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

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14 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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21 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 $\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is $\mathcal{N}(0,t^3/3)$. ...
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1answer
46 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
33 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 ...
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9 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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26 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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15 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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13 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 ] < ...
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52 views

prove $\frac{e^{iuX_t}} {\mathrm{E}{[e^{iuX_t}]}}$ is martingale

...knowing that $X_t$ has independent increments and is adapted to its natural filtration, $u \in \mathrm{R}$ My problem is in particular how to show this process has finite mean...(can I use the ...
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1answer
29 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
227 views

How do I find the stochastic differential equation given the solution?

Homework Exercise. I've been given the question, find the stochastic differential equations satisfied by the following processes and determine which are martingales? B_t is a standard Brownian ...
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37 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
60 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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69 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
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67 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
273 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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1answer
12 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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18 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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1answer
734 views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
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1answer
37 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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53 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
567 views

Stochastic Calc

(a) Consider the process $$ \mathrm d\sqrt{v} = (\alpha - \beta\sqrt{v})\mathrm dt + \delta \mathrm dW $$ Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that $$ \mathrm dv = ...
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1answer
46 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 ...
2
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1answer
39 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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1answer
16 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 ...
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30 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
21 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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35 views

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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9 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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47 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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1answer
40 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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1answer
52 views

double area integrals over coherence functions on circles

I am having trouble showing the following, which shows up from coherence theory: $\frac{\pi b^2}{\alpha^2}(1-J_0^2(\alpha b)-J_1^2(\alpha b))=\int_0^{2\pi}\int_0^b\int_0^b r_1r_2\frac{J_1\left ...
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23 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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20 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
47 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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25 views

generator of a function (stochastic) [closed]

How do I find a generator of $$g(Y_t)=Y_t^2-10Y_t+25 \, ,$$ where $Y_t$ is a geometric BM: $$dY_t=-1Y_tdt+2Y_tdW_t \, ,$$ and $W_t$ is white noise
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1answer
27 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
23 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
30 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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1answer
38 views

Integrated Ornstein-Uhlenbeck

Suppose we have an OU process given by the stochastic differential equation $dr_t = \kappa(\theta-r_t)dt + \sigma dW_t$. I think that the distribution of $D(t,T) := \int_t^T r_s\;ds$ is normal (I ...
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16 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. ...
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1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
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1answer
21 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
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1answer
35 views

Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
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1answer
15 views

Sequence solves inequalities

Suppose $Q(D)$ is a Markov chain with state space $E= \{0,1,...\}$. Further the transition matrix of $Q(D)$ is given by: $$P_D=\begin{pmatrix} \delta_0 & \delta_1 & \delta_2 & \delta_3 ...
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1answer
30 views

upper bound for Ito integral of deterministic integrand

It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of ...
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1answer
36 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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28 views

Matlab code for higher order scheme

Can somebody help me how to generate the code for the increment $\Delta$Z in the document I have attached? I know how to generate the rest of the increments but struggling in how to generate ...
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1answer
45 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
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2answers
24 views

Deriving Geometric Brownian Motion's solution?

The Black Scholes model assumes the following underlying dynamics, known as Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...