Questions tagged [stochastic-differential-equations]

Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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20 views

Equivalence of forward and backward laws on $C([0,T], \mathbb R^n)$ for hypoelliptic diffusions

Consider a time-homogeneous diffusion process on $\mathbb R^n$ $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ and suppose that it satisfies Hormander's Lie bracket condition. Suppose that it is stationary at a ...
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11 views

Some uniform property w.r.t the approximation of the SDE solution

I have a slightly general question that might be related to the Euler Approximation of SDE solution, could someone provide some thoughts on this or share some source could potentially helpful? ...
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21 views

Moments of Logistic SDE's solution

On this article starting from equation $(30)$ it's presented a derivation of the first moment for the solution the logistic SDE: $$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$...
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63 views

BSDE without volatility

Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE $$ dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t $$ with terminal condition ...
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1answer
125 views

Can we show that this transition semigroup preserves a certain Wasserstein space?

Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
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1answer
182 views

Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper. Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
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25 views

What is the Friedlin Wentzell rate function for $X(t)=\int_0^t f(B(s))ds+B(t)$?

Let $X(t)=\int_0^t f(B(s))ds+B(t)$ satisfy the SDE $dX(t)=\mu(X(t))dt+dB(t)$. Consider the process that solves $dX^\varepsilon(t)=\mu(X^\varepsilon(t))dt+\sqrt\varepsilon dB(t)$. We know that the ...
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54 views

Itō formula for the solution of a SPDE in the distributional sense

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
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36 views

Ito's Lemma (CVF) on product of Poisson processes

I have the following stochastic differential equation: $da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$, with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by: $dq(t)...
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28 views

Estimate for continuous Ito semimartingale

I'm reading an article in which one defines a continuous Ito semimartingale of the form $$\hat{V}_{i \Delta_n}' - \bar{V}_{i \Delta_n} = \frac{2}{k_n \Delta_n} \sum_{j=1}^{k_n} \int_{(i+j-1)\Delta_n}^{...
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134 views

What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?

What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$? I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$ But ...
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112 views

What is the drift for a convex combination of Girsanov measures?

Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a ...
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47 views

Differentiable dependence on the initial condition of the solution of a SDE

Let $b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-...
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22 views

If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...
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64 views

Large deviations estimate for arbitrary continuous function

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
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1answer
45 views

Probability that a stochastic flow is near $0$

Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form: $$ d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
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2answers
122 views

Intuition behind Gubinelli derivative

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject. On page 14 of "A Course on Rough Paths With an ...
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1answer
85 views

Test for OU-Process

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use? So far, everything I've seen is hand-...
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1answer
90 views

Diffeomorphism for mapping one SDE into another

Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...
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151 views

How to make sense of recursively defined SPDE solutions, like in Hairer's “Solving the KPZ equation” paper?

In Martin Hairer's 2013 paper "Solving the KPZ equation", the process $X_\epsilon^\bullet$ is defined as the stationary solution to $$ \partial_t X_\epsilon^{\bullet} = \partial_x^2 X_\epsilon^{\...
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24 views

Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\ X_0=x_0 \\ dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\ Y_0=y_0 \end{...
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22 views

Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors. It's a well ...
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139 views

Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$

We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$. Q: I am curious if ...
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210 views

On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
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24 views

SDE conditional expectation

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
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1answer
103 views

Absolute value of a diffusion

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below. Suppose, $X_0 = Y_0 = 0$ \...
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1answer
69 views

Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion: $$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$ where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...
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10 views

About the role of total variation measure on boundary reflected stochastic processes

I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that ...
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29 views

Expected value and variance and ode

I wonder if it is possible to compute statistics of a stochastic differential equation. I begin with a simple question about linear stochastic differential equation. Let $\dot{w} = z$, and $z$ a ...
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45 views

conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
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28 views

Hint/Clue on the solution and properties of a particular SDE

I hope you don't mind me asking for clues/hints/help on this particular SDE: $dX_{t}=(\mu + \gamma(Y(t)))X_{t}dt + \sigma X_{t}dW_{t}$ where $\mu , \sigma \in \mathbb{R}$, $W_{t}$ is a Brownian ...
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1answer
60 views

Stochastic invariant subset

Let us consider a stochastic differential equation (SDE), $$ dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}% $$ and a compact set $C\subset\mathbb{R}^{n}$. Given a stochastic ...
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1answer
119 views

Diverging solution to a SDE

I'm considering the SDE, with $B$ the brownian motion and $\beta$ a scalar (it can be negative) $$ X_t = x_0 + \int_0^t (\beta + X_s^2) ds + B_t $$ and I would like to show that $X_t$ almost surely ...
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42 views

How to find the PDE for the following transition density

Suppose I have the following two stochastic differential equations ($t\geq 0$) $$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$ where $X = (X_t)$, $Z = (Z_t).$ Note that $W=(...
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29 views

Associating “weak” solutions of stochastic differential equation on manifolds with real valued weak solutions

Let's say we are working on a (real) differentiable manifold $M$. For smooth vector fields $A_0,A_1,...,A_r$ on $M$ we define stochastic differential equations as $$dX_t = A_{\alpha}(X_t) \circ dB^{\...
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53 views

Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
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144 views

“Expected Value” of a solution to a differential equation

I'm going to write this question in a very informal way as I'm looking for guidance, rather than a specific answer to a specific problem. So I took a course on stochastic processes and Martingales ...
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59 views

McKean–Vlasov diffusion (strongly) Markov?

Consider the McKean–Vlasov diffusion $$dX(t) = b(X(t),\mu(t))dt + \sigma(X(t),\mu(t))dW(t),$$ in which $\mu(t)= \mathcal L_{X(t)}$ is the law of $X(t)$. Assume $b,\sigma$ are bounded and Lipschitz so ...
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26 views

Reference for an infinite system of SDEs

Consider system of the following form, \begin{align*} \mathrm{d} X_k(t) = \big(AX(t)\big)_k\mathrm{d}t + B_k(X_k(t))\mathrm{d}t+\mathrm{d}W_k(t),\quad k\in\mathbb{Z}, \end{align*} where $A$ is matrix, ...
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1answer
78 views

Rough paths theory for Non-Markovian processes

I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems. I would appreciate any example or also any other theory (...
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1answer
77 views

What's the role of commutation relations in stochastic mechanics?

In a stochastic context, we can understand a term like $$ \int_0^T \frac{d q(t)}{dt} dq $$ either as the (Ito) limit $$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$ or the (Anti-...
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48 views

Scaling Property Ito diffusion processes

It is well-known that a Brownian motion $W$ has the following scaling property $$ c^{-1/2}W_{ct} \qquad \mbox{ for any $c,t>0$}. $$ In particular this means that the increments of the processes $W$ ...
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1answer
91 views

Convergence rate estimates of Monte-Carlo first-passage time estimates

Setup Let $X_t$ be a $d$-dimensional diffusion process solving the Ito-stochastic differential equation $$ X_t = x+ \int_0^t f(X_t,u_t)dt + \int_0^t \sigma dW_t, $$ where $x \in \mathbb{R}^d$, $u_t$ ...
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510 views

Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit. In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...
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0answers
40 views

Large Deviations Principle for First Exit time of a Diffusion Process

Let $b:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be a smooth Lipschitz function, $x \in \mathbb{R}^d$, $\sigma >0$, and consider the solution to the SDE $X_t^x$ defined by $$ dX_t^x = b(X_t^x)dt + \...
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45 views

Numerical evaluation of KL divergence for SDE

Consider the SDE $$ dX_t = v(X_t)dt + dW_t $$ where $W_t$ is a standard Brownian motion. Girsanov's theorem tells us that the Radon-Nikodym derivative of the measure $\mathbb{P}_v$ of $X_t$ with ...
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55 views

Undergrad thesis topic involving stochastic differential equations

I am considering a topic for my undergraduate thesis. I want to study the estimation for the parameters in the SDEs. My instructor wants me to think of a generic and unique procedure to solve the ...
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58 views

Different solution to system of nonlinear second order ODEs

Given the system of the following two ODEs $$0=\frac{1}{2} \sigma^{2} \Phi_i''\left(x\right)+\left(\mu-\tilde{K} \Phi_1'(x)-\tilde{K} \Phi_2'(x)\right)\ \Phi_i'\left(x\right)-\delta \Phi_{i}\left(x\...
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151 views

Proving that $dX_t=a_tX_td\tilde{B}_t$ is a martingale

Fix $T>0. $Consider the probability space $(\Omega,F,Q)$ and a Brownian motion $\{B_t\}_{t\leq T}$ and filtration $\{F\}_t$ generated by the Brownian paths. Suppose $a_t,\gamma_t$ are random ...
3
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1answer
68 views

Hölder continuity for discrete time process

Let $(X_n)_{n\in\mathbb N}$ be a discrete time stochastic process taking values in a Banach space $E.$ Suppose there exist constants $C,\alpha,\beta>0$ such that $\mathbb E\|X_n-X_m\|^\alpha\leq C|...

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