-2
votes
0answers
68 views

Is there an example of an SDE that has no weak solution? [on hold]

So far I couldn't find an example for an SDE for which there exists no weak solution. Do you know one?
3
votes
1answer
107 views

Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
2
votes
0answers
63 views

Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
0
votes
0answers
72 views

Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns (http://www.math.upenn.edu/~pemantle/papers/burdzy.pdf). What ...
2
votes
1answer
104 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
0
votes
1answer
148 views

On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde: $dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $ $\partial_x X_t(0) = \partial_x X_t(1) = 0, $ $X_0 = 0, $ ...
1
vote
1answer
68 views

Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral. I have found that computing the Malliavin ...
1
vote
0answers
201 views

Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
5
votes
0answers
190 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
0
votes
0answers
45 views

question related to Tanaka Formulae

Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: ...
1
vote
0answers
36 views

question about the optimal decomposition of supermartingale

Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...
1
vote
0answers
122 views

Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type $$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$ converges to zero (in probability/distribution) for $h ...
1
vote
2answers
110 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
2
votes
3answers
70 views

a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...
0
votes
1answer
71 views

Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent. 1): $X$ is $L^p$-integrable. 2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...
0
votes
0answers
70 views

a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t. ...
0
votes
0answers
31 views

a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process ...
1
vote
1answer
79 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
3
votes
2answers
175 views

Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$, the sequence $(M^n)_n$ converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...
4
votes
0answers
222 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
1
vote
0answers
75 views

question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition: ...
12
votes
1answer
358 views

Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$: $$dX_t = f(X_t)\,dt + dW_t,$$ where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...
1
vote
0answers
90 views

On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows: $\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$, $u(t,0)=u(t,1)=0$, $u(0,x)=v(x)$, where $V$ is a bounded, smooth ...
1
vote
1answer
59 views

The probability of Levy process staying at a point

Assume $X_{t}$ is a 1-dimensional Levy process on a probability $(\Omega, \mathcal{F}, P)$. For a fixed point $x$ in the state space and fixed $t\neq 0$, what's the value of $ P(\omega: ...
7
votes
2answers
296 views

Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
1
vote
1answer
71 views

The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define $\Delta X_{t} = X_{t}- X_{t-}$. For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...
3
votes
1answer
107 views

The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$. This property is called the ...
3
votes
0answers
74 views

The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
3
votes
1answer
117 views

Can this two-dimensional process self intersect?

I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense) $$dX_t = f(X_t) dt + ...
0
votes
0answers
54 views

Ito formula for max(X,0) where X is a semimartingale

Has anyone ever applied the Ito formula on $|X^+|^2$ for $X^+ = \max(X,0)$ with $X(t) = X(0) + M(t) + V(t)$, where $M(t)$ is a local martingale and $V(t)$ is bounded variation process. I found it in ...
0
votes
1answer
121 views

a dominated convergence theorem for martingale (II)

The question is presented in a dominated convergence theorem for martingale Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means ...
3
votes
1answer
118 views

Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...
2
votes
1answer
93 views

Can $<.>$ of a martingale determine it only?

Let $\Omega$ be the space of continuous functions defined on $[0,1]$. Define the canonical process $B$ by $$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$ Let us equip $\Omega$ with the usual ...
3
votes
1answer
118 views

a $L^1$ convergence for backward martingale

I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...
1
vote
2answers
147 views

On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$ where $(t,x)\in {\mathbb R}_+\times ...
2
votes
2answers
204 views

Any suggestions on a rigorous stochastic differential equations book?

I have been looking through some books and they are not very rigorous. Any suggestions would be great.
2
votes
0answers
101 views

On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$: $\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...
0
votes
1answer
117 views

On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where: $X_s$ is a wide sense stationary process with known ...
-1
votes
1answer
83 views

Multiplicative version of Novikov inequality for Ito integral

It is clear that Ito isometry $E(∫^t_0fdW)^2=E(∫^t_0f^2dt)$ can be written in the multiplicative form as $E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt).$ Is it possible to obtain the multiplicative version ...
4
votes
3answers
302 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
1
vote
0answers
63 views

Time change for non-homogeneous Markov processes

Background: Let $C$ be the space of continuous function on $[0,T]$, $f, \sigma \in C$ bounded with $\sigma^2 \geq \varepsilon > 0$ and let $X=(X_t)_{t\in [0,T]}$ be a diffusion process of ...
0
votes
1answer
80 views

construction of a approximate martingale

everyone. Given a probabilistic space $(\Omega, \mathcal{F}_t, \mathbb{P})$ and a martingale $(M_t)_{t\leq 1}$ on it. Suppose $$M_1\stackrel{\mathbb{P}}{\sim}\mu$$ where $\mu$ is a probability ...
2
votes
1answer
174 views

Stochastic integral with respect to discontinuous martingale

in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely, $ \int_0^t f(X_t) dM_t,$ where $M(t) = N(t) - \int_0^t \lambda(s)ds$. The integrand ...
5
votes
1answer
419 views

On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
4
votes
2answers
196 views

Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
3
votes
1answer
121 views

Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift

Let $(\Omega, (\mathcal F_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W_t)_{t \geq 0}$ denotes the Wiener process. Let $Z \in L^1(\mathbb P)$ with $Z ...
2
votes
0answers
191 views

distribution of integral of exponential of wiener process

I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ...
1
vote
0answers
108 views

Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)?

Is there an existence and uniqueness theorem for SDEs of the following type: $dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$, where $\tilde{W}_{t}$ is say ...
0
votes
1answer
238 views

Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following: Firstly we have a Markov chain $\{Y_k\}$ with finite ...
2
votes
1answer
247 views

Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure

Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define $$\Delta(u)= \frac{\int u(h) \exp(-\eta ...