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.
20
votes
1answer
451 views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
14
votes
1answer
444 views
For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
12
votes
0answers
1k views
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 ...
11
votes
1answer
378 views
Does Brownian motion immediately visit both sides of a Jordan curve?
Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...
8
votes
1answer
468 views
Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?
Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...
7
votes
1answer
738 views
What is the idea behind interpolation spaces?
I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:
Definition. Let $H$ be an $\mathbb{R}$-...
7
votes
2answers
335 views
Good papers on stochastic differential equations with applications in finance
I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) ...
7
votes
1answer
253 views
Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$
I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in \...
6
votes
1answer
496 views
Why the term “geometric” rough path?
A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}_{s,t})=\frac{1}{2}...
6
votes
1answer
313 views
Is there any reason to use paracontrolled calculus over regularity structures?
Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
6
votes
2answers
355 views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
6
votes
3answers
348 views
Euler Schemes in Stochastic Differential Equations
So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations
I'll start with explicit. Say I have the following SDE known as ...
6
votes
0answers
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
votes
2answers
506 views
Analytic Solution to SDEs
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...
5
votes
2answers
273 views
How to define (and solve) the diffusion equation with a sticky boundary at the origin?
For the diffusion equation $\frac{\partial} {\partial t} P_t(x)=D \frac{\partial^2} {\partial x^2} P_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x} P_t(...
5
votes
2answers
311 views
Reference for Feynman-Kac
I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...
5
votes
1answer
395 views
English translation of “Les aspects probabilistes du contrôle stochastique”
I am looking for an English translation of "Les aspects probabilistes du contrôle stochastique" written by Nicole El Karoui, or knowledge whether it exists.
Other references with similar content on ...
5
votes
3answers
305 views
Perturbation of a stochastic differential equation
Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...
5
votes
2answers
467 views
Well-posedness of Fokker-Planck equation
Consider the following equation on $[0,T]\times\mathbb{R}^n$
\begin{eqnarray}
&\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\
&\rho|_{t=0}=\rho^0,
\end{eqnarray}
where $V\in C^2(\...
5
votes
1answer
237 views
Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?
Many books [see below for references] explore the connections between partial differential equations and expectation values.
Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
5
votes
1answer
267 views
Reference: Stochastic Analysis on Hilbert Manifolds
I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...
5
votes
1answer
355 views
Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation
Let
$T>0$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...
5
votes
0answers
139 views
Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
5
votes
0answers
305 views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
5
votes
0answers
243 views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
4
votes
1answer
318 views
Existence of normal number except random numbers
For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...
4
votes
2answers
321 views
A Stochastic Taylor Expansion/Asymptotics
Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...
4
votes
1answer
1k views
Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
4
votes
1answer
247 views
Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
4
votes
2answers
322 views
Tanaka-Meyer formula
I have a simple question about Tanaka-Meyer formula, I am having difficulty applying it. Let $X$ be a continous martingale vanishing at zero. From Tanaka-Meyer formula it holds $$d|X_t| = sgn(X_t)dX_t+...
4
votes
1answer
192 views
Infinite-time, Path-Dependent Expected Value of an Orstein-Uhlenbeck process
I am dealing with an Orstein-Uhlenbeck process $X_t$ with its stochastic differential equation being
$$dX_t=(\mu-X_t)dt+\sigma dW_t.$$
I want to show
$$\mathbb{E}\left[\frac{|X_\infty|}{\int_{0}^{\...
4
votes
1answer
246 views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
4
votes
1answer
326 views
Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
4
votes
1answer
320 views
Diffusion processes with different diffusion coefficients and absolute continuity
I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.
My initial situation is the following. Consider two stochastic ...
4
votes
2answers
182 views
Probabilistic interpretation for Fokker-Planck equation
It is well known that if $X_t$ is a stochastic process that solves the SDE
$$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$
with $W_t$ a Wiener process, then the associated ...
4
votes
1answer
126 views
Exponential of approximate quadratic variation of Brownian motion
Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...
4
votes
1answer
196 views
Uniqueness of a SDE with positivity constraint
We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
4
votes
1answer
79 views
Time Integral over the (positive) Innovations of a Stochastic Process
Consider the Itô-Process
$$X(t) = X_{0} + \int_{0}^{t}\mu(s,X(s))\,ds + \int_{0}^{t}\sigma(s,X(s))\,dW(s)$$
where you can safely assume that the drift $\mu$ and the volatility $\sigma$ satisfy the ...
4
votes
1answer
370 views
Limit of first passage time
I have a conjecture that seems rather obvious but the proof seems elusive.
Consider a diffusion given by,
$dX_t = \mu(X_t) dt + \sigma(X_t) db_t$
where $b_t$ is a standard Brownian motion.
$\mu,\...
4
votes
1answer
304 views
Malliavin derivative under change of measure
Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.
Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
4
votes
1answer
607 views
Onsager-Machlup function and most probable path of a diffusion process
Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...
4
votes
0answers
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
0answers
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
votes
0answers
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
votes
0answers
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
0answers
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
votes
1answer
612 views
Intuition about Skorohod integral
I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...
3
votes
1answer
308 views
When does the cumulative distribution function solve the Kolmogorov backward equation?
For a diffusion $X$ define the cumulative distribution function for $X_T$ started with $X_t=x$:
$$u(t,x):=E^{t,x}(1_{X_T\ge y})$$ Under what conditions does $u$ solve $X$'s Kolmogorov backward ...
3
votes
1answer
105 views
Under what condition we get back path from signatures in rough path theory?
A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path
It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...
3
votes
2answers
177 views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
