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

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1answer
36 views

Filtration right continuity completion

I have a question about filtration. Now fix a measurable space $(\Omega,\mathcal{M})$. Let $(\mathcal{M}_{t})_{t\in[0,\infty)}$ be a filtration on $(\Omega,\mathcal{M})$. We set \begin{eqnarray*} ...
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1answer
20 views

Theorem in finite case

I have found the following theorem in a book: Let $s \in S$ be any state of an irreducible Markov chain on state space $S=\{0,1,2,...\}$. The chain is recurrent if there exists a solution $\{ y_j ...
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1answer
39 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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0answers
10 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
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0answers
17 views

integrability condition stochastic process

Consider the finite time interval $[0,T]$ and the stochastic process $(X(t); t\leq s)$ Can the integral \begin{align} \int_{0}^{T}X(s)ds \end{align} de defined if the stochastic process $X$ is not ...
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1answer
59 views

Stopped sigma-algebra for a counting process

let $(\Omega, \mathcal{A}, P)$ be a probability space and $(N_t)_{t \geq 0}$ a right-continuous counting process with jumps of size 1, $N_0 = 0$ and canonical filtration $\mathcal{F}_t := \sigma( N_u ...
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1answer
43 views

Problem with understading “mixed” integration

Using standard notation: $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \:\:X_0=x \tag{1}$$ Now in my script it is said that if we integrate both sides, we get: $$X_t=x+\int_0^t ...
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0answers
18 views

Extension of martingale representation theorem.

It seems that the proof I am reading of the Martingale Representation Theorem, "A square integrable RCLL martingale which is adapted to the augmented filtration of a Brownian Motion must be an Ito ...
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0answers
27 views

Locate proof of Second Fundamental Theorem of Asset Pricing

Where can I find a $\textbf{rigorous}$ proof of the Second Fundamental Theorem of Asset Pricing. That is, A market is complete if and only if it has a unique risk neutral measure. Please do not ...
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1answer
34 views

Stochastic integral for local martingale

I have a question about local martingale. I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus(Second Edition). On page 146, it wrote ($M$ is a local continuous martingale ...
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1answer
34 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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1answer
26 views

Joint Quadratic variation

Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I ...
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1answer
52 views

Conditional Expectation of the Vasicek Model

The solution to$~~~~ dr_t=\alpha(\mu-r_t)dt+\sigma dW_t $ is given by: $$ r_t=r_0e^{-\alpha t} +\mu(1-e^{-\alpha t})+\sigma \int_0^t e^{-\alpha (t-s)}dW_s $$ I have been able to show that: $$ ...
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1answer
53 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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0answers
11 views

Approximation of the ito SDE using backward Euler approximation

I have the stochastic SDE $ dX_{t}=a X_{t} dt+ b X_{t} dW_{t}$ I succeeded to formulate a forward Euler approximation to approximate it but I have some problems to derive the right backward Euler ...
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0answers
23 views

Conditional Expected Value of Occurrence Time in Stochastic Process

I have a stochastic process defined by the intensity function $\lambda(t:F_t)$ where $t$ is time and $F_t$ is the filtration process. The stochastic process is self-exciting and models the occurrence ...
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1answer
26 views

Adapted and backward adapted?

I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
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1answer
70 views

Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Solve explicitly the 1-dimensional equation: $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$ I have hopelessly been guessing solutions to this. Does anyone know how to solve this ...
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2answers
58 views

Covariance of Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $ X(t)=x_{\infty}+e^{-at}(x_{0}-x_{\infty})+b \int_{0}^{t} e^{-a(t-s)} dW(s)$ where $a, b > 0 $ are given constants. I used the Itô Isometry to ...
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0answers
43 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
3
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1answer
74 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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0answers
43 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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0answers
20 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
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1answer
84 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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1answer
47 views

Stochastic integration by parts formula to prove identity between iterated integrals

if $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
3
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1answer
19 views

Independence of random variable and sum of iid random variables

Let $T_n=\sum_{i=1}^{n} X_i$ and $\{ X_i \} $ be a sequence of i.i.d. (strictly) positive random variables. So I know that $X_{n+1}$ is independent of $X_1,...,X_n$. Futher we have ...
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0answers
27 views

If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
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0answers
14 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
2
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1answer
38 views

Mean value theorem inside the Expectation

Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$ \operatorname{E} \left[ \int_0^t X_s \, ...
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1answer
81 views

Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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0answers
93 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
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1answer
19 views

Multi-dimensional Feynman Kac Theorem

I am trying to understand how to prove the multi-dimensional version of the Feynman-Kac formula. The single-dimensional version is proved on this page: en.wikipedia.org/wiki/Feynman–Kac_formula ...
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0answers
20 views

Change of variable in stochastic integral

Let $B$ be a standard Bronwian motion. Can we do a change of variable in the sense $s=\theta+h$ $$\int_{0}^{t+h}X_sdB_s=\int_{-h}^{t}X_{\theta+h}dY_\theta.$$ In this case what is the process ...
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0answers
59 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
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1answer
41 views

Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $ X_{n+1}=\max( 0 , X_n + \xi )$ $\xi $ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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0answers
58 views

Ito formula proof for bounded functions using stopping time

I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ...
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1answer
17 views

Size of the jumps in Itô-Lévy processes

I am trying to make sense of the Lévy Itô decomposition, in particular, of a note I have found regarding the size of the jumps. From the Lévy decomposition we know that any Lévy process is a ...
2
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1answer
86 views

The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
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0answers
35 views

Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
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1answer
50 views

Stochastic integral wrt the compensated Poisson random measure

I am solving the exercises in a book I have about Lévy processes ("Lévy Processes and Stochastic Calculus", Applebaum, 2003), and I cannot get my head around an exercise that seems rather simple. I ...
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1answer
22 views

Integration with respect to two different Brownian motions

Let $B$ be the standard Brownian motion. The process $W_s=B_{s+a}-B_a$ is also a Brownian motion. I just want an example of a process $X_s$ such that $$E\int_0^tX_sdB_s\neq E\int_0^tX_sdW_s.$$
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0answers
11 views

Time homogeneous asset dynamics model

I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ???
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52 views

Stochastic differential equation for a Fokker-Planck-type equation with a non-derivative term

I have something similar to a Fokker-Planck equation of the form $\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial ...
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2answers
65 views

independence two stochastic processes

being $X, Y$ two continuous processes, $\theta \in R$ $U_t=\sin{(\theta)}X_t+\cos{(\theta)}Y_t$ $V_t=\cos{(\theta)}X_t-\sin{(\theta)}Y_t$ I have to show that U and V are independent brownian ...
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0answers
25 views

Differential of stochastic process

How do I find the dynamics of $X_t=\int_0 ^t \sigma (s,t) dW_s$? It seems that the simple solution of $dX_t = \sigma(t,t)dW_t$ is not correct since I get $X_t = \int _0 ^t \sigma(s,s) dW_s$ if I ...
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0answers
41 views

Stochastic Leibniz rule Ito integral

Assume that $W$ is a Brownian motion and $f=f(t,u)$ is a function of 2 variables such that for all $t$, $f(t,\cdot)$ is adapted to the natural filtration of the Brownian motion and the Ito integral ...
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1answer
29 views

Given a process what is the stochastic differential equation it fulfils?

Given the process $X_t = (2+t+\exp(W_t))_t$ where $W_t$ is Brownian motion. How can I find the SDE that it fulfils. I am actually looking for two functions $\sigma, \tau$ such that $X_t = X_0 + ...
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1answer
24 views

How to find the dynamics of stochastic process?

We have $Y_t=e^{\int_0^t W_sds}$. How do I obtain the dynamics of $Y_t$ (i.e. $dY_t$)? It seems that we can't use Ito Lemma because $\int_0^t W_sds$ is not in the form $X_t = \int_0 ^t \sigma_s dW_s ...
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1answer
32 views

Dynamics of short rate in HJM

According to a simplified HJM framework, we have: Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion. Dynamics of forward rate: ...
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0answers
26 views

stochastic integration with respect to quadratic variation

I have been studying stochastic integral with respect to Brownian motion. At some point my professor generalized our approach such that we are able to integrate with respect to general Martingales. ...