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

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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 sum? ...
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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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37 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 C(\...
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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 $E^*...
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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 $(X_t)_{...
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1answer
64 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 $$k_n(Z)=\sum_{i=1}^{\infty}k_n(X_i),$$...
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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 ...
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93 views

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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61 views

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\\ X_0=\xi\end{array}\...
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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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23 views

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 s}...
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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 X_t-X_s,...
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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 hand-...
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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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29 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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48 views

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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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 $N_t^\lambda=(M_t^\lambda+M_t^{-\...
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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 = 2sgn(y_t)\sqrt{...
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1answer
46 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
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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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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 ...
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Proving the continuity of a function with respect to a measure

Let $\mathcal{M}([0,1])$ be the space of all real finite measures on $[0,1]$, with norm $\|\mu\|=|\mu|([0,1])$ and consider the function $$u(y)=\int_{[0,1]}\min\{x,y\}\mu(dx)$$ for $y\in[0,1]$. I ...
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Can I swap conditional expectation and limit

My problem is the following : let $B_t$ be a standard Brownian motion and $H_t$ a progressive measurable process such that $\mathbb{E}\left(\int_0^{+\infty} H_t^2\ dt \right)<+\infty$. Denote $X_t=\...
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Relation between $L^2$-equivalence and indistinguishability

Let $\{X_t\}_{t\in[0,T]}$ and $\{Y_t\}_{t\in[0,T]}$ be two stochastic processes defined on some probability space $(\Omega,\mathfrak{A},\mathbb{P})$. Assume that $X$ and $Y$ are $L^2$-equivalent, i.e. ...
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Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
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27 views

Characterizing superposition of two renewal processes

This is a follow-up question of "When superposition of two renewal processes is another renewal process?". How can we characterize the superposition of two renewal processes? The superposition ...
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Ito's formula for Poisson process

Suppose ($Y_t$) is a rate 1 Poisson process, and consider the jump process $Z_t=Y_{\int_0^tf(X_s)ds}$ for some non-negative process $X_s$. What would be the quadratic variation of $Z$, and how would ...
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stochastic integral with respect to a Brownian motion of a simple process

Given a simple process $H_u:=\sum_{i=0}^{n-1} h_i \mathbb{1}_{(t_i, t_{i+1}]} (u)$, where $h_i$ is bounded and $\mathcal{F}_{t_i}$ measurable, I have defined its stochastic integral with respect to a ...
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Relationship between a stochastic process and its P.D.F

So lets assume I numerically approximate a stochastic process $X(t)$ on an interval $0\leq t \leq 1$ I need the PDF of this process in order to find things such as expectation, variance, etc. So does ...
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Meaning of stochastic derivative vs deterministic derative

Suppose for a deterministic function $X(t)$, which represents distance traveled. We definite $\frac{dX(t)}{dt}$ as the rate of change of $X(t)$ with respect to $t$, with units $m/s$ (meter per second)....
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23 views

Exponential decay of a stopping time for an Ito diffusion process

Let $dX_t=dB_t + a \cot(X_t)dt$, with $X_0=x \in (0,\pi)$, where $a$ is a specific constant so that the lifetime of the process is infinite almost surely. The process has a transition density which ...
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2answers
23 views

Fourier transform with the derivative of a function

I have to identify the Fourier transform, defined as $\widehat f(x)=\displaystyle \int_{\mathbb R} e^{-ixy}f(y) dy$ As a task, I have to calculate the the fourier transform of $g(x)= \frac{32}{1875}...
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71 views

Distribution of stopping times

I encountered the following question in my research: Let the diffusion process $\{X_t\}_{t\ge 0}$ be governed by $$d X_t=s(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $s>0$, and $B_t$ is ...
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43 views

Why is $\{\omega \in \Omega:t\rightarrow X_t(\omega) \text{ is continuous on } [0,T]\}$ $\mathcal{F}_T^X$-measurable for a cadlag process?

I'm try to solving the following exercise: Let $X=(X_t)_{t\geq0}$ be a real valued stochastic process such that, for all $\omega\in \Omega$ the map $t\rightarrow X_t(\omega)$ is right continuous and ...