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, …).
925
questions
0
votes
0answers
16 views
Fokker Planck Equation
I was told in the lecture like the following:
"The Fokker-Planck equation is a deterministic partial differential equation that in general has to be solved numerically. For vector systems ...
1
vote
0answers
13 views
Kolmogorov Backward and Forward Equations: Why are there different derivations for the forward and backward dynamics?
Dear knowledgeable people of math.stackexchange :) ,
In Stochastic Analysis and Diffusion Processes by Kallianpur on pages 218 to 221 the derivations for the forward and backward Kolmogorov equations ...
0
votes
0answers
30 views
Show Geometric Brownian motion is the unique solution to $\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$.
Given
$$
X_t=\sigma W_t+\mu t,
$$
the SDE
$$
\frac{dS_t}{S_t}=\mu dt+\sigma dW_t
$$
has a unique solution
$$
Z_t=\exp\Big(X_t-\frac{1}{2}\langle X\rangle_t\Big).
$$
We know from various posts and ...
-1
votes
0answers
21 views
Linear transformation of SDE
Say that I define an SDE as
\begin{align}
dX(t) = F(x)dt + \sigma dW_t
\end{align}
where $F(x)$ is the deterministic part, $\sigma$ is the noise intensity and $dW_t$ is the Winer process. Then I show ...
4
votes
1answer
47 views
Probabilistic recursion
I have the following recursion:
$$p_{t+1} = \begin{cases} 1 \text{ with probability } 1-p_t \\ \alpha p_t \text{ with probability } p_t \end{cases}$$
for $0<\alpha<1$. Numerical simulations show ...
2
votes
0answers
18 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 ...
0
votes
1answer
26 views
Weiner Process Markovian property
Wiener process definition:
$W(t) \sim \mathcal{N}(0, \sigma\sqrt{t})$
$W(t)$ is Markovian
$W(t+t')-W(t)$ and $W(t)$ are independent.
I don't know of a case where $2$ is true but $3$ is not & ...
6
votes
2answers
118 views
A question about SDE and geometric Brownian motion.
In Bernt Oksendal's Stochastic Differential Equations, Chapter 4, one has the following stochastic differential equation (whose solution is geometric Brownian motion):
$$dN_t=rN_tdt+\alpha N_tdB_t\;\;\...
2
votes
0answers
41 views
SDE books for non-mathematicians
I want to learn about SDEs. Most books on this topic are very dry & mathematical with no intuition at all. Can someone recommend a good book or courses for non-mathematicians? I am from computer ...
2
votes
0answers
29 views
Can a stochastic process correspond to multiple SDE's?
In this question, and this question, it is clear that the solution of the one-dimensional SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ is $X_t-X_0=B_t$. This is also the Bessel process. I am confused by the ...
1
vote
0answers
28 views
What kind of stochastic process could this be?
I have been studying the fascinating subject of stochastic processes and have constructed various equations I can think of and then I try to look up literature for that process.
One process I have ...
1
vote
1answer
45 views
Finding the SDE
Let $X_t = e^{(\mu-\frac{\sigma^2}{2})t + \sigma B_t}$, where $B_t$ is a standard Brownian motion. How do I find the SDE satisfied by $X_t^{-1}?$ I know I must use Ito's formula but not sure how to ...
0
votes
2answers
48 views
Derivative of conditional expectation of integral of stochastic process
Let $T>0$. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathbb{F}=(\mathcal{F}_u)_{u\in[0;T]}$ be a filtration such that $\mathcal{F}_T=\mathcal{F}$. Let $(\alpha_u)_{u\...
3
votes
1answer
44 views
Dealing with SDE involving $\max\{X_t,0\}\mathrm{d}W_t$
I am considering the SDE $\mathrm{d}X_t=\ln(1+X_t^2)\mathrm{d}t+\max\{X_t,0\}\mathrm{d}W_t$, with the initial condition $X_0=a$. I am posed with the following daunting tasks:
prove $\exists$ aĀ unique ...
4
votes
0answers
70 views
Estimate to arrive at $\mathbb{E}[(|X^{i}_t|+\frac{1}{N}\sum_{j=1}^N|X^j_t|)^q\mid X_0^{i}=x]\leq C x^q$
Setup
We have a system of $N$ diffusion processes described by
$$X_t^{i}=X_0^{i}+\int_0^t \mu(s,X_s)ds+\int_0^t\sigma(s,X_s)dW^{i}_s,$$
with the Brownian motions $W^{i}$ as well as the drift $\mu$ and ...
1
vote
0answers
64 views
How to evaluate the integral of white noise multiplied by exponential (decaying) term
Consider a matrix ODE for the vector $\mathbf{y}$ of the form
\begin{equation}
\dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, ,
\end{equation}
where $A$ is a constant square matrix ...
1
vote
0answers
26 views
Divergence when calculating moments with a space-time Gaussian white noise
Consider a variable $M(x,t)$ driven by space-time Gaussian white noise:
$$ \partial_t M(x,t) = -k M + \xi(x,t).$$
The noise has mean $\langle \xi(x,t) \rangle = 0$ and correlation function $\langle \...
2
votes
0answers
44 views
Intuition of Kolmogorov equations and infinitesimal generators
So if we have the infinitesimal generator of a Markov process $\{X_t\}_{t\geq0}$, the infinitesimal generator is given by $$Af(x) = \lim_{t\rightarrow 0} \frac{1}{t} \big(P_tf(x) - f(x)\big)$$ where $...
2
votes
0answers
42 views
On an example of SDE where strong and weak solutions are different
I was wondering about explicit examples of SDEs where strong and weak solutions are indeed different. I found online that the following is an example of and SDE where the two solutions are different:
...
2
votes
1answer
38 views
Have trouble understanding Matlab code of stochastic integrals
Hell, I have trouble understanding the matlab code of approximating stochastic integrals.
...
0
votes
0answers
26 views
Find an SDE given a probability density function p(x,t)
Usually, we start with an SDE for $X(t)$ and then try to solve it and find the corresponding probability density function $p(x,t)$. But what if we start with a pdf - can we find a corresponding SDE? ...
0
votes
0answers
27 views
Simplifying terms in Ito's Lemma like $Y_t dX_t$ and $dX_tdY_t$
Suppose that I have two stochastic processes that can be represented as
$$dX_t= udt + \sigma dB_{1,t}$$
$$dY_t= vdt + \nu dB_{2,t}$$
I know by Ito's Lemma that
$$d(X_tY_t)=X_tdY_t + Y_tdX_t + dX_tdY_t ...
0
votes
0answers
19 views
Diffusion in 1D with spatially varying diffusion coefficient, Fokker Planck equation
I am trying to find the steady state densities to the following problem. We have a random walk in 1D with spatially varying diffusion constant, which i numerically solve as
$$ x_t = x_{t-1} + \sqrt{2 ...
2
votes
1answer
66 views
Solving SDE with sign function in drift term?
Consider the following SDE with $X_0 = 1$,
$$
dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t,
$$
where $\operatorname{sign}(x) = \mathbb{1}_{\{x \ge 0\}} -\mathbb{1}_{\{x < 0\}}$. How am I ...
1
vote
1answer
14 views
Apply Ito's formula to $d(X_t)$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$
I'd like to find $dX_t$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$. I have no idea how to apply Ito's formula here.
1
vote
0answers
16 views
Expected Discount Factor with respect to First Passage Time
I am curious about and have been trying for a long time the following question. I was wondering if there will be any hints or references that could help.
Assume that $B_t$ represents a standard ...
2
votes
0answers
26 views
Scaling of white noise from scaling of brownian motion
It is a well-known fact that
$$\lambda^{1/2}B(\lambda^{-1}t)\stackrel{d}{=}B(t) \ ,$$
for $B$ a $1$-dimensional brownian motion. Also, we know that for $\xi$ a $1\text{D}$ white noise, we have
$$\...
3
votes
1answer
36 views
Linearization of a SDE and comparison to deterministic setting
I am a bit puzzled having no experience in this area on the following problem.
It is well know that if we consider a system as:
\begin{equation}
\dot x = f(x(t)), x \in R
\end{equation}
with $f$ of ...
0
votes
0answers
24 views
Write stochastic integral equation in differential form.
If I have a stochastic process given by
$$Y(t)=x+\int_s^t\frac{a(\tau)-\sigma(\tau)\gamma(\tau)}{Z(\tau)}d\tau + \int_s^t\frac{\gamma(\tau)}{Z(\tau)}dW(\tau)$$
where $X(s)=x$, how would I write it in ...
0
votes
0answers
42 views
Itô's formula for an expectation result
I applied ItƓ's formula to $f(x)=x^q$ for $q>1$ with a process defined by the SDE
\begin{align*}
dX_t = \mu(t,X_t)dt + \sigma(t,X_t) dW_s,
\end{align*}
where $W_s$ is a Brownian motion, $\mu(t,X_t)$...
1
vote
1answer
138 views
Application of Dynkin's formula
Let $x\in\mathbb{R}^d, (W_t)_{t\geq0}$ be a $d$-dim. Brownian motion. I have the following processes
$$
Z_{t, x} = b(Z_{t, x})dt + \sigma(Z_{t, x})dW_t, \qquad Z_{0, x} = x \\
V_{t, v} = \nabla b(Z_{t,...
0
votes
0answers
31 views
Independence of the components of an Ito process
If $X_t = (X^i_t)_{i \leq n}\in \mathbb{R}^n$ solves an SDE of the form
$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$
Does it hold that each of the real valued random variables $X^i_t$ and $X^j_t$ are ...
1
vote
0answers
12 views
Probability density function (PDF) of a two-dimensional linear stochastic differential equation (SDE) with additive noise
I am considering the same two-dimensional linear SDE as in a previous question:
\begin{cases}
dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1,
\\
dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \...
1
vote
1answer
36 views
Form of Ito's formula for Ito diffusions
Suppose we have an Ito diffusion ($Z_{t, z}$) satisfying
$$
Z_{t, z} = z + \int_0^t b(Z_{s,z})ds + \int_0^t \sigma(Z_{s,z})dW_s
$$
where $z\in\mathbb{R}^d, b:\mathbb{R}^d\to\mathbb{R}^d, \sigma:\...
0
votes
0answers
65 views
Conditional expectation on the product of squared Brownian motion with Doléans-Dade exponential (Radon-Nikodym derivatives)
I try to compute the conditional expectation $E_{t}^{P}\left[B_{T}^{2} \ e^{-\frac{1}{2} \int_{0}^{T} \left( a + b B_{u}^{2}\right)^{2} d u+\int_{0}^{T} \left( a + b B_{u}^{2}\right) d B_{u}}\right]$ ...
1
vote
1answer
33 views
Stochastic calculus of Gaussian white noise
I see a strange stochastic integral and don't know how to proceed?
Denote $\xi$ a Gaussian white noise,
Denote
$$I(n)=\int_t^{t+n\Delta t}ds\int_t^sd\xi$$
How to prove that $<I(1)I(1)>= \frac{1}{...
0
votes
0answers
30 views
Name for class of functions from Øksendal's Introduction to SDEs
From Ćksendal's Introduction to Stochastic Differential Equations we have the following:
Definition $3.1.4.$
Let $\mathcal{V} = \mathcal{V}(S, T)$ be the class of functions
$f(t, Ļ): [0, ā) Ć ā¦ ā R$
...
0
votes
0answers
13 views
Power spectral density (PSD) of a two-dimensional linear stochastic differential equation (SDE) with additive noise
I am considering a two-dimensional linear SDE of the form
$$\begin{cases}
dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1,
\\
dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \bigr)\,dt + \sigma \,...
1
vote
1answer
34 views
What is the expression $E[X_t dt]$ using stochastic calculus?
I am trying to finish a calculation using ItƓ calculus but do not know the final step.
What is the expression $E[X_t dt]$ equal to?
Note that the $X_t$ is drifted Brownian motion $X_t = \mu_x t + \...
0
votes
0answers
40 views
Backward Stochastic Riccati Differential Equation
Let's say $(P,\Lambda)$ is an adaptive solution to the following backward stochastic riccati differential equation:
$$ \begin{aligned}
dP_t &= \left[a - \frac{2}{\kappa_t} \left(P_t-\lambda a^2\...
0
votes
0answers
80 views
Dynkin Operator explained to a finance student
Can somone gives me an introduction about the so-called Dynkin Operator ?
(finance student means that I know stochastic calculus only at an introductory level without a solid background in funtional ...
0
votes
1answer
27 views
moments of a Geometric brownian motion using Ito's lemma
Can someone explain me what is wrong in my derivation of the formula for the moments of a GBM using Ito's lemma (I am not interested in other methods) ?
\begin{equation}
dX=\mu Xdt+\sigma XdW_t
\end{...
0
votes
0answers
35 views
Distributions of stopping/hitting times of SDE
Let $W_t$ be an $m$-dimensional Wiener process on a complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following SDE:
\begin{equation}
dX_t = b(X_t,t)dt + \sigma(X_t,t)dW_t
\...
0
votes
0answers
10 views
Girsanov theorem for bounded variation process
I have the path:
$$X_t = B^H_t + H_t $$
where $(B^H_t)_{t \geq 0}$ is the fractional Brownian Motion (if necessary, we can consider ordinary Brownian Motion, that is $H=1/2$) and $H_t$ is a bounded ...
0
votes
0answers
25 views
Infinitesimal generator of smooth SDE
Suppose we have the SDE $dX_t = \mu_t dt + \sigma_t dS_t$ where $S_t$ is a.s. continuously differentiable. More precisely, I want to put $S_t = \sum_{i = 1}^N X_ib_i(t)$ where the $X_i$ are i.i.d ...
0
votes
0answers
39 views
How to differentiate this exponential process?
I understand how to apply Ito's formula to differentiate the Radon-Nikodym density $Z_t = \exp \bigg(-\int_0^t \theta_sdW_s - \frac{1}{2} \int_0^t \theta^2_sds \bigg)$ and get the SDE $dZ_t = -\...
2
votes
0answers
70 views
Expected value of stopping time
Let $X = (X_t)_{t\geq 0}$ be a non-negative stochastic process solving
$$
d X_t = 3dt + 2\sqrt{X_t}d B_t, \quad\quad X_0=0
$$
with $B = (B_t)_{t\geq 0}$ is a standard Brownian motion. I want to ...
0
votes
0answers
13 views
Weak error versus strong error
Currently I'm learning about SDE's. I understand the conceps weak and strong error mathematically, but I find it hard to see in which context each error is used.
The strong error tells something about ...
2
votes
0answers
35 views
Why are SDEs given in the form $dX_t=\mu dt+\sigma dB_t$?
All the resources I have found state that a Stochastic Differential Equation has the form:
$$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dB_t$$
Where $B_t$ is Brownian motion. My question has two parts. The first ...
1
vote
1answer
90 views
$M = (M_t)_{t\geq 0}$ obtained by Itô's formula is a martingale
Here I defined a non-negative stochastic process. Now, taking $F(t,x) = tx^2$, I would like to find that the continuous local martingale ``part'' obtained by ItƓ's formula is indeed a Martingale. To ...
