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

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2
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0answers
27 views

Counterintuitive result on quadratic variation

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then ...
0
votes
0answers
35 views

Gaussian Random Walk

Suppose $\{Z_{i}\}_{i=1,2,\ldots}$ are iid normally distributed with mean zero and positive variance random variables ($\sim N(0,\sigma^{2})$). Write the discrete time stochastic process as: ...
0
votes
0answers
12 views

Can a Brownian motion be defined for negative time?

I was just looking at fractional brownian motions on this page. The definition of $B_H(t)$ requires integrating on a negative time domain on $dB(t)$ where $B(t)$ is a Brownian motion! Could you please ...
1
vote
1answer
32 views

self financing strategy

how could one prove the following proposition from stochastic calculus applied to finance? Proposition : Let $\Phi$ a trading strategy. Then, $\Phi$ is self financing if and only if ...
0
votes
1answer
8 views

Tower property of conditional expectations - Application Question

How could I prove the folowing using the tower property of conditional expectations? ...
1
vote
2answers
23 views

Reference for stochastic calculus with jumps

All the standard books I know on stochastic calculus work almost exclusively with continuous martingales. What are the standard references for the general theory (with jumps)?
3
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0answers
28 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the probability distribution function of the ...
0
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0answers
25 views

Why $d\langle X \rangle_t = d X_t dX_t$ if $X_t$ is a semimartingale?

Following this question, proving the equivalence between equation $(1)$ and $(2)$, I deduced that $$d\langle X \rangle_t = d X_t dX_t$$ (where $X_t$ was an Ito's process, hence a semimartingale). I ...
0
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0answers
27 views

How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, ...
5
votes
1answer
61 views

$dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
3
votes
1answer
1k 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 ...
1
vote
1answer
20 views

An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
0
votes
0answers
12 views

Manipulating a log normal variable

I am wondering given: and is it possible to state: $$\text{Jdq}_t s_t-\text{dq}_t s_t=\text{dq}_t \log (J) s_t$$ And if it is the case can we show how this argument is done?
2
votes
1answer
54 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. ...
0
votes
0answers
13 views

Brownian Motion Hitting Time?

So my problem is the following. Take a 2D Brownian motion $(W_{1t}, W_{2t})$ such that it starts at $(1,1)$. With probability 1 it will hit the x-axis. What is the probability that it will hit the ...
0
votes
0answers
23 views

Does Ito isometry hold pointwise?

It is known that the stochastic integral satisfies the following property $$ \mathbb{E}\left[\left\langle \int_0^{\cdot}X(s)\,dM(s) \right\rangle_t\right]= \mathbb{E}\left[ \int_0^t X^2(s) \, ...
2
votes
0answers
50 views

Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
0
votes
0answers
21 views

How to evaluate the expectation of the exponential of reflected brownian motion

How do you compute this expectation $\mathbb{E} \left [ e^{\varepsilon|W_t|} \right] $ where $W_t$ is a Brownian Motion Do I need to expand the absolute value? Can I use the standard Taylor series ...
0
votes
1answer
15 views

Independent stochastic processes

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks
0
votes
1answer
29 views

Exponential martingale and change of measure

$\newcommand{\qq}{\mathbb{Q}}\newcommand{\ee}{\mathbb{E}}$ Denote $Z_t= \exp( \theta B_t - \frac{1}{2}\theta^2t )$ Given the probability measure $\qq(A) := \ee[ Z_t \mathbb{1}_A ]$ I must ...
0
votes
0answers
17 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
0
votes
2answers
69 views

Linear non-homogenous SDE

I'm struggling to understand how to resolve the following SDE: $$dX(t)=(\sin(t)-2X(t)) dt + (1+X(t))dB(t)$$ I understand that I should use the Ito formula but I have no idea how the $F(X(t),t)$ should ...
1
vote
1answer
35 views

A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and ...
3
votes
0answers
29 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space ...
2
votes
1answer
46 views

Rigorous meaning of conditional expectation in Feynman-Kac formula/in general

In Wikipedia https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula and plenty of other books/sources, Feynman-Kac formula is expressed in a form of the type $$f(t,x)=E(f(T,X_T)\mid X_t=x)$$ What ...
1
vote
1answer
51 views

How to solve a SDE defined via a Markov Process?

I have to solve the following SDE. $$ \mathrm dY_t= f(X_t) \mathrm dt, \tag{1} $$ where $X_t$ is a two-state Markov Process possesses states $a$ and $b$. Moreover, I would like to solve $$ \mathrm ...
1
vote
0answers
16 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
1
vote
1answer
66 views

Change stochastic integral on a null set

This is a bit vague but something I have been wondering about. Consider a Lebesgue integral $\int f \;\text{d}\mu$ - then we know that we can change $f$ on a $\mu$-null set without changing the ...
0
votes
0answers
4 views

Testing numerical solvers for multidimensional SDEs with non-commutative noise

I am implementing the multidimensional Milstein scheme to solve SDEs. I am trying to test the solver on benchmark equations but I cannot find an analytic solution for the case with a 2-d state vector ...
0
votes
0answers
16 views

Proving bounded expectation and switching order of stochastic integrals

I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the ...
0
votes
0answers
37 views

Pathwise definition of stochastic integral consistent with the Ito isometry

My definition of the stochastic integral is that it it is the image of the Ito isometry. Now we also prove Ito's formula and then apply it pathwise and get a pathwise definition in some cases. But in ...
1
vote
0answers
51 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
4
votes
1answer
29 views

Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$ \int_0^T W(t) dt = \int_0^T (T-t) dW(t) $$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ ...
2
votes
0answers
18 views

Distribution MSR

We have $Y_i = \beta_0 +\beta_1(X_i -\bar X )+\epsilon_i$ for i=1,...,n $$\epsilon_i \sim N(0,\sigma^2)$$ We know that $SSR= Y^T P_xY - n\bar Y^2=Y^T (P_x -n^{-1} J_nJ_n^T)Y$ ...
1
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0answers
55 views

Kolmogorov's continuity criterion Ornstein-Uhlenbeck process

Let $(X_t, t \in \mathbb{R})$ be an Ornstein-Uhlenbeck process, i.e. $X_t$ is defined by $$X_t = \sigma \int_{-\infty}^t e^{-\theta(t-s)} dW_s$$ for $t \in \mathbb{R}$ for parameters $\theta, \sigma ...
0
votes
0answers
29 views

Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} ...
2
votes
1answer
30 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...
3
votes
0answers
80 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
0
votes
0answers
31 views

Can we apply the Itō formula to find an expression for ${\rm d}\eta_t(X_t)$ where ${\rm d}X_t=v_t(X_t){\rm d}t+\xi_t(X_t){\rm d}B_t$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(B_t)_{t\ge 0}$ be a $d$-dimensional $\mathcal F$-Brownian motion ...
1
vote
1answer
36 views

Law of total expectation well-defined?

Wikipedia states that this is a special case of the law of total expectation click me. Given a partition $A_1,...,A_n$ of the outcome space, we have for a random variable $X$ that ...
1
vote
0answers
41 views

Show that the solution to a stochastic differential equation is satisfied by the following

I am confused on how to get from the first statement to the second. Getting from the second statement to the third would just a simple case of substituting s=0. The solution sheet says to use ...
3
votes
1answer
52 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
0
votes
1answer
45 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
4
votes
0answers
49 views

Stochastic Integral of Particle Scattering

I have a stochastic process that describes a particle moving through a field of randomly distributed particles and undergoing scattering collisions (modeled simplistically) off of them. In its ...
2
votes
0answers
42 views

Expectation over Markov Process and discrete Ito integral

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends ...
5
votes
1answer
98 views

Help integrating the transition probability of the Brownian Motion density function.

1. Problem: Given the Brownian Motion with Drift: $$ dx = \mu \, dt+\sigma \, dW $$ It can be shown that the transition density function is the following: $$ p(x, t) = \frac{e^{-\frac{(x-\mu ...
1
vote
1answer
39 views

What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for ...
6
votes
1answer
110 views

Given Q and $X_t$ is Q-Brownian, find $\frac{dQ}{dP}$ / Uniqueness of Brownian motion or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t ...
2
votes
1answer
21 views

Strong solution SDE - independence of initial conditiion

I am currently studying the existence and uniqueness of strong solutions of SDEs of type $$\left[\begin{array}{l} \, dX_t=\mu(t,X_t)\,\mathrm{d}t+\sigma(t,X_t)\,\mathrm{d}W_t\\ ...
0
votes
0answers
47 views

Does this make sense?

Can I write this? Let $W_s$ be a Wiener process and let $x_s$ be a stochastic square integrable process adapted to the filtration generated by $W$. Is such an expectation nonsensical? And if not, how ...