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, …).
574
questions
0
votes
1answer
9 views
Convergence of an Ornstein-Uhlenbeck Process
Given the stochastic differential equation:
$dx=-xdt+dW$
and the initial condition $\left( t_{0},x_{0}\right) $, the solution
trajectory $x\left( t;t_{0},x_{0}\right) $ can be derived by ...
0
votes
0answers
17 views
How to compute the quadratic variation of $\langle X,Y \rangle_t$?
Let $(\Omega,(\mathcal{F}_t)_{t≥0}, \mathcal{P})$ be a filtered probability space. Let $(B_1(t), B_2(t))_{t≥0}$ be a two-dimensional Brownian
motion. Let $ θ\in \mathbb{R}$ and let
$$ X_t^θ = B_1(t) \...
0
votes
0answers
17 views
Does Causation imply dependency in the Euler scheme discretization for SDEs?
I have a confusion concerning the following statement:
Assume we have n-observations $(X_0,\dots,X_n)$ that are coming from a discretization of an SDE using the following Euler scheme :
$$
X_i=X_{i-1}...
0
votes
0answers
15 views
what linear stochastic equation will fractional Brownian motion satisfy? [closed]
Is there a linear stochastic equation driven by Brownian motion of which solution is the fractional Brownian motion?
1
vote
1answer
24 views
Existence and uniqueness for SDE 2D linear system
I have the following SDE $$\ddot{x} +x = \dot{B(t)}$$
with some given initial condition $(x_0,\dot{x_0})$ and where $B(t)$ is a standard Brownian motion. I can reduce it to first order by ...
0
votes
0answers
10 views
Which of these equations is the backward Kolmogorov Equation?
I am reading Stochastic Processes and Applications by Pavliotis. At some point he defines Kolmogorov backward equation as the differential equations
$$\frac{\partial u}{\partial t} = \mathcal{L}u$$
$$...
0
votes
0answers
29 views
Finding the drift coefficient
$X_t$ is a solution of SDL
$$dX_t=b(X_t)dt+X_tdB_t.$$
What is the the drift coefficient $b(x)$ of this equation, if the square of its solution
$$M_t=X^2_t, t\geq0,$$
is a martingale?
So far, I ...
2
votes
0answers
55 views
Solve following SDE
Any ideas on how to solve the following SDE
$$X_t=\int_0^te^{-X_s}dW_s+\frac{1}{2}\int_0^te^{-2X_s}ds$$
It does not appear there is a strong solution, but what about weak solution?
Consider the ...
0
votes
1answer
19 views
Discrepancy between initial value and closed form solution of linear 2D SDE
Kloeden Platen Schurz 1994 states the linear 2D SDE:
$$dX_t = A X_t dt + B X_t dW_t$$
where $a=5$ and $b=0.01$ and
$$A = \Big( \begin{matrix} -a & a \\ a & -a \end{matrix} \Big)$$
$$B = \...
0
votes
1answer
28 views
Variance of $n$-dimensional Brownian motion (Oksendal)
I apologize if this is a trivial question but I am confused by eq. $2.2.9$ in Oksendal (see image below). If $B_t$ is $n$-dimensional Brownian motion then isn't $B_t\sim\mathcal N(\mathbf x,tI_n)$ a $...
0
votes
1answer
21 views
Preserving symplectic structure iff matrix has unitary determinant
Let us consider a linear stochastic oscillator system:
$$ d \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x(t) \\y(t)\end{bmatrix}...
0
votes
0answers
41 views
Multi-factor Ito Lemma & Girsanov Theorem
Assume $X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma_1 X(h)dW_1(h)+\int^{t}_{0}\sigma_2 X(h)dW_2(h)$
FIRST PART OF THE QUESTION:
Assume first that $\mu$ and $\sigma_1$ & $\sigma_2$ are ...
1
vote
1answer
110 views
+100
Show that a certain solution of an ODE is a stochastic process.
I am working on an exercise as follows:
Let $v_{1}, v_{2}:\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ be $C^{\infty}$ vector fields over $\mathbb{R}^{2}$. Let $(X_{n})_{n\in\mathbb{N}}$ be an i.i....
1
vote
0answers
15 views
How can I prove that the solution to this SDE is a Markov process?
Consider the two-dimensional SDE
\begin{align}
dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\
dV_t &= \kappa(\theta - V_t)dt + \sigma ...
-1
votes
0answers
26 views
$\int_0^t e^{a\cdot s + b\cdot W_s}\, ds$
I'm trying to solve the following integral,
$$\int_0^t e^{a\cdot s + b\cdot W_s}\, ds $$
where $W_s$ is a standard brownian motion and $a$ and $b$ are constants. I've thought about using integration ...
3
votes
1answer
31 views
Solve the SDE $dX_t=dB_t+\frac{c-X_t}{T-t}dt$
How to solve the following SDE characterizing Brownian motion with fixed end point $c$ at time $T$?
$$dX_t=dB_t+\frac{c-X_t}{T-t}dt$$
I do not believe a strong solution exists, by ito lemma it would ...
2
votes
0answers
27 views
Stochastic differential equation form
A few days ago, I disccussed my Phd thesis (thesis defense), one of the mathematical mistakes that the committe members alerted me to it, is how to write correctley an ordinary differential equation ...
2
votes
0answers
22 views
What is a Markov kernel of a Euler-Maruyama discretization?
Let $B_t$ be a $d-$dimensional Gaussian process and we consider being given the SDE, $dY_t = - \nabla U(Y_t)dt + \sqrt{2}dB_t$. A Euler-Maruyama discretization of this will look like the Markov Chain, ...
0
votes
0answers
12 views
Steady-state distribution for a 2-dimensional SDE
Consider the following 2-D SDE
\begin{align*}
\dot x_1(t) &= \frac{1}{1+x_2(t)}+x_1(t) + w_1(t)\\
\dot x_2(t) &= \frac{1}{1+x_1(t)}+x_2(t) + w_2(t)
\end{align*}
where $w_1(t)$ and $w_2(t)$ are ...
0
votes
0answers
13 views
What's the role of commutation relations in stochastic mechanics?
In a stochastic context, we can understand a term like
$$ \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} $$
or the (Anti-...
0
votes
0answers
40 views
quadratic covariation of SDE
consider the following SDE for the 2-dimensional stochastic processes $ X_t=( ( X_1(t) ),( X_2(t) ) ) $ driven by a 2-dimensional continuous standard Brownian motion $ B_t=( ( B_1(t) ),( B_2(t) ) ) $
...
1
vote
0answers
32 views
Solution to the SDE
i'm confused with the first integral (not the stochastic part) of following SDE:
$$dX_t = (X_t - X_t^3)dt + \int kX_t dB_t.$$
The solution can be written as
$$X_t = x_0 + \int_0^{t} X_s - X_s^3 ds + \...
2
votes
1answer
38 views
Show that $\left(\phi\left(X_{t}\right)\right)_{t \geq 0}$ is a martingale
For some point $x \in \mathbb{R},$ we consider the SDE:
$$
\forall t \geq 0, \quad X_{t}=x+\int_{0}^{t} \sin \left(2 X_{s}\right) d s+\int_{0}^{t}\left(1+\cos ^{2}\left(X_{s}\right)\right)^{1 / 2} d ...
1
vote
0answers
20 views
is $\int_{0}^{t} dB_s = B_t$?
$(B_t)_{t \geq 0}$ is a standard Brownian motion, first of all I know that they are equal in distribution but as I was solving the Geometric Brownian motion SDE I got a solution of the form :
$$X_t =...
1
vote
1answer
25 views
Why is the expectation value of a stochastic integral equal to $\sum_k (\tau_k - t_k)$ where $\tau_k$ denotes the partitioning points?
In the book "Handbook of Stochastic Methods for Physics" by Crispin and Gardiner, I found the following calculation to show that stochastic integrals depend on the choice of partitioning points.
...
1
vote
1answer
50 views
Differential equation to Ito equation
I have a differential equation:
$$\frac{dX}{dt}=f(X)+\epsilon X n(t)$$
where $f$ is a deterministic function, $\epsilon$ is a constant, $n(t)$ is a white Gaussian noise and $X$ is a random process.
I ...
2
votes
2answers
67 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 ...
1
vote
0answers
22 views
Centre the solution of an SDE
In the following $W_t$ is a Brownian motion say in $\mathbb{R}$ or $\mathbb{R}^d$ and $y_0$ is deterministic.
If you have an SDE say $Y_t\in \mathbb{R}^d$
$$ dY_t=b(Y_t)dt+dW_t,~~Y_0=y_0$$
you can ...
0
votes
1answer
18 views
how to compute the cross variation process here?
let $C = (C_t = C_0 e^{\alpha W_t})_{t\geq 0}$ where $C_0 \geq 0$ (constant) and $W_t$ is a standard Brownian motion
let $X_t$ be a stochastic process such that :
$dX_t = \mu X_t dt + \sigma X_t d ...
0
votes
0answers
12 views
Solution of a Simple Discrete-time SDE
I'm looking for the solution of the following SDE:
$p_{t+1} = a \cdot p_t + b + c \cdot w_t$,
where $a, b$ and $c$ are constant scalars, $w_t \sim N(0, \sigma^2)$ is a white noise and $p_t$ is ...
1
vote
0answers
29 views
Solve $dX_t=(a+bW_t)dt+cdW_t.$
Question: If $a,b,c$ are constants and $W_t$ is a Brownian motion, solve the following SDE
$$dX_t=(a+bW_t)dt+cdW_t.$$
First of all, this is not a linear SDE due to presence of $W_t$.
I attempted ...
1
vote
0answers
28 views
How to solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \mathrm{d}t + c X_t \mathrm{d}W_t$
Disclaimer: I am a total beginner in stochastic calculus
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation (this SDE is the variance ...
0
votes
0answers
22 views
Prerequisite for “Limit Theorems for Stochastic Processes”
Just wondering if anyone had read this book: "Limit Theorems for Stochastic Processes" by Albert Shiryaev and Jean Jacod. For one, I know this book is not at beginner level. I have introductory to ...
2
votes
0answers
132 views
First hitting time density of Ornstein-Uhlenbeck model
Let $x(t)$ be an standard Wienner process defined by the SDE:
$$dx(t) = \mu dt + \sigma \, dW $$
where the constants $\mu$ ,$\lambda$ and $\sigma$ are non-negative and constants.
Let be $\tau$ the ...
3
votes
0answers
67 views
PDE for holding a stock
Let's suppose we owe an asset and its path is given by the following SDE:
$$ S(t) = A(S, t) dt + B(S, t) d\widetilde{W}$$
where $\widetilde{W}$ is a Brownian motion (under risk neutral measure).
What ...
2
votes
0answers
21 views
Moments of Multivariate SDEs
Consider the following SDE:
$$ \mathbf{y}_{t} = \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} $$
where the subscripts indicate dependence on $t$, $s$, and $u$ ...
4
votes
1answer
42 views
What sort of qualitative behaviour does a stock following a process of the form $dS_t=α(μ-S_t )dt+S_t \sigma dW_t$ exhibit?
The following is a question taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $5.5$
Question: What sort of qualitative behaviour does a stock following ...
2
votes
0answers
39 views
Why do we use $\mathbb P^x(X_t^0\in A)$ to denote $\mathbb P(X_t^x\in A)$.
Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. Consider the following SDE $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t.\tag{E}$$
Suppose that $\mu$ and $\sigma $ are nice enough so that it has a ...
0
votes
0answers
32 views
Derive $\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$ from the SDE $dX{t} = -\alpha X_{t} dt + \beta dW_{t}$
I want to derive the following relation:
$$\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$$
from the following SDE:
$$ dX{t} = -\alpha X_{t} dt + \beta dW_{t} $$
Motivation:
In the paper, "...
1
vote
2answers
67 views
Find the covariance of $x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$
I want to find the covariance of the following SDE:
$$x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$$
To start, I find the mean, it is simply:
$$ E[x_{t}] = E[x_{0}] e^{-\...
2
votes
1answer
77 views
Prove that portfolio is self financing?
Let assume we have a portfolio with strategy described by $θ_t = \int_0^t S_udu$ (position in stock) and $ψ_t = -\int_0^t \frac{S_u^2}{B_u}du$ (position in bond). How to prove that this strategy is ...
4
votes
1answer
64 views
Show that a SDE has no solution
Let $B$ be a standard Brownian motion. I want to prove that the SDE
\begin{equation}
X_t = \int_0^t 1_{\{ X_s \geq 0 \}} \, dB_s
\end{equation} has no solution on any set-up.
A hint I was given was ...
0
votes
0answers
15 views
Comparison theorem for the following linear SDE
Let $W$ be a two-dimensional Brownian motion. Let $A^a$ and $A^b$ be two adapted (w.r.t the natural filtration of $W$), cadlag, non-decreasing processes with the convention that $A^a_{0-}=A^b_{0-}=0$. ...
2
votes
1answer
89 views
Martingale representation of European option.
Let stock price $S$ satisfy
$$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$
I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
0
votes
0answers
37 views
Sample Second Moment of SDE
Consider a simple sde
$$\mathrm{d}X_t = \mu \mathrm{d}t + \sigma \mathrm{d}W_t.$$
Assuming $X_0 = x$,
$$\mathbb{E}\left[X^2_t\right] = \left(x +\mu t\right)^2 + \sigma^2 t.$$
I have been seeking to ...
4
votes
0answers
44 views
Ito Derivative of White Noise
We know that white noise $w_{t}$ is given by the time derivative of Brownian motion $\beta_{t}$, ie that:
$$ w_{t} = \frac{d \beta_{t}}{dt} $$
Now I want to define a new process, called blue noise $...
0
votes
0answers
21 views
Simulating a simple SDE (geometric Brownian motion) - problems with Wiener Process
I am trying to simulate a simple SDE
$$dS = \mu Sdt + \sigma S dW_t$$
with $\mu,\sigma \in\mathbb{R}$. When I use the scheme
\begin{align}
S_{n+1} &= S_n + \mu h S_n + \sigma S_n N(0,h)
\end{...
0
votes
1answer
75 views
multivariate Ito isometry
I wonder whether there exists a straightforward extension of the Ito isometry to multidimensional processes.
In the one-dimensional case the Ito isometry can be written as
$\mathbb{E}[ (\int_0^T X_t ...
1
vote
0answers
43 views
Deriving Stochastic Differential Equations from Autocorrelation
Suppose I know my stationary stochastic process has the following autocorrelation function:
$$ R(\tau) = \sigma^{2} e^{-\alpha |\tau|} \cos(\omega \tau) $$
How can I derive the stochastic ...
1
vote
0answers
10 views
What resources can I read to learn how the norm of the distance to a stationary distribution evolves over time in a Fokker-Planck solution?
Let's say we have an initial distribution $p_0$, which we evolve according the Fokker-Planck equation with stationary distribution $p_\infty$ (which need not be log-concave).
$$
\frac{\partial p_t}{\...
