# Tagged Questions

Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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### A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
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### Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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### Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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### Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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### $L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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### What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
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### What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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### What kind of decomposition is $X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$?

In one of the papers I was reading for my masters thesis I came across a theorem with no references. Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ ...
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### Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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### Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
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### Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
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### Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
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### Why is the expression $E[M_L^*] \leq \inf_{\mu >0}\{\mu^{1/{1-r}} \varphi_r(1)+\mu E[\langle M \rangle_L^{r/2}]\}$ an explicit function of $C_r$

Why is the expression $E[M_L^*] \leq$ $\inf_{\mu >0}\{\mu^{1/{1-r}}\varphi_r(1)+\mu E[\langle M \rangle_L^{r/2}]\}$ an explicit function of $C_r$ where $C_r$ is a function of $\varphi_r(1)$. In ...
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### What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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### linear combination of infinitely divisible random variables

If $X$ and $Y$ are real valued random variables with infinitely divisible distributions, does $aX + bY$ also have an infinitely distribution ($a, b \in \mathbb{R}$). I've seen this stated in several ...
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### Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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### What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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### How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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### Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \rho_n^2(t)=2 \int_0^...
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### Rigorous meaning of conditional expectation in Feynman-Kac formula/in general

In Wikipedia https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula and plenty of other books/sources, Feynman-Kac formula is expressed in a form of the type $$f(t,x)=E(f(T,X_T)\mid X_t=x)$$ What ...
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### Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
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### Why does $1 \leq \sup \limits_{0\leq t \leq 1}( C|B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold?

Why does $1 \leq C\sup \limits_{0\leq t \leq 1}( |B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold ? I am trying to show by contradiction that the Burkholder-Gundy ...
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### cumulant of infinite sum of random variables

Could you help me the following question? Let $X_i$ are identical independent random variables. Putting $Z:=\sum_{i=1}^{\infty}X_i$. Which conditions do we have $$k_n(Z)=\sum_{i=1}^{\infty}k_n(X_i),$$...
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### What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...