# 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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### Matrix-valued stochastic DE to a vector-valued stochastic DE

Suppose we have a matrix-valued ODE: $$i\tag{1} \frac{d}{dt}U(t) = HU(t)$$ where $U(t)$ and $H$ are complex square matrices. Given a time-independent vector $\psi$, we can convert (1) into a vector-...
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### "Generalized" Geometric Brownian Motion as a SDE system

It is very well known that the equation $$d X_t = \mu X_t dt+\sigma X_tdW_t$$ has a solution $$X_t = X_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t},$$ and we say that $X_t$ follows Geometric Brownian ...
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### Mass matrix in underdamped Langevin dynamics and convergence

I have two doubts regarding the underdamped Langevin dynamic. The first one is that I noticed that the underdamped Langevin dynamic can be written in the following two ways: \begin{align}\label{eq:...
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### Comparison of Solutions to SDEs

Let $B(t)$ be a Brownian motion. If we assume that $b(t, x), b_i(t, x)$, $\sigma(t, x)$, and $\sigma_i(t, x)$ are Lipschitz functions for $i = 1, 2$ then it is known that if $X_i(t)$ is the unique ...
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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. ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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\...