Questions on the calculus of stochastic processes, or processes that have a random component.

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Evaluating Stratonovich integral from definition

$\bf 3.9.$ Suppose $f\in\mathcal V(0,T)$ and that $t\to f(t,\omega)$ is continuous for a.a. $\omega$. Then we have shown that $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ...
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56 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
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40 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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41 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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33 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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33 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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46 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
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1answer
58 views

Solve Itô integral with power

$$\int_0^t e^{Ws} W_s^r dW_s$$ where $W_s$ is Wiener process and r> in $\mathbb{Z}$ My first approach would be to use Ito's lemma, however, coming up with the function $g(t,x)$ is difficult The ...
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1answer
50 views

is $(x-6)^2$ in $C_0^2$?

My math problem involves using a theorem that requires $f(x)=(x-6)^2$ to be in $C_0^2$. I'm trying to understand what $C_0^2$ means and how to check whether a function belongs to it. The course I'm ...
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2answers
781 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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59 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
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42 views

Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
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60 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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41 views

inverse function type SDE

SDE $dX_t=-a^2\sin X_t\cos^3X_tdt+a\cos^2X_tdW_t$ with $X_0=x_0$ I think this is inverse type of SDE, refer to Itô's formula and SDE. However, I can't find the inverse funcition. My try ...
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72 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
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1answer
292 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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1answer
67 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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106 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} ...
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1answer
58 views

Help with integral (inner product of stochastic and deterministic process)

i need to calculate an integral of the form $$ X = \int_0^T w(t) \sin (\omega t) dt $$ where $w(t)$ is a stochastic normal process (white noise), $\sin(\omega t)$ is deterministic. How do I do that? ...
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56 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
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61 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
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62 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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247 views

Girsanov theorem conditions

If we have an adapted function $f(t)$ such that $\int_0^t f(s)ds\,<\infty$, then the Girsanov exponent can be defined: $$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
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1answer
62 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
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1answer
38 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
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1answer
128 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
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1answer
139 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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1answer
74 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
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1answer
53 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion. Now look at $B^x_t = x + B_t$ which is a BM starting ...
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48 views

IID implies Ergodicity

The environment space is given by $\Omega:=P^{\mathbb{Z}^{d}}$, where P contains the 2d-vectors serving as admissible transition probabilities. An Element $\omega \in \Omega$ is defined as ...
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92 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
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45 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
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1answer
123 views

Solve the linear SDE $dX_t = aX_t \, dt +(b+cX_t) \, dW_t$

I am trying to find the solution to the SDE: $$ dX_t=aX_tdt+(b+cX_t)dW_t $$ for $t\ge0$, $X_0>0$, constants $a,b,c$ Would appreciate any hints as to how to approach this using ito's formula, I'm ...
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1answer
73 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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2answers
102 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
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27 views

Stochastic differential equation of a falling body

It's well known the motion of a falling body in a constant gravity model, for high speed is given by: $$m\ddot{x}(t)=g-\beta\dot{x}(t)^2$$ where $\beta$ is he drag coefficient. In a turbolent flow we ...
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2answers
29 views

Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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1answer
86 views

Proving a Self Financing Portfolio

Question: Let $(S_t)_{t\ge 0}$ be a stock price process. Assume $u(.,.)$ satisfies the Black Scholes PDE with short rate $r=0$. Assume that under a risk neutral measure P: $$ dS_t=\sigma_tS_tdW_t $$ ...
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1answer
127 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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206 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
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131 views

Problem on Solving Stochastic Differential Equation

Let $(Xt)$ be a solution to the equation $dX_t = aX_t dt + \sqrt{(1+X_t^2)} dW_t$ where $W_t$ is a Brownian motion process at time t Let $Y = F(X_t)$ for a certain function $F$. Find $F$ for which ...
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2answers
64 views

Using Ito's Lemma with more than one brownian motion term

Question : Let $$ dY_t=c_tdt+d_tdW^1_t+e_tdW^2_t $$ Where $W^1_t,~~W^2_t$ are standard independent brownian motions. I am trying to apply Ito's formula to this, say for example trying to find ...
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1answer
33 views

Pricing a claim dependent on two stock processes

QUESTION Consider two stock processes: $$ dS^1_t=S^1_t(r\,dt+\sigma_1\,dW^1_t) $$ $$ dS^1_t=S^2_t(r\,dt+\sigma_2\,dW^2_t) $$ $$ t,S^1_0,S^2_0\ge0 $$ and $$ W^1_t,W^2_t $$ are standard independent ...
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1answer
250 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
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1answer
87 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
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44 views

Plot histogram and density function

I need to plot a histogram for the data: ...
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38 views

I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
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63 views

Black Scholes Pricing of a claim

Question: Let H(x)=1/x be the payoff function for a European style derivative security. Find a closed form expression for the price: $$ u(t,x)=e^{-r(t-t)}E[H(S_T)|S_t=x] $$ for this claim using Black ...
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1answer
187 views

Solving a Stochastic Differential Equation (SDE)

Question: Solve the stochastic differential equation: $$ dX_t=X^3_t\,dt-X^2_t\,dW_t $$ where: $$ X_0=1 $$ My Attempt: Using Ito's with: $$ f(x)=\log(x) $$ I get that: $$ ...
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1answer
48 views

Prove a P Martingale

If: $$ \sigma_t $$ is a bounded function of both time and sample path, show that: $$ dX_t=\sigma_tX_tdW_t $$ is a P Martingale. *Does this question make sense, that is, should the question be: is ...