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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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18 views

Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$ \sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
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0answers
15 views

Is this random Lebesgue-integral well-defined?

Let $$ X : [0,T] \times \Omega \rightarrow \mathbb{R} $$ be an almost-surely continuous stochastic process. Then how is the random Lebesgue-integral $$ \omega \mapsto \int_{0}^{T} X_t(\omega ) dt \...
5
votes
1answer
50 views

Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations?

I am reading some introductory material on stochastic differential equations at the moment. In almost all cases, the equations which are presented are of the form $$ dX_t = \mu(t,X_t) dt + \sigma(t, ...
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0answers
17 views

lower bound for advection diffusion equation on the Torus

The motivation for this question comes from the Kolmogorov equation for basic SDE. Consider the PDE $$ \partial_t f + u \cdot \nabla f = \Delta f. $$ posed on $\mathbb{T}^2$, where the drift $u$ is ...
1
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1answer
55 views

Expected Solution of a Stochastic Differential Equation as a Conditional Expectation (this is a tough one).

On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \...
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0answers
33 views

How to solve $\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$ holds, where $X_{0} = x?$ [duplicate]

I'm learning about stochastic processes, and I want to solve $$\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$$ where $X_{0} = x$. I think that the solution uses Ito's Lemma; ...
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0answers
35 views

How can I find the distribution of an itô diffusion given it's generator?

I'm having a lot of truble with those generators. So if we have an Itô diffusion (in $\mathbb R$ and $f,\sigma $ nice enough to have uniqueness of solution) $$dX_t=b(X_t)dt+\sigma (X_t)dB_t,$$ it's ...
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0answers
20 views

Non existence of limit of Gibbs distribution

We consider an N-particle system given by the gradient dynamics $dX(t)= -N\nabla H_N (X(t)) dt + \sigma d\beta(t)$ in $(\mathbb{R}^d)^N$, where $\sigma$ is a positive constant. We assume that for $x=...
1
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0answers
29 views

Implementing Ito Taylor Method to SDE

I am trying to apply the Ito Taylor method to the SDE. I have attached what I'm using for the Ito Taylor because I would like for it to be double checked please. I'm supposed to get a strong speed of ...
0
votes
1answer
20 views

Differential of traders wealth function equals $0$. Correct?

Calculating the differential of the trader's wealth function I get $dV_t \equiv 0$, which not only surprises me but also stops me from going forward with the current financial model I am looking at (I ...
0
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0answers
13 views

How can I simulate increments of a two dimensional brownian motion?

I am attempting to simulate an sde system of the following form $$ dX_t=\sqrt{\vert aX+bY\vert}dW^1_t \\ dY_t=\sqrt{\vert cX+dY \vert}dW^2_t $$ where $W=(W^1,W^2)$ is a standard two dimensional ...
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0answers
12 views

Meaning of dummy calibration procedure of GBM

Ciao, I was doing a dummy check on calibration of GBM. Let me explain. We will consider the usual process: $$ dX_t = \mu X_t dt + \sigma X_t dW_t $$ with $X_0$ initial data and $t\in [0, T]$ Suppose ...
0
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0answers
48 views

How to write a system of stochastic differential equations (SDE) mathematically correct?

Aim I am trying to write down a system of stochastic differential equations, however, since I lack a background in mathematics I am not sure how to do this. Let's say I have the following system of ...
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0answers
19 views

Expectation to SDE in filtration problem

I am considering the following system of SDE $\frac{dS_t}{S_t}=Y_tdt+\sigma dB_t$ (observed) $dY_t=\frac{1}{\epsilon}(\theta-Y_t)dt+\frac{\beta}{\sqrt{\epsilon}}dB_t$ (hidden) where $S_t$ denotes a ...
0
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1answer
31 views

Is ODE a superset of SDE and how are they diffrent?

Im currently starting my master studies in mathematics and have found PDEs fascination, but more on the applied side rather than the theoretical. Im wondering if you should study SDE in order to ...
1
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0answers
13 views

Under what conditions does the solution to a mean reverting sde satisfy $E[\sup_{[0,T]} r(t)^2]<\infty$ for all $T>0$

Consider the mean reverting square root SDE $dr(t)=\alpha(\mu-r(t))dt+\sigma \sqrt{r(t)}dW(t)$ Under what conditions on the coefficients does the solution to a mean reverting sde satisfy $E[\sup_{[0,...
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0answers
23 views

Finding the unique solution of an SDE

For a given Weiner process W consider the 1-d interest rate model \begin{align*} dr_t&=4(4-r_t)dt+\sqrt{|r_t-6||r_t-2|}dW_t, t\in[0,T] \\ r_0&=4. \end{align*} Show that this equation has a ...
0
votes
1answer
47 views

Partial Derivative for Stochastic Integral

Good day, I am trying to apply Ito's lemma to find an integral but I am struggling with my choice of functions. $\int^ T _0 tdW(t) = T W(T)- \int^T_ 0W(t)dt$ Our version of Itos lemma states the ...
0
votes
1answer
34 views

How interpret that $\mathbb E[|X_t^e-x_t|^2]\leq e^2a(t)$, if $\dot X_t^e=b(X_t^e)+e\sigma (X_t^e)dB_t$

Consider the stochastic differential equation $$\dot X_t^\varepsilon =b(X_t^\varepsilon )+\varepsilon \sigma (X_t)dB_t,$$ where $X_0^\varepsilon =x_0$. I have a theorem that says that if $$|b(x)-b(y)|...
1
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0answers
19 views

SDE with stationary Log-normal distribution

Is there a stochastic differential equation whose solution follows a stationary Log-normal distribution? I was thinking in the geometric Brownian motion $$dx = (\alpha x )dt + (\sigma x )db, \quad \...
0
votes
1answer
37 views

SDE Integration

Does anyone know how to get the integration of the SDE below (Assume $\sigma \to 0$)? $$\dfrac{\mathrm dS_t}{S_t}=(r_d-r_f)\mathrm dt+\sigma(t, S_t)\mathrm dW_t$$ Thank you in advance! Image Link ...
2
votes
1answer
75 views

Expectation of solution to SDE $dX_t=-\tanh(X_t) dt + dW_t$

How do I calculate the expectation of the process given by the SDE $$dX_t=-\tanh(X_t) dt + dW_t, \qquad X_0=x_0$$ and $W_t$ a Wiener process? If I start with $$ d\left(e^{t/2}\sinh(X_t)\right) = e^{...
2
votes
1answer
96 views

Solving second order nonhomogeneous ODE where the RHS is a random process

Context: I'm trying to characterize the metastability behavior of a digital latch. I'm modeling two cross-coupled inverters as RC circuits with negative gain. One of the inverters has a source of ...
1
vote
1answer
109 views

Show that the larger $c$ is the faster ${\rm d}U_t^c=\frac c2h'(U_t^c){\rm d}t+\sqrt c{\rm d}W_t$ converges to its stationary distribution

Given two Markov chains $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ and $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ with transition kernel $\kappa_1$ and $\kappa_2$, respectively, and a common ...
0
votes
0answers
17 views

What is the connection between regularity structure and rough path theory?

This refers to the page https://en.wikipedia.org/wiki/Rough_path where it mentions about rough path and regularity structure as explains in the page:https://en.wikipedia.org/wiki/Regularity_structure ...
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0answers
35 views

Exact (!) relation between Martingale Problem, SDEs and Markov processes

I am currently trying to understand the big picture/connections of Martingale Problem, Fokker-Planck-equations (although, until now, I have mostly kept these out of my considerations), SDEs and Markov ...
0
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0answers
26 views

SDEs: Find the value of $\Bbb E[V(t_1)V(t_2)]$

Question: Consider the following SDE: $$V(t+\Delta t) = V(t)-V(t)\Delta t + \sqrt{\Delta t}\xi \qquad , \qquad V(0) = v_0$$ where $\xi \sim \mathcal N(0,1)$ (standard normal distribution) is ...
0
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0answers
27 views

SDE: How to calculate this kind expectation of SDE solution?

For a Stochastic Differential Equation model: $$ dx = bx(1-x/F)dt + cxdB $$ It has exact solution with initial $x(t_0)$: $$ x(t) = \frac{x(t_0)\exp((b-\frac{c^2}{2})\,t + c\,B(t))}{1+b\,\frac{x(t_0)...
0
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0answers
35 views

Why isn't it trivial that pathwise uniqueness implies uniqueness in law?

Consider an SDE $${\rm d}X_t=b(t,X_t){\rm d}t+\sigma(t,X_t){\rm d}W_t\tag1$$ with Lipschitz continous $b,\sigma:\mathbb R\to\mathbb R$ and a Brownian motion $(W_t)_{t\ge0}$. We know that, given a ...
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0answers
16 views

Numerically approximate a 1+4 dimensional parabolic PDE by two weakly coupled 1+2 dimensional PDEs?

In modeling population genetics, I derived a 1+4 dimensional parabolic equation of the form: \begin{align} \frac{\partial u}{\partial t}=&\frac{1}{2}\sum_{i=1}^4a(x_i)^2\frac{\partial^2u}{\partial ...
1
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0answers
37 views

Markov property of SDE's solution

Considering the SDE $dX_t=b(t,X_t)dt+\sigma (t,X_t)dW_t$ ($W$ is Brownian motion)  If there exists weak solution $(X,W),(\Omega ,\mathscr{F} ,P),\{\mathscr{F}_t\}$, is $X$ Markov process? I know ...
0
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0answers
15 views

Converting Stochastic Difference Equation to a Stochastic Differential Equation

I have a discrete time stochastic difference equation of the form $$x(k+1)=f(x(k),w(k)).$$ I need to convert this to a stochastic differential equation (continuous) of the form $$d(x(t))=f(x(t))dt+...
0
votes
1answer
65 views

How can we show that this nonnegative symmetric bilinear form is closable?

Let $(E,\mathcal E)$ be a measurable space $\mu$ be a measure on $(E,\mathcal E)$ and $$\mu f:=\int f\:{\rm d}\mu$$ for Borel measurable $f:\mathbb R\to\mathbb R$ with $f\ge0$ or $\mu|f|<\infty$ $\...
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0answers
22 views

non-trivial weak solution of SDE

Let consider the SDE $dX_t=|X_t|^adW_t$ for $0<a<\frac{1}{2}$  I know that this SDE has non-zero weak solution by Engelbrt & Schmidt's theorem. But I cannot find weak solution $(X,W)$ with $...
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votes
2answers
45 views

Expectation of a function of Ito diffusion

Given an Ito Diffusion i.e.: $$ dX(t) = \mu dt + \sigma dW(t) $$ and a function $$ k(x) = \lambda x^2 $$ and I want to find the expected value $E[k(X(t)]$ of the function - the only way I know ...
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0answers
54 views

If $L$ is a diffusion operator with corresponding carré du champ operator $\Gamma$, then $Lf^2=2fLf+2\Gamma(f)$

Let $(E,\mathcal E,\mu)$ be a measure space and $$\mu f:=\int f\:{\rm d}\mu\;\;\;\text{for }f\in L^1(\mu)$$ $\mathcal A_0$ be a subspace of $\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\...
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0answers
75 views

Show that $C_c^∞(\mathbb R)$ is a core of the generator of the Feller semigroup induced by the strong solution of an SDE with Lipschitz coefficients

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
1
vote
1answer
57 views

Solving $dS^x(t)=S^x(t)(r(t)dt+\gamma(t,S^x(t))dW(t))$

Consider the stochastic process $S^x(t)$ with $S^x(0)=x$ $$dS^x(t)=S^x(t)(r(t)dt+\gamma(t,S^x(t))dW(t))$$and define $D^x(t)=(\partial/\partial x)S^x(t)$. Then $$dD^x(t)=D^x(t)\Big[r(t)dt+\frac{\...
1
vote
1answer
74 views

Is the transition semigroup of the solution of an SDE with Lipschitz coefficients strongly continuous on $C_b$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous (and hence at most of linear growth) and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\...
1
vote
0answers
21 views

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 ...
1
vote
1answer
63 views

Show $\text P\left[|X^x_t|<r\right]\xrightarrow{|x|\to\infty}0$ for strong solutions of SDEs

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous $(X_t^x)_{t\ge0}$ be a continuous process on $(\Omega,\mathcal A,\...
2
votes
1answer
36 views

Estimate for the distance from the initial value of a strong solution of an SDE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous $(X_t^x)_{t\ge0}$ be a continuous process on $(\Omega,\mathcal A,\...
2
votes
0answers
28 views

Convergence of martingale (reference)

I have a sequence of processes $\{X^N(t)\}_{t\in [0,T]}$, $N\in\mathbb N$ such that $X^N(t)=x+M^N(t)$, where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $\langle M^N \...
1
vote
1answer
26 views

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?
2
votes
1answer
48 views

Derivation of the Fokker-Planck equation

Let $b\in C^1(\mathbb R)$ be Lipschitz continuous $\sigma\in C^2(\mathbb R)$ be Lipschitz continuous with $\sigma(\mathbb R)\subseteq\mathbb R\setminus\left\{0\right\}$ and $\sigma''$ being bounded $...
1
vote
0answers
21 views

integrability of system of SDE relative to Langevin dynamics

Similar to the system in this question and a follow up to here, consider the system (S) defined as \begin{align} {\rm d}X_t&=V_t\,{\rm d}t+\sigma_x\,{\rm d}W_t^1\\ {\rm d}V_t&=F_t\,{\rm d}t+\...
3
votes
0answers
62 views

Show that if $(κ_t)_{t≥0}$ is the transition semigroup of a strong solution to an SDE, $t↦(κ_tf)(x)$ is continuous for all $x$ and suitable $f:ℝ→ℝ$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
1
vote
0answers
34 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 ...
0
votes
2answers
134 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 ...
2
votes
1answer
53 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 ...