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, …).
1,186
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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
$$ dX_t ...
1
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0
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20
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Non-linear backward Kolmogorov equation
The backward Kolmogorov equation (BKE):
$$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$
If $f\equiv 0$ then ...
- 1,404
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0
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22
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Compare a computer simulation (of a chemical reaction network) to an SDE
I am currently reading Anderson & Kurtz's Stochastic Analysis of Biochemical Systems. In this book, one studies chemical reaction networks and also how to simulate these with MATLAB. I know, that ...
- 1,722
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9
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How to determine the reflecting horizontal hidden barrier for Ito diffusion
The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
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0
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22
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When is an SDE solution differentiable in its starting value?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ ...
- 495
1
vote
1
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34
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Distribution of solution to SDE
Let $X_0$ be a standard normal random variable and suppose that $$dX_t=-\frac{1}{2}X_tdt+dB_t.$$ $X_0$ is independent of the Brownian motion. Find the distribution of $X_t$ for $t\geq0$ and find $\...
- 833
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0
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36
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Proving uniqueness of weak solution to SDE
This is the SDE:
$$dX_t=\operatorname{sign}(X_t+1)dt + dB_t$$
This is a $\mathbb{Q}$-Brownian motion:
$$W_t=B_t -\int_{0}^{t}\operatorname{sign}(B_s+1)ds$$
I've already shown that $B_t$ under measure $...
3
votes
1
answer
38
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Question on use of Ito's formula with integral in the function
This question is being asked in the context of the Feynman-Kac formula. Suppose the real-valued process $Z$ satisfies the SDE $$dZ_t=b(Z_t)dt+\sigma(Z_t)dW_t.$$ Suppose we have functions $f:\mathbb{R}\...
- 833
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1
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36
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Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$
Suppose that $X(t)$ satisfies
$\hspace{5cm}$
$dX(t)=-1.5X(t)dt+0.85dW(t)$
with $X(0)=0.7.$ Find the mean and the variance of $X(1).$
I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
- 363
3
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1
answer
54
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Black-Scholes model with a derivative with payoff $S_{T}^{3}$
Given a Black-Scholes Model and a derivative with payoff $S_{T}^{3}$ at time $T$. Check that the value of that derivative at time t is $V_{t} = g(t, T)S_{t}^{3}$, where $g(t, T)$ has to be determined.
...
- 33
2
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1
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76
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Can I get the following estimate for a family of solutions of SDE's?
Fix a time horizon $T>0$. Consider a fixed discretization parameter $\Delta>0$ and divide $[0,T]$ into intervals of the form $[K \Delta, (K+1)\Delta]$ for $K=0...\lfloor T/ \Delta \rfloor - 1$. ...
- 51
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1
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Using SDE uniqueness in law to show that two processes have the same distribution
Let $B$ and $\tilde{B}$ be independent standard Brownian motions defined on the same probability space with $B_0=\tilde{B}_0=0$. Let $$X_t=e^{B_t}\int_0^te^{-B_s}d\tilde{B}_s,\hspace{1cm}Y_t=\sinh(B_t)...
- 833
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0
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28
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Finding a change of measure so that rescaled local martingale is a local martingale
Let $\mu,\sigma:[0,\infty)\to\mathbb{R}$ be deterministic continuous functions, assume that $\sigma$ is bounded below by a strictly positive constant and that $\mu$ has compact support. Suppose that $...
- 833
2
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1
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46
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Finding a weak solution to an SDE
Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to ...
- 833
1
vote
1
answer
28
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Using the BDG inequality to show a process is uniformly integrable
For a BSDE :
$$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$
Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$.
There ...
- 141
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53
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Can this system of stochastic differential equation be solved algebraically?
Can the following system of stochastic differential equation be solved algebraically? I was trying to solve numerically but I wonder this can be done algebraically.
$dX_t=(a-Y_t)X_tdt+\...
- 3
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1
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Integrating factor and homogeneous equation for SDEs.
Can someone explain how an integrating factor is obtained when solving SDEs.
An example would be when finding the solution for a general linear SDE: $dX_t = (a(t)X_t +b(t))dt + (c(t)X_t +d(t))dB_t, ...
- 307
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1
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Using BDG inequality to show the solution to a BSDE belongs to $S^2_{\mathscr{F}}$
For a BSDE:
$$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$
Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$.
There ...
- 141
2
votes
1
answer
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Stochastic Differential Equation after a Change in the Time Parameter
How does the stochastic differential equation for a stochastic process change under a change in the time parameter? For example, consider the Bessel Squared Process
$$ dR_t = m \, dt + 2 \sqrt{R_t} \, ...
- 1,353
2
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1
answer
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Show that $\mathbb{E}(X|\mathcal{F}_t)$ is a square-integrable martingale
For a BSDE :
$$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$
Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$.
There ...
- 141
3
votes
1
answer
43
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Question about Ito formula and BSDE
When I was reading the paper from Peng, I saw an equation which I had no idea about how to get it. The details are shown below:
For a BSDE :
$$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$
...
- 141
2
votes
1
answer
83
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The stochastic differential of $\cos (B_t^{(1)}B_t^{(2)})$
Let ($B_1$, $B_2$) be a bi-dimensional correlated Brownian motions Calculate the stochastic differential equation of the process $\cos(B_{1,t}B_{2,t})$.
Attempt:
Let $X_t$ be the stochastic process ...
- 41
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0
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17
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Showing that a composition of a non-explicit function is Lipschitz
Let $b$ be a bounded and continuous function and let $W$ be a scalar Brownian motion. Consider the SDE $$dX_t=b(X_t)dt+dW_t.$$
I was tasked with showing that there exists a strictly increasing ...
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4
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0
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61
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Radon-Nikodym derivative of pushforward measures and Girsanov theorem
Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
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2
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1
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Solution of a Stochastic Differential Equation
I am trying to prove that the solution of the SDE:
$$
Z_t = Y_t + \int_0^t Z_s dX_s
$$
is:
$$
Z_t = \mathcal{E}(X)_t \left(y_0 + \int_0^t \mathcal{E}(X_s)^{-1}dY_s - \int_0^t \mathcal{E}(X)_s^{-1}d\...
- 184
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1
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How to show that $(X,B)$ and $(Y,W)$ satisfy the same SDE if their joint law is equal?
Let $(X,B)$, where $B$ is a standard BM and $X$ is process, satisfy the SDE $dX_t = b(t,X_t) + \sigma(t,X_t)dB_t$. Suppose that for some other standard BM $W$ and process $Y$ we have that the joint ...
- 343
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0
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How can I apply the Euler–Maruyama approximation to the following SDE?
I'm trying to apply the Euler–Maruyama discretization to a python code using this Wikipedia page Wikipedia page where it says that the SDE
$$dX_t = a(X_t, t) dt + b(X_t, t) dW_t, \quad X_0 = x_0 $$
...
0
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0
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13
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Stochastic difference equation selecting solution
I'm hoping someone can point me in the right direction for the following "problem":
start with a difference equation using lag operator:
$E_t[(1-\lambda_1^{-1}L)(1-\lambda_2L^{-1})\pi_{t+1} =...
- 1
2
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0
answers
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Is a Riccati BSDE explicitly solvable?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
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1
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$dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and $Y_t=S^{2-\alpha}$, can one simulate exact paths for Y_t?
The task states that $dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and the question is if one can generate (simulate) exact paths for $S_t$ by taking the transformation s.t. $Y_t=S_t^{2-\alpha}$.
I ...
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2
answers
47
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Solving a linear backward stochastic differential equation
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $\{B_t\}_{t\in[0;T]}$ be an adapted process and ...
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0
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Correlation function in the Langevin equation
So the Langevin equation of Brownian motion is a stochastic differential equation defined as
$$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$
where the noise function $\eta(t)$ has ...
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Stochastic Integral and Standard Brownian motion
Let $B_t (t\geq 0)$ be a standard Brownian Motion, where $B_0=0$. I have $d(B_s^2)= 2B_sdB_s$. I want to find the integral $ \int_0^tB_sdB_s$. Therefore, I get $ \int_0^tB_sdB_s= \frac{1}{2}\int_0^td(...
2
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0
answers
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How are anisotropic heat-equation and Fokker-Planck equation related?
Consider first a rescaled brownian motion $X_t$, in $\mathbb{R}^n$ which fulfills the SDE
$$dX_t = \sqrt{2} dB_t,$$
where $B_t$ is brownian motion. Then the density $p(t,x,x_0)$ of the process ...
- 1,019
3
votes
1
answer
90
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Properties of the time integral of Ito process
Consider the Ito process $X_t$ defined by
$$
dX_t = a(t,X_t) dt + b(t,X_t) dW_t
$$
where $W_t$ is the standard continuous-time Wiener process. Let's define the process $Y_t$ to be some integral of $...
- 1,904
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0
answers
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Verify process is an Ito process
Let $(\xi(t),t\ge 0)$ be an Ito process with $\xi(0)=\theta$ and $$ d\xi(t)=\kappa(\theta-\xi(t))dt+\sigma\sqrt{\xi(t)}dW(t)$$ for $\kappa, \theta, \sigma \in \mathbb{R}$. Show that $(\eta(t),t\ge 0)$,...
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Deriving Stochastic Differential Equations about Population Dynamics
This is somewhat of a high-level, I-hardly-know-much-about-the-topic-yet question, but here goes.
I came across online a presentation in which part of it documented deriving an SDE for a population ...
0
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0
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46
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Calculating the limit of a Wasserstein distance of two SDE's
I am trying to prove that:
$\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with
$$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$
$$dZ_t = -h(...
- 1
2
votes
1
answer
66
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Expectation of modified Ornstein-Uhlenbeck process with random long run mean
I would like to compute the expectation of a modified Ornstein-Uhlenbeck process of the form
$$ dx_t = \theta(\mu_t-x_t)dt + \sigma x_t dW_t \ ,$$
where $\kappa, \sigma>0$ and $W_t$ is a Brownian ...
- 25
1
vote
0
answers
37
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Is there any way to “analytically” compute the following conditional expectation?
Let me define the following SDE: $$\begin{cases}
dX_{t}=X_{t}dW_{t}\\
X_{0}=1
\end{cases},$$ where $W$ is an appropriate Wiener process. My question is how we can compute
$$f\left(z\right)=\mathbb{E}\...
- 828
3
votes
1
answer
36
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Infinite variation of Brownian motion and Continuity
Let $C>0$ be a constant. Brownian motion is Hölder continuous for $\alpha=1/2$:
$$| B(t+h) - B(t) | \leq C \sqrt{h \log(1/h)} \leq C h^\alpha,$$
for every sufficiently small $h$.
But Brownian ...
- 41
3
votes
1
answer
30
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A definition of Stochastic integral that is both a martingale and preserves chain rule?
my question is straightforward: is there any definition of Stochastic integral (so I assume at least some kind of Riemann sum compatibility and linearity) that is both a local martingale and preserves ...
- 83
1
vote
0
answers
25
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Confusion about the covariance of the Wiener process
When studying the Wiener process, I learned that the variance of this process is $Var(W_t) = t$, (which can be proven by calculating the quadratic variation) and furthermore that $\mathbb{E}W_tW_s = \...
- 417
0
votes
0
answers
23
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the mean of a stochastic differential equation
Let $(X_{t} , t \geq 0)$ be a processus and solution of the sifferential equation : \
$ X_{t}= X_{0} + \int_{0}^{t} \mu (1-2X_s)ds+ \int_{0}^{t} \sqrt{X_s(1- X_s)} dB_s$ with
$ \mu > 0$ and B a ...
- 1
2
votes
0
answers
31
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Calculate the expectation of random differential equation?
I have a differential equation
$$\frac{dx(t)}{dt}=F(g(x(t),\omega)),\;\;x_0=x,$$
where $\omega$ is standard normal variable.
Given that for each fixed $y\in\mathbb{R}$, $$\mathbb{E}_\omega[g(y,\omega)]...
- 27
1
vote
0
answers
31
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Showing that a diffusion process is positive recurrent
Consider an one-dimensional diffusion, with values in $(0,\infty)$, solution of the SDE
$$dX_t = \mu(m-X_t)dt + \sigma X^{\psi}_tdB_t$$
Where $B_t$ is a standard one-dimensional brownian motion, $m>...
- 35
0
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0
answers
19
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Finding the stochastic differential equation using multidimensional Ito's formula
I've been working on this question and have been having a little difficulty with it.
I'm trying to find the SDE for Y, where:
$$
Y_t=X_te^{W_{2,t}}
$$
and
$$
X_t=X_0+\int_0^{t}a_{1,s}ds+\int_{0}^{t}b_{...
1
vote
2
answers
32
views
Solving SDE $dY = Adt + BYdW, Y(0) = Y_0$?
I am trying to solve the following SDE:
$dY = Adt + BYdW$, where $Y(0)=Y_0$.
Thus far, I have done the following.
First, from Ito's Lemma, we have:
$d(ln(Y))=\frac{1}{Y}dY-\frac{1}{2Y^2}dY^2$.
...
- 37
1
vote
1
answer
27
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Integrability of linear SDE
Consider a linear SDE of the form
$$dX_t = X_t(\alpha_tdt + \beta_t dW_t), \ X_0=x, $$
where $\alpha_t$, $\beta_t$ are $L^p$ integrable stochastic process:
$$\mathbb{E}\int_0^t |\alpha_s|^p ds , \...
- 329
0
votes
1
answer
52
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Ornstein-Uhlenbeck process with random long run mean
I am considering an OU process of the form
$$ dx_t = \theta(\mu-x_t)dt + \sigma dW_t $$
where $\kappa, \sigma>0$ and $W_t$ is a Brownian motion. I know that $x_t$ has expectation given by:
$$ \...
- 25