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

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9 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
4
votes
0answers
29 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
3
votes
0answers
14 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
1
vote
0answers
37 views

Girsanov theorem for Ito diffusion process

I am getting confused about some important point of Girsanov theorem used for diffusion process. Starting with the diffusion $$dX_t=a(X_t)dt+b(X_t)dW_t$$ where $W_t$ is a P-Brownian motion. One can ...
0
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0answers
25 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
-2
votes
1answer
24 views

How to evaluate a stochastic integral

By taking a=0 and b=1 (basically that s=b), how would you work out the stochastic integral of $B^2$ I have the step $$df=d(s^3)=3Bdt+3B^2dt$$ Where does this step come from?
2
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0answers
25 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
0
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0answers
5 views

Distribution of absolute sqaured variable [on hold]

Given $P = \vert r \vert ^2$, where $P\sim N(\mu,\sigma)$, $r \in \mathbb{C}$ and $\vert * \vert $ denotes the absolute value of $*$, what is the distribution of $r$? If there is a solution to the ...
1
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0answers
16 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
1
vote
1answer
21 views

Applying Ito's formula to a complicated expression

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t ...
0
votes
0answers
12 views

local martingales and dividend processes

Consider a $d+1$ asset, continuous-time model where asset $0$ is a riskless numeraire. Assume that the asset prices are modelled by a $(d+1)-$dimensional Ito process (B,S). Further, let $D$ be the ...
0
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0answers
15 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
0
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0answers
19 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
0
votes
0answers
15 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
0
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0answers
29 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
0
votes
0answers
22 views

Deciding if a measure is dominated by the Lebesgue measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
1
vote
1answer
27 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
5
votes
1answer
46 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
1
vote
1answer
31 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
2
votes
1answer
31 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
0
votes
0answers
18 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
votes
1answer
41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
1
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1answer
27 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
0
votes
2answers
24 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
0
votes
0answers
30 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
1
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0answers
19 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
0
votes
1answer
34 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
0
votes
0answers
30 views

Fail of reverse implication of completeness.

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
1
vote
0answers
27 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
0
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0answers
19 views

Definition of Simple Predictable Process

I am reading Protter's book "Stochastic Integration and Differential Equations". He (page 51) defines $H$ to be a simple predictable processes if it has a representation ...
2
votes
1answer
50 views

Completeness of a statistic. Implication of equivalent probability classes [duplicate]

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
0
votes
0answers
17 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
1
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0answers
70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
0
votes
1answer
10 views

What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion?

What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion? $B_t$ is the standard Brownian Motion, what's $\operatorname{Cov}(B_t,B_{t^2})$?
1
vote
1answer
24 views

What is the difference between “filtration for a Brownian motion” and “filtration generated by a Brownian motion”?

I'm reading Shreve's book "Stochastic Calculus for Finance: Vol II". In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: "The assumption that the filtration ...
0
votes
1answer
19 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
0
votes
1answer
69 views

Expected value and variance of a stochastic process

Having trouble finding expected value and variance of a stochastic process defined by SDE: $dX_{t} = a X_{t} dt + b dB_{t}$ $X_0 = x$, $a$ and $b$ are constant values, $B_t$~$N(0,t)$ Thank you for ...
0
votes
0answers
22 views

How to identify the future distribution of a stochastic variable from its SDE

I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE. Would you study it empirically (like generating Monte-Carlo ...
1
vote
1answer
21 views

Find a process $f=f(t,W_t)$ such that another process is a martingale

Find a process $f=f(t,W_t)$ such that process: $$X_t=\exp(W_t^2-2tW_t^2)+\int_0^tf(s,W_s)ds$$ is a martingale. Justify the fact that $X_t$ is martingale. I think I should find a process such that ...
1
vote
1answer
18 views

A question on integration wr.t to a local martingale

In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ ...
1
vote
1answer
39 views

Find conditional expectation of $\int_1^2W_t^2dt$ with respect to $F_1$

$$\mathbb{E}(\int_1^2W_t^2dt|F_1)=\int_1^2\mathbb{E}((W_t-W_1+W_1)^2|F_1)dt=\int_1^2\mathbb{E}((W_t-W_1)^2|F_1)dt+\int_1^2 2\mathbb{E}((W_t-W_1)W_1|F_1)dt+\int_1^2 ...
1
vote
2answers
36 views

Calculate conditional expectation of integral

I have to calculate $$\mathbb{E}\left(\int_1^2 (t^2W_t+t^3 )\,dt\mid F_1\right)$$ My attempt: $$\int_1^2 (t^2W_t+t^3 )\,dt=\int_1^2t^2W_t\,dt+\frac{15}{4}$$ Now I will focus on: ...
0
votes
1answer
28 views

partial derivative of $f(X(t),t)$ with respect to $t$

Suppose that $f(x,t) = x^2$. Clearly, $\frac{\partial f}{\partial t} = 0$. However, let us now consider $f(X(t),t) = X(t)^2$. The book I am reading claims that $\frac{\partial f}{\partial t}(X(t),t) ...
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votes
0answers
75 views

Find the relation between 2 stochastic integral

$g(s,t)(\omega)$ is an adapted stochastic process on $\mathbb R^2$ define: $$X=\int_0^1\int_0^1g(s,t) \,dW_s\,dt$$ $$Y=\int_0^1\int_0^1g(s,t) \,dt\,dW_s$$ Could we conclude that "$X=Y$ a.s"? I ...
3
votes
0answers
25 views

Regarding proof of converse to Girsanovs theorem

This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space ...
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0answers
67 views

How to derive this step in a book called Brownian motion calculus?

How to derive the step which result in the magnitude of slope of the path in section 1.8.2? I know it is not the definition itself.
1
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1answer
48 views

Conditional likelihood of continuously-combounded returns

The simplest possible asset pricing model ist the geometric brownian motion for asset price. Here the price $S_t$ solve the familar $$dS_t = (\mu +0.5 \sigma^2)S_t \, dt + \sigma S_t \, ...
0
votes
1answer
16 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
0
votes
0answers
19 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
0
votes
0answers
17 views

Stochastic Calculus Research topics

If I just finished taking a Stochastic Calculus course and wanted to do research on it, what are some hot topics currently going on in this area? I know there's a lot but if I wanted to investigate ...