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

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42 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 $E(X)=\sum_{i=1}^{...
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74 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 &...
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46 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 ...
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1answer
105 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 ...
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1answer
79 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 ...
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93 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 ...
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1answer
61 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 }\...
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1answer
29 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\\ X_0=\xi\end{array}\...
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44 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 $n-...
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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 ...
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53 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 ...
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50 views

Kolmogorov extension theorem

I have attached to this post a short treatment of the Kolmogorov extension theorem for measures. In the following, I did not understand what is meant by the $A$ that I circled in red. I suppose that $...
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14 views

How can I show the existence of a right continuous version to the supermartingale $\{P(L>u|\mathcal{F}_u),u \geq 0\}$?

I was reading a paper by Marc Yor for my thesis and in the statement of one of the theorem he mentions Consider the super martingale $\{P(L>u|\mathcal{F}_u),u \geq 0\}$ where $L$ is a random time ...
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35 views

Feynman-Kac for dividend stream

Suppose an asset value process $V_t$ that solves the PDE $$dV_t=\mu V_tdt+\sigma V_tdW_t \text{ with }\mu\in\mathbb{R},\sigma>0, W \text{ Brownian Motion}.$$ I want to price a dividend stream $D_T=\...
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30 views

What is the formula for a likelihood ratio $L$ that transforms martingale Geometric BM to Geometric BM with positive growth?

What is the formula for a likelihood ratio $L$ that transforms martingale Geometric BM given by $dS = \sigma S \,dW_t$ to Geometric BM with positive growth $dS = \mu S \, dt + \sigma S \, dW_t$? My ...
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34 views

Quadratic variation along a sequence of subpartitions

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ with $|\pi_n|=\max\limits_{t^n_i,t^n_{i+1}\in \pi^n}|t^n_{i+1}-t^n_i|\to_{n\to +\infty} 0$ the quadratic variation of a path $x\...
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33 views

exercise 3.3.34 from Karatza and Shreve [duplicate]

In the exercise, W is a standard, one-dimensional Brownian motion and $0 \lt T \lt \infty$. We are asked to show that $$\lim_{\beta\rightarrow\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^t e^{\beta s}...
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26 views

Is $d \langle X,Y \rangle = \langle dX,dY \rangle$ where X,Y are continous semi-martingales

Is $d \langle X,Y \rangle = \langle dX,dY \rangle$. I think the answer is yes because $ d \langle X,Y \rangle=\langle X,Y \rangle_t- \langle X,Y \rangle_s$ and $\langle dX,dY \rangle=\langle X_t-X_s,...
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22 views

Fractional moments of stochastic integrals

I want to bound the moments of stochastic integrals as $$E\left|\int_0^1 f(s)d L_s\right|^\alpha,\alpha\in[0,1],$$ where $(L_s)_{s\ge0}$ is a Lévy process with Gaussian part $\sigma^2$ and Lévy ...
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31 views

What is a “stochastically stationary solution” to a stochastic differential equation?

I was recently reading Numerical Solution of Stochastic Differential Equations by Kloeden and Platen and trying the understand the linearisation of an SDE to determine its Lyapunov exponents. However,...
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1answer
132 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 \...
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29 views

Ito rule for a given ratio and exponential

Helo, I have trouble performing the following differentiation following Ito calculus $$d(e^Z/B)$$ Given that $Z_t$ is a logarithm of a certain process and follows $$dZ=mu_zdt+sigma_zdW$$ $$dB=rBdt$$ ...
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19 views

Number of combinations for specific problem

I'm currently programming a small script and have a statistical problem: I have 24 sequences, consisting of 4 different characters and a length of 8 characters. It is necessary to group the sequences ...
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10 views

Question about the bounded requirement of the simple function of definition of stochastic integration

Let $W_t$ be one-dimensional Brownian motion, to calculate $\int_0^tW_sdW_s$ by the definition of stochastic integration, one way is to use the integration of $W^{(n)}_t=W_{[nt]/n}$ to approximate $\...
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42 views

Numerical method for SDEs

I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the ...
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41 views

Integrability condition of stochastic Fubini's theorem

This is a special case of stochastic Fubini's theorem for deterministic integrands: Let $f : [0,t] \times [0,t] \to\mathbb{R}$ be measurable. Assume that $$ \int_0^t \left( \int_0^t |f(r,s)|^2 dr \...
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18 views

Where does right continuity of the filtration play a role in the poofs of the classical results of stochastic calculus?

Suppose that I define processes in continuous time as mesurable applications from $\Omega$ to the space of continuous functions (with Borel $\sigma$-algebra under the supremum norm), does the right ...
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28 views

Mean value of stochastic process with random variable as the index

Let $\{X_t: t\geq 0\}$ be a stochastic process on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with values in $\mathbb{R}$ and $T:(\Omega,\mathcal{A},\mathbb{P})\rightarrow\mathbb{R}^{+}_0$ ...
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89 views

hitting time for a continuous time markov chain

Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confusion, and Depression according to the following transition rates when t is the time in months. They are ...
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31 views

Black Scholes partial differential equation; Derivation

I have an exam tomorrow and the issue is, my notes just really briefly mentions it. It doesn't even take a full 2 pages to mention the partial differential equation. I haven't even seen it in hand-...
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2answers
68 views

Ito integral for Brownian motion

I know that because $W_t$ is a martingale, $$E\left[\int_{0}^{T} W_t dW_t\right] = 0$$ then what should the value for this equation be: $$E\left[\int_{0}^{T} W_t^{n}dW_t\right]?$$ $n$ is the power of ...
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78 views

Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense?

Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense? Hint: calculate $E[ dY | \mathcal{F}_s]$ where $dY = Y_{s-ds} - Y_s$. This is ...
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1answer
19 views

Ito's formula; when to use one and when to use the other form

I have seen $2$ "forms" of the Ito formula which are essentially, in the end, equivalent. But my question is, having seen quite a few questions on stochastic differential equations, I am wondering ...
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29 views

Stochastic calculus for continuous time Markov chains

I have absolved a course on stochastic analysis, i.e. integrals with respect to the brownian motion. Now I know that there is a theory of stochastic calculus for diskrete matringales, however I was ...
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1answer
50 views

Check solution to the SDE $dX_t = - \mu X_t \, dt+ \sigma \, dW_t$

I get stuck in this problem. I just can't get the hang of how we need to "guess" a function first and almost everything along the process of solving depends on it; It's not entirely logical to me when ...
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1answer
33 views

Probability of exponential of brownian motion

$W_t$ is a brownian motion, I have this exponential value: $$v(t)= e^{0.00025 + 0.3W_t}$$ what's the probability that $v(1)<0.5$? By taking natural log on both size, I got $0.00025 + 0.3W_1 < ...
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13 views

geometric brownian motion start with 0

Say I have an asset that has price 0 today, and $1 at 1 year. Assume it follows GBM, what should the price be? Is the following statement correct? If the asset price follows brownian motion, the ...
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1answer
14 views

Find E(X^-1) for stochastic variable

Let $X$ be a stochastic variable with density function: $f(x)=x\exp(-x)$ if $x>0$ and $0$ otherwise. Show that $E(X^{-1} )=1$. I believe I have to integrate but is it simple $x\exp(-x)$ I ...
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22 views

Markov chain jump

It follows from applying the Markov property that if we start in some point $x \in S$ ($S$ is assumed to be finite here) where $(X_t)_{t \ge 0}$ is a Markov chain that the stopping time $\tau_x:=\inf\...
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2answers
59 views

Compute expectation of the cube of stochastic integral

I want to compute: $$\mathbb{E} \left( \left(\int_0^tudW_u \right)^3 \mid \mathcal{F_s} \right),$$ hence I write $$\int_0^t \text{as} \int_0^s + \int_s^t.$$ Then I need to compute: $$\mathbb{E} \...
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48 views

Difficult problem on Markov chains

Assume that we have a continuous time Markov chain $(X_t)$ on $\{0,1\}$ and $f(t):=P(X(l)=0 \text{ for all } l\in [0,t]|X(0)=0),$ then I want to show that under the assumption that $f'(0)$ exists, we ...
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48 views

Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
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1answer
38 views

Check process is a martingale

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a ...
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1answer
22 views

Ito's formula but not given $\mu$ and $\sigma$

I have a little question from one of my worksheets(the solution I was given was almost not even a solution, super brief). let $f(t,x)=t\cos(x)$. Use Ito's formula to calculate $df(t,W_t)$. Well, ...
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33 views

Integration wrt BM

How do I integrate: $\int_{\mathbb{R}} (S_t - K)^+ \phi(t) dt$ where $\phi$ is a normal density and $S_t$ is a geometric brownian motion? I know my answer should be $\Phi(d_1)$, where $\Phi$ is the ...
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1answer
42 views

Calculating expectation using martingales

Could anyone help me with this exercise or show me similiar example? Any help appreciated. Using the martingales $M_t^\lambda=\exp(\lambda W_t-\lambda^2t/2)$ and $N_t^\lambda=(M_t^\lambda+M_t^{-\...
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36 views

Most recent jump probability

Assume I have two independent Poisson processes with respective parameters$$ \sim\text{Poisson}(\alpha_1),\sim \text{Poisson}(\alpha_2)$$ that I observe over a time interval $[0,t].$ What is the ...
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20 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = 2sgn(y_t)\sqrt{...
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1answer
46 views

Conditional expectation w.r.t Lebesgue measure

Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})=((0,1)^{2},\mathcal{B}((0,1)^{2}),\lambda_{2})$, where $\lambda_{2}$ is the Lebesgue measure in $\Omega=(0,1)^{2}$. Then, for $\...
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0answers
35 views

Versions of Tanaka's SDE

Consider the following versions: $$dX_t=x_0+sgn(X_t)dW_t \tag1$$ $$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$ $$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$ SDE (1) is a classical example of SDE with ...