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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Average velocity of a process with additive noise conditional on endpoint

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE $$dX_t = \mu(X_t) \, dt + \sigma \, dW_t$$ with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \...
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Stochastic dynamics: how do the random matrix $J_{ij}$ and coupling strengh $g$ affect the variance of the local field $h_i$?

Context: Q3 in How to understand the largest Lyapunov exponent? We know $g$ is proportional to (square root of) the variance of $J$'s every entry ($J_{ij}\sim \mathcal{N}(0,g^2/N)$). Why is it also ...
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Stochastic differential equation [closed]

$$dY_t = (-2\alpha Y_t + \sigma^2)\,dt + 2 \sigma\sqrt{Y_t}\,dB_t$$ Hint here is letting $X_t = \sqrt{Y_t},$ find that $dX_t = -\alpha X_t\,dt +\sigma \,dB_t$ Question is how do you derive the 2nd ...
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How to use Itô's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\left<\mathbf{J}x_u, x_t\right>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?

I am reading a lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it shows that for $s\ge t$, $$ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}...
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Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient

Consider the SDE below $$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$ where $X_0>0$ is square integrable, $b:\mathbb R_+\...
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Probability that a geometric Brownian motion with additional determinstic drift ever hits zero

Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE $$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$ where $\mu, \sigma, C, k > 0$ are constants, ...
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Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
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Chow's theorem for time one flows

Chow's theorem gives a criterion for the reachable set of the points of a manifold $M$ to be full. Specifically, if we have a distribution $D$ whose iterated commutators span the tangent bundle then ...
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Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE. Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
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Can integrals with respect to time-changed Brownian motion be seen as integrals with respect to Brownian motion?

Let $X_t:=W_{t\wedge \tau}$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $\tau:=\inf\{t\ge 0: |W_t|=1\}$. It holds $$X_t=\int_0^t {\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\...
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Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality?

I have a question on Lawler – Notes on the Bessel process, on page 4. Let $X_t$ be one-dimensional Brownian motion, and we want to use $N_t$ as a measure-changing (local) martingale, defined as $$N_t=\...
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Feynman-Kac formula with non-zero boundary condition

Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by ...
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How does the probability of staying positive depend on the diffusion coefficient?

Let $X$ and $Y$ be two continuous martingales given as $$X_t=z + \int_0^t a(s,X_s)\, dW_s,\quad \quad Y_t=z + \int_0^t b(s,Y_s) \, dW_s,$$ where $z>0$, $a,b$ are Lipschitz and bounded functions s....
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Uniform bound for the occupation time of a diffusion

Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$. Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions. Suppose the ...
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The ODE modeling for gradient descent with decreasing step sizes

The gradient descent (GD) with constant stepsize $\alpha^{k}=\alpha$ takes the form $$x^{k+1} = x^{k} -\alpha\nabla f(x^{k}).$$ Then, by constructing a continuous-time version of GD iterates ...
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Autocorrelation function of Itô process

I'm working with a time independent (vector) Itô SDE such as: $$ dX = a(X) dt + b(X) dW. $$ I've looked (numerically) at several examples and it seems that the autocovariance function $r_{xx}(\Delta t)...
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Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define $$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
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Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$

Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
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Uniqueness of the solution to some SDE of state-dependent coefficient

This is a continuation of my question posted in Uniqueness of the solution to some SDE Consider $$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$ ...
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Has this "stochastic differential equation" been studied?

Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\...
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6 votes
1 answer
316 views

A singular stochastic differential equation

We consider the following SDE: $$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be ...
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Dependence of the density on the coefficients

Consider a parametric SDEs $$dX_t = b\big(t,X_t,\alpha(t)\big)dt + dW_t,\quad \forall t\ge 0,\quad \quad \quad \quad (\ast)$$ where $\alpha=(\alpha(t))_{t\ge 0}$ be a parameter taking values in some ...
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Solution to a fully nonlinear SDE

Let $W$ be a standard one dimensional Brownian motion. Does the following (fully nonlinear) SDE admit a strong/weak solution? $$dX_t = X_{t + W_t} \, dt \, ,\, X_0 = 1 \text{ a.s.}$$ Explictly, we ...
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On the Lipschitz constant of $\Gamma$

Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
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Regularity of Fokker-Planck equation

Consider solutions $\rho_{1,2}$ of the Fokker-Planck equation $$\begin{cases}\partial_t \rho_i = \Delta \rho_i + \nabla \cdot (\rho_i \nabla \Phi_{1,2})\\ \rho_i(0,\cdot) = \rho^0 \end{cases}$$ for ...
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Stochastic filtering with time delayed observation

Let $X_t$ be a suitably nice real valued Markov process. The primary two cases I have in mind are a finite state space Markov process, and a Ito diffusion. Define the observation process $Y_t$ by $$...
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What are the optimal times to sample a process?

Let $X$ be a one dimensional Ito diffusion given by $$X_t = b \,W_t$$ where $b$ is a constant, and $W$ is a standard Brownian motion. Let $B$ be another Brownian motion independent of $W$, and define ...
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Search for conditions of the positive probability that a stochastic process never hits zero

Consider a stochastic process $X$ defined by $$X_t:=1+\int_0^t b(s,X_s) \, ds+ W_t,\quad \forall t\ge 0,$$ where $(W_t)_{t\ge 0}$ is a standard Brownian motion. Suppose that $b:\mathbb R_+ \times \...
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3 votes
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126 views

What is the formal definition of a stochastic PDE and a solution to a stochastic PDE?

While searching through this Wikipedia article, I have stumbled uopn the following 'stochastic' heat equation $$\partial_tu=\Delta u+\xi,$$ where $\xi$ is the space-time white noise. However, I don't ...
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PDE interpretation of some properties of the solution to Fokker–Planck equations

Consider $$X_t=X_0 + \int_0^t b(s)ds+ \int_0^t \sigma(s)dW_s,\quad \forall t\ge 0,$$ where $X_0\ge 0$ is a random variable of density $\rho$, $(W_t)_{t\ge 0}$ is an independent Brownian motion and $b,\...
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When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
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5 votes
1 answer
224 views

Uniqueness of the solution to some SDE

Consider the stochastic differential equation as follows: $$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$ where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
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Dependency of first hittimg time on coefficients of SDE

Let $b: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline b,\overline b]$ and $a: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline a,\overline a]$ be Lipschitz, where $\overline b>\...
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A variant to the Fokker–Planck equation

Consider the PDE of $p(t,x)\ge 0$ given as $$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$ with initial and boundary conditions $p(0,\cdot)=\rho$...
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Fokker–Planck equation for very degenerate diffusion processes

Consider a diffusion process $$X_t=X_0+\int_0^t {\bf 1}_{\{X_s>0\}}b(s,X_s)ds+ \int_0^t {\bf 1}_{\{X_s>0\}} a(s,X_s)dW_s,\quad \forall t\ge 0,$$ where $a: \mathbb R_+\times \mathbb R\to [1,2]$ ...
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Bounded Solution for a two-dimensional SDE

Good evening, I was thinking about the following situation: Let $I \subset \mathbb{R}^2$ be a bounded subset and $X$ be a stochastic process such that $$dX_t = b(X_t) dt + \sigma(X_t)dW_t,$$ where $W$ ...
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Occupation time of a diffusion with drift bounded away from 0

Let $\sigma, \mu: \mathbb R_+ \times \mathbb R \to \mathbb R$ be Lipschitz continuous functions, with $\mu > C > 0$ for some constant $C$. Let $W$ be a standard Brownian motion, and let $X$ be ...
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Reference request : Is there "equilibrium point" for (martingale) stochastic differential equations?

It is known that ODEs have the so-called equilibrium point, i.e. $x(t)\equiv x^*\in\mathbb R$ is an equilibrium point of $x'(t)=f(t,x(t))$ if $f(t,x^*)\equiv 0$. Consequently, an study of the ...
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Existence/Uniqueness of the solutions to SDEs of locally Lipschitz coefficients

I look for references on the existence/uniqueness of the solution to SDE $$dX_t = b(t,X_t)dt + a(t,X_t)dW_t,\quad \forall t\ge 0,$$ where $b :\mathbb R_+\times\mathbb R\to\mathbb R$, $a :\mathbb R_+\...
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5 votes
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Long list of exactly solvable nonlinear SDEs

In P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations. pg.118, they go over some special cases of nonlinear SDEs $dX_t=\alpha(t,X_t)\,dt+\sigma(t,X_t)\,dB_t$ ...
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Explicit solution for a simple SDE?

I'm not well-versed in stochastic calculus so I assume the question might be trivial. Consider the one dimensional SDE : $$dX_t = (1-X_t^2)dB_t $$ $$X_0 = x_0 \in [-1,1] $$ Where $B_t$ is a standard ...
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Can one compute explicitly the following occupation probability?

Let $X_t$ be a geometric Brownian motion, that is, a solution of the SDE $$dX_t = X_t (\mu \, dt + \sigma \, dW_t)$$ for $a, b > 0$ constants, $W_t$ a standard Brownian motion, and with initial ...
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On the definition of a certain probability measure

I’m having trouble understanding a detail in a paper to do with McKean Vlasov equations and mean field game theory. In the paper Non-asymptotic convergence rates for mean-field games: weak formulation ...
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Martingale representation of time-changed Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion. Let $\phi: [0,1)\to [0,\infty)$ be defined by $ \phi(t):=t/(1-t)$. Then $(M_t)_{0\le t<1}$ is a continuous Markov martingale with $M_t:=B_{\phi(...
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4 votes
1 answer
134 views

Conditions for the SDE be transitive

This question was previously posted on MSE. Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...
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70 views

Is a Levy diffusion square integrable with respect to the associated compensated Poisson measure?

Let $X_t$ be a one dimensional Levy diffusion of the form $$dX_t = \mu(t, X_t) \, dt + \sigma(t, X_t) \, dW_t + \int_{\mathbb R} \, c(t, z) \, \overline N (dt, dz)$$ with $c > -1 + \delta$ for some ...
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1 vote
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Existence and uniqueness for Levy BSDE with random terminal time

Consider the following setup. Suppose we have: $(\Omega, \mathcal F, \mathbb P)$ a probability space, $\mu, \sigma: \mathbb R_+ \to \mathbb R$ Lipschitz continuous functions with $\sigma(t) \neq 0$ ...
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Joint distribution of the time changed Brownian motion associated to a diffusion and the quadratic variation

This question was motivated by the considerations in the following post - Bounded density for diffusions with diffusion coefficients bounded away from $0$. Let $a: \mathbb R \times \mathbb R^n \to \...
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3 votes
2 answers
188 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
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5 votes
2 answers
215 views

A comparison of diffusions

Consider two diffusions given by $$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$ for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
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