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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.

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188 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 ...
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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 ...
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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 ...
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106 views

Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes: [1] F. Soucaliuc, Réflexion entre deux diffusions ...
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1answer
161 views

Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE $dX_t = dW_t + dL_t^0(X_t)$, where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$....
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80 views

hitting time of an Ornstein-Ulhenbeck

If we consider a nice Ornstein Uhlenbeck process $d x (t) = - \gamma x(t) dt + \sigma d w (t)$ with $x(0) = x_0 \in (-L,L)$. Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener process....
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74 views

Weak solutions of linear parabolic PDEs and corresponding SDEs

It is well known that for an Stochastic differential equation (on the real line) of the form: $dX_t = \mu(X_t)dt + \sigma(X_t)dW$ where $W$ is the standard Wiener process, the transition probability ...
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1answer
807 views

using Feynman-Kac formula

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
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1answer
110 views

Differentiability of value function

Suppose $X$ is a process given by - $dX_t = db_t$ where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$. Suppose an agent earns a payoff given by $V(x) = \mathbb{E} [\...
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1answer
319 views

General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
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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*} ...
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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 ...
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1answer
348 views

What is the derivative of this integral?

I have asked this question here https://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral but still has no response. Might I ask it here ? Let $\alpha(t)\in\{0,1\}: ...
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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 ...
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2answers
332 views

Understanding the limits of the Ito Process Defintion

I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form $$dX_t=g(t)dt+f(t)dW(t),$$ where $W(t)$ is a ...
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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 ...
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56 views

$X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations ...
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72 views

Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function: $$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$ such that $$dX_t=\mu(t,X_t)...
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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 \...
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1answer
1k views

Time Change of a Brownian motion

We know that for if $X$ is a stochastic integral of the form below - $X_t = \int_0^t v(s,\omega) db(s,\omega)$. then we can use time change formula to claim that $X_t = W_{\alpha(t)}$ where $W$ is ...
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1answer
383 views

Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation $$dg_t = dB_t \circ g_t$$ where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
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94 views

Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
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141 views

Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution. The SDE is - $ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$ where $b_1$ and $b_2$ are two ...
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122 views

Convergence of approximate quadratic variation in $L^p$

For a diffusion $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...
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192 views

The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
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110 views

probabilistic interpretation of a finite difference scheme

Let me start with some simple background. Consider the heat equation : $ \frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} \quad \mbox{in} \quad \mathbb{R}\times (0,\...
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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}...
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366 views

Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is $$ P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
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1answer
365 views

Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...
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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 ...
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71 views

The Stratonovich formulation of the Double Mean Reverting Model

I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice ...
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2answers
104 views

SDEs: Bounding the variance of a solution

I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it. Let $$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} \...
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2answers
478 views

Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution. I want to know if we assume $b$ ...
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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(\...
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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 ...
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87 views

Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...
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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 ...
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916 views

Expected value of a stochastic integral expression

I am wondering if the following expression can be processed a bit analytically, $$ E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right], $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
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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 $...
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1answer
344 views

Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) ...
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0answers
119 views

Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-...
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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(...
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1answer
165 views

Certain construction of the Itô integral on manifolds

Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral $$ I(X) = \int_0^T \langle X(t, ...
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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 $\mathbb{R}$-...
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1answer
131 views

Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem: Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation: $Z_{k+1}-Z_k=P_k(1-2Z_k)$ where $P_k=0$ with probability $...
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1answer
353 views

Question on viscosity solution through stochastic differential equations

I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with $\epsilon&...
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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 ...
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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). ...
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84 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ $$\mathcal{F}^{\...
3
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1answer
90 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...