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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.

1
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1answer
266 views

Approximate an exponential martingale through its kernel

Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale \begin{align} M_t = \exp\left[\int_{0}^{t} h_s \,dB_s - \frac{1}{2}\int_{0}^{t} h_s^2\...
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2answers
354 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 ...
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1answer
164 views

Reflected SDE with non-Lipschitz coefficients

I have an equation of the form: $$dX_t=\mu(X_t)dt+\sigma(X_t)dZ_t+dL_t, \quad X_0=x_0\in (-\infty,a]$$ where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
2
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1answer
207 views

Existence of solution for reflected SDE

I have an equation of the form: $$dX_t=\mu(X_t)X_tdt+\sigma(X_t)X_tdZ_t+dL_t, \quad X_0=x_0\in (0,a]$$ where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
3
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1answer
431 views

Strong solution for geometric brownian motion with varying drift and volatility

I have an equation of the form: $$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$ I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the ...
2
votes
1answer
149 views

Itô Formula for Hilbert space-valued Lévy processes

I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is: does there exist an Itô formula ...
2
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1answer
647 views

On a reflecting Brownian motion and its boundary local time

I have a question about a reflecting Brownian motion and its boundary local time. Bass and Hsu that the existence of Reflecting Brownian motion and boundary local time on a bounded Lipschitz domain ...
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1answer
49 views

relationship between transition semigroup and first order spatial derivative

In my research, all the proof comes down to an estimate of the following term $$\int_t^{t+h} E|\partial_x P_{t+h-\tau}f(X_\tau)-P_{t+h-\tau}\partial_xf(X_\tau)|^2\,d\tau,\tag{1}$$ where $t>0$ is ...
2
votes
1answer
452 views

Existence of a solution to an infinite dimensional Stratonovich SDE

Let $U,H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
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1answer
74 views

Hilbert-Space Values SDE in terms of Basis

Suppose: $$ dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t $$ is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
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1answer
528 views

Properties of the trace term in the Itō formula

Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where $U,H$ are separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ is nonnegative ...
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1answer
214 views

Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces

I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand. In the notation of the paper, let $H,H_1$ be separable $\...
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1answer
84 views

Reference for convergence of Hilbert-space valued SDEs

I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However ...
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1answer
141 views

Sufficient condition of continuity of the expected stopping time

Let $\sigma \in C(\mathbb R)$, and $X$ be a solution of \begin{equation}\label{eq:1} X_{t} = x + t + \int_{0}^{t} \sigma(X_{s}) dB_{s} \end{equation} where $B$ is 1-d Brownian motion under filtered ...
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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 $M$...
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3answers
199 views

Existence of solution to SDE with perscribed initial and terminal conditions

The SDEs \begin{equation} dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t \end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on ...
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0answers
77 views

Continuity of solution map to Stratonovich Integral

For paths $x:[0, T] \rightarrow \mathbb{R}^n$, the Stratonovich integral along a one form $\omega$ on $\mathbb{R}^n$ can be defined by $$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\...
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1answer
210 views

Asymptotic behavior of an integral of OU process

Let $X=(X_t)_{t\ge 0}$ be a stochastic process (Ornstein-Uhlenbeck process) determined by $$dX_t=-aX_tdt+\sigma dW_t,$$ where $X_0=0$, $a>0$ and $\sigma>0$ are constants, and $W=(W_t)_{t\ge 0}$...
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0answers
95 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
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0answers
79 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
3
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1answer
391 views

Calculate Moments of SDE

I have posted a similar question on math.stackexchange (https://math.stackexchange.com/questions/1848492/calculate-mean-of-sde), but didn't find anyone who could help. I'm interested in the one-...
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1answer
304 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
2
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0answers
137 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 ...
3
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1answer
138 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
3
votes
1answer
245 views

Stochastic process with discontinuous drift

While studying a portfolio optimization problem, I came across the process $$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$ which has a discountinuous ...
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0answers
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 $$ \dot u = -u + f(a) $$ $$ \...
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votes
1answer
67 views

Nonparametric estimation in diffusion

Fan and Wang In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$. My question is how to find estimators for the process ...
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0answers
133 views

Feynman–Kac formula terminal condition [closed]

Hi I am trying to calculate $E(\phi(X(1))$ with $X(t)$ satisfies the following $$d(X(t))=\sigma(X(t))dW(t)$$ $$X(0)=x_0$$ where $\phi$ and $\sigma$ are arbitrary functions and $W(t)$ is Brownian ...
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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 ...
1
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1answer
122 views

The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \begin{equation}\left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
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1answer
346 views

Girsanov theorem and the density of a process

I am coming across a paper ( Proposition $1.1$ from http://www.sciencedirect.com/science/article/pii/0304414987901840 ) that claims the following fact which I don't understand why: On a probability ...
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,\...
1
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1answer
105 views

Existence of strong solution in SDEs and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, b:...
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 ...
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1answer
142 views

Computing transition operators for Markov processes

Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$ (or given by ...
3
votes
1answer
145 views

Optimal control / Portoflio optimization: Maximize expected utility of total consumption

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
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 ...
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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 $\...
1
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1answer
215 views

Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded ...
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0answers
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:\...
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 $T$ (...
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0answers
83 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
3
votes
0answers
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
2answers
220 views

Decouple system of SDEs / handle scaling problem

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
6
votes
3answers
340 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 ...
3
votes
0answers
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
0answers
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 ...
1
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1answer
735 views

Linking Wasserstein and total variation distances

I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...
20
votes
1answer
450 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, ...
1
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0answers
187 views

Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...