# 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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### American put option pricing by “binomial trees”

I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar.
I'll try and give a description ...

**6**

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**0**answers

83 views

### Reference Request: Vector-Valued Ito Formula

I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...

**5**

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139 views

### Second order calculus and rough paths

In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...

**5**

votes

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305 views

### Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...

**5**

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243 views

### Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...

**4**

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**0**answers

32 views

### Reference request on theory about Stochastic Riemann problem

I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit.
On one hand, in the literature it is not hard to find books or papers that deal ...

**4**

votes

**0**answers

92 views

### What are morphisms between regularity structures?

In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here:
Is there any way of extending this to morphisms ...

**4**

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87 views

### Exit time of a stochastic process defined by a SDE

Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...

**4**

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201 views

### Uniqueness of a SDE with non-negativity constraint

I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...

**4**

votes

**0**answers

76 views

### Feynman-Kac formula and time-ordering for vector bundles

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ (...

**3**

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**0**answers

63 views

### Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...

**3**

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**0**answers

91 views

### Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...

**3**

votes

**0**answers

80 views

### “Expanding” around the invariant measure

In the spde literature we have results of the form
$$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$
where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $...

**3**

votes

**0**answers

117 views

### Optimal control of SDEs

I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...

**3**

votes

**0**answers

63 views

### Generalisation of Lyapunov time to stochastic dynamical systems

Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...

**3**

votes

**0**answers

148 views

### Reference request on connection between PDE problems

I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...

**3**

votes

**0**answers

46 views

### Assertion of Local Martingale

I am currently reading a proof of the Feynman-Kac representation theorem. The main step in the proof is to consider an "interpolation martingale" which has the form $$M_s := \varphi(t-s, x+B_s)\exp \...

**3**

votes

**0**answers

62 views

### Uniqueness of a stationary measure for a simple ODE

How to prove uniqueness of the stationary measure using coupling approach to the following process: $u_{n+1}=S(1)u_n+\eta_{n+1}(\omega)$, where
a) $S(1)y_0:=y(1)$, $y(t)$ solves the following Cauchy ...

**3**

votes

**0**answers

88 views

### Monte-carlo estimation on the drift of SDE

On any probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $W$, we consider the following one-dimensional SDE:
$$dX_t = F(t, X_t) \, dt + dW_t,$$
where $F(t,x) = \mathbb{...

**3**

votes

**0**answers

51 views

### Perscribed/Inverting Conditional Expectation

I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...

**3**

votes

**0**answers

90 views

### When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?

When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...

**3**

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115 views

### Hypergeometric function

Suppose that $V$ follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$
I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that $F(V)$ must ...

**3**

votes

**0**answers

96 views

### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$?
For examples: are there:
Ito Isometry(-types)...

**2**

votes

**0**answers

54 views

### Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...

**2**

votes

**0**answers

39 views

### Floquet stochastic process

Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...

**2**

votes

**0**answers

99 views

### Is there a distinct Ito-Sasaki version of Riemannian stochastic development?

Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...

**2**

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62 views

### Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...

**2**

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**0**answers

100 views

### Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain

Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...

**2**

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**0**answers

125 views

### An SDE version of a Fokker Planck Equation

Assume $\rho$ is probability density defined on $\mathbb{R}^d\times\mathbb{R}^d$. I am interested in the Wasserstein gradient flow of a functional:
\begin{equation*}
\mathcal{E}(\rho)=\iint_{\mathbb{...

**2**

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**0**answers

92 views

### What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?

Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...

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51 views

### Convergence of empirical measure in case of proliferation

I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:
I don't understand why he calls $u(t,x)$ a time dependent ...

**2**

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**0**answers

99 views

### Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...

**2**

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74 views

### Stochastic stability of “open” continuous-time stochastic systems: reference request

I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...

**2**

votes

**0**answers

84 views

### Ito formula between manifolds

I have seen many Ito formulae giving dynamics for $f(X_t)$ where $f:M\to \mathbb{R}$ is a smooth function from a manifold $M$ and $X_t$ is a (continous) manifold-valued semi-martingale.
My question ...

**2**

votes

**0**answers

119 views

### Ito lemma for manifold semimartingales

I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...

**2**

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**0**answers

65 views

### stochastic differential equations via Hida distributions?

Can one prove the solvability of stochastic Ito differential equations via Hida distributions?

**2**

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**0**answers

85 views

### Gaussian Processes and Paths

As we know, the only stationary Gaussian Markov process with continuous
autocorrelation function is the stationary OU process (Doob's theorem). To show this, one shows that the autocorrelation ...

**2**

votes

**0**answers

84 views

### Markov chain approximates a fractional diffusion

Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...

**2**

votes

**0**answers

116 views

### Feynman-Kac formula for *general* Sturm-Liouville operator

One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...

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**0**answers

88 views

### I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a “trace”

Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...

**2**

votes

**0**answers

182 views

### Definition of the Stratonovich integral in Hilbert spaces

Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...

**2**

votes

**0**answers

89 views

### Well-posedness of a stochastic differential equation in the Stratonovich sense

Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
$(B_t)_{t\in[0,\:T]}$ be an $(\mathcal F_t)_{...

**2**

votes

**0**answers

43 views

### Minimizer of a class of SDEs

Setup
Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...

**2**

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**0**answers

138 views

### Boundary behavior for Ito diffusions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...

**2**

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**0**answers

207 views

### Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...

**2**

votes

**0**answers

61 views

### Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...

**2**

votes

**0**answers

160 views

### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...

**2**

votes

**0**answers

62 views

### Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$.
Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...

**2**

votes

**0**answers

172 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...

**2**

votes

**0**answers

84 views

### Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...