# Tagged Questions

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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### Evaluate $\mathbb{E}\left(\left[W\left(\frac{k}{n}\right)-W(t)\right]^2\right)$ for all $t\in\left(\frac{k}{n},\frac{k+1}{n}\right]$

I am trying to do a past exam paper to practice, but I don't know if I have answered this question properly... I would really appreciate it if someone could double check it. Thanks a lot! QUESTION: ...
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### Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
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### Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
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### Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
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### Using Feynman-Kac, compute the following: [closed]

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
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### Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
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### Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
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### Stationary distribution for Kolmogorov Forward Equation

Given $X_t$ which satisfies the following SDE, $$dX_t = f'(X_t)dt + \sigma dW_t$$ where f is an infinitely differentiable function, and $f'$ above is the first derivative of $f$. We know that ...
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### How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$\mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0,$$ where $a\neq 0$ and $b\geq 1/2$, is ...
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### Ito's formula and Infinitesmal generator

Consider an Ito process $$dX_t = \sigma_t dB_t$$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
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From Wikipedia the multidimensional Ito lemma is: If $\mathbf{X}_t = (X^1_t, X^2_t, \ldots, X^n_t)^T$ is a vector of Itō processes such that d\mathbf{X}_t = \boldsymbol{\mu}_t\, dt + \mathbf{G}... 0answers 33 views ### Cross variation I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals : \begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf ... 0answers 77 views ### Stochastic Integral of Simple Predictable Process is a Martingale TakeH\in S$to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and$X$a Martingale. I ... 0answers 26 views ### A Question About Probability of ratio of$\max(\cdot)$? In My field , I reached to this problem. Assumptions: Consider$x_i,\hat{x}_i$are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have$N$... 1answer 35 views ### what would be power series of$x_t = e^{\beta_t} $if$\beta_t$is a Brownian motion process? In general the power series of$e^x =1+x/1!+x^2/2!+x^3/3!+...$but because the process is random we can't apply the direct differentiation than how can i write it's power series.In the book stochastic ... 2answers 92 views ### Stochastic Exponential - Protter I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ... 1answer 74 views ### Distribution of stochastic integral w.r. to brownian motion Let$B=(B_t)_{t \geq 0}$be a standard brownian motion,$T > 0$and$f : [0,T] \rightarrow \mathbb{R}$a continuous function. I want to determine the distribution of the following integral:$\int_{...
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Consider a good integrator $X$ (semi-martingale) and the relative quadratic variation process indicated by: $Y_t:=[X,X]_t$. Why is that: $$[Y,Y]_t=0 \ \ \ \ \ and \ \ \ \ \ \ [X,Y]_t=0 \ \ ?$$ ...
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### How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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### Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$S_t = \exp\left( W_t - \frac{1}{2} t \right)$$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
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### Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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### Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): {{P}_{t}}+(r-\delta )S{{P}...
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### Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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### Prove that the Itô integral for elementary predictable processes builds a martingale

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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### Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...