# 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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### How to take the average of a stochastic differential equation?

I am solving a set of stochastic differential equations and I need some feedback about if what I am doing is correct. Given a vector $\boldsymbol{C}(t)=(C_+(t),C_-(t))^T$, we can writte a set of ...
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### Calculation of Autocorrelation Function in a System with Harmonic Oscillations in a Fluid: A Study of Langevin Dynamics

Introduction Explanation of the system being studied Objective of the study Presentation of the Langevin equation governing the dynamics of the particle in the system I am studying a system that ...
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### Solution of a GBM

Looking to check my solution to the below : Suppose that $X$ satisfies the SDE $dX_t = αX_tdt+σX_tdW_t$ Now define $Y$ by $Y_t=X_t^{\beta}$ ⁠, where β is a real number. Then $Y$ is also a GBM process. ...
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### Integral of "white noise" multiplied by exponential term (multivariate Ornstein-Uhlenbeck)

Consider a matrix differential equation for the vector $\mathbf{y}$ of the form $$\dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, ,$$ where $A$ is a ...
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### Can I verify the weak order of convergence of a numerical scheme for SDEs by checking convergence in distribution?

let $X_{t=0}^T$ be a continuous-time stochastic process defined by a certain stochastic differential equation $$X_t=a(t,X_t)dt+b(y,X_t)dW_t$$ where W_t is a Wiener process. Let $\tilde X_{i=0}^{N}$ ...
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### Technical difficulties with degenerate PDEs

Crossposted at Quantitative Finance SE I have seen lot of discussions in this Math. S.E. platform about 'degenerate partial differential equations'. But I still unclear about the 'technical ...
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### Use Ito's formula for calculate the dynamics of Ornstein-Uhlenbeck process [duplicate]

Consider this stochastic differential equation: $$𝑑𝑋(𝑡)=−a𝑋(𝑡)𝑑𝑡+σ𝑑𝑊(𝑡),\quad 𝑋(0)=𝑥_0∈ \Bbb R$$ where $a$ and $σ$ are constants and $𝑊(t)$ is a Brownian motion. Can someone show me how ...