Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [sde]

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, …), ...

0
votes
0answers
12 views

Show that the solution of an autonomous SDE is a time-homogeneous Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space and $$\mathcal N:=\left\{N\in\mathcal A:\operatorname P[N]=0\right\}$$ $(W_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\...
1
vote
0answers
26 views

When the following expected value is finite?

Let us consider the stochastic process $(X_t)_{t\geq0}$ that can be described by the following SDE: $$ dX_t = \alpha(X_t, t) dt + \sigma(X_t, t) dB_t $$ Now I consider the following expected value: $$...
0
votes
0answers
33 views
+50

Dependence on the initial datum of the strong solution of a SDE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ $W$ be a $\mathcal F$-Brownian motion ...
0
votes
0answers
13 views

Relation between time-changed solutions to SDEs

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a $\mathcal F$-...
0
votes
0answers
11 views

Stratonovich Differential: $Y_t\circ dB_t( dB_t) = Y_t\circ dt = Y_t\, dt$?

I have the following where $B_t$ is standard Brownian Motion $$Y_t\circ dB_t( dB_t)$$ I assume I can convert it to $$Y_t\circ dB_t( dB_t) = Y_t\circ dt$$ Then I further assume I can do this: $$...
4
votes
1answer
40 views

Difference between weak ( or martingale ) and strong solutions to SDEs

Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$ ...
0
votes
0answers
12 views

What does a 'pathwise' solution mean

I have not done very much stochastic calculus but know the basics. (Ito formula, Ito lemma, Stochastic Integral construction, the properties of B.M, Some properties of the stochastic integral, I have ...
0
votes
0answers
7 views

A Question about the Fixed Point Method for SDEs

We assume a filtered probability space satisfying the usual conditions. Let $\mathbb{H}^2$ denote the space of $dt \otimes P$ square integrable and progressively measurable processes with the norm $$ |...
0
votes
0answers
11 views

Explicit solution of SDE with linear drift and constant diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(B_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b:\mathbb R\to\mathbb R$ be linear $\sigma>0$ We ...
1
vote
0answers
54 views

Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
0
votes
0answers
17 views

What is the difference between a global solution and strong solution of an SDE?

Can someone guide me to know if there is a difference between a global solution of a SDE and a strong solution? I'm a little bit confused.
1
vote
0answers
35 views

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 ...
3
votes
0answers
35 views

Variance spectrum of sde

I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal. Consider the Ito SDE. $dX_t=−λX_tdt+σdB_t$ Now state the variance spectrum of $\{X_t\}$ and ...
3
votes
0answers
37 views

Stratonovich equation

I am having the following problem: Consider the following stratonovich equation: $dY_t=-\sin(Y_t)\cdot dt+s\cos(Y_t)\circ dB_t$ where $B_t$ is a regular brownian motion. Then define the process $...
2
votes
0answers
30 views

SDE existence and uniqueness

I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal. Consider the two coupled It SDEs. $dXt=−λX_tdt+σdB_t$ $dYt=−\sin Y_tdt+sX_t\cos Ytdt$ Where ...
0
votes
0answers
18 views

Stochastic Differential Equation and Runge-Kutta

I have to solve the Black-Scholes equation, $\textrm{d}X\left(t\right)=\lambda X\left(t\right)\textrm{d}t+\mu X\left(t\right)\textrm{d}W\left(t\right),$ by making use of a RK method (in Python). ...
2
votes
1answer
27 views

Does each PDE's have a strong link to a Particular Stochastic Process

Brownian Motion has a deep link to the heat PDE. Studying the dynamics of a particle moving (as if undergoing Brownian Motion) one can derive the heat equation. Also looking at the generator of the ...
1
vote
0answers
39 views

Ito or Stratonovich equation

Consider the two coupled Ito SDEs $dX_t=-\lambda X_t\cdot dt+\sigma\cdot dB_t$ $dY_t=-\sin Y_t\cdot dt+s\cdot X_t\cdot \cos Y_t dt$ I assume that $X_t$ is an Ornstein-Uhlenbeck process, and that $...
0
votes
0answers
43 views

Some problems on OU process

From Marc Yor's Continuous Martingale and Brownian Motion Page 38, we know that the process $X_t=e^{-\lambda t}B_{e^{2\lambda t}}$ is an OU process, where $B_{t}$ is an one-dimensional standard ...
1
vote
0answers
21 views

Does the dynamics of this Fund follow a GBM?

I have a Fund $F_t$ made of a bond ($B_t$) combined with an equity ($S_t$). $$F_t=S_t + B_t$$The dynamics are : $ dB_t = r B_tdt$ and $dS_t=rS_tdt + \sigma S_tdW_t$ where $r$ and $\sigma$ are constant ...
0
votes
0answers
18 views

Itō diffusion as a solution to a martingale problem

Let $b,\sigma\in C_b([0,\infty)\times\mathbb R)$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $B$ be an $\mathcal F$-...
0
votes
0answers
29 views

Understanding the positivity to the solution of the SDE $dX_t= adt+\sqrt{ \vert X_t \vert}dW_t$

I am trying to reason that the(I am quite sure that my reasoning is fallacious but I can't see why) solution of the SDE $$dX_t= adt+\sqrt{\vert X_t\vert }dW_t$$ with $X_0=2a$ and $a>0$ is non-...
0
votes
0answers
24 views

Sum of independent Brownian Motions: $\sqrt{a}dW_{1,t}+\sqrt{b}dW_{2,t}=\sqrt{a+b}dW_{3,t}$?

Consider two stochastic differential equations: a) $$ dX=\sqrt{a}dW_{1,t}+\sqrt{b}dW_{2,t} $$ b) $$ dX=\sqrt{a+b}dW_{3,t} $$ where $W_{1,t},W_{2,t},W_{3,t}$ are independent brownian motions. Do ...
0
votes
0answers
22 views

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 ...
10
votes
1answer
373 views

Is an SDE really equal to an integral equation, or is it rather “its integral” that is?

Ive been told and been reading in some textbooks on SDE's that an SDE or stochastic differential really is an integral equation. In other words, that $ dX= \beta dt + \sigma dW$ $\,$ "really means" $...
0
votes
1answer
54 views

What does the drift and diffusion coefficent of a solution to an SDE tell us in general?

Solutions of SDE's are most often written of the form $$X(t)=X(0)+\int_{0}^{t} \beta (X,s)ds + \int_{0}^{t} \alpha(X,s) dW(s).$$ I think about this as essentially the same thing as $$dX(t)=\beta (s)...
0
votes
2answers
35 views

In probability, what is exactly a noise and a white noise?

I'm reading a book on SDE, and the author often talk about Noise or white noise. But what is it exactly ? I saw on wikipedia that a Gaussian noise is a Normal r.v. but it doesn't really help to ...
1
vote
0answers
35 views

Text for stochastic processes

I’ve been reading a paper where they derive a Fokker Planck equation starting with some transition equations (the underlying system is a system of chemical equations). In the process, they state some ...
0
votes
1answer
34 views

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 = ...
4
votes
1answer
77 views

Differential of determinant in the Itô calculus

I'm trying to answer a question about a particular Itô equation $$\mathrm{d}X_t = AX_t\mathrm{d}t+X_tB(\mathrm{d}W_t)$$ where $X_t$ and $A$ are $n\times n$ square matrices and $B : M_n\to M_n$ ($M_n$ ...
0
votes
0answers
11 views

Norm for Progressive Processes

Let $(\Omega, \mathbb{F} = (\mathcal{F}_t)_{t \in [0,T] }, P)$ be a filtered probability space. We denote by $\mathbb{H}^2$ the space of real-valued progressively measurable processes $H$ such that $$ ...
1
vote
1answer
46 views

Solution of SDE with additive noise

I have a question about stochastic differential equations with additive noise. My question is: Is the solution of a SDE with additive noise almost surely equal to the solution of the corresponding ...
0
votes
0answers
50 views

Finding stationary distribution of a 2D stochastic process

Consider the following SDE for $z_t = (x_t,v_t)\in \mathbb{R}\times [-1,1]$: $$dx_t=-\mu x_t dt+ v_0 v_t dt + \sigma dW_t,$$ $$dv_t=-\frac{a^2}{2} v_t dt - a\sqrt{1-v_t^2} dB_t,$$ where $\mu, v_0, a, ...
2
votes
0answers
26 views

background and interpretation of langevin dynamics

I came across a stochastic dynamical system which is modeled with a conservative Hamiltonian component and an Ornstein-Uhlenbeck component. It is meant to represent small perturbations around a ...
2
votes
0answers
35 views

How to solve this stochastic differential equation, $dx_t=\mu dt+(\sigma-\bar{\sigma}x_t)dW_t$?

I have this stochastic differential equation which I would like to solve: $$ dx_t=\mu dt+(\sigma-\bar{\sigma}x_t)dW_t $$ It is similar to the mean-reverting process, but variance-reverting kind. No ...
0
votes
0answers
23 views

Is there any reference of application tensor norm on signatures?

This questions is related to https://en.wikipedia.org/wiki/Rough_path The signatures deriving from the path is used as features in machine learning areas. As I understand signature is equipped with a ...
1
vote
0answers
20 views

A progressively measurable process.

Let $\mathcal{P}_2(\mathbb{R})$ be the space of measures on $\mathbb{R}$ with finite second moment. We equip this space with the Wasserstein metric $W^2.$ We recall tat this metric space is separable ...
3
votes
0answers
139 views

Equality in law of two stochastic processes

Let $\lambda(t)$ be a CIR process, i.e. the strong solution of the SDE $$ \mathrm{d}\lambda(t)=\kappa(\theta-\lambda)\mathrm{d}t+\sigma\sqrt{\lambda(t)}\mathrm{d}W^1(t) $$ The integrated CIR is ...
1
vote
1answer
75 views

Progressively measurable process.

Let $b: [0,T] \times \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ with the properties: For all $ x \in \mathbb{R}$ the process $(t,\omega) \mapsto b(t,\omega, x)$ is progressively measurable. ...
1
vote
1answer
25 views

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=\...
1
vote
0answers
31 views

Solving Matrix RODEs with 1/f noise

I've been trying to solve differential equations of the form $$\dot{\rho}=\omega L\rho$$ where $\omega=\omega(t)$ is scalar 1/f (or pink) noise and $L$ and $\rho$ are matrices. I wanted to know if ...
2
votes
0answers
41 views

Substitution in two coupled Ito SDEs

Suppose we have the set of simple coupled stochastic differential equations (SDEs) in Ito form \begin{align*} \mathrm{d}X & =\left\{ c_{x}X+c_{xz}Z\right\} \text{d}t+\left\{ c_{z}Z\right\} \mathrm{...
3
votes
0answers
63 views

Are SDE's really “differential”?

An SDE of the form $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$ is really short-hand notation for an equation involving Ito integrals: $$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,...
0
votes
0answers
19 views

Joint stationarity of two diffusions driven by the same Wiener process

I'm looking at two $d$-dimensional diffusions represented by the following SDEs: $$dX_t = \nabla (\log p_1) (X_t) dt + \sqrt{2} dW_t$$ $$dY_t = \nabla (\log p_2) (Y_t) dt + \sqrt{2} dW_t$$ where $...
4
votes
0answers
43 views

Discretizing a Stochastic Volatility SDE

I have the following continuous time SDE for a stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process. $$ dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{1t} \\ dv_t = (\theta - \alpha ...
0
votes
1answer
65 views

A transformation function to be used with Ito's lemma for a specific SDE

So I am trying to find a closed-form solution for the following SDE. $$dX_t=a(e^{-t}-X_t)dt+be^{-t}dB \quad X_0=0\quad (I)$$ where $B$ represents a Brownian motion. My approach, so far, is to use ...
1
vote
0answers
25 views

Are Holder Condition and signal to noise ratio (SNR) related?

This question has evolved from watching the video: https://www.youtube.com/watch?v=dh5hHpJ79jc In the video the speaker talked about white noise case and $\alpha$ related to the Holder condition, (...
2
votes
1answer
44 views

Integral operation on a SDE

Suppose I have an SDE, for example $dN=rdt + adB$ for constants a and r, as in Oksendal chapter $5$. He take the "integral" of this and ends up with, $N=rt + aB$. without regard or mentioning what ...
1
vote
0answers
30 views

Dynamics of a future option

I have trouble understanding why $$V_s = \exp\left(\int_t^s r_u\;du\right) \int_t^s \exp\left(-\int_t^v r_u\;du\right) \theta_v dW_v$$ solves the SDE $$dV_S = \theta_s dW_s + r_sV_sds$$ I tried using ...
2
votes
1answer
41 views

convergence to SDE

How do I show that $\Delta Z_t = \theta \, \Delta t+\Delta B_t\rightarrow dZ_t = \theta \, dt+dB_t$ when $\Delta t\rightarrow0$, where $B_t$ is a standard brownian motion? Here the sde $dZ_t = \...