# 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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### Discretization formula for system of differential equations. "Solution to one of these is the initial condition of the other". In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
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Let $(\Omega,\mathcal{F},\mathbb{P})$ be a nice filtered probability space with an $m$-dimensional standard Brownian motion $W$. Fix a time horizon $T>0$. Let $\mu \colon [0,T] \times \mathbb{R}^d \... • 1,192 8 votes 0 answers 215 views ### Reference request for a Riemannian Fokker-Planck equation I am looking for any reference which states, and proves, a Fokker-Planck equation for Riemannian manifolds. In particular, if$\mathrm{d}X_t=\mu(X_t)~\mathrm{d}t + \sigma(X_t)~\mathrm{d}B_t$is a ... • 450 8 votes 0 answers 452 views ### Asymptotic behaviour of integral. How should I proceed? Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with$b, \sigma: (l, r)\to\mathbb{R}$,$−\infty \leq l < r \leq \infty$bounded functions on compact intervals of$(l, r)$... 7 votes 0 answers 142 views ### Exit and hitting times for the Bessel process$\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$I am trying to analyse the exit time$T_1:=\inf\{t:X_t\notin[\alpha,2]\}$and hitting time$T_2:=\inf\{t:X_t=0\}$, where$\alpha<1$is a constant, and$X_t$follows the Bessel process defined by ... • 2,158 7 votes 0 answers 230 views ### Solving root stochastic differential equation I'm concerned with the following SDE: $$d Y_t= v \,dt + \sqrt {|Y_t|} \,d W_t$$ with$Y_0=-a$,$v>0$being a constant,$a>0$and$W_t$as standard Brownian Motion. Do you have hints how to solve ... 7 votes 0 answers 191 views ### Recast the scalar SPDE$du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$into a SDE in an infinite dimensional function space. Let$^1(\Omega,\mathcal A,\operatorname P)$be a probability space$U$be a separable Hilbert space$Q\in\mathfrak L(U)$be nonnegative and symmetric operator on$U$with finite trace$(W_t)_{t\ge0}...
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I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z$~$N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...