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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, …).

SDEs are used to model various phenomena such as stock price diffusions or physical systems subject to thermal fluctuations. Typically, SDEs are often driven by Brownian motion or other continuous martingales. However, other types of random behavior are possible, such as jump processes--for instance a Poisson process.

Early work on SDEs was done to describe Brownian motion in Einstein's famous paper, and at the same time by Smoluchowski. However, one of the earlier works related to Brownian motion is credited to Bachelier (1900) in his thesis 'Théorie de la spéculation'. This work was followed upon by Langevin. Later Itô and Stratonovich put SDEs on more solid mathematical footing.

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Solving SDE $dx_t = (a) x_t dt + (b) dZ_t$

I am new to stochastic differential equations. I would like to solve something like this: $dx_t = (a) x_t dt + (b) dZ_t$ The solution is: $x_t = e^{at} x_0 + b e^{at} \int_0^t e^{-au} dZ_u$ I would like to understand the steps to find the solution.…
NC520
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Integrability of linear SDE

Consider a linear SDE of the form $$dX_t = X_t(\alpha_tdt + \beta_t dW_t), \ X_0=x, $$ where $\alpha_t$, $\beta_t$ are $L^p$ integrable stochastic process: $$\mathbb{E}\int_0^t |\alpha_s|^p ds , \mathbb{E}\int_0^t |\beta_s|^p ds < \infty . $$ Can…
Kenneth Ng
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Simpler for of Euler-Maruyama or am I missing something

The stochastic part of the Euler-Maruyama method is often written with random increments with zero mean and variance $\Delta t$. For example for the SDE (drift diffusion) $$ dX(T)=\delta dt + sdW(t) $$ one could simulate this as $$ X(t+\Delta…
LiKao
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How can I solve $dX_t=|X_t|^\alpha dW_t$?

How can I solve $$dX_t=|X_t|^\alpha dW_t\ \ ?$$ What I tried is to set $X_t=u_tdt+v_tdW_t$. Then, by Itô formula $$df(X_t)=f'(X_t)u_tdt+f'(X_t)v_tdW_t+\frac{1}{2}f''(X_t)v_t^2dt.$$ So, we should have $$f'(X_t)u_t=\frac{1}{2}f''(X_t)v_t^2$$ and…
joshua
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Solving SDE example

Let's assume $f_{B_t}(t,B_t) = t*B_t$ ($B_t$ is Brownian motion) $f(t,B_t) = \frac{1}{2}t*B_t^2$ $f_t(t,B_t) = \frac{1}{2}*B_t^2$ $∫_0^Tf_t(t,B_t)dt = ∫_0^T\frac{1}{2}*B_t^2dt$ Above those are solution. First, what I confused is that How Brownian…
user13232877
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Apply Ito's formula to $d(X_t)$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$

I'd like to find $dX_t$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$. I have no idea how to apply Ito's formula here.
Van Tom
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Quotient of two geometric Brownian motions

Given $X=X_t$, $Y=Y_t$ and \begin{align} dX_t &= μX_t\,dt + σX_t\,dB_t, \\ dY_t &= σY_t\,dt + μY_t\,dB_t, \\ V_t &= \frac{X_t}{Y_t} \end{align} How to calculate $dV_t$? I'm new to SDE and calculus as well, so I'm not sure if I…
Ksdes219
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How to compute the global error of euler maruyama in order to illustrate the order of convergence?

Currently I'm learning about SDE's. For an assignment I need do realise tracks of the Black-Scholes equation. The realisation of these tracks is not the problem. I need to show that the rate of convergence in the weak sense is dt and in the strong…
Tim
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How is the simulation done of the black scholes model?

This post contains additional questions from: How do I implement the Euler scheme for this SDE? I think it is more appropriate to start a new post. In an assignment the following SDE needs to be simulated: \begin{array}{l} d S_{t}=\alpha S_{t} d…
Tim
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Mean reverting Ornstein-Uhlenbeck SDE

A mean reverting Ornstein-Uhlenbeck SDE is given by $$=(−)+;_0=,$$ where m and are positive constants and is a standard Brownian motion in 1 dimension. I have obtained the solution of this equation, $$X_t = xe^{-t} + m(1-e^{-t}) + \sigma \int_0^t…
Sybil
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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 solution (for instance Lipschitz with linear growth).…
Todd
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Hull and White SDE solution for r

I am trying to understand how does one define $f(t,r(t))=exp(α*t)r(t)$ to solve the SDE for Hull and White short rate model : $dr(t) = [v(t) − a*r(t)]dt+σdW(t)$, using Ito's Lemma. Any help is appreciated
Pranav
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Does solution exit for the stochastic differential equation (SDE) $d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$

Let $W_t$ be a standard Brownian Motion. Solve the stochastic differential equation: $$d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$$ Does the solution exist for all times t?
GOAT Messi
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second order SDE vs general diffusion stationary distribution

A quick follow-on to this question. Consider the following SDE: $$ \ddot{x} = f(x) - \gamma g(x)\, \dot{x} + \sigma h(x)\, \xi(t) \tag{1} $$ Based on [1], we can represent (1) as a system of two first-order equations: $$ \begin{cases} \mathrm{d} X_t…
jjjjjj
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Stochastic differential equation without diffusion

I have the following SDE $X_t=\int_0^t \sqrt{1+B_s^2}\cdot dB_s$ or $dX_t=\sqrt{1+B_t^2}\cdot dB_t$ If I define the stochastic variable $Y_t=X_t^2$. How can I determine the Ito equation that would satisfy $Y_t$? Can I apply Ito's lemma? Thanks
Jonathan Kiersch
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