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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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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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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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)$$
...
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Has this "stochastic differential equation" been studied?
Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\...
user420828
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A singular stochastic differential equation
We consider the following SDE:
$$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$
where $(B_t, \, t \ge 0)$ is linear Brownian motion.
Let $\tau: = \inf\{t >0: X_t = 0\}$ be ...
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On the Lipschitz constant of $\Gamma$
Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
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Regularity of Fokker-Planck equation
Consider solutions $\rho_{1,2}$ of the Fokker-Planck equation
$$\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}$$
for ...
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What are the optimal times to sample a process?
Let $X$ be a one dimensional Ito diffusion given by
$$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,$$
where $(W_t)_{t\ge 0}$ is a standard Brownian motion. Suppose that $b:\mathbb R_+ \times \...
user420828
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What is the formal definition of a stochastic PDE and a solution to a stochastic PDE?
While searching through this Wikipedia article, I have stumbled uopn the following 'stochastic' heat equation
$$\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,$$
where $X_0\ge 0$ is a random variable of density $\rho$, $(W_t)_{t\ge 0}$ is an independent Brownian motion and $b,\...
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When enlarging a filtration makes a stochastic processes into a solution to an SDE
Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
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Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$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)$$
with initial and boundary conditions $p(0,\cdot)=\rho$...
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Probability that a drifted Gaussian process does not hit zero
Let $m: \mathbb R_+\to [0,1]$ be continuous and decreasing. Consider
$$X_t=1+bt+\int_0^t\frac{\sigma}{1+m(s)}dW_s,\quad \forall t\ge 0,$$
where $b, \sigma>0$ are given and $(W_t)_{t\ge 0}$ is a ...
user128095
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Fokker–Planck equation for very degenerate diffusion processes
Consider a diffusion process
$$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,$$
where $a: \mathbb R_+\times \mathbb R\to [1,2]$ ...
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Existence/Uniqueness of the solutions to SDEs of locally Lipschitz coefficients
I look for references on the existence/uniqueness of the solution to SDE
$$dX_t = b(t,X_t)dt + a(t,X_t)dW_t,\quad \forall t\ge 0,$$
where $b :\mathbb R_+\times\mathbb R\to\mathbb R$, $a :\mathbb R_+\...
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Long list of exactly solvable nonlinear SDEs
In P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations.
pg.118, they go over some special cases of nonlinear SDEs $dX_t=\alpha(t,X_t)\,dt+\sigma(t,X_t)\,dB_t$ ...
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Explicit solution for a simple SDE?
I'm not well-versed in stochastic calculus so I assume the question might be trivial.
Consider the one dimensional SDE :
$$dX_t = (1-X_t^2)dB_t $$
$$X_0 = x_0 \in [-1,1] $$
Where $B_t$ is a standard ...
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Martingale representation of time-changed Brownian motion
Let $(B_t)_{t\geq 0}$ be a standard Brownian motion. Let $\phi: [0,1)\to [0,\infty)$ be defined by $
\phi(t):=t/(1-t)$. Then $(M_t)_{0\le t<1}$ is a continuous Markov martingale with $M_t:=B_{\phi(...
user420828
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Conditions for the SDE be transitive
This question was previously posted on MSE.
Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...
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Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
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A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
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Existence and uniqueness for a path dependent SDE depending on the moving average
Let $\mu: \mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R_+ \times \mathbb R^{d} \to \mathbb R^{d \times m}$ be Lipschitz continuous in both variables and uniformly bounded.
...
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First hitting time for non-homogeneous diffusion martingale
This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$
Consider the martingale given as
$$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$
...
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Conditioning an SDE on the event that the driving noise is small
Let $X$ be the solution to the one dimensional SDE
$dX_t = \mu(t, X_t)dt + \sigma(t, X_t) dW_t$, for $t \in [0, T]$.
with $X_0= x_0$ a.s. for some $x_0 \in \mathbb R$.
Here $W_t$ denotes a standard ...
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How to get speed measure $m(dx)$, scale function $s$, and killing measure $k(dx)$ of a diffusion from the infinitesimal generator? [closed]
This question comes from P13 and P17 of the book Andrei N.Borodin and Paavo Salminen.
Page P13 defines the speed measure $m(dx)$, the scale function $s$, and the killing measure $k(dx)$.
Case 9 on P17:...
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Perturbation of volatility term in an SDE
Suppose $X, X^{\varepsilon}$, for $\varepsilon > 0$ are real valued stochastic processes satisfying the following SDE on $[0, T]$:
$dX = \mu(t, X_t) dt + \sigma (t, X_t) dW_t,$
$dX^{\varepsilon} = \...
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A bound for the occupation time of a diffusion
Let $\sigma: \mathbb R \times \mathbb R \to \mathbb R$ be a Lipschitz continuous function bounded below by some $M > 0$.
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the ...
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Existence/uniqueness of the solution to some SDE with discontinuous coefficient
Consider a SDE
$$dX_t = b(t,X_t)dt + f\big(a(t,X_t)\big)dW_t,\quad \quad\quad\quad\quad\quad\quad\quad\quad(\ast)$$
where $(W_t)_{t\ge 0}$ is a Brownian motion and
$$f(z):={\bf 1}_{\{z>0\}} +\frac{...
user128095
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Implicit function theorem for stochastic differential equation
For each $\theta\in \mathbb{R}$,
we consider a stochastic differential equation (SDE):
$$
d X_t =b(t,X_t,\theta)dt+\sigma dW_t,\; t\in [0,T];\quad X_0=x_0\in \mathbb{R},
$$
where $\sigma\ge 0$ and ...
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How to understand the transition density of reflected Brownian motion
We can see from the above picture the transition density of reflecting Browninan motion is given by (19). As we know, the first part ($2p(t,x,y)$) is the transition density of a Brownian motion (from $...
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existence/uniqueness of the (weak) solution to SDEs with discontinuous volatility
Consider a sequence of parametrized SDEs :
$$X^{a}_t = z + \int_0^t b(a,s,X^a_s)ds+\int_0^t\frac{\sigma(a,s,X^a_s)}{1+{\bf 1}_{\{b(a,s,X^a_s)>0\}}}dW_s,\quad \forall t\ge 0,~~~~~~~~~~~~~~~~~~~~~(\...
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Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
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Is the integral against a Brownian motion conditioned to stay bounded a local martingale?
Let $W$ be a standard Brownian motion on a probability space $(X, \mathcal F, \mathbb P)$ let and $\mathcal F_t$ its natural filtration.
For $\varepsilon > 0, T \in [0, \infty)$ let $A_{\varepsilon,...
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Random time change from Oksendal's SDE textbook
I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in ...
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Question on the martingale representation theorem
Let $(X_t)_{0\le t\le 1}$ be a continuous Markov martingale (with respect to its natural filtration) s.t. $X_0=0$ and $X_1\in\{-1,1\}$. Can we prove the existence of some measurable function $\sigma: [...
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