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, …).
1,358
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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
2
votes
0
answers
41
views
Ito formula for non $C^{2}$ function
In the book Brownian motion and Stochastic Calculus by Karatzas and Shreve, page 215, there is a problem that says the following:
Let $a_{1}, ..., a_{n}$ be real numbers and denote $D=\lbrace a_{1}, .....
- 43
0
votes
0
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23
views
How to choose $g(t,x) $ when solving SDEs using Itô formula
I'm learning to solve SDEs through several examples and I understand how to use Ito's lemma once I have defined $ g(t,x)$. Nevertheless, I can't seem to find a way to derive the function $g(t,x)$. In ...
0
votes
0
answers
64
views
Stochastic Integrals over infinite time horizon
The stochastic Integral w.r.t Semimartingales is defined for predictable Processes $\phi$ where $\int_{0}^T \phi_t dt < \infty$. How would I extend the Definition, when I want to integrate over $[0,...
- 51
1
vote
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15
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Are these conditions sufficient for the process to be continuous?
The text book I’m working on considers the following SDE:
$dX(t) = \mu(t)dt + \sigma(t)dW(t)$, where $\mu$ is defined to be a càdlàg predictable and finite variation process, while $\sigma$ is a ...
0
votes
0
answers
24
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Pushforward of Gaussian Measure by Solution Operator to SDE
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dW(s)
$$
driven by an n-dimensional Brownian motion; where $\alpha,\beta$ are Lipschitz functions with ...
1
vote
1
answer
38
views
Verify Langevin equation
Consider the Langevin equation:
$$dX_t = -bX_tdt +adB_t, ~~~ X_0 = x_0,$$
where $a,b>0$.
We know that the solution is
$$X_t = e^{-bt}x_0 + ae^{-bt} \int_0^t e^{bs} dB_s.$$
Now I want to verify the ...
- 39
0
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0
answers
28
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Calculating the unconditional expectation of an Ito process
Suppose that:
$$dX_t =\mu(t,X_t)dt+\sigma(t,X_t)dW_t,$$
where $X_t$ is a vector valued stochastic process, $W_t$ is a vector of Brownian motions, $\mu$ is a vector valued function and $\sigma$ is a ...
- 673
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0
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38
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Stochastic differential equation and generalization to more dimensions
Given deterministic functions of one variable $f(t)$ and $g(t)$, the stochastic differential equation
$$
dX= f(t) X dt + g(t) X dW
$$
has the following solution:
$$
X(t) = \exp \left \{ \int_0^t f(s) -...
- 211
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0
answers
22
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Convergence of an integral of a stochastic process
I am not a mathematician so I apologize for the sloppy language in advance. I am dealing with a random variable $z(t)=\int_{0}^{t} r(\tau) d\tau$ where $r(t)$ is some hitherto unknown random variable. ...
0
votes
1
answer
67
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Is this a convention? Correlated Brownian Motions in SDE
Sometimes I see the following in certain papers for SDE in $\mathbb{R}^n$:
$$dX = \mu dt + \sigma dB$$
But they specify $\mathbb{E}[B^i B^j] = D^{ij} \neq \delta^{ij}$ for some symmetric matrix $D$.
I ...
- 911
1
vote
1
answer
77
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Evaluating multiple integrals after Itô-Taylor expansion
Consider an autonomous, scalar stochastic differential equation (SDE):
$$ d[X(t)] = f[X(t)]\textrm{d}t + g[X(t)]\textrm{d}W(t) $$
Consider also a scalar function $U[X(t)]$ of the solution of the SDE.
...
1
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0
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26
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Solving chemical master equation using Kolmogorov system
I am recently studying about chemical reaction networks. And I have problem about application of Kolmogorov system to actual examples.
So, here's a chemical reaction
$S_1 + S_2 \rightarrow S_3$ with $...
- 187
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votes
0
answers
81
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Find probability distribution of stopping time consisting of a two-sided barrior and a time constrained for Brownian motion with drift?
For the following Brownian motion with drift $X_t = X_0 + \mu t + \sigma B_t$ where $\mu \in \mathbb{R}$, $ \sigma > 0$ and $X_0 \in (a,b)$ which is a solution to the stochastic differential ...
- 117
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0
answers
53
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Existence Uniqueness theorem for SDEs
Is there a uniqueness and existence result for solutions to systems of Stochastic Differential Equations of diffusion type where coefficients are $\mathcal F_t$-measurable, i.e., could depend on the ...
- 23
0
votes
1
answer
27
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Calculating different expected values of a normal distribution
Be $X ∼ N$ $(µ, σ^2)$ a normally distributed random variable on a probability space $(Ω, \cal F, \cal P)$ Can someone help me to calculate the following expected values:
i) $\mathbb E[X^{2k+1}]$ for $...
- 11
-2
votes
1
answer
41
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How to make program for five realizations of ito process X = exp(t + 0.2W(t)) [closed]
I want to plot the following figures in octave.
I prepared the following program to plot one realization of ito process X =exp(t + 0.2W(t)) where W(t) is a Wiener process= $ \displaystyle\sum_{j=0}^{\...
2
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0
answers
83
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Stochastic Lorenz model
Consider Lorenz model
$$
\begin{align*}
\frac{dx}{dt}&=\sigma(y-x)\\
\frac{dy}{dt}&=\rho x-y-xz\\
\frac{dz}{dt}&=xy-\beta z
\end{align*}
$$
with $\sigma=10$, $\rho=28$ and $\beta=\frac{8}{...
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1
vote
1
answer
127
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SDE driven by Poisson Process
Suppose that $(N_t)_{t\in \mathbb{R}^+}$ is a Poisson process with intensity $\lambda$>0 and that $a\in\mathbb{R}$ and $X$ being a stochastic process which solves the following SDE:$$dX_t=aX_t^-...
- 358
0
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0
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27
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Reference for Mckean-Vlasov stochastic differential equations
Could someone please provide me with some reference books on nonlinear stochastic differential equations in the sense of McKean-Vlasov, the propagation of chaos in corresponding systems of particles, ...
- 791
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25
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Numerical schemes for simulating path dependent SDEs
Background:
The Euler-Maruyama scheme for SDEs of the form
$$dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dB_t,$$
is well studied and easy to implement. It is given by
$$X_{t_i}=X_{t_{i-1}}+\mu(t_{i-1}, X_{...
- 1,763
2
votes
0
answers
42
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Does this class of stochastic differential equations exist a unique solution?
There is a stochastic differential equations:
$$dX_t=a(t,X_t)dB_t+b(t,X_t)dt\quad t\in[0,T]$$
where $B_t$ is the one-dimensional Brownian motion.The "a" and "b" are functions:
$$a,...
- 21
2
votes
0
answers
62
views
Ito's formula and Feynman Kac formula for path-dependent SDEs
In Rogers and Williams Volume 2, Chapter V Section 2 Subsection 8, we have introduced there the notion of a general form of SDEs where the coefficients are taken to be previsible path functionals:
$$...
- 1,763
0
votes
0
answers
23
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A SDE depending on its running maximum
Let $X_t$ be a process which satisfies the SDE
$$dX_t=(aX_t+bM_t)dt+M_tdB_t$$
where $a,b$ are constants, $B$ is a standard Brownian motion and $M_t=\sup_{s\leq t} X_s$. Since the coefficients are ...
- 369
3
votes
1
answer
110
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Understanding HJB equation for the infinite horizon consumption control problem
Context
Given the following maximization problem as well as wealth dynamics
$$\max\mathbb{E}\left[\int_0^{\infty} \frac{1}{\gamma} e^{-\beta t} c_t^\gamma \mathrm{d} t\right]$$
$$\mathrm{d} X_t=X_t\...
- 312
1
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0
answers
30
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Ito rule in backward difference integrals
While defining the Ito integral, we generally take forward difference. Then we go on to prove many properties, one of them being Ito rule. Formally, we could write the Ito rule as
$$
df(X_t) = f'(X_t)...
- 1,044
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0
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16
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Definition of stochastic differential equations in infinite dimensions without traceclass
In infinite dimensions, the main idea to define a stochastic differential equation is to consider a trace class (cylindrical) Brownian motion. However, I've been wondering whether there exists a body ...
- 308
4
votes
1
answer
103
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Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces
Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
- 1,763
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0
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16
views
Estimates of supremum distance between diffusion and a given curve
Let $g\in C^1[0,1]$ be a given smooth curve. Consider a random path $X\in C[0,1]$ that solves the SDE
$$dX_t = \mu(t) dt+\sigma dB_t,$$
i.e.
$$X_t = X_0+\int_0^t \mu(s) ds +\sigma B_t.$$
We will be ...
- 1,763
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votes
1
answer
54
views
Textbook definition for path measure /probability measure over paths
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
0
votes
0
answers
51
views
Non-trivial examples of mapping between SDEs and PDEs?
The Feynman-Kac formula (or Kolmogorov backward equation) describes a link between SDEs and PDEs. This answer introduces Ricci flows on Riemann manifolds which also seems to provide a mapping between ...
- 11
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0
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51
views
Product rule for multi-dimensional Ito processes
Suppose we have d-dimensional Ito process $X_t$ with:
$dX_t = b(t, X_t)dt + \sigma(t, X_t)dW_t$,
where $W_t$ is d-dimensional standard Brownian motion.
And one-dimensional Ito process $Y_t = f(X_t)$:
$...
- 1
0
votes
1
answer
17
views
Solving SDE $dx_t = (A - a x_t) dt + (b) dZ_t$
I am new to stochastic differential equations. I would like to solve something like this:
$dx_t = (A - a x_t) dt + (b) dZ_t$
where:
$A = \frac{ - \delta k }{\delta + a} $
The solution is:
$x_t = e^{-...
- 193
1
vote
1
answer
65
views
Solve the SDE: $dX = \cos(X)\sin^3(X)dt +\sin^2(X)dW$
Solve the SDE:
$$
dX = \cos(X)\sin^3(X)dt +\sin^2(X)dW
$$
where $W$ is a Brownian motion. The SDE has initial condition $X(0) = \frac{\pi}{2}$. I am given the hint that $\int dx/\sin^2(x) = -\cot(x) + ...
- 305
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vote
0
answers
44
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Cross Covariance Matrix of Multidimensional Ornstein-Uhlenbeck Processes
The multivariate Ornstein–Uhlenbeck process is defined as the following
\begin{equation}
dX(t) = - I_p X(t) dt + \sqrt{2}I_p dW(t)
\end{equation}
where $I_p$ is an $p \times p$ identity matrix, ...
- 77
0
votes
1
answer
37
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Variance of Wiener processes in Geometric Brownian Motion
The analytical solution to the Geometric Brownian Motion (GBM) SDE is given by
$
S_t = S_0 \exp( (\mu - \frac{\sigma^2}{2})t + \sigma W_t )
$
where $W_t$ is a Wiener process. One of the properties of ...
0
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0
answers
28
views
how to proof the stability of the solution of this SDE equation
I want to proof the stability of the following SDE equation:
$$dx_t=(1-clnX_t)X_tdt+\sigma X_t dW_t$$
with the solution in the form
$$ X_t=\exp\left[\frac{1}{c}+(\ln(x_0)-\frac{1}{c})e^{-ct}+\sigma\...
- 9
0
votes
0
answers
27
views
Is this stochastic equation numerically solvable?
My knowledge of Stochastic ODEs is poor at best, but I have a problem I need to solve. I've boiled it down to a more straightforward scaler form.
$dX_t = e^{-X_t}X_t+dW_t e^{-X_t}$
I was wondering if ...
- 1
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0
answers
17
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Optimal filtering with changing variance
I have seen on some books (e.g. Lipster and Shiryayev (1977)) some concepts from optimal filtering. One idea is that, given a hidden state $\theta$ (say time invariant and taking finitely many values) ...
- 71
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0
answers
63
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The Riemannian metric induced by an SDE
Following this paper, a diffusion process in $\mathbb{R}^d$
$$dX_t = f(X_t) dt + \sigma(X_t) dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be considered ...
1
vote
0
answers
22
views
Is there an inverse Lamperti transformation for diffusions?
The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...
0
votes
1
answer
40
views
Two methods to solve SDE yield different answers
The Questions is the following:
Consider the system of SDE:
be a 1-d Brownian Motion, issued from the origin. For every $c>0$ and $\alpha, \beta \in \mathbb{R}$, consider the system of SDE:
$$
\...
- 1,107
0
votes
0
answers
25
views
X(t) denote arithmetic Brownian motion, for each future time T, what is VAR[X(T)] as a function of T?
Let X(t) denote arithmetic Brownian motion with no drift:
dX = s dW, X(0) = 0, s is a constant.
For each future time T … X(T) is a random variable.
[a] What is VAR[X(T)] as a function of T?
Let Y(...
- 1
0
votes
1
answer
85
views
How to derive the following in Ho Lee Model?
I am trying to understand the proof of the zero bond price $Z(t)$ of the Ho-Lee model which is the unique solution of the following SDE:
$$ dZ(t) = -Z(t) [ \sigma(T-t)dW(t) + [ \int_t^T \alpha(t,u)du -...
- 145
2
votes
1
answer
85
views
How to solve SDE: $ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$
Problem:
I would like to find a solution to the following SDE:
$$ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$$
Calculations:
by Ito's lemma: $$df(t,X_t) = \Bigg(f_t(t,...
- 85
1
vote
1
answer
61
views
Ito Stochastic Differential Equation Construction
How do you construct an Ito Differential Equation with solution $Y_p$, where $p \in \mathbb{Z}+$ and
$Y_p = (X(t) - \mathbb{E}(X(t)))^p$
Where $X(t)$ is the Ornstein- Uhlenbeck process, i.e
$dX = - \...
- 29
1
vote
0
answers
65
views
Prove that the stochastic process $s_t$ follows a normal distribution where the mean and the variance are functions of time in each case.
The two basic models of finance are the following:
$\textbf{The Samuelson SDE (aka Black - Scholes - Merton model):}$ Suppose that $Z=\left(Z_t, t\in\mathbb{R}^{+}\right)$ is a Wiener process (aka ...
- 85
0
votes
0
answers
30
views
Euler method for ODE with noisy derivative
I am interested in numerically integrating a noisy differential equation:
$\frac{dx}{dt} = f(x,t) + \epsilon(t)$ where $\epsilon(t) \sim \mathcal{N}(\mu, \sigma^2)$. Is this a RODE or SODE?
How does ...
- 217
0
votes
0
answers
54
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SDE/brownian motion with drift
Let
$dX_t = \mu dt + \sigma dW_t$. I need to calculate the solution of this SDE.
So I know that the solution of this sde is the brownian motion with drift:
$X_t = x_0 + \mu t +\sigma B(t) $
Now I need ...
- 493
0
votes
0
answers
20
views
Transform system of AR processes into SDEs
I have the system of AR processes
$$
x_t = q x_{t-1} + \epsilon_t
$$
$$
y_t = f_t + x_{t} + \eta_t
$$
where $f_t$ is deterministic and known, $q$ is constant, deterministic and known and
$$
\epsilon_t ...
- 105