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, …).
1,357
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Applying Ito's rule to: $Y(t)=f(M(t),⟨M⟩(t))$ (a function of Martingale and its quadratic variation)
I am attempting to compute $dY(t)$ where $Y(t)=f(M(t),⟨M⟩(t))$ and $M(t)=\int_0^t\sigma(s,\omega)dB(s,\omega)$.
My attempt is that $dY(t)=\frac{\partial f(M,⟨M⟩}{\partial ⟨M⟩} d⟨M⟩ + f'(M,⟨M⟩)dM + \...
- 995
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18
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Stochastic average of square root correlation
I am solving a stochastic differential equation and at one step I need to take the following average
\begin{equation}
\langle \sqrt{\alpha^*(t)\alpha(t)} \rangle
\end{equation}
where $\alpha(t)$ is ...
- 99
0
votes
1
answer
47
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Asymptotic variance of stochastic differential equation
Consider the stochastic differential equation (SDE)
\begin{align}
dX_t = \alpha \left( \theta - X_t + \sqrt{2\mu} \left(\int_0^t e^{\mu(s-t) }dW_s \right) \right) dt
\end{align}
where $W_t$ is a ...
- 8,579
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0
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34
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On Langevin equation and Ito's lemma
Citing from Wikipedia:
For an Itô drift-diffusion process
$$dX_{t}=\mu _{t}\,dt+\sigma _{t}\,dB_{t}$$
and any twice differentiable scalar function $f(t,x)$ of two real variables $t$ and $x$, one has
$...
- 51
2
votes
0
answers
38
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How to take the average of a stochastic differential equation?
I am solving a set of stochastic differential equations and I need some feedback about if what I am doing is correct.
Given a vector $\boldsymbol{C}(t)=(C_+(t),C_-(t))^T$, we can writte a set of ...
- 99
1
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0
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+100
probabilistic interpretation of eigenvalues of the Laplacian
Problem: Let $D \subset R^n$ be open, bounded and let $\lambda \in R$. Suppose there exists a solution $u \in C^2(D) \cap C(\overline{D})$, u not identically zero, so that $$\begin{cases} -\frac{1}{2}\...
- 695
6
votes
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60
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Intuition for expression of most likely trajectory of an SDE
Consider a stochastic differential equation evolving on $\mathbb R$
\begin{equation}
dx_t = f(x_t)dt + c dw_t ,\quad x_0 = y \in \mathbb R
\end{equation}
where $f: \mathbb R \to \mathbb R, c \in \...
- 1,014
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0
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20
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Transition distribution of a stochastic process
I am currently reading papers about diffusion models (such as https://arxiv.org/abs/2101.09258, bottom of page 2), where we are given a stochastic process $x(t)$ (which is a diffusion process, i.e. of ...
- 209
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Stochastic differential equations in economathematics [closed]
Prove that the stock price, governed by the binomial tree, converges to the stock prices of black scholes model.
What kind of assumptions should be made?
Are they unique?
...
- 772
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38
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Expectation formula of a stock price process
Still new to stochastic integral so would love some guidance on how to approach this.
Consider the stochastic differential
$$dX_t = \alpha X_t dt + \sigma_t dW_t $$
Where $\sigma_t\$ is an integrable ...
- 17
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0
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26
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SDE in 2-dimensional time domain
Usually an SDE $dX_t = f(X_t,t) + g(X_t,t)dW_t$ is formulated in 1-dimensional time domain, $t\in \mathbb{R}$. However, in principle, the time could be also a subset of $\mathbb{R}^2$, $t=(t_1,t_2) \...
- 63
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Estimate parameters in system of correlated SDEs [migrated]
I have the following system of SDEs
$$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$
$$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$
$$dX_t^3 = \mu_3 dt + \sigma_3 dW_t^3$$
where $dW_t$ is a standard ...
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0
answers
44
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Applying Ito's formula for log to Ornstein-Uhlenbeck process
My question is about applying Ito's formula to the radial equation for a $2D$ Ornstein-Uhlenbeck process.
Radial equation
I'm solving a system of SDEs of the following form:
$$
\begin{cases}
dx=-xdt+\...
- 2,327
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votes
0
answers
21
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Calculation of Autocorrelation Function in a System with Harmonic Oscillations in a Fluid: A Study of Langevin Dynamics
Introduction
Explanation of the system being studied
Objective of the study
Presentation of the Langevin equation governing the dynamics of the particle in the system
I am studying a system that ...
1
vote
1
answer
43
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Solution of a GBM
Looking to check my solution to the below :
Suppose that $X$ satisfies the SDE
$dX_t = αX_tdt+σX_tdW_t$
Now define $Y$ by $Y_t=X_t^{\beta}$
, where β is a real number.
Then $Y$ is also a GBM process.
...
- 8,579
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47
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Trajectories of Stochastic Differential Equations driven by shear flows
We consider the following autonomous vector field $u(x, y)=(y, 0)$ and we consider the backward trajectories
$$ d \left(X_{t,s}(x,y),Y_{t,s}(x,y)\right)=u\left((X_{t,s}(x,y),Y_{t,s}(x,y))\right)ds + \...
- 31
1
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1
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33
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"Infinitely Small Quantity" in SDE
The original question is
Solve the stochastic differential equation $dY_t=rd_t+αY_tdB_t$
where $B_t$ is a Brownian motion.
In Theoretical's answer,
\begin{align*}
\mathrm d(FY)_t &= F_t\,\...
- 11
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1
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Show the solution of a scalar SDE
I am currently self-studying some financial mathematics and am stuck on a practice problem:
Question
I have defined the below as hinted
\begin{align}
Y_t &= x_0 e^{\alpha t} \\
Z_t &= \sigma e^...
- 8,579
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0
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28
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Ergodic theorem for stationary and ergodic process
This is a result stated in the paper "Asymptotic Compactness and Absorbing Sets for 2D Stochastic Navier-Stokes Equations on Some Unbounded Domains" by Zdzisław Brzeźniak and Yuhong Li. It ...
- 894
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1
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57
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The solution $X$ of the SDE $\mathrm d X_t = f(t, X_t) \mathrm d t + g(t, X_t) \mathrm d B_t$ is a Markov process
I'm reading Section 5.4 Markov property from these notes, i.e.,
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\,\mathrm d}$Let $B$ be a standard Brownian motion and $(\mathcal F_t, t \ge 0)$ its ...
- 12.3k
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votes
1
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What are some good reference on Stochastic Differential Equations and Functional Analysis [closed]
There are quite a few books that use functional-analytical tools to study ODE/PDE. However, I haven't seen such a book on SDE. Some books seem to allude to this, but I cannot find a good enough ...
- 524
1
vote
1
answer
224
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Numerical simulation of SDE's driven by Lévy processes (particularly stable processes)
I'm trying to learn how to numerically simulate SDE's of the form
$$
dX(t) = f(t,X)dt + g(t,X)dZ(t)
$$
Where Z(t) is a Lévy process with triplet $(a,0,\nu)$.
My question: what is currently considered ...
- 61
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29
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Proof of the mass conservation property for the Fokker-Planck equation
I am trying to prove that the mass is conserved for the Fokker-Planck equation.
I know that the Fokker-Planck equation can be written as a continuity equation:
$$
\frac{\partial p}{\partial t}+\nabla \...
- 61
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0
answers
58
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Experimentally verifying order of convergence of SDE
I am trying to experimentally verify strong and weak orders of convergence of Euler-Maruyama and Milstein methods for the following SDE.
$$
\begin{cases}
\begin{align}
dX_t&=9.5(Y_x-X_t)dt+(Y_t-...
- 33
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votes
0
answers
44
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Application of Itô formula to stochastic heat equation
Let $\Omega \subset \mathbb{R}^n$ be a bounded set with boundary $\partial \Omega,$ and consider the following stochastic heat equation
$$
\begin{cases}
du(x,t)=(\Delta u(x,t)+f(x,t))dt+\sigma u(x,t)...
- 139
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Integral of "white noise" multiplied by exponential term (multivariate Ornstein-Uhlenbeck)
Consider a matrix differential equation for the vector $\mathbf{y}$ of the form
\begin{equation}
\dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, ,
\end{equation}
where $A$ is a ...
- 1,943
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1
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How is exponential martingale the solution of $d Y_t=Y_t d M_t$?
I'm reading about exponential martingale from these notes.
4.1 Exponential martingale
Let $M$ be a continuous square-integrable martingale and let $Y$ be the process defined as
$$
Y_t=\exp \left(M_t-\...
- 10.9k
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1
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Is there a closed expression for a LTI SDE covariance matrix?
I am studying stochastic differential equations and I still don't have a very firm grasp on all the concepts. My question has to do with the following SDE:
$$
\frac{d\textbf x (t)}{dt} = A\textbf ...
- 5,447
1
vote
2
answers
57
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Compute expectation of SDE [closed]
Given $dY_t = 5Y_tdt + 3dW_t$ and $Y_0 = 1$. Compute $E[Y_t]$ for $t\geq 0$.
This is what I did so far
I integrate both sides from $0$ to $t$
$E[Y_t - Y_0] = 5E[\int_0^t Y_s\,ds] + 3E[\int_0^t Y_s\,...
- 12.2k
1
vote
0
answers
34
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Is stochastic differential equation a continuous mapping?
We have a well-defined SDE:
$$
{\rm d}X_t=\mu(X_t){\rm d}t+\sigma(X_t) {\rm d}B_t,
$$
where the initial condition $X_0$ is a known r.v., and $B_t$ is a standard Brownian motion.
Can we say the above ...
- 61
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votes
3
answers
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Infinitesimal generator of the Brownian motion on a sphere
As explained here, the infinitesimal generator of a 1D Brownian motion is $\frac{1}{2}\Delta$. As discussed here, for the Brownian motion on circle we can write
$$Y_1=\cos(B) \\
Y_2= \sin(B)$$
and ...
1
vote
1
answer
32
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Verifying the Bakry-Émery criterion on spheres
Suppose that I am interested in a probability measure $\mu$ defined on the sphere $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$, whose density with respect to the normalised surface measure can be written ...
- 13.7k
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SDE with non-diagonal noise
In high-energy physics, one has the following type of SDE:
$$\partial_t Z_t(\vec{x},\vec{y}) = \int d^2\vec{u} d^2\vec{v}d^2\vec{z} f(\vec{x},\vec{y},\vec{u},\vec{v},\vec{z})\sqrt{\nabla^2_{\vec{u}}\...
- 169
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1
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40
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Is geometric brownian motion, $X_t + c$, shift invariant? And what is the expected value?
Consider a geometric brownian motion (GBM),
$$X_t = X_0 \exp \left[\left(\alpha - \frac{1}{2} \sigma^2\right)t+\sigma W(t)\right].$$
Using Ito's lemma, we find that it has dynamics given by
$$dX_t = \...
0
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0
answers
14
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What is the filtered probability space used to study linear SDEs with constant coeficients?
Context:
I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as:
$$
dx(t) = Ax(t)dt + Bdw(t)
$$
...
- 1,674
3
votes
1
answer
75
views
Stochastic model of a RL circuit
I would like to solve stochastic electric circuits numerically. I already know that I should use Milstein method to solve those systems. However, I'm not sure if the stochastic differential equations ...
- 456
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0
answers
36
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Itô differential of a stochastic integral
Let $K$ be a convolution kernel defined by :
$$\forall \ \theta \in \mathbb{R}_+^*: \quad K(\theta) := \omega \theta^{H - \frac{1}{2}}, \quad \omega >0, \ H \in (0, \frac{1}{2})$$
Define for $t\...
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0
answers
18
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Can I verify the weak order of convergence of a numerical scheme for SDEs by checking convergence in distribution?
let $X_{t=0}^T$ be a continuous-time stochastic process defined by a certain stochastic differential equation
$$
X_t=a(t,X_t)dt+b(y,X_t)dW_t
$$
where W_t is a Wiener process.
Let $\tilde X_{i=0}^{N}$ ...
- 237
3
votes
1
answer
181
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Technical difficulties with degenerate PDEs
Crossposted at Quantitative Finance SE
I have seen lot of discussions in this Math. S.E. platform about 'degenerate partial differential equations'. But I still unclear about the 'technical ...
- 701
0
votes
1
answer
87
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Use Ito's formula for calculate the dynamics of Ornstein-Uhlenbeck process [duplicate]
Consider this stochastic differential equation:
$$𝑑𝑋(𝑡)=−a𝑋(𝑡)𝑑𝑡+σ𝑑𝑊(𝑡),\quad 𝑋(0)=𝑥_0∈ \Bbb R$$
where $a$ and $σ$ are constants and $𝑊(t)$ is a Brownian motion.
Can someone show me how ...
- 3,155
0
votes
0
answers
35
views
Expectation of system of stochastic differential equations
I have following system of stochastic differential equations.
$$
\begin{equation}
\begin{cases}
dx_t=\sigma_t(y_t-x_t)dt\\
dy_t=(\rho x_t-y_t-x_tz_t)dt\\
dz_t=(x_ty_t-\beta z_t)dt\\
d\sigma_t=-\alpha\...
- 33
0
votes
0
answers
25
views
Can we write the Ito differential dBt into some "differential form" like we do for df where f is smooth?
There is a standard definition that $\mathrm{d} f$ means a cotangent vector field (1-form) when $f$ is a differentiable function on $\mathbb{R}^n$ or a smooth $n$-manifold $M$, by which we can make ...
- 470
0
votes
1
answer
45
views
"Generalized" Geometric Brownian Motion as a SDE system
It is very well known that the equation
$$d X_t = \mu X_t dt+\sigma X_tdW_t$$
has a solution
$$X_t = X_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t},$$
and we say that $X_t$ follows Geometric Brownian ...
- 452
0
votes
0
answers
19
views
Matrix-valued stochastic DE to a vector-valued stochastic DE
Suppose we have a matrix-valued ODE:
$$
i\tag{1} \frac{d}{dt}U(t) = HU(t)
$$
where $U(t)$ and $H$ are complex square matrices. Given a time-independent vector $\psi$, we can convert (1) into a vector-...
- 211
0
votes
0
answers
25
views
Numerical methods to solve a Stochastic Differential Equation (SDE) with the noise term inside a nonlinear function.
I'm trying to computationally solve the following SDE:
$\frac{dx(t)}{dt} = DeterministicTerm - \epsilon \, sin(x(t)-\sigma \eta(t)) $.
Here, $\epsilon$ and $\sigma$ are two parameters, and $\eta(t)$ ...
- 1
0
votes
0
answers
40
views
Find a good substitution for Stochastic Differential equation
Is there a good substitution to solve this SDE, for I could not figure out :
$$ dX_t =X^β_t dt+t^αX_t dB_t, X_{0} =1.$$
4
votes
1
answer
102
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Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces
Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
- 1,763
2
votes
1
answer
70
views
Comparison of Solutions to SDEs
Let $B(t)$ be a Brownian motion. If we assume that $b(t, x), b_i(t, x)$, $\sigma(t, x)$, and $\sigma_i(t, x)$ are Lipschitz functions for $i = 1, 2$ then it is known that if $X_i(t)$ is the unique ...
- 8,579
2
votes
1
answer
52
views
Brownian motion and geometric series convergence
I'm confused with the property of Brownian motion that it is of infinite variation.
Consider the sequence
$$\sum_{i=0}^{t-1} a^{t-i} (B_{i+1} - B_i)$$
where $a \in [0,1)$ and $B_t$ is Brownian motion.
...
- 21.3k
0
votes
0
answers
18
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Mass matrix in underdamped Langevin dynamics and convergence
I have two doubts regarding the underdamped Langevin dynamic.
The first one is that I noticed that the underdamped Langevin dynamic can be written in the following two ways:
\begin{align}\label{eq:...
- 1