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

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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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30 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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1answer
37 views

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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23 views

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
17 views

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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24 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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42 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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40 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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17 views

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
43 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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26 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 ...
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44 views

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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14 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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22 views

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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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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23 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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10 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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20 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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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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70 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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22 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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37 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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15 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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71 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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30 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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65 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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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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93 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 ...
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103 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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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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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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21 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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29 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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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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26 views

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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43 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
31 views

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 ...