Motion in ...
- 1,337
3
votes
0
answers
78
views
Decompose the Laplace transform of autocorrelation function of a stachastic process using Wiener-Kolmogorov whitening procedure
Here I read that it is possible to use the Wiener-Kolmogorov whitening procedure to decompose the Laplace transform of autocorrelation function into the product of white noise and a system function:
$$...
- 7,659
3
votes
0
answers
32
views
Question on Markov property
I am self-studying "Introduction to Stochastic Integration" by Hui-Hsiung Kuo and have some doubts on the lemma. In the book it first defines that the Markov property on page 198 as follow
...
- 1,098
3
votes
0
answers
52
views
How to make sense of SDE notation
I have been trying to understand SDE notation and what it means but I cant seem to figure it out. Im sorry to post this question here, since this is probably elementary.
It is known to me that
$$ dX_t ...
