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

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20 views

Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
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How to prove that the following process is a Martingale using Ito's formula?

I am asked to prove that $Y_t$ is a martingale where $Y_t=\exp\left(\int_0^tf(s)\,dW_s-1/2\int_0^tf(s)^2\,dt\right)$ using Ito's formula. After applying Ito's formula (I hope I made no mistake) I get ...
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13 views

Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
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17 views

Distribution of Hitting Times

I am curious about the following problem: Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard ...
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8 views

How is the form of the transition functions of a process define as the solution of a stochastic differential equation?

I´m searching for a prove that solutions of SDE are markovian and i'm trying to find (or understand) in this case (SDE) what form have the transition functions associated to this kind of process. I´ve ...
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30 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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Compound Poisson Process - calculate [on hold]

Compound Poisson Process $\{Y_t : 0 \le t \}$ with the intensity $\lambda$ there are breaks with distribution $exp(a)$, a>0. Calculate: $$P \{ Y_t >2 |N_t = 3\}$$. Any ideas? I know that $Y_t$ is ...
2
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0answers
19 views

Stock Price Dynamics correlated with Bond market returns

I am currently working on to derive the following form of the stock price dynamics: $$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$ where the ...
2
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31 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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9 views

Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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1answer
27 views

$L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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1answer
12 views

Proving that $X^2- [X]$ is a local martingale given that $X$ is a cadlag locally square-integrable martingale

Suppose that $X$ is a cadlag locally square-integrable martingale. Let $[X]$ denote the quadratic variation of $X$. My textbook claims, by Ito's formula that $$ X^2 _t = X^2_0 + [X]_t + 2 ...
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1answer
21 views

symmetrical random walk P[M(n)=k]

On a symmetrical random walk, I am trying to deduce P[$M_{n}$ = k] = $(\frac{1}2)^n$ ${n \choose \frac{n+k}2}$ where n is the total number of steps and ${n \choose \frac{n+k}2}$ is the number ...
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2answers
66 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
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15 views

What kind of decomposition is $X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$?

In one of the papers I was reading for my masters thesis I came across a theorem with no references. Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ ...
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1answer
16 views

Probability Brownian motion is positive at two points

Let $0<s<t$ and $(B_r)_r$ is Brownian motion. Does anybody know what $P(B_s>0,B_t>0)$ is? I think I remember it was some $arctan$-law but I don't know the exact form. So I do not need a ...
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13 views

local martingales/ Ito formula

I have a problem with following task. Find $(A_t)_{t\ge0}$ a process of bounded variation on bounded intervals, such that $A_0=0$ and process $M_t=W_tsin(\int^t_0W_s^3dW_s)-A_t$ is a local martingale. ...
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1answer
25 views

Let $X(t) = e^{r(T-t)}/S(t)$. Find the SDE of $X(t)$ provided that $S(t)$ satisfies the BSM model.

This is the last part to a 3 part question! I am nearly done going through the questions I had difficulty with while studying, again, anyone's help would be greatly appreciated! Let $X(t) = ...
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1answer
20 views

If $S(t)$ is the stock price that satisfies BSM model in SDE form how can I derive an SDE for $S^n (t)$ for some positive integer n

If $S(t)$ is the stock price that satisfies BSM model in SDE form where $dS(t) = \mu S(t) dt + \sigma S(t) d W(t)$ where $\mu >0$ and $\sigma>0$ are two constants. how can I derive an SDE for ...
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2answers
23 views

Use Ito's Lemma to compute $d(\log S(t)$ and use this to find the closed form solution of S(t)

I am having issues with this practise problem. If someone could help me solve it that would be greatly appreciated! Let $S(t)$ be the stock price that satisfies the BSM model in SDE form $dS(t) = ...
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2answers
25 views

derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
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0answers
21 views

Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
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Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
2
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0answers
12 views

Calculate expectation of stochastic integrals

I am trying to calculate $$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$ where $(B_t)_{t\geq 0}$ is a brownian motion, $h>0$ and $\lambda > 0$ is some ...
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0answers
27 views

Distribution of “range” of a process

Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value $X_t$ and a past value $X_{t-100}$ is $$M_t ...
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1answer
32 views

What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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1answer
48 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
2
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1answer
38 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that ...
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Parallel Shifts of Forward Curve and Arbitrage Opportunities

I came accross a phrase in the Paul Glasserman, Monte Carlo Methods In Financial Engineering, page 153 : "a model in which the forward curve makes only parallel shifts admits arbitrage opportunities ...
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Is the following modification of a martingale still martingale? [on hold]

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
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17 views

How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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8 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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1answer
23 views

Martingale Poisson [closed]

Can somebody help me with working out: $$E[(N_{t}-\lambda t)^2\mid F_{s}]$$ where $N_{t}$ is a Poisson process and $F_{s}$ the $\sigma$-algebra generated by $N_{s}$, $0 \leq s < t$.
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Showing martigale, submartingale or supermartingale with log

Can somebody help me with determining whether $Z_{n}=\log(2n+S_{n})$ is a martingale, supermartingale or submartingale with $S_{n}=\sum_{i=1}^{n}X_{i}$ and the are i.i.d. random variables with $P(X_i ...
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1answer
47 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
3
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31 views

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
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49 views

Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...
2
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1answer
50 views

HJM model - Differentiation Problem

starting from the folowing SDE (HJM model): $$df(t,T)=\left(\sigma(t,T)'\int_t^T{\sigma(t,u)du}\right)dt+\sigma(t,T)'dW_t$$ And having $r(t)=f(t,t)$, I have two questions : 1) how do we obtain the ...
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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 ...
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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: ...
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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 ...
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0answers
35 views

Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 ...
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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 ...
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1answer
8 views

Tower property of conditional expectations - Application Question

How could I prove the folowing using the tower property of conditional expectations? ...
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0answers
96 views

How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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2answers
22 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)?
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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 ...
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0answers
24 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 ...
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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
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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 $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...