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, …).
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Is there a way to solve linear 1st order ODEs with time-varying coefficients driven by geometric Brownian motion?
Let $X_t$ be a stochastic process satisfying,
$$ dX_t = \mu(t)X_t dt + \sigma(t)X_t dW_t $$
where $W_t$ is a Brownian motion. Is there a way to solve the following for $y$?
$$ y'(t) a(t) + y(t) b(t) + ...
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Solution to SDE as measurable function of initial value
In Schilling's "Brownian Motion", it is argued in Remark 21.24 that if the stochastic process $X^x$ is the solution to an SDE with initial value $x\in\mathbb{R}$, then it depends measurably ...
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Ito’s method to construct diffusion
I was reading a lecture slide online. In the lecture slide, it says that "Kolmogorov’s
method is useful to construct processes in distribution’s sense. Ito introduced stochastic differential ...
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Stochastic process with Holder continuous trajectories s.t. Holder norm over $[0,\infty)$ has finite moments
Typically, KCC allows us to show that a stochastic process has Holder-continuous trajectories and that
\begin{equation}
\mathbb{E}\left[\sup_{0<s<t<T}\frac{|X_t-X_s|}{|t-s|^{\alpha}}\right]&...
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How does the Ito calculus chain rule preserve the norm of a unit vector?
Consider a 2-dimensional vector (inspired by (active) Brownian particle dynamics with rotational diffusion constant $D_r$) that is given by
$\vec{n}(\theta) = \cos(\theta) \hat{x} + \sin(\theta) \hat{...
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"Derivative" of a random process experiment of Itos Lemma
I though that the meaning of saying that the stochastic process given by $S_t = e^{0.5t + W_t}$ satisfies the SDE, $dS = dt+dW$ meant in particular that the best linear approximation to S around $t =0$...
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Yamada--Watanabe theorem up to a stopping time holds
I wonder if a Localized Yamada--Watanabe theorem up to a stopping time holds.
Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space satisfying the usual ...
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Connecting two forms of the Kolmogorov Backward Equation
I am struggling to connect the standard formulation of the Kolmogorov Backward Equation (KBE) from Oksendal with the one found in a paper by Andersson (1982) on reverse stochastic differential ...
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"Stochastic midpoint method" is the Milstein method?
I tried to "generalize" the midpoint method, known from the topic of numerical solutions of ODEs, to a stochastic framework. Analogously to the midpoint method, I derived a formula for SDEs ...
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Density function of reverse diffusion process
All the prcoesses involved are continuous Markov process. The reverse diffusion and forward diffusion traverse identical trajectories in reverse temporal order.
In the Machine Learning paper Deep ...
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Regions created by a reflecting random curve
Consider a continuous unit-speed curve $X:[0,L]\to\mathcal{S}$ in the unit square $\mathcal{S}=[0,1]^2$ with specular reflection at the boundary (we can also unfold to the flat torus $\mathbb T^2$). ...
user1675092
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Basic question on the definition of stochastic PDE.
I just started learning stochastic PDE. At the moment I would like to understand the scope of this notion. In textbooks one considers equations of the form
$$dX_t=\sigma(t,X_t)dB_t+b(t,X_t)dt, \tag 1$$...
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Are there any results on the numerical method for multi-valued SDEs driven by fractional Brownian motion?
I am studying the numerical solution of multi-valued stochastic differential equations driven by the fractional Brownian motion (fractional white noise).
The multi-valued SDEs can be written as the ...
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The Cameron Martin Space of the Wiener Measure
I am going through some notes on SPDEs and I am having some difficulties with the following problem:
The definition I am working with is
My attempt:
I know that the space $\mathring{\mathcal{H}}_\mu$...
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Strong uniquness of $\mathrm dX_t = \operatorname{sign}(X_t) \sqrt{|X_t|} + \mathrm d W_t$, $X_0 =0$ in 1D
I am a little rusty on S.D.E., and I remember that if we have the S.D.E.
$$
\begin{cases}
\mathrm dX_t = \sqrt{|X_t|} + \mathrm d W_t \\
X_0=0
\end{cases}
$$
then there is a unique strong solution to ...
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