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

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1answer
42 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
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0answers
28 views

Asymptotic results for functions of order statistics

There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...
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0answers
24 views

Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
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0answers
16 views

What does it mean for a stochastic process to be measurable?

In my first class of Stochastic calculus the professor said that a process X is measurable if the map $(t,\omega) \mapsto X_t(\omega)$ is measurable from $(\mathbb{R^+ \times ...
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1answer
172 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
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0answers
37 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...
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30 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
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1answer
41 views

What is the Stochastic Differential Equation representation of a Poisson Process

What is the Stochastic Differential Equation representation of a Poisson Process? Can it be expressed as a function of $dW$ where $W$ is a Wiener process?
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1answer
33 views

Is exit probability monotonic in drift and diffusion coefficient?

Let $W$ be Brownian motion. Let $b_t$ and $\sigma_t$ be adapted to $\mathcal{F}_t^W$. Consider the SDE $$dx_t=b_tdt+\sigma_tdW_t.$$ Assume that $b$, $\sigma$ are such that $x$ stays non-negative. Fix ...
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1answer
90 views

Deriving Black Scholes using CAPM

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from ...
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1answer
24 views

An equality in SDE.

I read an example in Shreve: How to get the equality in the last line?
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1answer
66 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
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0answers
18 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
0
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1answer
78 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
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0answers
15 views

Empirical Intensity Function

I would like to ask for help determining what other ways are there to compute the "empirical intensity function" of a process. In essence, given that I observe the occurrences of an event in time ...
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1answer
337 views

Final Step in calculating option prices under the Heston Stochastic Volatility Model

Let: $$ \alpha = -\frac{u^2}{2}-\frac{iu}{2}+jiu\\ \beta = \lambda-\rho \eta i u - j \rho \eta\\ \gamma = \frac{\eta ^2}{2}\\ $$ where $j \in \{0,1\}$ and $i^2=-1$, $g=\frac{r_-}{r_+}$ and ...
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0answers
45 views

Ito formula for integral function

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where $W_t$ is a Wiener process. Let $$Z_t = e^{-r(T-t)} \int_{t}^{T}{h(u,S(u))du} = g(t,S)$$ where $h$ is a known function of $t$ and $S$. How can we ...
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1answer
61 views

Rewriting Diffusion Processes: Combining Independent Wiener Processes

In stochastic calculus, a rule of thumb for computations is $(dW_t)^2 = dt$ for a Wiener process $W_t$. Say we have a diffusion process $dX_t = dW^1_t + X_t dW^2_t$, with $W_t^1, W_t^2$ independent ...
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1answer
43 views

Ito Isometry - Definitions

Three different lecturers have provided three different definitions of Ito Isometry. These are: Lecturer A \begin{align*} \mathbb{E}\left[ \left(\int_{0}^{\infty} h_{s}\,dW_{s}\right)^{2} \right] = ...
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0answers
37 views

Stochastic integral in Tanaka formula

Tanaka's formula is the following result $$|B_t| = \int_0^t \text{sgn}(B_s)\, dB_s + L_t$$ I can see how to show that the stochastic integral $$M_t = \int_0^t \text{sgn}(B_s)\, dB_s$$ is a martingale ...
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2answers
96 views

Itô process and covariance of two Brownian motion

I'm a novice in studying the stochastic different equation, and didn't know whether I have describe the question correctly. Here is the question: Suppose $$\begin{array}{rcl} ...
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1answer
90 views

Feynman-Kac representation for a PDE

I have the following PDE: $$ u_t + r x u_x + \frac{\sigma^2 x^2}{2} u_{xx} + h(t,x) u_y - ru =0 $$ $$ u(x,T,y) = y $$ I wanted to check whether the following representation is correct (I used ...
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1answer
38 views

Multi variable Langevin equation

I need to solve the following system $\frac{\partial f(t)}{\partial t}=a_1 f(t)+a_1 g(t)+s_1(t) \\ \frac{\partial g(t)}{\partial t}=a_3 f(t)+a_4 g(t)+s_2(t) $ with $s_i$ being a noise, ...
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2answers
127 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
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0answers
33 views

Independent Brownian motions question

Let $B$ and $W$ be independent Brownian Motions and let $\tau$ be a stopping time of $W$. Is it true that $E[\int_0^{\tau} B_s \, dW_s] = 0\text{ ?}$ So far I have tried the following: The integral ...
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0answers
32 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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1answer
55 views

Isometric in stochastic integral.

If $\{X_t\}_{t\ge 0}$ is a simple process. i.e.$0=t_0\le t_1\le\cdots\le t_n=T$ $\exists \xi_i\in\mathcal F_{t_i}$ s.t.$X_t(\omega)=\xi_i(\omega)$ when $t\in[t_i,t_{i+1}].$ $\{W_t\}_{t\ge 0}$ is a ...
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1answer
21 views

Theorem in finite case

I have found the following theorem in a book: Let $s \in S$ be any state of an irreducible Markov chain on state space $S=\{0,1,2,...\}$. The chain is recurrent if there exists a solution $\{ y_j ...
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1answer
48 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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0answers
11 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
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0answers
26 views

integrability condition stochastic process

Consider the finite time interval $[0,T]$ and the stochastic process $(X(t); t\leq s)$ Can the integral \begin{align} \int_{0}^{T}X(s)ds \end{align} de defined if the stochastic process $X$ is not ...
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1answer
73 views

Stopped sigma-algebra for a counting process

let $(\Omega, \mathcal{A}, P)$ be a probability space and $(N_t)_{t \geq 0}$ a right-continuous counting process with jumps of size 1, $N_0 = 0$ and canonical filtration $\mathcal{F}_t := \sigma( N_u ...
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1answer
53 views

Problem with understading “mixed” integration

Using standard notation: $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \:\:X_0=x \tag{1}$$ Now in my script it is said that if we integrate both sides, we get: $$X_t=x+\int_0^t ...
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45 views

Extension of martingale representation theorem.

It seems that the proof I am reading of the Martingale Representation Theorem, "A square integrable RCLL martingale which is adapted to the augmented filtration of a Brownian Motion must be an Ito ...
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0answers
42 views

Locate proof of Second Fundamental Theorem of Asset Pricing

Where can I find a $\textbf{rigorous}$ proof of the Second Fundamental Theorem of Asset Pricing. That is, A market is complete if and only if it has a unique risk neutral measure. Please do not ...
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1answer
51 views

Stochastic integral for local martingale

I have a question about local martingale. I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus(Second Edition). On page 146, it wrote ($M$ is a local continuous martingale ...
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1answer
43 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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1answer
37 views

Joint Quadratic variation

Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I ...
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1answer
89 views

Conditional Expectation of the Vasicek Model

The solution to$~~~~ dr_t=\alpha(\mu-r_t)dt+\sigma dW_t $ is given by: $$ r_t=r_0e^{-\alpha t} +\mu(1-e^{-\alpha t})+\sigma \int_0^t e^{-\alpha (t-s)}dW_s $$ I have been able to show that: $$ ...
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1answer
57 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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0answers
19 views

Approximation of the ito SDE using backward Euler approximation

I have the stochastic SDE $ dX_{t}=a X_{t} dt+ b X_{t} dW_{t}$ I succeeded to formulate a forward Euler approximation to approximate it but I have some problems to derive the right backward Euler ...
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0answers
42 views

Conditional Expected Value of Occurrence Time in Stochastic Process

I have a stochastic process defined by the intensity function $\lambda(t:F_t)$ where $t$ is time and $F_t$ is the filtration process. The stochastic process is self-exciting and models the occurrence ...
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1answer
31 views

Adapted and backward adapted?

I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
2
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1answer
95 views

Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Solve explicitly the 1-dimensional equation: $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$ I have hopelessly been guessing solutions to this. Does anyone know how to solve this ...
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2answers
140 views

Covariance of Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $ X(t)=x_{\infty}+e^{-at}(x_{0}-x_{\infty})+b \int_{0}^{t} e^{-a(t-s)} dW(s)$ where $a, b > 0 $ are given constants. I used the Itô Isometry to ...
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0answers
49 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
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1answer
85 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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0answers
53 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
2
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0answers
37 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
1
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1answer
239 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...