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

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Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
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Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ...
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2answers
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Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
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20 views

Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
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1answer
74 views

Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
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Why are functions of semimartingales again semimartingales?

I am trying to prove the Itō's lemma, and need to show that if $X$ is a semimartingale and $f$ is a $\mathcal{C}^2$-function, then $f(X_t)$ is again a semimartingale. How do I do that? I cannot see ...
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How to compute the solution of a differential equation involving Brownian local time

My problem is to compute numerically a function F. F is known to be convex and have kinks. It's also known to satisfy a "second order differential equation". Since the function is not everywhere ...
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25 views

Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: ...
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DRIFT MATRIX in Ornstein Uhlenbeck Process

The Weiner Process was unable to explain Brownian Motion and then there was the need of Ornstein-Uhlenbeck Process. The Ornstein-Uhlenbeck Process describes the Brownian Motion in the presence of ...
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Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...
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Weakening mean integral requirements of stochastic integrals.

Consider the Ito integral. It is well known that adapted and measurable processes $f(s,\omega)$ that satisfy \begin{align} E \Big[ \int_0^T \big| f(s,\cdot) \big|^2 ds \Big] < \infty \end{align} ...
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How to compute solutions to differential equations coming for Ito's lemma for convex functions

I want to solve computationally for a function $V: \mathbb{R^2} \rightarrow \mathbb{R}$ which is known to be convex. When V is $C^2$ I know the function satisfies, $$\rho V(x,z) = \max_{x} f(x) + ...
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2answers
51 views

Stochastic Exponential - Protter

I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ...
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1answer
21 views

Equivalent definitions of reflecting brownian motion

In what follows, $D$ is a precompact subset of $\mathbb{R}^n$ with say $C^3$ boundary, $\eta(a)$ is the inward unit normal vector to the boundary at $a$.I'm trying to prove the equivalence of the ...
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47 views

Semigroup associated to a Markov process

I'm studying the transition semigroup associated to a Markov Process, in particular the Hille-Yosida theorem and the Martingale Problem. In my notes I found : "If $\{T_t\}_t$ is a strongly continuous ...
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1answer
21 views

Expectation of Poisson integral for random integrand

Let $N$ be the Poisson random measure associated to a Levy process $X$, with intensity $\nu$. Furthermore, let $A$ be bounded from below and $f \in L^1(A,\nu)$ be measurable. It is well known that ...
2
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1answer
49 views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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Understanding Quadratic Variation

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much ...
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Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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2answers
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Mean Square Error of Monte Carlo

Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself a bit lost when going through a simple proof in the literature. I am working in the context of ...
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1answer
27 views

Expected Value and Variance of a GBM Function

What is the the expected value of the process $Y = X^{3}$, where X satises the SDE $$ dXt = −X_tdt + σX_tdB_t $$ $(σ > 0)$ and $X_0 = 1$ I have two different answers: 1) I know that $X_t$ is a ...
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36 views

Transformation Stratonovich to Itô SDE (for BM on a surface)

The question arises from a section to Stochastic Differential Geometry in Rogers L.C.G., Williams D. Diffusion, Markov processes and martingales. Vol.2. Itô calculus. (31.22) Brownian motion on a ...
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1answer
16 views

Expectation of Integral of Brownian Motion

I'm working through some stochastic analysis problems at the moment and I've come across a problem that is a bit tricky (to me) - does anyone know how to calculate this expecation? I'm not sure what ...
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65 views

Problem including SDE

I have following problem. Let $Y_{t}$ be an exponential Lévy Process. That is: $$Y_{t} = Y_{0}e^{X_{t}}$$ Where $X_{t}$ is Lévy process. I have a function of $Y_{t}$, $f$ :$\mathbb{R}_{+} \times ...
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62 views

Derivation of Backward Kolmogorov Equation

I'm following Kallianpur-Gopinath's textbook "Stochastic analysis and diffusion processes" to study Kolmogorov equations and I got stuck in a step of the derivation of the backward equation. In ...
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How can we deduce uniqueness for SDEs by Girsanov's theorem?

Let $\mu\in L^{\infty}(\mathbb{R};\mathbb{R})$ be a bounded deterministic function. Then my understanding is that by using Girsanov's theorem, we can deduce uniqueness (in law) for the following ...
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Cross Variation of two stochastic processes

I am currently working on a stochastic calculus exercise at the moment and I am slightly confused when it comes to finding cross variation. We are given that the process $X_t = W_t^3$ ($W_t$ is ...
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3answers
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Showing time changed brownian motion is martingale.

Let $W$ be a one dimensional Brownian motion and define, $$ X_t=W_{(\text{exp}(\beta t)-1)}\\ \hat{W}_t=\frac{1}{\sqrt{\beta}}\int_0^te^{-\frac{\beta s}{2}}dX_s $$ Show that $\hat{W}_t$ is a local ...
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1answer
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Construction on Ito Integral with Brownian Motion

I have just started learning stochastic calculus and my professor posed the following as exercises to help understand how we construct the Ito Integral. Let $B$ be a standard Brownian motion. Fix ...
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1answer
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Itô's formula yields an Itô process

In our course on stochastic analysis, we proved the following version of the one-dimensional Itô formula: Let $\{W_t\}_{t\ge 0}$ be a one-dimensional Brownian motion w.r.t. some (right-continuous and ...
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Partial differential equation involving a random process (literature advice)

In articles like this one (end of page one and page two), physicists often tend to treat a random process with discrete time and countable space set as a differentiable function (whose domains are ...
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1answer
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Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...
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Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
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1answer
39 views

Scaling in utility maximisation

If I have the wealth process $$dw_t=rw_tdt+n_tS_t(\sigma dB_t+(\mu-r)dt)-c_tdt,$$ where $n$ is number of $S_t$ and $B_t$ is Brownian motion. If we define the admissible set $A$ as follows: ...
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15 views

Image of 1-dimensional Brownian motion

I'm suppose to show that, given a function $g\in C^2(\mathbb{R})$ and a 1-dimensional Brownian motion $\{B_t\}_{t\geq 0}$, we have that $g(B_t)=g(0)+\int_0^t g'(B_s)dB_s + \frac{1}{2} \int_0^t ...
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Can these random variables be seen as products of indicator functions?

Spin-off from here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ How about using indicator functions? I was thinking that $X_n = 2^n 1_{A_1}$, but I guess ...
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1answer
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Reasoning in “Prove X is a martingale” [duplicate]

From here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ Why exactly? In retrospect, I'm not sure I really got it. I'm trying to think about it in terms of ...
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1answer
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$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$

Under what conditions does it hold that $$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$$ if we are given a stochastic process $X = (X_n)_{n \geq 0}$ on a filtered probability space $(\Omega, ...
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1answer
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Proving existence of Itō Integral

Here's an extract from some Continuous Martingales notes I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that ...
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1answer
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What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...
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1answer
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Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
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1answer
34 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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1answer
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Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
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Doleans measure for local martingales

I came across the following question in my textbook and something in it doesn't quite make sense to me. Let $M$ be a local $L^2$ martingale. Then $X,Y \in \mathcal{L}(M,\mathcal{P})$ are ...
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1answer
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An application of Itô's lemma

I found this question in a past exam for a course on Financial Economics. Given the function $f(t,x)$, let $F(t,x)$ be a function such that $∂F/∂x = f$. (a) By writing Itô’s formula in ...
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1answer
41 views

Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
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2answers
41 views

Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
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1answer
61 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...