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

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Ito stochastic integral vs Skorohod integral

I'm new in stochastic calculus and I'm confused about specific, but interesting topic. Skorohod integral is an extension of Ito integral for non-adapted processes, but how should I think about this ...
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29 views

Application of Stochastic Calculus to Interest Rate Model (Ito's Formula)

Above is my question. Now, the setting is of mathematical finance, but the part that I'm stuck on isn't directly related to finance, but stochastic calculus (hence posting on this site). We have the ...
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1answer
17 views

Finding a solution to the SDE of $dX_t = -2 (1-t)^{-1}X_tdt + \sqrt{2t(1-t)} dW_t$.

I am trying to find the solutions to the SDE: The solution of the following SDE $$dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2t(1-t)} dW_t, \quad X_0 = 0 $$ where $W_t$ is a Wiener process. I know that ...
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7 views

Find Itˆo diffusions $X_t = (t-2)^2_+W_2^4W_t$ in the differential form

I have $Y_t = (t-2)^2_+W_2^4W_t$. (The notation $x_+$ means the positive part of x, i.e. max(x, 0)) I try to write $Y_t$ in the differential form, that is: $$dX_t = U_tdt + V_tdW_t$$ In order to ...
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Backward stochastic differential equation

I am interested by this problem Find a solution to this backward stochastic differential equation : $\ y(t) = (ry(t) + az(t))*dt + z(t)dW_t$ with the terminal condition $y(T) = \xi$ with $\xi$ ...
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29 views

Itô diffusion and Kolmogorov backward and forward equations

For the Kolmogorov backward and forward (aka Fokker-Planck) equations to hold, and also for the Feynman-Kac formula, is it necessary for the terms in the stochastic differential equation $$ dX_t = ...
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21 views

Intuition behind “stochastic orthogonality”

Whilst doing an exercise on the Brownian Motion on a sphere I came across this identity: $$ \langle Z\times B,Z\times B\rangle = 2|Z|^2dt $$ where $\times$ denotes the cross product and $Z$ is a ...
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1answer
23 views

How many types of martingale related stochastic processes are there?

Previously I had thought that the only concepts in this direction were martingales, submartingales, and supermartingales. However, at least when discussing quadratic variation and stochastic ...
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1answer
35 views

Construct a martingale with a given distribution?

Given a random variable Y, is it possible to construct a martingale M such that $$M_1 \stackrel{D}{=} Y$$ I'm not sure how to go about proving that such an M exists under such general conditions, but ...
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24 views

Finding the mean of $X_t = \int_0^t sW_sdW_s$

For the stochastic integral, where $W_t$ is a Wiener process, I am trying to find the mean of $X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with $dWt$ has mean zero, but I ...
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1answer
25 views

Will Ito's Isometry result in $E\left(\int_0^t \cos(u)\,dB_u \int_0^t \sin(u)\, dB_u \right) = E\left(\int_0^t \cos(u) \sin(u)\, du \right)$?

If I have two integrals, $X_t = \int_0^t \cos(u)\,dB_u$and $Y_t = \int_0^t \sin(u)\, dB_u$ , where $B_u$ is a Wiener Process and I am trying to find: $$ E\left(\int_0^t \cos(u)\,dB_u \int_0^t ...
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21 views

Is this solution of the PDE correct?

http://www.impa.br/opencms/pt/ensino/downloads/mestrado_profissional_projeto_fim_curso/projetos_fim_cursos_2010/Diogo_Duarte.pdf Pages 24-25 How do they get from 2.30 to 2.32 using boundary ...
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35 views

Generating a list of numbers

A set of numbers is generated starting from $0$ in the following way: Add the current number to the resultset In a chance of 50:50, do Either add $2$ to the current number Or subtract $1$ from the ...
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1answer
42 views

Coefficient matching proof that $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\alpha^n$, where $H_n(x)$ are Hermite poly.?

Hermite polynomials can be defined as (from wikipedia): $$ H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}. $$ I am trying to show that: $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} ...
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18 views

inverse Stochastic differential equation

SDE are really new for me, so I'm sorry if this is a silly question. Let $W_t$ be a Wiener process and let $x_0$ denote the initial value of the process. If I'm correct, for $\text{d}X_t = -(\beta X_t ...
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1answer
19 views

Stochastic Different Equation

Consider the stochastic differential equation $\frac{dX_t}{X_t}=adt+bdW_t$ for the diffusion $X_t$ . The parameters $a,b$ are constant.Using Ito's lemma and suitable integration over $[0,T]$, show ...
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1answer
84 views

Discrete and continuous Girsanov

I'm trying to write a proof of the Girsanov theorem based on a discrete version of it. Discrete version Suppose that I have a random vector $X$ and two equivalent probability measures $\mathbb{P}, ...
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62 views

Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: \begin{equation} X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1}, ...
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1answer
92 views

Stopping times and hitting times for càdlàg processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable ...
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19 views

Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
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1answer
41 views

Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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42 views

Relations between Call and Put

I am trying to solve a question in finance but I am pretty much stuck and would need your help :) Suppose you know the following information about a market: Future is at 66 70 strike straddle is ...
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1answer
40 views

Martingale property for two stochastic processes

Let $(\Omega,F,P)$ be a probability space with filtration $\left\{F_{t}\right\}_{t\geq 0}$ generated by one dimensional Brownian motion $(B_{t})_{t\geq 0}$ defined on $(\Omega,F,P)$, assume that ...
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21 views

Compound Process and its compensator

I have always implicitly thought that for a counting process $N_t$, defining the compound process $$\sum_{i=1}^{N_t} X_i,$$ where $X_i$ are i.i.d, was pretty much equivalent to constructing a ...
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1answer
37 views

Itô formula question

My question is at the end of the problem statement. Solve the following stochastic differential equation. $dX_t = (\beta - \alpha X_t)dt + \sigma dB_t$, $X_0 = x_0$ where $\alpha$, $\beta$, ...
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32 views

Are Ito Integrals adapted to the Brownian Motion Filtration

Give a probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_t, P)$, we could define a 1-dim Brownian motion $W_t$ adapted to $\{\mathcal{F}_t\}_t$ with its own filtration $\mathcal{F}_t^W$. For ...
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2answers
65 views

Good book that contains stochastic integeration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading: ...
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3answers
180 views

Expectation of absolute value of stationary time series

Let $Y_t$ be a stochastic process (time series). We consider stationarity as follows: $Y_t$ is said to stationary if the mean $\mu_t = \mathbb{E}(Y_t)$ is constant (given $\mathbb{E}|Y_t|<\infty$) ...
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1answer
57 views

SDE Integration: Normal-Mean Reverting Process - Question

I am trying to figure out how a particular SDE can be integrated. The SDE is the normal mean-reverting model: $dX_t = \theta(\mu - X_t)dt + \sigma dW_t$ (1) Where $W_t - N(0,t)$. So far, I have ...
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35 views

Distribution of stochastic process and Ito's lemma

Consider an arithmetic Brownian motion $X_t$ which follows $dX_t=\mu dt+\sigma dZ_t$ where $\mu$ and $\sigma$ are constants and $r$ is the discount rate. Assume an asset price $S_t=X_t^2$. I need to ...
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1answer
33 views

Stopping Times, the $\inf$ is not a stopping time

I'm having a hard time figuring out why the infimum of a sequence of stopping times is not necessarily a stopping time itself. Indeed, the justification my book gives me is that: Given $(\mathcal ...
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19 views

Why is the solution to a Stratonovich SDE coordinate invariant while Itô SDEs are not?

This question most probably stems from my very poor understanding of manifold theory. I suppose it has something to do with the fact that the solution $Y$ to the Stratonovich SDE$$dY=f(Y)\circ dB$$ ...
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35 views

Integral w.r.t. a Martingale

Consider the stochastic integral $$ Z_t = 1+\int_0^tZ_{s^{-}}\,dX_s $$ where $X$ is a Martingale. In the textbook by Shreve (see here pages 493-493) it is said that since $Z_{s^{-}}$ is ...
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2answers
407 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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25 views

Solve this simple Linear SDE?

How do I solve the following BSDE? $$ \left\{ \begin{aligned} dX_t &=(rX_t+\theta Z_t ) \, dt + Z_t \, dW_t \\ X_T &=\xi \end{aligned} \right. $$ There appears to be nothing online about ...
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16 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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43 views

expected value problem for a normalized integral of sign

Say we have a Gausian process $X_s=\int_0^sh(x)\,dW(x)$ where $W(x)$ is a Wiener process. Now define $$Z=\frac{\int_0^1\operatorname{sign}(X_s) \, dW(s)}{\int_0^1|X_s| \, ds}$$ Intuitively we have ...
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41 views

Kummer equation, solution to find optimal value

Suppose V follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that F(V) must satisfy $$ ...
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1answer
21 views

Multi-derivative containing standard normal CDF

I've got a question about the following multi-derivative $$\frac{\text{d}^m}{\text{d}a^m} \left(e^{-2\mu a}\Phi \left(\frac{a-\mu u}{\sqrt{u}}\right)\right),$$ where $m> 0$ is an integer , ...
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1answer
40 views

Why is Backward SDE more difficult than forward SDE?

I need to explain Backward Stochastic Differential Equation (BSDE) for some non-mathematicians. The audiences are most likely familiar with ODE/PDE as physicists. One concern is probably that why ...
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26 views

Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , ...
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1answer
72 views

Following a derivation using Ito's lemma

I am trying to follow a derivation, but I get stuck could someone take me take me through the rest: We start with, $$s(t,x_t)=e^{g(t)+x_t}$$ where $$dX_t=\log (J) dq_t+\left(-\text{$\alpha ...
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1answer
45 views

How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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23 views

PDE for Brownian Bridge Expectation?

Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} ...
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2answers
33 views

Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
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1answer
31 views

Expected value of product of an ito integral and a random variable

I want to compute $$E[\int_0^t W_r dr \int_0^s W_r^2 dW_r].$$ Here $t,s$ are arbitrary. I have thought about this a lot but not sure how to proceed. I tried to apply Ito's formula to one of the ...
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1answer
39 views

Use Ito formula to compute expected value

Let $W_t$ be a standard brownian motion. I am trying to compute $E[(\int_0^t s^2 dW_S)^4]$. I applied Ito's formula and got $$t^2 W_t = \int_0^t s^2 dWs + \int_0^t 2s W_s ds$$. This gives us ...
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13 views

Deciding which version of Ito's lemma to use

The equation above is the baby version of Ito's lemma that we are given. The equation below is the generalised version of Ito's lemma that we are given. Now from what I understand, we use the ...
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1answer
18 views

Using Ito's lemma to compute a SDE

This is the version of Ito's lemma that we are given in our notes. Now I'm just not able to understand how to begin this problem and arrive at the given solution. The g(x) integral function that ...