# 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.

364
questions

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20 views

### What can be said about the coefficients of the expansion of a drift corresponding to a Girsanov measure?

Let $\mu_0$ be classical Wiener measure on $(C_0[0,T]), \mathcal B(\|\cdot\|_\infty))$. Let $\mu$ be another Borel measure so that $\mu\sim\mu_0$ are equivalent. Then by Girsanov there is a process $F(...

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36 views

### Exit probability on a finite interval

I have a question about the estimate of the exit probability on a finite interval. Given a $q$ function bounded and continuous, given the following SDE
\begin{cases}
dX_s=(\beta-q(s))X_sds+\frac{1}{2}...

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38 views

### Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...

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37 views

### A doubt on the derivation of the Wiener Chaos expansion propagator

So I have seen the following calculation in a number of articles (for instance 1, 2 3) and I just can't get my head around it.
The idea is basically as follows, let $\mathcal L$ be some differential ...

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65 views

### Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...

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43 views

### convexity of a function inside expected value

Assume that $g(x)=\int_{y\in Y}h(y)f(x,y)dy$. I know the properties of function $g(\cdot)$, e.g. first- and second-order derivatives signs. Is there a way to drive the same properties i.e. convexity ...

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130 views

### Associativity rule for integration against fractional Brownian motion

In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...

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90 views

### Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure

I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...

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59 views

### Local inverse bound of Cameron Martin and Banach norms

Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...

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68 views

### Onsager--Machlup functional as the density across a mesh of discrete points

It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...

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87 views

### Derivative of the function of random variable

Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...

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63 views

### Are SDE adapted to the natural filtration?

Let $(B^H_t)_{t\in [0,T]}$ be a fractional Brownian motion. We consider the following SDE where $b$ and $\sigma$ are Lipschitz
$$X_t=x+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB^H_s.$$
When $H>1/2$, ...

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38 views

### Solving SDE with sign function in drift term?

Consider the following SDE with $X_0 = 1$,
$$
dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t,
$$
where $\operatorname{sign}(x) = \mathbb{1}\{x \ge 0\}$. How am I supposed to solve this SDE?

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47 views

### Choice of Banach space for stochastic processes

In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used:
$$
\sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p}
$$
and
$$
\mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p]^...

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19 views

### When is the dual infinitesimal generator of a S.D.E self-adjoint and negative definite?

Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be,
self-adjoint i.e $\int f ({\...

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32 views

### Quadratic variation of generalized stochastic integrals

My question is based on this paper: https://pdfs.semanticscholar.org/0b5a/e41096a3b16d0756a1d36da55143d861ed7c.pdf.
In summary, this talks about the generalization of stochastic integrals to a two ...

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170 views

### Path integral presentation of solutions of Dirac equation

It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).
Is there a way to present solutions of the Dirac equation using path ...

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82 views

### Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?

From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...

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58 views

### Good proof of vector version of Ito Formula

does anyone have a good proof of the vector version of the Ito formula?
$$f(t,B_t)=f(0,0)+\int_0^t \frac{\partial f}{\partial t}(s,B_s)ds+\int_0^t \frac{\partial f}{\partial x}(s,B_s)dB_s+\frac{1}{2}\...

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62 views

### Conditions for Gaussianity of SDE

Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...

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23 views

### Nested subspaces of measurable functions through noise

Let $(X_t)_t$ be a Markovian semi-martingale generating the filtration for the stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ on which a Brownian motion $(W_t)_t$ is defined. ...

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38 views

### existence and uniqueness of solution to CEV model sde

Suppose that you have the CEV model for a stock price following the sde
$$dS_t = r S_t dt + \sigma S_t^{\eta} dw_t$$
where $ 0 \leq \eta\leq 1$, $S_0=s_0$ and $w$ is a Brownian motion.
How do you ...

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66 views

### Time-Reversal of BSDE = SDE

Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$:
$$
Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;
$$
...

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48 views

### Reference on infinitesimal generators for functional SDEs

I am trying to solve a problem using convergence of infinitesimal generators of functional SDEs. I havt yet found good material on this.
The setting is that of Wanatbe Ikeda -88 page 167. It looks ...

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134 views

### Discrete random walk and SDEs

My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them.
To be more precise, ( if I understand correctly what my ...

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83 views

### How to get the mean, skewness of a Itō integral?

If $B_t$ denotes a standard Brownian motion, and let $X_t = \int f(s)dB_s$, f(s) is a deterministic integrand. I known $B_t$ is a martingale, Is $X_t$ also a martingale? And how can i get the formula ...

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64 views

### About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E

We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...

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43 views

### Invariant measures of Levy S.D.Es

Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...

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104 views

### How to adapt an equation from a paper to a Cox–Ingersoll–Ross stochastic differential equation, and why?

I quote from Delbaen and Shirakawa - An interest-rate model with upper and lower bounds.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\...

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271 views

### Hitting probability for mean-reverting stochastic process

I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...

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182 views

### Some doubts on proof of pathwise uniqueness of a stochastic differential equation

I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...

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27 views

### Generator of a Hilbert space valued Wiener process from the solution of a martingale problem

Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...

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79 views

### Estimating the hitting time for a SDE solution

Consider a the following OU process in one dimension,
$$dX = -\theta(X -x_0)dt + \sqrt{s}dW $$
Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$.
Then ...

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33 views

### Convergence of the probability that hitting times being infinity

Let $X^n=(X^n_t)_{t\ge 0}$ and $X=(X_t)_{t\ge 0}$ be RCLL (right-continuous with left limits) processes such that
$$\lim_{n\to\infty}X^n=X,\quad \quad \mbox{almost surely},$$
where this convergence ...

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88 views

### Path dependent Markov property

Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...

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68 views

### Show an SDE's solution has positive probability to visit every set in the state space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...

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211 views

### Probability of a particle surviving forever

Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...

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52 views

### Literature on convergence of BSDE sequences?

Is there any paper/book literature on convergence behaviour of solutions to any general or specific BSDE sequences? More precisely, consider the following general problem:
Assume that, for all $n=1,2,\...

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50 views

### Can derivatives of 2 stochastic processes be multiplied?

We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is ...

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28 views

### Does there exist some analytical result for Wright-Fisher SDE?

Are there already known analytical results (e.g. Laplace transform) about the Wright-Fisher SDE
$$dY=(A-(A+B)Y)dt+C\sqrt{Y(1-Y)}dW$$
with $A$, $B$ and $C$ parameters?
If so, could you please quote ...

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308 views

### Asymptotic behaviour of an integral. How should I proceed?

Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$...

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105 views

### Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field

Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...

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51 views

### Reference request for invariance principles

In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...

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54 views

### Kernel of the adjoint of the infinitesimal generator of Levy SDE

Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the ...

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24 views

### Extension of the solution to a nonlinear SDE

Let $Y_0$ be a random variable and $(B_t)_{t\ge 0}$ be a standard Brownian motion independent of $Y_0$. For each $T>0$, consider the process $(Y_t^T)_{t\in [0,\tau_0)\cap [0,T]}$ defined as the ...

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117 views

### When and why do we require the condition that :“a subset bounded from below and has no accumulation points?”

I have been tyring to understand the first condition given in the link https://en.wikipedia.org/wiki/Regularity_structure for quite some time now, at least a year. I have posted a similar question in ...

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35 views

### Are the densities of a continuous stochastic process locally positive in time?

Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...

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68 views

### Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?

Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...

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32 views

### Backwards Regulated Branching Process with Browning Motion; duality

I am working on a problem which I have not well understood completely, so I can only give the intuition of it. Imagine that we have a population on the (unit) torus $\Bbb T\subseteq\Bbb R$ distributed ...

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54 views

### Reference to log-transition-density of a diffusion process

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely ...