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

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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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35 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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8 views

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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39 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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5 views

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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1answer
21 views

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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42 views

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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15 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 ...
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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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27 views

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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23 views

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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18 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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29 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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38 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
37 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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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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Critical point of contact model

Let $d\geq 2$ and let $\Pi:\mathbb{Z}^d \rightarrow \mathbb{Z}$ be given by $$\Pi(x_1, x_2, ...,x_d)=\sum_{i=1}^d x_i$$ Let $(A_t:t\geq 0)$ denote a contact process on $\mathbb{Z}^d$ with parameter ...
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1answer
13 views

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 ...
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12 views

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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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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39 views

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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8 views

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 ...
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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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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)= ...
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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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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 ...
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Expectation of exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion. Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...
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30 views

Why the measure of the Cameron-Martin space is zero?

I am studying the construction of an abstract Wiener space on "Kuo - Gaussian measures in Banach spaces". Consider an abstract Wiener space $(X, H, \mu)$ where $X$ is a real separable Banach space, ...
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Weak convergence of stochastic integral and its quadratic variation

Consider a sequence of stochastic processes $Z^{(n)}$ and a stochastic process $Z$ on $[0,1]$ such that $$ \int_0^\cdot Z^{(n)}_t dW_t \overset{d}{\longrightarrow} \int_0^\cdot Z_t dW_t $$ converge in ...
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Probability kernel for the Galton-Watson process

Consider the Galton-Watson process $\{X_n\}$ defined by $X_0=1$ and $X_{n+1}=\sum_{j=1}^{X_n}\xi_j^{(n)}$ for all $n\geq 1$, where $\{\xi_j^{(n)}:n,j\in \mathbb{N}\}$ are i.i.d. natural number-valuad ...
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Dual space $L^p$

Take a probability space $(\Omega,\mathscr{E},\mathbb{P}).$ Then it is known that $L^\infty \subset L^p \subset L^q \subset L^1$ for $\infty \ge p \ge q \ge 1.$ Let $l: L^p \rightarrow \mathbb{R}$ be ...
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What practical applications do SDEs and SPDEs have?

I will study from the probability thory to its application to stochastic differential equations with my friend. Of cource I'm looking forward to study them but would be a littel discouraging because I ...
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38 views

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ ...
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Modify process to semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...
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Is it true that $ \cap_{k \in \mathbb{N}}\mathcal{F}_{\frac{1}{k}}=\cap_{\epsilon >0} \mathcal{F}_{\epsilon}$

Is it true that $ \cap_{k \in \mathbb{N}}\mathcal{F}_{\frac{1}{k}}=\cap_{\epsilon >0} \mathcal{F}_{\epsilon}$ where $(\mathcal{F}_t)_{t \in \mathbb{R}_+}$ is a flitration(increasing family of ...
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19 views

Is $\int_s^t {B_u}^2 du$ independent of $ F_s$

In order to prove that $(B_t^4)_{t \in \mathbb{R+}}$ is a continuous semi-martingale I need to compute. $E(\int_s^t B_u^2 du |F_s)$ I was first thinking that $\int_s^t B_u^2 du$ independent of $ ...
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Transition density for reverse stochastic process

Consider an Ito process $$dX_t = a(t,X_t)dt + b(t,X_t)dW_t.$$ Assume that the functions for drift and diffusion, $a$ and $b$ are continuous and differentiable. Also assume that we know the transition ...
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23 views

Proof of the stochastic Fubini's theorem

I am trying to prove the Stochastic Fubini's theorem which is an exercise of An Introduction to Stochastic Calculus Applied to Finance. Let $(W_t)_{t\in[0,T]}$ be a Brownian motion and $H(t,s)$ has ...
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53 views

How to calculate analytical solution to fokker planck equation in finite domain?

I am looking for the solution to the following fokker planck equation. $\frac{\partial f(x,t)}{\partial t}= (k_1 x - v)\frac{\partial f(x,t)}{\partial x} + (k_2 x + D) \frac{\partial^2 ...