# 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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### Convergence of an Ornstein-Uhlenbeck Process

Given the stochastic differential equation: $dx=-xdt+dW$ and the initial condition $\left( t_{0},x_{0}\right)$, the solution trajectory $x\left( t;t_{0},x_{0}\right)$ can be derived by ...
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### what linear stochastic equation will fractional Brownian motion satisfy? [closed]

Is there a linear stochastic equation driven by Brownian motion of which solution is the fractional Brownian motion?
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### Existence and uniqueness for SDE 2D linear system

I have the following SDE $$\ddot{x} +x = \dot{B(t)}$$ with some given initial condition $(x_0,\dot{x_0})$ and where $B(t)$ is a standard Brownian motion. I can reduce it to first order by ...
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### Solution of a Simple Discrete-time SDE

I'm looking for the solution of the following SDE: $p_{t+1} = a \cdot p_t + b + c \cdot w_t$, where $a, b$ and $c$ are constant scalars, $w_t \sim N(0, \sigma^2)$ is a white noise and $p_t$ is ...
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### Solve $dX_t=(a+bW_t)dt+cdW_t.$

Question: If $a,b,c$ are constants and $W_t$ is a Brownian motion, solve the following SDE $$dX_t=(a+bW_t)dt+cdW_t.$$ First of all, this is not a linear SDE due to presence of $W_t$. I attempted ...
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### How to solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \mathrm{d}t + c X_t \mathrm{d}W_t$

Disclaimer: I am a total beginner in stochastic calculus Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation (this SDE is the variance ...
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### Prerequisite for “Limit Theorems for Stochastic Processes”

Just wondering if anyone had read this book: "Limit Theorems for Stochastic Processes" by Albert Shiryaev and Jean Jacod. For one, I know this book is not at beginner level. I have introductory to ...
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### First hitting time density of Ornstein-Uhlenbeck model

Let $x(t)$ be an standard Wienner process defined by the SDE: $$dx(t) = \mu dt + \sigma \, dW$$ where the constants $\mu$ ,$\lambda$ and $\sigma$ are non-negative and constants. Let be $\tau$ the ...
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### PDE for holding a stock

Let's suppose we owe an asset and its path is given by the following SDE: $$S(t) = A(S, t) dt + B(S, t) d\widetilde{W}$$ where $\widetilde{W}$ is a Brownian motion (under risk neutral measure). What ...
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### Moments of Multivariate SDEs

Consider the following SDE: $$\mathbf{y}_{t} = \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u}$$ where the subscripts indicate dependence on $t$, $s$, and $u$ ...
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### What sort of qualitative behaviour does a stock following a process of the form $dS_t=α(μ-S_t )dt+S_t \sigma dW_t$ exhibit?

The following is a question taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $5.5$ Question: What sort of qualitative behaviour does a stock following ...
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### Why do we use $\mathbb P^x(X_t^0\in A)$ to denote $\mathbb P(X_t^x\in A)$.

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. Consider the following SDE $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t.\tag{E}$$ Suppose that $\mu$ and $\sigma$ are nice enough so that it has a ...
### Derive $\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$ from the SDE $dX{t} = -\alpha X_{t} dt + \beta dW_{t}$
I want to derive the following relation: $$\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$$ from the following SDE: $$dX{t} = -\alpha X_{t} dt + \beta dW_{t}$$ Motivation: In the paper, "...