# 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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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 $x_0 \in \... 1 vote 0 answers 42 views ### Stochastic dynamics: how do the random matrix$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 ... 0 votes 0 answers 43 views ### 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 ... 0 votes 0 answers 29 views ### 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$, $$K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}... 3 votes 0 answers 37 views ### Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient Consider the SDE below$$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)$$where X_0>0 is square integrable, b:\mathbb R_+\... 0 votes 1 answer 74 views ### Probability that a geometric Brownian motion with additional determinstic drift ever hits zero Let W be a standard Brownian motion, and let X_t be the solution to the following SDE$$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$where \mu, \sigma, C, k > 0 are constants, ... 2 votes 0 answers 59 views ### Is a Riccati BSDE explicitly solvable? Let W=\{W_t\}_{t\in[0;T]} be a real-valued Brownian motion, \{F_t\}_{t\in [0;T]} the filtration generated by W, augmented with the nullsets, let C\in (0;\infty) and \{a_t\}_{t\in[0;T]} be a ... 2 votes 0 answers 63 views ### Chow's theorem for time one flows Chow's theorem gives a criterion for the reachable set of the points of a manifold M to be full. Specifically, if we have a distribution D whose iterated commutators span the tangent bundle then ... 1 vote 0 answers 30 views ### Let (X, W) be a weak solution to a SDE. Is W a Brownian motion w.r.t. \sigma(X_s : s \le t)? Let (X, W), (\Omega, \mathcal{F}, \mathbb{P}), \{\mathcal{F}_t\} be a weak solution to an SDE. Per definition W is an \mathcal{F}_t-Brownian motion and both X, W are adapted to \mathcal{... 2 votes 0 answers 47 views ### Can integrals with respect to time-changed Brownian motion be seen as integrals with respect to Brownian motion? Let X_t:=W_{t\wedge \tau} for t\ge 0, where (W_t)_{t\ge 0} is a standard Brownian motion and \tau:=\inf\{t\ge 0: |W_t|=1\}. It holds$$X_t=\int_0^t {\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\... 0 votes 2 answers 83 views ### 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 $$N_t=\... 0 votes 0 answers 66 views ### Feynman-Kac formula with non-zero boundary condition Let D \subseteq \mathbb{R}^m be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition u(t, x) = f(x) and boundary condition u(t, x)|_{\partial D} = 0 is given by ... 4 votes 0 answers 100 views ### How does the probability of staying positive depend on the diffusion coefficient? Let X and Y be two continuous martingales given as$$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,$$where z>0, a,b are Lipschitz and bounded functions s.... 3 votes 0 answers 198 views ### Uniform bound for the occupation time of a diffusion Note: We denote by \mathcal L(U) the Lebesgue measure of a set U. Let \mu: \mathbb R^d \to \mathbb R^d and \sigma: \mathbb R^{d} \to \mathbb R^{d \times d} be Borel functions. Suppose the ... 4 votes 1 answer 123 views ### The ODE modeling for gradient descent with decreasing step sizes The gradient descent (GD) with constant stepsize \alpha^{k}=\alpha takes the form$$x^{k+1} = x^{k} -\alpha\nabla f(x^{k}).$$Then, by constructing a continuous-time version of GD iterates ... 0 votes 1 answer 83 views ### Autocorrelation function of Itô process I'm working with a time independent (vector) Itô SDE such as:$$ dX = a(X) dt + b(X) dW. $$I've looked (numerically) at several examples and it seems that the autocovariance function r_{xx}(\Delta t)... 3 votes 0 answers 210 views ### Malliavin calculus and geometric interpretation of \nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}}), with regards to the surface S = \{F = 0\} Let F:\mathbb R^n \to \mathbb R be a "sufficiently regular" function. For any k \ge 1 and x \in \mathbb R^n, define$$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(... 0 votes 2 answers 161 views ### 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, ... 0 votes 0 answers 78 views ### 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 $$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$ ... 0 votes 1 answer 187 views ### 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:\... 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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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 ...