0
votes
1answer
18 views

Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
2
votes
1answer
33 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
23 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 ...
2
votes
1answer
86 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
1
vote
0answers
34 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
29 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 ...
1
vote
1answer
94 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
2
votes
1answer
63 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
4
votes
1answer
160 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 ...
2
votes
1answer
53 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 ...
1
vote
1answer
35 views

Prove a P Martingale

If: $$ \sigma_t $$ is a bounded function of both time and sample path, show that: $$ dX_t=\sigma_tX_tdW_t $$ is a P Martingale. *Does this question make sense, that is, should the question be: is ...
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 $M_t^n$ ...
0
votes
1answer
54 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
0
votes
1answer
39 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) = ...
0
votes
1answer
49 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
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?
0
votes
0answers
44 views

Martingale and Stochastic equation

Using the Ito formula, I can show that the martingale $$ Z_{t}=\frac{1}{\sqrt{1-t}}\exp -\frac{B_t^2}{2(1-t)}\qquad 0\leq t<1 $$ admits the following differential $$ dZ_t=-\frac{B_t}{1-t}Z_tdB_t. ...
4
votes
0answers
91 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 ...
0
votes
1answer
41 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
0answers
65 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 ...
0
votes
1answer
43 views

Prove that a process is a martingale

Let $W_t$ be a Wiener process, and let $N_t$ be a Poisson process with intensity $\lambda$. We define a process $Z_t = \lambda Wt^2 − N_t$ Prove that the process $Z_t$ is a martingale
0
votes
0answers
31 views

Defining the Radon-Nikodym as a solution to an SDE

Can someone please clarify this to me: If I have the Radon-Nikodym $L_t=\frac{dQ}{dP}$, on $\mathcal{F}_t$, then I know that $L_t$ is a non-negative P-martingale. So in many textbooks they say it is ...
0
votes
0answers
137 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 ...
1
vote
1answer
62 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
0
votes
0answers
82 views

Jump diffusion process with sum of Poisson processes a martingale?

Hi Mathematics community, assume you have dynamics of a jump diffusion process consisting of a Brownian motion and a sum of compensated (not necessarily independent) Poisson processes, i.e. ...
0
votes
1answer
51 views

Show the following is Local Martingale

$X_t$ bessel square process which satisfies $$\mathop{dx_t}= 2(a+1) \mathop{dt} +2 \sqrt{x_t} \mathop{dB_t}$$ and $u$ is a function which satisfies $x^2 u'' +x u' -u(a^2 + b x^{2p+2})= 0$. How can I ...
4
votes
1answer
109 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...
1
vote
2answers
92 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
1
vote
1answer
139 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
0
votes
0answers
51 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
1
vote
1answer
82 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
3
votes
0answers
43 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
1
vote
0answers
48 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
0
votes
1answer
43 views

sup of a submartingale until $t$

My problem: For any submartingale $(X_s)_{s\geq0}$ and for all $t\geq0$ show that $\sup_{s\in[0,t]}\mathbb{E}[|X_s]|]$ is a.s. finite. What I have until now: I know that $\mathbb{E}[X_s]$ is ...
1
vote
1answer
101 views

Basic (continuous) martingale properties

I just learned about martingales in continuous time and solved some basic exercises. But unfortunately there are some seemingly easy and surely basic things I still have problems with. 1) Let ...
1
vote
0answers
104 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
5
votes
2answers
198 views

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?
1
vote
0answers
61 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
2
votes
1answer
125 views

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t - \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability ...
0
votes
1answer
130 views

If quadratic variation of a local martingale is zero then it is itself zero

Let $M$ be a local martingale, if we need it, we can assume that $M$ is continuous. We know that $\langle M\rangle =0$. This implies that $M$ and $M^2$ are local martingale. Can we conclude that ...
2
votes
1answer
133 views

Girsanov transformation and preservation of independence

If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original ...
8
votes
0answers
153 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
1
vote
1answer
708 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 ...
0
votes
1answer
74 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
1answer
105 views

Martingale inequality

Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, ...
1
vote
1answer
60 views

Discontinuous Martingales on the interval $[0,T]$

Does there exist a Martingale on continuous time $[0,T]$ such that it is discontinuous for every $t \in [0,T]$?
1
vote
1answer
160 views

Backward martingale property of quadratic variation

Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
3
votes
2answers
304 views

Can we prove directly that $M_t$ is a martingale

Suppose we define the stochastic process $$M_t:=e^{\int_0^t\phi_s dW_s -\frac{1}{2}\int_0^t\phi_s^2ds}$$ where $\phi\in L^2[0,T]$, $t\in [0,T]$. Note that $M_t$ is just the stochastic exponential of ...
1
vote
1answer
77 views

Asymptotic Behaviour of a Continuous Square Integrable Martingale

Consider $\left( M^\alpha _t\right)_{t \geq 0} \in \mathcal M_c ^2$ such that $$ \mathbb E \left\{ \sup_{0\leq s \leq t} \left |M^\alpha _s \right| ^2\right \} \leq C_\alpha t^{1-\alpha} $$ How to ...