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1 vote
0 answers
44 views

What do we know about Poisson boundaries of arbitrary Riemannian manifolds?

For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
3 votes
1 answer
289 views

Smoothness of expectation

Suppose that $X_t$ is a strong solution to the SDE, $$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
3 votes
2 answers
490 views

SDE driven by fractional Brownian motion

Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below: $$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$ I am looking for references that ...
1 vote
0 answers
134 views

Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1

Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
3 votes
1 answer
545 views

Each diffusion SDE is associated to a *unique* family of transition kernels

I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$. How can I prove that there exists a unique family of transition ...
7 votes
2 answers
613 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 $...
1 vote
1 answer
739 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}{\...
1 vote
0 answers
89 views

Comparison of the numbers of particles surviving forever

Consider two $N\text{-}$particle systems as follows : for $1\le i\le N$, $$X^i_t=1+\int_0^t(b+\phi^i_s) \, ds+W^i_t \quad\mbox{and} \quad Y^i_t=1+ct+W^i_t,\quad \forall t\ge 0,$$ where $c>b>0$ ...
4 votes
0 answers
167 views

Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation $$ X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
2 votes
1 answer
538 views

Generalized Fokker-Planck equation

Consider the diffusion process $$ d X = \mu(X, t) dt + \sigma(X, t) dY. $$ When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...
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 ...
4 votes
1 answer
509 views

Conditional stochastic integration

Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g. $$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$ What is the ...
2 votes
0 answers
75 views

Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?

Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...
2 votes
0 answers
146 views

Exit time for Brownian motion with stochastic barriers

I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows. Context: If $L_t$ and $R_t$ denote the distance to the left and ...
2 votes
1 answer
596 views

Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion: $$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$ where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...
2 votes
1 answer
490 views

Absolute value of a diffusion

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below. Suppose, $X_0 = Y_0 = 0$ \...
0 votes
0 answers
59 views

How to find the PDE for the following transition density

Suppose I have the following two stochastic differential equations ($t\geq 0$) $$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$ where $X = (X_t)$, $Z = (Z_t).$ Note that $W=(...
3 votes
0 answers
90 views

Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
2 votes
0 answers
74 views

Floquet stochastic process

Let $X_t$ be defined by the SDE $$ dX_t = A(t, X_t)dt + dW_t $$ where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
1 vote
0 answers
57 views

Matching Numbers in Ito McKean

Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as $e_1 = \lim_{b \...
2 votes
1 answer
528 views

Any modern/recent version of Ito & McKean?

This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
3 votes
0 answers
186 views

When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?

When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...
1 vote
1 answer
460 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 votes
1 answer
880 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 votes
1 answer
1k 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
0 answers
161 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_{|\...
1 vote
0 answers
66 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 ...
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)...
2 votes
1 answer
3k 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 ...
3 votes
1 answer
604 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 ...
7 votes
1 answer
4k 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 $...