For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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2
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1answer
25 views

Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
2
votes
1answer
27 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
0
votes
0answers
20 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
3
votes
1answer
57 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
-1
votes
0answers
33 views

$\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$

Let $(B_t)_{t \ge 0}$ and $(C_t)_{t \ge 0}$ be two standardized brownian motions. Now why is $\sup_{t \ge 0} B_t$ distributed same as $\sup_{t \ge 0} C_t$? This is a result we assumed as trivial ...
9
votes
1answer
98 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
0
votes
1answer
20 views

construct a martingale process from any process [closed]

If ${Z_n, n \geq 0}$ is any sequence of integrable random variables, then ${\sum_{i=1}^{n}[Z_i-E(Z_i|Z_{i-1},...,Z_1)]}$ is a martingale relative to the sequence of $\sigma$-fields generated by $Z_i$, ...
-2
votes
0answers
51 views

about martingale

The definition about martingale process is $E(Z_{n+1}\mid \mathcal F(X_n))=Z_n$, where $\mathcal F(X_n)$ is the $\sigma$ field generated by $X_n$. My question is if $E(Z_{n+1}\mid \mathcal F(X_n) ) = ...
2
votes
1answer
32 views

Why use stopping times rather than a deterministic sequence to localise a martingale?

I am a beginner on stochastic processes I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the ...
1
vote
0answers
13 views

Strong law of large numbers for continuous time martingales?

Is there a theorem/reference that states if $M(t)$ is a martingale, then under certain mild conditions, $M(t)/t\to 0$ a.s. as $t\to\infty$?
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votes
0answers
29 views

Prove continuous stopped process $X_{T\wedge t}$ is a martingale if $X_t$ is a martingale [closed]

Looking for help proving that a continuous stopped process $X_{T\wedge t}$ is a martingale if the underlying process is a martingale. Any help is appreciated!
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votes
0answers
16 views

Convergence proof of square integrable martingale.

I was reading the convergence proof of square integrable martingale. The theorem is that it converges on the set where the limit of the increasing process is finite. The proof starts by defining a ...
1
vote
1answer
42 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
1
vote
1answer
28 views

One Martingale problem

In the setting of Kolmogorov's maximal inequality, I need to prove the following $$P(\max_{1\leq m \leq n}|S_m| \leq x) \leq \frac{(x+K)^2}{var(S_n)}$$ Hint: Use the fact that $S_n^2 -s_n^2$ is a ...
-1
votes
0answers
30 views

Hitting time vs supremum of a càdlàg process

A question on stopping times that came up while I was trying to prove Doob's inequality. Let $X$ be a càdlàg, nonnegative submartingale. Define $X^*_t = \sup_{0\le s\le t} X_s$ and, for $K \ge 0$, ...
-1
votes
1answer
31 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
1
vote
0answers
22 views

Discrete stochastic integral and optional sampling theorem

I want to prove the optional sampling theorem using the fact that discrete stochastic integrals for martingale integrators are still martingale. To prove: if $(M_t)_{t\in T}$ is a martingale and ...
0
votes
1answer
10 views

Show that for martingale and predictable process, it is not possible to gain almost surely in some step

Let $X_t, t = 0, 1,\ldots, T$ be a martingale and $V_t, t = 1,2,\ldots, T$ a predictable process, I want to show that for $t = 1,2,\ldots, T$ we have $$ V_t\cdot (X_t - X_{t-1}) \ge 0 \textrm{ ...
0
votes
0answers
14 views

probabilty of maximum of stochastic process

Given, $$ M_t=exp\left( \int_0^t f(s) dW_s - \frac{1}{2}\int_0^t f(s)^2ds \right) $$ where $W_t$ is a brownian motion. Let $Z_t=W_t-\int_0^tf(s)ds$. How do i show that the above may be used with ...
2
votes
1answer
30 views

BMO martingales

Let $(Y_t)_{t\leq 0}$ be a continuous uniformly integrable martingale. It can be shown that for any $p\geq 1$, the following two properties are equivalent : there is a constant $C$ such that for any ...
0
votes
0answers
23 views

Doob's optional sampling theorem

Say we have a right continues super-martingale $(X_t)_{t\geq 0}$ with filtration $F_t$ and a stopping time $\tau$ for which $P( \tau < \infty)=1$ why is it true to claim that $(X_{\tau\wedge ...
3
votes
3answers
63 views

Showing time changed brownian motion is martingale.

Let $W$ be a one dimensional Brownian motion and define, $$ X_t=W_{(\text{exp}(\beta t)-1)}\\ \hat{W}_t=\frac{1}{\sqrt{\beta}}\int_0^te^{-\frac{\beta s}{2}}dX_s $$ Show that $\hat{W}_t$ is a local ...
2
votes
2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
0
votes
0answers
25 views

Probability of hitting zero

Suppose time is discrete. $X_{t+1} = X_t + x_t$. $x_t$ is of continuous value, iid with mean zero and finite variance. Let initial condition $X_0>0$, how can I prove that the probability of $X_t$ ...
0
votes
2answers
62 views

Show that an expected value of a sum of random variables is finite [on hold]

Let $A_1, A_2,...$ are i. i. d. non-negative random variables, $B_1,B_2,...$ are i. i. d. non-negative variables and $A_1,B_1,A_2,B_2,...$ are mutually independent. We also know that ...
0
votes
1answer
51 views

$e^{X_t - \frac{t^3}{6}}$ is a martingale - show it [closed]

I am trying to use Ito's integral properties to prove it is a martingale, but am getting stuck in the preliminaries. More so, I wanted to confirm, do I use this formula?
2
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0answers
17 views

Discrete-Time Stochastic Calculus and Stopping Times: Resources

In my measure-theoretic probability course we covered what the professor called "discrete-time stochastic calculus". Essentially, it was a three part method for computing certain quantities such as ...
1
vote
1answer
31 views

Proof that $\left<M\right>_{S_n}$ has finite expectation

Let $M$ be a continuous martingale with $M_0=0$ and $S_n:=\inf \{t:|M_t|>n\}$. Show using Ito's Isometry that $\langle M\rangle_{S_n}$ has finite expectation for each $n\in\mathbb{N}$. I know ...
1
vote
1answer
27 views

Martingale Strategy and Infinity

Given a game in which a gambler has the opportunity to bet that a fair coin, flipped randomly, will have the outcome of heads or tails... The gambler decides to always bet on heads and double their ...
0
votes
1answer
24 views

A uniformly bounded local martingale is a martingale

I was trying to prove that A uniformly bounded local martingale is a martingale. Clearly a bounded local martingale is integrable I know how to show that a lower bounded local martingale is a ...
2
votes
1answer
17 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
1
vote
1answer
43 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
0
votes
1answer
15 views

Why are local Martingales (in general) no true Martingales

What is wrong in following argumentation? Let $M$ be a local martingale (w.r.t. the filtration $(F_t)_{t\ge0}$). And $ T_n $ the localizing sequence of stopping times, such that $ M^{Tn} $ the ...
1
vote
0answers
29 views

Showing $e^{2B_t - 2t}$ is a martingale

A process $M_t$ is a martingale, if $1(a): \space \space \mathbb{E}[M_t | \mathcal{F}_s] = M_s$ for all $s \leq t$ Or equivalently, $1(b): \space \space \mathbb{E}[M_{t+s}| \mathcal{F}_t] = M_t$ ...
0
votes
0answers
25 views

Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
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vote
0answers
49 views

Can these random variables be seen as products of indicator functions? [duplicate]

Spin-off from here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ How about using indicator functions? I was thinking that $X_n = 2^n 1_{A_1}$, but I guess ...
0
votes
1answer
37 views

Reasoning in “Prove X is a martingale” [duplicate]

From here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ Why exactly? In retrospect, I'm not sure I really got it. I'm trying to think about it in terms of ...
0
votes
1answer
32 views

Proving existence of Itō Integral

Here's an extract from some Continuous Martingales notes I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that ...
1
vote
1answer
15 views

Questions on proving a stochastic process to be a martingale

I need to prove that a stochastic process $M_{t} $to be a martingale, is it necessary and sufficient to prove that $E[M_{t}]=M_{0}$ and if so, can it be proved rigorously? Thank you!
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votes
1answer
41 views

Calculate a differenciation [closed]

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
5
votes
1answer
63 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: ...
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votes
0answers
25 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
2
votes
2answers
52 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
3
votes
1answer
30 views

Can this sequence be a martingale?

Consider the following sequence of random variables: $X_1$ has only values $0$ and $1$ with positive probability $X_2$ only $0,1,2$ $X_3$ only $0,1,2,3$. Let's stop here. Can this sequence be a ...
2
votes
0answers
33 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
0
votes
1answer
33 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
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votes
1answer
66 views

Lemme itô and Martingale [closed]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
2
votes
0answers
34 views

Why do we use an exponential Martingale for the stopping time of a BIASED random walk?

The following is a passage from the lecture notes: Let a simple random move to the right with probability $p$ and to the left with probability $q = 1 − p$. We want the probability that it hits ...
0
votes
1answer
33 views

Locally lipschitz implies zero quadratic variation? [closed]

How can I prove that a locally Lipschitz function has zero quadratic variation? Thanks.
2
votes
1answer
23 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...