# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...