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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What phenomena are better modelled by SDE instead of ODE?

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
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23 votes
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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, ...
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Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit. In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...
jak's user avatar
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14 votes
1 answer
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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 $$B^\top - T B^\top = B + B T$$ ...
stochastic's user avatar
12 votes
0 answers
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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 ...
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11 votes
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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). ...
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10 votes
2 answers
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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(\...
UPS's user avatar
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10 votes
2 answers
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Parameter estimation for stochastic differential equation from discrete observations

Suppose we have a time-series $x(t_i)$ at discrete times $t_i$ and we want to estimate the parameters of an underlying SDE corresponding to this time-series: $$dx_t = f(x_t,\theta)dt + \sigma(x_t,\...
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9 votes
1 answer
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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 ...
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8 votes
2 answers
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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}...
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8 votes
1 answer
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total variation distance between two solutions of SDE

Suppose we have two stochastic differential equations with the same initial conditions: $$d X_t^1= b_1(t,X_t^1)dt + dW_t$$ $$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$ $X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
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8 votes
1 answer
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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 ...
Josh Burby's user avatar
7 votes
2 answers
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Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$. It happens that the ...
Linus Hamilton's user avatar
7 votes
2 answers
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Interpretation of second order term in Fokker-Planck equation

Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by $$ \begin{align*} \nabla^2\cdot G(x) &=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...
Peter's user avatar
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2 answers
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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.) ...
Heisenberg's user avatar
7 votes
1 answer
904 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 $\mathbb{R}$-...
alexlo's user avatar
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7 votes
2 answers
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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 ...
waaat's user avatar
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7 votes
1 answer
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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 ...
KDD's user avatar
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7 votes
1 answer
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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 $...
chuse's user avatar
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7 votes
2 answers
388 views

Fractional Brownian motion of Riemann-Liouville type is not a semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
El_mago's user avatar
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3 answers
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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 ...
Trelokoritso's user avatar
7 votes
1 answer
781 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 \...
Julian Karch's user avatar
7 votes
1 answer
363 views

What happens when the diffusion term in an SDE becomes zero?

Consider this time-homogeneous SDE, in the Ito sense: $$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$ where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...
ColorfulLion's user avatar
7 votes
1 answer
219 views

Onsager-Machlup functional when drift is time-dependent

Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by \begin{align} \mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}, \end{align} where $b_i(x) \in \mathcal{C}_b^2(...
Enforce's user avatar
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7 votes
2 answers
694 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 ...
Sam's user avatar
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7 votes
0 answers
312 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 ...
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7 votes
0 answers
299 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 $\...
0xbadf00d's user avatar
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6 votes
1 answer
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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 ...
user3353819's user avatar
6 votes
2 answers
651 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 ...
ABIM's user avatar
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6 votes
2 answers
2k 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+...
Johny's user avatar
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6 votes
1 answer
372 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 ...
JSG's user avatar
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6 votes
1 answer
496 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 ...
Abakus's user avatar
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6 votes
1 answer
428 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\;\...
0xbadf00d's user avatar
  • 161
6 votes
1 answer
350 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 ...
ABIM's user avatar
  • 5,019
6 votes
0 answers
70 views

Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)

Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
ABIM's user avatar
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6 votes
0 answers
163 views

Fokker-Planck equation for SDEs on manifold

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate ...
Fawen90's user avatar
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6 votes
0 answers
726 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 ...
0xbadf00d's user avatar
  • 161
5 votes
2 answers
620 views

Intuition behind Gubinelli derivative

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject. On page 14 of "A Course on Rough Paths With an ...
Furdzik Zbignew's user avatar
5 votes
2 answers
896 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(...
user2379888's user avatar
5 votes
2 answers
273 views

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 ...
Iosif Pinelis's user avatar
5 votes
3 answers
987 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(...
user1611107's user avatar
5 votes
2 answers
1k views

Textbooks or lecture notes about mean field games

I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...
Anita Poller's user avatar
5 votes
1 answer
554 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 ...
user123124's user avatar
5 votes
1 answer
442 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 ...
Creator's user avatar
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5 votes
3 answers
758 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 ...
Hans's user avatar
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5 votes
1 answer
2k views

Variance of Multi-Dimensional Ornstein Uhlenbeck process

I am trying to compute the asymptotic variance of OU process $$ d X_t = - H X_t dt + S dW_t $$ where $X_t$ takes value in $R^d$. $H$ and $S$ are $d\times d$ matrices that does not have $HS = SH$ in ...
Nikolayevich's user avatar
5 votes
2 answers
414 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 ...
Hans's user avatar
  • 2,169
5 votes
1 answer
311 views

Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
ABIM's user avatar
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5 votes
1 answer
367 views

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[...
GJC20's user avatar
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5 votes
2 answers
503 views

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 ...
Nate River's user avatar
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