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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, …), and control and signal processing (controller, filtering, …).

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16 views

Fokker-Planck equation for a Markov semigroup with densities

Let $(E,\mathcal E)$ be a measurable space $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ and $$\kappa_tf:=\int\kappa_t(\;\cdot\;,{\rm d}y)f(y)$$ for $$f\in F_0:=\left\{f:E\to\mathbb ...
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2answers
50 views

Solve SDE for Brownian Bridge

Let $(B_t)$ be a one-dimensional Brownian motion and $y \in \mathbb{R}$. Show that the solution to the SDE $$dX_t^y=dB_t + \frac{y-X_t^y}{1-t}dt$$ with initial value $X_0^y = 0$ on $[0,1)$ is given by ...
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1answer
40 views

How is Grönwall's inequality applied here?

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ Now, let $(X_t)_{t\ge0}$ be the unique strong ...
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1answer
21 views

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 ...
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17 views

decoupling and integrating second-order SDE with different noise models

I'm considering a second-order Hamiltonian ODE and am trying to understand different approaches for introducing noise (and corresponding numerical integration techniques). Given that the deterministic ...
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1answer
22 views

Calculation of variance of complicated random variable ( white noise discretization )

so I have been doing some state estimation and in one part of my work it is necessary to discretize a continous time differential equation with white noise. I understood the discretization process for ...
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0answers
24 views

Obtaining derivatives for the moments of random variables

I am currently working through a problem and am having trouble interpreting my results. I start with a system whose dynamics are described by the equation: $$ \frac{dC}{dt} = f(C(t),\theta) $$ ...
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14 views

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, \...
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1answer
127 views

Convergence of the distribution of the Langevin diffusion to its invariant measure

Let $(X_t)_{t\ge0}$ be a solution of $${\rm d}X_t=-h'(X_t){\rm d}t+\sqrt 2W_t,\tag1$$ where $(W_t)_{t\ge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. Assume ...
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1answer
61 views

Nonlinear term in the KPZ equation

I'm reading up on the KPZ equation through the article by Bertini and Giacomin from 1997 and some lecture notes by Jeremy Quastel, the equation in 1+1 dimensions is stated as (for $h_t$ the height of ...
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1answer
34 views

How to decrease number of noises in stochastic differential equation

I have a Quantum SDE containing both white and color noises (open quantum system). $$ \dot\rho(t) = A\rho_s + (\nu_{1t}\hat{c}^\dagger \hat{X}^-_1 + \omega_{1t} \hat{X}^+_1 \hat{c})\rho_s +(\nu_{2t} \...
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0answers
26 views

If $(X^x_t)$ is the stochastic flow generated by a SDE and $(X_t)$ is the strong solution with $X_0=ξ$, is $X_t=X^ξ_t$ for all $t$ a.s.?

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

Is the strong solution of a SDE adapted to the filtration generated by the driving Brownian motion?

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

Optimal strategy - HJB equation - Change to Mathematics

I am sitting with the following control problem. Given know the controlled Markov equation \begin{align} dX_t&=-\lambda X_t\cdot dt+ U_t\cdot dt+\sigma\sqrt{1+X_t^2}\cdot dB_t \end{align} with ...
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1answer
24 views

Dynkin's Theorem and expectation

Suppose I have the following SDE. $dX_t=-k\cdot X_tdt+\sigma\sqrt{X_t}dB_t$ If I want to find a bound at any $t$ ofthe expectation of $X_t^2$, given $X_0=0$, is it legitimate to do the following? I ...
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0answers
30 views

Dynkin formula and expectation

I have a question about the use of Dynkin's formula. Suppose I have a stochastic process $X_t$, from an SDE that I cannot solve analytically. I want to find the expectation $\mathbb{E}X_t$. By Dynkin ...
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1answer
48 views

Invariant measure for Itō diffusion

Let $f\in C^2(\mathbb R)$ be positive and $h\ge 0$. Assume that $g:=f'/f$ is Lipschitz continuous and let $U$ be a strong solution of $${\rm d}U_t=\frac h2g(U_t){\rm d}t+\sqrt h{\rm d}W_t$$ ($W$ being ...
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0answers
17 views

Stability of parameters in sde

From lecture notes on SDE's. Consider the Stratonovich equation $dX_t=rX_tdt+\sigma X_t\circ dB_t$. It has initial condition $X_0=x$. What are the conditions for the parameters $(\sigma,r)$, for $\...
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0answers
13 views

Advection and diffusion for 2d sde

When looking at a 2d sde, for processes $Z_t=(X_t,Y_t)^T$. $dZ_t=f(X_t,Y_t)dt+g(X_t,Y_t)dB_t$ My textbook states that the diffusion is found by calculating $D=1/2\cdot g(X_t,Y_t)g(X_t,Y_t)^T$. For ...
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0answers
5 views

Prove that a process (given through rsdes) is a martingale.

i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely, let $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t $ ...
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1answer
77 views

Solve the SDE $dX_t=\sqrt t(X_t+\sin t)dW_t$

I am new to stochastic differential equation and ran into a question of solving $$dX_t=\sqrt t(X_t+\sin t)dW_t$$ where $W_t$ is the standard Wiener Process and $X_0 \equiv K\in \mathbb R$. I know Ito'...
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0answers
23 views

Explicit solutions of SDE

I am trying to use random variables to represent variation in population growth models such that growth rates can be represented as random variables themselves. In the simple exponential growth case ...
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0answers
14 views

The property of coercivity in stochastic analysis

Given an SDE $$ dX_{t}=b(t,X_{t})dt+\sigma (t,X_{t}) dW_{t} $$ With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,\sigma$ such as : i) ...
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0answers
34 views

Backward Kolmogorov equation for simple markov process

The following exercise is from a course on SDE's and I am a bit stumped. Consider the process. $dX_t=\lambda\left(\xi-X_t \right)dt+\gamma\sqrt{|X_t|}dB_t$ $\lambda,\xi,\gamma>0$ Find $\mathbb{P}...
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0answers
6 views

SDE of Mean-Field Type

We consider the system of mean-field type SDEs \begin{align} dX_t &= \sigma(t,X_t, P(X_t) ) dB_t, \\ dY_t &= E[ \phi(Y_t) X_t ] d B_t \end{align} In a paper I am studying it is claimed that a ...
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0answers
30 views

Backward Kolmogorov equation to find probability

From lecture notes in a course on SDE's. We are tasked with using the backward Kolmogorov equation to find. $\mathbb{P}^{X_t=x}\left(X_T\geq2 \right)$ I am confused by the terminology here. We are ...
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1answer
37 views

Brownian motion on the n-sphere

From course notes on SDE's. We consider a Stratonovich equation. $dX_t=\left(I-\frac{1}{|X_t|^2}X_tX_t^T \right)\circ dB_t$ With $X_t\in \mathbb{R}^n$ and $\{B_t\}$ being n-dimensional brownian ...
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0answers
19 views

Mode of the stationary distribution

From a course on SDE's. We consider a general, stationary advection-diffusion equation. $\left(uC-DC'\right)'=0$ We wish to show that stationary points are those at which $u=0$ and use this to find ...
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1answer
53 views

Stationary distribution of Cox-Ingersoll-Ross process

I am uncertain how to go about the following problem from the lecture notes on a course in SDE's. We are given the following SDE. $dX_t=\lambda\left(\xi-X_t\right) dt+\gamma\sqrt{|X_t|}dB_t$ Where $\...
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1answer
33 views

When is the local martingale in the Itō formula a (strict) martingale?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
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0answers
61 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,\...
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0answers
28 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: $$...
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0answers
60 views

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 ...
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0answers
44 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 $\mathcal F$-...
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0answers
15 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: $$...
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1answer
128 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 : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$ ...
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0answers
16 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 ...
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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 $$ |...
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0answers
13 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 ...
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0answers
59 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 ...
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0answers
19 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.
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0answers
39 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 ...
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0answers
40 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 ...
2
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0answers
39 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 $...
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0answers
43 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 ...
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0answers
47 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
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1answer
33 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 ...
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0answers
56 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 $...
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0answers
44 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 ...
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0answers
23 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 ...