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

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Expectation of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $\mathbb R$ and $$\langle ...
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27 views

Is $\phi B(\omega,\;\cdot\;)$ Lebesgue integrable over $[0,\infty)$ for a real-valued Brownian motion $B$ and $\phi\in C_c^\infty(\mathbb R)$?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\lambda$ be the Lebesgue measure on $\mathbb R$. Is ...
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34 views

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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37 views

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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29 views

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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17 views

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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44 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb ...
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54 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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Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
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Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...
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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 \begin{equation} \rho_n^2(t)=2 ...
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How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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Reference for stochastic calculus with jumps

All the standard books I know on stochastic calculus work almost exclusively with continuous martingales. What are the standard references for the general theory (with jumps)?
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$\lim_{N\to \infty} \frac{1}{N+1}\sum_{n=0}^N f_n=\lim_{N\to \infty} \frac{1}{N+1}\sum_{n=0}^N f_{n+m}$ for any $f_n$ and $m\in \mathbb N$

Show that $\lim_{N\to \infty} \frac{1}{N+1}\sum_{n=0}^N f_n=\lim_{N\to \infty} \frac{1}{N+1}\sum_{n=0}^N f_{n+m}$ for any $f_n$ and $m\in \mathbb N$, where the limit exists. Can I split the ...
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Why $d\langle X \rangle_t = d X_t dX_t$ if $X_t$ is a semimartingale?

Following this question, proving the equivalence between equation $(1)$ and $(2)$, I deduced that $$d\langle X \rangle_t = d X_t dX_t$$ (where $X_t$ was an Ito's process, hence a semimartingale). I ...
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How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, ...
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If one stochastic process is a modification of another, then they have the same finite probability distribution.

On page 2 in Karatzas and Shreve: Brownian Motion and Stochastic Calculus it is said that a stochastic process $Y$ is a modification of $X$ if for all $t$: $P(X_t=Y_t)=1$. If both are stochastic ...
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33 views

Brownian Motion Hitting Time?

So my problem is the following. Take a 2D Brownian motion $(W_{1t}, W_{2t})$ such that it starts at $(1,1)$. With probability 1 it will hit the x-axis. What is the probability that it will hit the ...
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Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
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Independent stochastic processes

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks
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modulus of continuity of Ito process

We know from Levy's (uniform) modulus of continuity that for Brownian Motion, almost surely any sample path is locally Holder continuous for any $\rho <\frac{1}{2}$, i.e. $$ |W_t - W_s | \leq ...
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1answer
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A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and ...
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How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space ...
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1answer
60 views

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 ...
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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 ...
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Pathwise definition of stochastic integral consistent with the Ito isometry

My definition of the stochastic integral is that it it is the image of the Ito isometry. Now we also prove Ito's formula and then apply it pathwise and get a pathwise definition in some cases. But in ...
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Diffusion Process Expectation Smoothness Condition

Consider a diffusion process on a sample space $\Omega$ $$dx_t = \mu(\omega,t)dt+\sigma(\omega,t)dB_t,\, \forall\omega\in\Omega$$ where $B_t$ is the standard Brownian motion on the filtration ...
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1answer
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What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for ...
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Adjoint of evaluation operator: Inverse Bayesian Analysis

I'm reading "Inverse Problems - A Bayesian Perspective" by Andrew Stuart and I'm stuck with working out an application (an easier form of section 3.2): Consider a random process $u: (0,1) \to \mathbb ...
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Strong solution SDE - independence of initial conditiion

I am currently studying the existence and uniqueness of strong solutions of SDEs of type $$\left[\begin{array}{l} \, dX_t=\mu(t,X_t)\,\mathrm{d}t+\sigma(t,X_t)\,\mathrm{d}W_t\\ ...
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Kolmogorov extension theorem

I have attached to this post a short treatment of the Kolmogorov extension theorem for measures. In the following, I did not understand what is meant by the $A$ that I circled in red. I suppose that ...
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Brownian Motion and Poisson's problem

Let $U\subset \mathbb{R}^d$ be a bounded domain and $g: U\to \mathbb{R}$ be continuous. A continuous function $u:\overline{U}\to \mathbb{R}$, $u\in \mathcal{C}^2(U)$ is said to be a solution of ...
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How can I show the existence of a right continuous version to the supermartingale $\{P(L>u|\mathcal{F}_u),u \geq 0\}$?

I was reading a paper by Marc Yor for my thesis and in the statement of one of the theorem he mentions Consider the super martingale $\{P(L>u|\mathcal{F}_u),u \geq 0\}$ where $L$ is a random time ...
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exercise 3.3.34 from Karatza and Shreve [duplicate]

In the exercise, W is a standard, one-dimensional Brownian motion and $0 \lt T \lt \infty$. We are asked to show that $$\lim_{\beta\rightarrow\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^t e^{\beta ...
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Is $d \langle X,Y \rangle = \langle dX,dY \rangle$ where X,Y are continous semi-martingales

Is $d \langle X,Y \rangle = \langle dX,dY \rangle$. I think the answer is yes because $ d \langle X,Y \rangle=\langle X,Y \rangle_t- \langle X,Y \rangle_s$ and $\langle dX,dY \rangle=\langle ...
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Ito integral via simple process when the integrand is C^1

I have the following problem. Let $H_t$ be an adapted process with trajectories a.s. of class $C^1$ on $\mathbb{R}_{+}$. Compute using simple process $\int_o^t H_s d B_s$. My idea is to firstly set ...
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Black Scholes partial differential equation; Derivation

I have an exam tomorrow and the issue is, my notes just really briefly mentions it. It doesn't even take a full 2 pages to mention the partial differential equation. I haven't even seen it in ...
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How to price a supershare option; expected value of a payoff function?

I thought I'd be able to do this but evidently not. Let $S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_t}$ for all $t$. $W_t$ is a standard brownian motion. We have the following function for payoff ...
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Find a random variable Z (i.e a process Z_t) that maximizes f(Z)

Consider the following problem: find/construct a process $(Z_t)_{0\leq t \leq T}$ on some prob. space s.t \begin{equation} \max_Z\mathbb{E}\Big[(e^{Z_T}-K)^+\Big] \end{equation} Given that ...
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1answer
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Ito's formula; when to use one and when to use the other form

I have seen $2$ "forms" of the Ito formula which are essentially, in the end, equivalent. But my question is, having seen quite a few questions on stochastic differential equations, I am wondering ...
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25 views

Stochastic calculus for continuous time Markov chains

I have absolved a course on stochastic analysis, i.e. integrals with respect to the brownian motion. Now I know that there is a theory of stochastic calculus for diskrete matringales, however I was ...
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Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
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1answer
42 views

Calculating expectation using martingales

Could anyone help me with this exercise or show me similiar example? Any help appreciated. Using the martingales $M_t^\lambda=\exp(\lambda W_t-\lambda^2t/2)$ and ...
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What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = ...
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1answer
45 views

Conditional expectation w.r.t Lebesgue measure

Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})=((0,1)^{2},\mathcal{B}((0,1)^{2}),\lambda_{2})$, where $\lambda_{2}$ is the Lebesgue measure in $\Omega=(0,1)^{2}$. Then, for ...
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1answer
19 views

Parameter of Ornstein-Uhlenbeck (O-U) process

I am considering the following O-U process $$\mathrm{d}X(t)=-gX(t)\mathrm{d}t+\mathrm{d}B_t$$ From my dataset, I can estimate value of $g$ and it is very small. If covariance of this process is ...
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28 views

Malliavan Derivative of a Geometric Brownian Motion

I'm trying to understand a proof that requires Malliavan Calculus, but have no experience with the topic. My question revolves around showing that the Malliavan derivative of a geometric brownian ...