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Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE. Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
Lochend's user avatar
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0 answers
124 views

On the Lipschitz constant of $\Gamma$

Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
GJC20's user avatar
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1 vote
0 answers
91 views

When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
ABIM's user avatar
  • 5,405
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0 answers
54 views

Conditions ensuring that conditional law of a process belongs to a given exponential family

Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
Joe_Affine's user avatar
1 vote
0 answers
76 views

Gronwall type lemma for an Ito process

For all $t\in \mathbb{R}$ let $h_t = \frac{1}{2} + \int_0^t v_s\cdot dB_s$ be an Itô process, where $B_s$ is a standard Brownian of $\mathbb{R}^d$ and $v_t$ an $\mathbb{R}^d$ valued adapted process, ...
Gericault's user avatar
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1 vote
0 answers
78 views

If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
vaoy's user avatar
  • 309
1 vote
0 answers
222 views

Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
Kolodez's user avatar
  • 335
1 vote
0 answers
766 views

Derivative of the function of random variable

Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...
Xu Shan's user avatar
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95 views

Generator of a Hilbert space valued Wiener process from the solution of a martingale problem

Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
276 views

Path dependent Markov property

Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded \begin{align*} \Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty) \end{align*} Then my question is:...
defex95's user avatar
  • 159
1 vote
0 answers
185 views

Ito's Lemma (CVF) on product of Poisson processes

I have the following stochastic differential equation: $da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$, with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by: $dq(t)...
Beatrice's user avatar
1 vote
0 answers
80 views

Large deviations estimate for arbitrary continuous function

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
237 views

On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
user156337's user avatar
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0 answers
73 views

conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
defex95's user avatar
  • 159
1 vote
0 answers
59 views

Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$

Consider the linear discrete-time stochastic systems: \begin{equation} x_{k+1} = Ax_k + v_k, \end{equation} with time-instants $k \in \mathbb{N}$, state $x_k \in \mathbb{R}^n$, stochastic process $v_k ...
OliVer's user avatar
  • 53
1 vote
0 answers
235 views

Associative law of the stochastic integral in Hilbert spaces

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
106 views

Domain of a reflected stochastic differential equation

I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
fast_and_fourier's user avatar
1 vote
0 answers
90 views

Onsager-Machlup Function of a Killed Diffusion Process

Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
user3658307's user avatar
1 vote
0 answers
340 views

Construction of the quadratic variation for Hilbert space valued local martingales

Let $H$ be a separable $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
79 views

Stochastic Control with Stochastic Cost-functional

Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also? That is, let $X_t^u$ is the solution to a controlled SDE $$ dX_t = \mu(t,u_t,X_t^u)dt ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
331 views

Mean and Variance of SDE

What is the mean and the variance of $y_t$, given the following SDE: $dy_t = -x_t y_t dt + \sigma_1 dW^1_t$ $dx_t = -\sigma_2 y_t dW^2_t$ $W^1$ and $W^2$ are (possibly correlated) Wiener processes.
Posch79's user avatar
  • 111
1 vote
0 answers
124 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,\...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
118 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 ...
ysys's user avatar
  • 43
0 votes
2 answers
313 views

Some doubts on proof of pathwise uniqueness of a stochastic differential equation

I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions. Starting from a stochastic differential equation of the form: $$dr_t=\alpha\left(r_{\mu}-r_t\...
Strictly_increasing's user avatar
0 votes
3 answers
639 views

Non-smooth Ito lemma for semi-martingales

Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth? I've been looking but have not found much, any ...
ABIM's user avatar
  • 5,405
0 votes
2 answers
182 views

Distribution of local martingale is absolutly continuous to that of the Brownian motion?

Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
null's user avatar
  • 227
0 votes
1 answer
379 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\}: ...
Nguyen's user avatar
  • 131
0 votes
1 answer
154 views

Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time

Consider the following stochastic integral: $$ X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s. $$ Is $X_t$ almost-surely non-negative? Using this answer, it seems that $$ X_t = \max( W_t, 0) - \...
oswinso's user avatar
  • 109
0 votes
1 answer
463 views

Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure

I'm trying to figure out the connections between two contructions of Gaussian measure. Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
null's user avatar
  • 227
0 votes
1 answer
152 views

About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E

We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...
gradstudent's user avatar
  • 2,246
0 votes
1 answer
349 views

Probability that a geometric Brownian motion with additional determinstic drift ever hits zero

Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE $$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$ where $\mu, \sigma, C, k > 0$ are constants, ...
Nate River's user avatar
  • 6,223
0 votes
1 answer
898 views

How to understand the transition density of reflected Brownian motion

We can see from the above picture the transition density of reflecting Browninan motion is given by (19). As we know, the first part ($2p(t,x,y)$) is the transition density of a Brownian motion (from $...
Ailiy Evan's user avatar
0 votes
1 answer
341 views

Hitting probability for mean-reverting stochastic process

I quote Delbaen and Shirakawa (2002). Starting from a stochastic differential equation of the form: $$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
Strictly_increasing's user avatar
0 votes
1 answer
206 views

Stochastic invariant subset

Let us consider a stochastic differential equation (SDE), $$ dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}% $$ and a compact set $C\subset\mathbb{R}^{n}$. Given a stochastic ...
UnclePetros's user avatar
0 votes
0 answers
15 views

Conditions on SDE coefficients for well-posedness of Fokker-Planck equation

Consider the following $n$-dimensional Ito-SDE: \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} What are the necessary regularity conditions on $\mu$ and $\sigma$ to ensure that the ...
GigaByte123's user avatar
0 votes
0 answers
42 views

Bound on the radon-nikodym derivative between two stochastic processes at a time point

I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution: \begin{align*} dX_t &= b(t, X_t)...
optimal_transport_fan's user avatar
0 votes
0 answers
76 views

When we should integrate on both side over a SDE?

Maybe I am quite stupid, I am quite confused about, when we should use ito formula to solve SDE and when it is appropriate to integrate directly to get the solution? Specifically, let us consider the ...
XZCDRMS's user avatar
0 votes
0 answers
122 views

Laplace transform of a stochastic process

Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE: $$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
Greyearl's user avatar
0 votes
0 answers
120 views

Predictability of the mild solution of a SPDE

Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
mathex's user avatar
  • 573
0 votes
0 answers
75 views

Regularity of solutions to forward-backward stochastic differential equations

Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy $$ \left\{ \begin{aligned} d X_t &= aP_t dt +bdB_t;\\ X_0 &= x_0;\\ d P_t &=cP_t dt + c^*Z_t dB_t; \\ P_T &...
mnmn1993's user avatar
0 votes
0 answers
468 views

The relationship between measurability and weak measurability

For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple functions, measurability (the ...
Guomin Liu's user avatar
0 votes
0 answers
294 views

Malliavin derivative of Ito process

Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
76 views

Ornstein-Uhlenbeck type process with thresholding

(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding: $$ dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0, $$ where $...
Nick's user avatar
  • 31
0 votes
0 answers
70 views

If $(Φ^x)_{x∈ℝ}$ is a family of real-valued stochastic processes and $B$ is a Brownian motion, then $\int_0^tΦ^x_s\:dB_s=(\int_0^t\Phi_s\:dB_s)(x)$

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$ $B$ be a (standard, real-valued) $\mathcal F$...
0xbadf00d's user avatar
  • 167
0 votes
0 answers
153 views

Embedding a martingale by SDE

Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE: $$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...
CodeGolf's user avatar
  • 1,835
0 votes
1 answer
360 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 $0$....
ysys's user avatar
  • 43
0 votes
0 answers
77 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)...
skillfeedback's user avatar
-1 votes
1 answer
169 views

joint density of two relevant random variables

It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
Wang Jing's user avatar
-1 votes
1 answer
2k views

The probability distribution of "derivative" of a random variable

Disclaimer: Cross-posted in math.SE. Let me set the stage; Consider a stochastic PDE, which has to following form $$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$ where $H$ is a deterministic function, ...
Our's user avatar
  • 133
-2 votes
1 answer
138 views

Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
YT_learning_math's user avatar

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