Mathematics Stack Exchange
Mathematics Stack Exchange

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.

1569 questions
0
votes
1 answer

When is the solution to the SDE $dX_t = f(t, X_t)dt + \sigma dW_t$ Gaussian?

I am very new to SDEs, so this question may be basic, but I'm trying to figure it out. Suppose we have and SDE $$dX_t = f(t, X_t)dt + \sigma dW_t$$ where $X_t$ is a 1-dimensional variable, $\sigma$ is a constant, $f(t, X_t)$ is some arbitrary drift…
Alex
  • 1
0
votes
0 answers

How can I write the Gompertz SDE in matlab?

I have the sde as SDE=(ax(t)-cx(t)*log(x(t)))dt+bx(t)*dW(t); with a=0.1; b=0.2;c=0.3. I was wondering how to implement that into matlab and get a graph as I have been unable to get anywhere so far?
George Hendry
  • 1
0
votes
1 answer

Solving SDE (Geometric Brownian motion)

$$S_t=S_0\exp\bigg(\bigg(\mu-\frac{\sigma^2}{2}\bigg)t+\sigma W_t\bigg).$$ What I tried to do is applying Ito's formula to above equation to derive SDE. If I define $g = S_t$ then, $dg =…
user13232877
  • 155
0
votes
0 answers

Find an SDE given a probability density function p(x,t)

Usually, we start with an SDE for $X(t)$ and then try to solve it and find the corresponding probability density function $p(x,t)$. But what if we start with a pdf - can we find a corresponding SDE? How can this be done? Would we use the…
Confounded
  • 583
0
votes
0 answers

What is the difference between the exact solution of the black scholes equation and the numerical euler scheme?

I'm learning about SDE's and I need to simulate the black scholes equation. I asked already several things about it, but I'm still puzzeled by a few things. I obtained (with the help of forum members) the following equations for the black scholes…
Tim
  • 684
0
votes
1 answer

Discrepancy between initial value and closed form solution of linear 2D SDE

Kloeden Platen Schurz 1994 states the linear 2D SDE: $$dX_t = A X_t dt + B X_t dW_t$$ where $a=5$ and $b=0.01$ and $$A = \Big( \begin{matrix} -a & a \\ a & -a \end{matrix} \Big)$$ $$B = \Big( \begin{matrix} b & 0 \\ 0 & b \end{matrix} \Big)$$ They…
Lars Ericson
  • 160
0
votes
0 answers

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 discrete-time 1D stochastic process. The solution is…
Dimka Kopitkov
  • 85
0
votes
0 answers

Moment Estimate for Simple Linear SDE

In a paper I am reading, it is claimed that the SDE $$ dX_t = b X_t dt + (\sigma X_t + \beta_t) dB_t , \quad t \in [0,T],$$ satisfies $$ E\left[ \sup_{t \in [0,T]} |X_t|^p \right] < \infty, \forall p \in [2,\infty), $$ where $b, \sigma \in…
White
  • 183
  • 9
0
votes
0 answers

Solution for Linear SDE with 2-dimentional noise and static coefficients

I need to find solutions for SDE of types: $$ dX_t=a\,dt+b^1\,dW_t^1+b^2\,dW_t^2\\ dX_t=a\,dt+b^1\,dW_t^1+b^2\,X_t\,dW_t^2 $$ As far I've found solutions for SDE with 2-dimensional noise in an amazing book of P. Kloeden - Numerical Solution For…
mirik
  • 111
0
votes
1 answer

how to solve SDE

How to solve this SDE: $dX(t)=X^\alpha(t)dt+\sigma X(t)dW(t), for X(0)=x_0$ $dW(t)$ is Wiener process. Also I have to use $f(t)=X(t)exp(-\alpha W(t)+1/2\alpha^2t)$ as integration factor. I tried to solve it looking at solutions of other SDE, but…
Sally
  • 23
0
votes
1 answer

How to show that $I_tdX_t + I_tP_tX_t dt = d(I_tX_t)$ using Ito's Lemma?

$dX_t +P_tX_tdt = dB_t$ and the integrating factor is $I_t = e^{\int{P_t dt}}$
Anonymous
  • 9
0
votes
1 answer

Negative volatility of Ito Diffusion?

This might be a silly question. But I wonder if the volatility or diffusion parameter in Ito diffusion must be positive or not. I.e. dX=$\mu dt$+$\sigma dz$, where z is a standard brownian motion. Does $\sigma$ have to be positive? From the…
ProbNerd
  • 1
  • 1
0
votes
1 answer

Determine the dynamics (SDE) of X_t^4

Suppose X and Y evolve according to: dXt= (2 + 5t + Xt)dt + 3 dz_1t dYt= 4Ytdt + 8Ytdz_1t + 6dz_2t where z_1t and z_2t are Brownian motions with (dz_1t )(dz_2t )=0.1dt Can you give me some starting point how to transform the equations to the initial…
Slavia Sofia
  • 1
1 2 3
4