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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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Recovering the SDE of Vasicek model.

Suppose we have the solution to the ordinary Vasicek model: $$r_t = r_0 e^{-a t} + b(1 - e^{-a t}) + \sigma \int^{t}_0 e^{-a (t-s) } dW_s$$ How do I use the Ito's lemma to recover the SDE $$dr_t = a(b - r_t)dt + \sigma dW_t$$ Thank you for your…
INiCOLAl
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SDE multiplied by previsible process

I'm reading "Financial Calculus" of Baxter and Rennie, and have a question regarding some substitution in SDE. Let's suppose that $\Phi_t$ and $\Psi_t$ are previsible processes. We have SDE: $$dS_t=\sigma S_t dW_t + rS_t dt $$ where $\sigma$ and $r$…
siwy9
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Uniqueness of solution for SDE with random coefficients

Suppose we have a SDE in the form of $$dX_t = f(t, X_t, W_t)dt + g(t)dW_t$$ In my case, I am able to find one solution for this equation. Is there any theorem which can prove that this solution is unique? More specifically, my SDE is $$dX_t = (\mu…
Alexander Azbil
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Distribution of solution to linear SDE

Consider a linear SDE of the form: $$dX_t = \alpha X_tdt + \beta dW_t,\\ X_t = x$$ I would like to determine an explicit solution, as well as that solution's distribution if possible. My issue with the first task is that we are given $X_t = x$, and…
Jaood
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System of SDE with 3 Brownian Motions

$dY_1 =\beta_1dt +1dB_1+2dB_2+3dB_3$ $dY_2 =\beta_2dt +1dB_1+2dB_2+2dB_3$ $\beta_{1,2} $ bounded, $B_{1,2,3}$ Brownian Motions. System of SDEs. I know how to solve a linear SDE with 1 Brownian motion. I can (or at least think) solve a system of…
JonesY
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Coefficient matching method for SDEs

I am using the book by M.Steele, Stochastic calculus and financial applications and came across the following solution method in chapter 9. The method is called coefficient matching and starts like: $$dX(t) = \mu X(t)dt + \sigma X(t)dB(t) \quad…
Gabor Bakos
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Solve simple system of SDEs

I want so solve the following SDE \begin{align}\dot{y}(t)=r(t)y(t)+\epsilon_1(t) \\ \dot{r}(t)=r(t)+\epsilon_2(t)\end{align} with $r,y$ both being stochastic processes and $e_1,e_2$ both being gaussian white noise processes with variance…
Julian Karch
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Predictable cross variation - SDE

Suppose I have two continuous semimartingales $X$ and $Y$. By Ito's formula for semimartingales, we have $$ X_tY_t=X_0Y_0+\int_0^t X_sdY_s+\int_0^tY_sdX_s+\langle X,Y\rangle_t $$ where $\langle X,Y\rangle_t$ is the predictable cross-variation…
Sam Wong
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How do we prove conditional probability distribution for stochastic equation

Back Ground I read books "Diffusion Models" written by Okanohara Daisuke. I cannot understand bleow problem. Please tell me how to prove this problem. Problem We consider below stochastic differential…
user18000142
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Linear stochastic differential equation solution proof

I'm studying from Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion book. In particular, I'm having a hard time understanding a particular step into the proof of the Proposition 2.3 of the IX chapter - Stochastic Differential…
user515933
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Forward discretization of the BSDE in the Deep BSDE method

In the Deep BSDE method proposed by E et al., (2017) and Han et al., (2018), the backward stochastic differential equation (BSDE): $\begin{cases} dY_t &= -f(t, \mathbf{X}_t, Y_t, \mathbf{Z}_t)\ dt + \mathbf{Z}_t^\intercal \…
poglhar
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Solution to Black-Scholes (geometric Brownian motion)

For a basic stochastic differential equation $$dS_t=S_t(μdt+σdW_t),$$ a classic method to solve it is by letting $f = \ln S$, and when using Ito formula to this $$df=d\left(\ln(Y(t)\right)=\left(\frac{∂f}{∂t}+\frac{∂f}{∂Y}+\frac…
Wang Jing
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Exact solution to a system of Ito SDE's in order to study convergence behaviour

I'm working on a simple stock pricing model described by the following model of Ito SDE's: $$ d S_t = \mu S_t dt + \sigma_t S_t dB^1_t \\ d \sigma_t = -(\sigma_t - \xi_t)dt + p \sigma_t dB^2 _t \\ d \xi_t = \frac{1}{\alpha}(\sigma_t - \xi_t)…
Lennard
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definition of stochastic diferential equation

What is mean by: $X_Y(t)$ satisfies the stochastic differential equation $$ dX_Y(t) = \sigma X_Y(t) dW(t). $$ What is definition of $dX_Y(t)$ and $dW(t)$?
franta96
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Existence of Solution to SDE - Oksendal Theorem 5.2.1

I am trying to understand a few lines in Oksendal's Theorem 5.2.1 (5th edition). This is the existence and uniqueness theorem for SDEs. The relevant lines are: Oksendal bottom of page 68 and Oksendal top of page 69. I understand the application of…
Peter
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