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

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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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1answer
15 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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13 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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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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1answer
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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24 views

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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13 views

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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11 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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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)= ...
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1answer
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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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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23 views

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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39 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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17 views

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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72 views

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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36 views

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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72 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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28 views

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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21 views

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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25 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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59 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 ...
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1answer
111 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
7
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1answer
107 views

Weak convergence of stochastic integral

Consider a sequence of processes $Z_t^n$ and a procoss $Z_t$, $t\in[0,1]$ such that all $\int_0^1 Z^n dW$ and $\int_0^1 Z dW$ are martingales. Assume $$\int_0^1 Z_t^n \mathrm dW_t \xrightarrow{d} ...
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Can we find the boundary condition of a function of diffusion process?

given $dx_t=b(t,x_t)dt+\sigma(t,x_t)dW_t$, if it is in one dimensional case, one can use Feller non-explosion test to see if $x_t$ attains a particular boundary. How about $f(x_t)$, $f$ is any ...
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41 views

Canonical probability space for Brownian motion [closed]

Let $\Omega$ be the space of continuous functions $\omega: [0,T]\to \mathbb{R}^{d}$, $\mathcal{F}=\mathcal{B}(C[0,T))$ and $\mathbb{P}$ be the Wiener measure. Therefore the coordinate processes $W$ ...
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35 views

Is a martingale conditioned to return in the future a supermartingale?

Let $M_t$ be a martingale on $\mathbb{Z}^d$ and assume $M_0=0$. Let $n \in \mathbb{N}$. For any $t \leq n$, let $Q_t^n$ be distributed as $|M_t|$ conditioned on $M_n=0$, where $|M_t|$ is some norm. Is ...
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Generalization of the Ito formula

I have a question concerning Ito’s formula for semimartingales with jumps. I am familiar with Ito’s formula in the following setting: Let $X_t=X_0+M_t+A_t$ be an $\mathbb{R}^d$-valued continuous ...
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23 views

Condition for covariance of Ornstein-Uhlenbeck processes

I am considering this Ornstein-Uhlenbeck process: $$X_t=e^{-at}X_0+e^{-at}\int_{0}^{t}e^{as}\mathrm{d}W_s$$ in which, $a$ is a constant, and $W_s$ is a standard Brownian motion. Expected value of ...
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1answer
32 views

Why for a continous local martingale ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$

Why for a continous local martingale $(M_t)_{t\in \mathbb{R}_+}$ ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$ where $(B_u, u \geq 0)$ is Brownian motion . ...
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1answer
24 views

Dominant and Monotone Convergence With Expectation

I've tried dotting around this article to little avail: ...
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18 views

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$ and $E\{\mid \int_{\alpha}^{\beta}f(t)d\omega (t)\mid^2|\mathscr{F}_\alpha ...
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Solving a Stochastic PDE with two variables in time

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} ...
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45 views

Complete orthonormal system of eigenfunctions for trace-class nonnegative operator on a Hilbert space

In Da Prato/Zabczyk's book "Stochastic equations in inifinite dimension" I stumbled over the following paragraph: Let $Q$ be a trace class nonnegative operator on a Hilbert space $U$. [...] Note that ...
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1answer
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Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid \int^{\tau}_{0} f(t)d\omega(t)\mid^2=E[\int^{\tau}_{0} f^2(t)dt]$.

Suppose $f \in L^{2}_{\omega} [0, \infty]$, and $\tau$ is a stopping time such that $E[\int^{\tau}_{0} f^2(t)dt]<\infty$. Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid ...
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Moments of the integrated Bessel process

I am trying to compute the moments of the integrated and the integrated-inverse Bessel process. For simplicity, if $X_t$ is a BES$(d)$ assuming $d>2$, I am trying then to compute $$\mathbb E ...
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1answer
55 views

Proof of white noise expansion in Banach spaces

We're reading through Da Prato/Zabczyk's proof of the existence of a white noise expansion of Gaussian measures and there's a step where we are stuck: Theorem 2.12 [Da Prato/Zabczyk, Stochastic ...
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Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
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1answer
30 views

Tensor products of $L^2$ spaces

Consider a probability space, $(\Omega,\Sigma,P)$, and some arbitrary Hilbert space $H$ (in my case, a space of functions on $\mathbb{R}^d$). On an intuitive level is ...
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1answer
44 views

the convergence in probability of the stochastic integral

In Jacod's Limit Theorems for Stochastic Processes:page 47,thm 4.31 (iii) $X$ is a semimartingale , $H_n$ are predictable process converge pointwise to $H$ , and $|H^n|\le K$ , where $K$ is a locally ...
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1answer
132 views

Representation theorem for local martingales

I want to prove the following local martingale representation theorem. For the statement of the theorems to come we fix a filtered probability space $(\Omega,\mathcal{A},\mathcal{F},\mathbb{P})$ where ...
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Multiple Wiener Integral by Ito

In the context of Wiener-Ito chaos expansion, I had a look at Ito's paper "Multiple Wiener Integral", 1951. I am puzzled by his last result, theorem 5.1, that a multiple Wiener integral ...
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Are the elements of the set of adapted, almost surely increasing processes with $A_0=0$ finite variation processes?

$\mathbb{A}_c=\{A_t \mid A \text{ adapted ,a.s continuous and increasing process,} A_0=0\}$ $\mathbb{V}_C=\{V_t \mid V \text{ adapted ,a.s continuous and finite variation process,} V_0=0\}$ Is ...
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Verify a stochastic integral has normal distribution.

A well-known result is that:if $\sigma$ is a non-random process,then $$\int_0^T\sigma_t\,dW_t\sim N(0,\int_0^T\sigma_t^2\,dt)$$ ( from Shreve's "Stochastic Calculus for Finance" thm 4.4.9) by means of ...
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1answer
62 views

Why do we need optional stopping theorem?

For martingale,optional stopping theorem states: Let $(M_n)_{n\in \mathbb{N}}$ be adapted with $M_n\in L^1$ for all $n$ and if $(M_n)_{n\in \mathbb{N}}$ is a martingale, then $E[M_T]=E[M_0]$, for all ...
3
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1answer
38 views

A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If ...
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26 views

proving $C_0$ group (strong continuity on the whole $\mathbb{R}$

Let $T$ be a $C_0$-semigroup on the Banach space $E$. Then $\exists$ $M\ge 0$ and $\omega\in\mathbb{R}$ such that a) \begin{equation*} \|T(t)\|\le M e^{\omega t}\quad\forall\quad t\ge 0 ...