0
votes
0answers
16 views

martingale difference

I am trying to solve the following question. {$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0 $ a.s. ) Also, for every $n$ , $E[ξ_n^2]<\infty$ Show that ...
2
votes
1answer
16 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
2
votes
0answers
18 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
4
votes
2answers
40 views

Why are stochastic processes with decreasing expected value called supermartingales?

I am curious to know why a process which has decreasing expected value is called a supermartingale. From a beginners perspective it would seem reasonable to have the following picture: ...
1
vote
1answer
59 views

Proving a property of hitting times of a simple random walk on $\mathbb{Z}$

I'm reading the course notes of a probability course about martingales currently and I'm trying to solve some of the exercises, however I'm very much stuck with the following exercise: Let $\left\{ ...
0
votes
0answers
30 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
1
vote
0answers
33 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 ...
0
votes
1answer
18 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
0
votes
1answer
28 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 ...
2
votes
1answer
56 views

Strict local martingale implies $\mathbb E_t[S_u]<S_t\ \forall t<u$

Is it true that if $S$ is a strict local martingale (i.e. it is a local martingale but not a true martingale) such that $S_t\ge 0\ \forall t$, then we have $$\mathbb E_t[S_u]<S_t\quad \forall ...
5
votes
1answer
61 views

Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
2
votes
0answers
25 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
5
votes
1answer
159 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 ...
0
votes
0answers
24 views

Decomposition of noisy measurements

What can be a good intuition behind decomposing a sequence $\{Y_n\}$ of noisy measurements (i.e. random variables) into two parts: one unpredictable and the other depending on the past. $$Y_n = ...
2
votes
1answer
52 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 ...
4
votes
1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
3
votes
1answer
27 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
2
votes
2answers
39 views

Making a non-Martingale process a Martingale

Stuck on this question for a very long time: was wondering if any kind soul could help me out: Suppose $B_t$ is a standard Brownian Motion under measure P. Question: Create a martingale process that ...
2
votes
1answer
31 views

Is the following Markov Chain a martingale?

Say I have a finite, ergodic Markov chain with states ${0,1,2,3}$ and with the following transition matrix: $$\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 &0\\ \frac{1}{10} & ...
0
votes
1answer
37 views

Question on complex valued local martingales

So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = ...
2
votes
1answer
35 views

Is the sum of two compensated poisson processes always a martingale?

Let $M^{1}_t=N^{1}_t-t\lambda^{1}$ and $M^{2}_t=N^{2}_t-t\lambda^{2}$ be two compensated poisson processes, where $\lambda^{1}$ and $\lambda^{2}$ are the constant intensities of $N^{1}_t$ and ...
2
votes
1answer
42 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
1
vote
1answer
76 views

Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
1
vote
1answer
59 views

supremum and expectation of a martingale

Let $X_{t}$ a right continuous $\textbf{F}_{t}$ martingale and $\textbf{F}_{t}$ satisfying the usual condition Show that $ \sup_{t\geq 0}\textbf{E}(X^{2}_{t})<\infty$. I know that $X^{2}_{t}$ is ...
0
votes
1answer
64 views

Ornstein-Uhlenbeck processs: Markov, but not martingale?

I'm puzzled about properties of the Ornstein-Uhlenbeck process, given by the Itō integral $$ X_t = x e^{-\lambda t} + \sigma \int_0^t e^{-\lambda(t-s)} d W_s \,. $$ I compute that $\{X_t\}$ is not ...
2
votes
1answer
40 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
0
votes
0answers
112 views

understanding submartingale proof with discrete state space

I am reading a text about branching markov chains: My question is about the first half of page 8 where $Q(t)$ is proven to be a submartingale. Briefly the used notation: $t$ is discrete time, $n(t)$ ...
1
vote
0answers
41 views

Stochastic Differential Equation- When martingale?

Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others?
4
votes
0answers
89 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
votes
1answer
62 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
0
votes
1answer
34 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
0
votes
1answer
43 views

Squared Poisson Martingale

I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for ...
0
votes
0answers
64 views

Finding dynamics of a dividend paying stock under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
2
votes
0answers
142 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
1
vote
1answer
77 views

Is this process a martingale

I was solving some practice problems in stochastics and faced the following exercise: Given Brownian motion $W(t)$ and a stochastic process $B(t)$ defined as: $$B(t) = \begin{cases} W(t), & ...
1
vote
1answer
65 views

Find a function f(t) such that Y is a martingale

Let $(X_t)$ be a process with independent increments such that $X_0=0$ and $E(X_t)=0$ Let $F_t$ be a natural filtration of $X_t$ Let $a$ and $b$ be arbitrary real numbers and let $(Y_t)$ be a random ...
3
votes
0answers
108 views

Levy's extension of the Borel-Cantelli Lemmas

Following is the statement and proof of Levy's extension of the Borel-Cantelli Lemmas, as given in Williams' "Probability with Martingales" (1991), in section 12.15 on page 124. I understand most of ...
1
vote
1answer
40 views

Infinite oscillation of random signs

Suppose that $\left(a_n\right)$ is a sequence of real numbers and that $\left(\varepsilon_n\right)$ is a sequence of IID RVs with $$P\left(\varepsilon_n = \pm 1\right) = \frac{1}{2}$$ According to ...
0
votes
0answers
29 views

square integrable martingale gaussian conditionally on its quadratic variation

First of all, I recall that if $M = (M_t)_{t \in [0,T]}$ is a cadlag square-integrable martingale, its quadratic variation process $[M]$ is defined as the unique cad increasing process such that ...
0
votes
2answers
94 views

Find a parameter for which a process is martingale

Find $\beta \in \mathbb{R}$ for which $$2W_t^3+\beta tW_t$$ is a martingale, where $W_t$ is standard Wiener process. My attempt: $$E(2W_t^3+\beta tW_t|F_s)=2E(W_t^3|F_s)+\beta ...
2
votes
1answer
50 views

Showing that a nonnegative integer-valued random variable is NOT a stopping time

Suppose that $\left(A_n\right)$ is an adapted process, and that $B\in\mathcal{B}$. Let $L = \sup\left\{n:n\leq10;A_n\in B\right\}$, $\sup\left(\emptyset\right)=0$. Convince yourself that $L$ is NOT ...
3
votes
1answer
65 views

The branching process $\mu^{-n}Z_n = \mu^{-n}\sum_{k=1}^{Z_n{-1}}X_{n,k}$ is a martingale

Let $\{X_{n,k} : n,k \geq 1\}$ be a collection of i.i.d. $\mathbb{Z}_+$-random variables with finite variance $\sigma^2 > 0$ and mean $\mu > 0$. Define $(Z_n)_{n\geq 0}$ recursively by ...
0
votes
1answer
42 views

Bilinearity of quadratic variation

Fix a filtered probability space satisfying the usual conditions. Let $\mathcal{M}^2_0$ be the vector space of cadlag martingales null at $0$ bounded in $L^2$. We state without proof the following ...
0
votes
1answer
38 views

product of martingales bounded in $L^2$

Let $(M_t)_t$ and $(M_t)_t$ be two càdlàg martingales on the same filtered probability space. We know that $M_{\infty}$ and $N_{\infty}$ are orthogonal in $L^2$. Is it true that $(M_t N_t)_t$ is a ...
2
votes
1answer
148 views

What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
3
votes
1answer
97 views

Continuous local martingale of finite variation is constant

Is a continuous local martingale $M$ of finite variation constant? We know that there exists a sequence of stopping times $T_n\nearrow \infty$ a.s. as $n\to\infty$ such that the stopped process ...
0
votes
0answers
131 views

Proving the martingale property of stochastic exponentials of pure jump processes

I am playing with different versions of compound-Poisson like processes with regime-switching features. Then I take stochastic exponentials of these to define a change of measure process. However, how ...
2
votes
0answers
147 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
2
votes
1answer
202 views

Conditional Expectation of Poisson Process

I have a Poisson Process with stationary and independent increments. Therefore I know: $$P(N_T - N_t = r) = \dfrac{\exp(-\lambda(T-t))(\lambda(T-t))^r}{r!} \mbox{ where } T>t.$$ Now suppose I am ...
3
votes
0answers
64 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...