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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48 views

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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1answer
43 views

Absolute continuity counterexample of a stochastic process

This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358. The setup is the following: Let $\{Z_{t}\}$ be stochastic process on a Skorokhod space $D$ and a $\...
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31 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
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1answer
55 views

Itō Integral multiplied by Riemann Integral

I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is $$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$ where $W$ ...
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39 views

Impose initial condition on partial differential equation

After solving a Fokker-Planck equation (using expansion in eigenfunctions) I have obtained the following, general solution for the probability density: \begin{equation} p(x,t) = \int_0^\infty dk~ c(k)...
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61 views

Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let $u\in\...
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227 views

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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53 views

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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1answer
69 views

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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98 views

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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1answer
41 views

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 ...
2
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1answer
55 views

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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97 views

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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0answers
46 views

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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38 views

Is the following counter example correct?

In the book "Malliavin Calculus and related topics", the author states that $\|F\|_{k,p}=((E(|F|^p)+\sum_{n=1}^k E(\|D^n F\|^p_{H^k}))^{\frac{1}{p}}$ has monotonicity property, i.e. $\|F\|_{k,p}\leq \|...
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40 views

How to apply Holder inequality to prove the following?

In the book "Malliavin Calculus and related topics", the author states that $||F||_{k,p}=((E(|F|^p)+\sum_{n=1}^k E(||D^n F||^p_{H^k}))^{\frac{1}{p}}$ has monotonicity property, i.e. $||F||_{k,p}\leq ||...
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76 views

Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: $\...
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45 views

Quadratic Variation of Stochastic Integral of Simple Predictable process

Take $H\in S$ 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 Good Integrator. ...
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47 views

How to make the following conclusion?

There is a statement as follow: $E(|X_1(t)-X_2(t)|)\leq\int_0^t \kappa[E(|X_1(t)-X_2(t)|)]ds$, where $\kappa$ is a strictly increasing concave function such that $\kappa(0)=0$ and $\int_{0+}\kappa^{-...
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130 views

Proving Ito's product rule

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}...
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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 ...
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77 views

Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ 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 ...
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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 $N$ ...
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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 ...
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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 ...
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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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1answer
117 views

Quadratic Variation of Quadratic Variation

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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115 views

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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70 views

Properties of Stochastic Differential Equations

Suppose I have an SDE of the form: $$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$ By defining $y_i = \log x_i$, I can change the SDE to: $$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$ Both $...
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1answer
46 views

Show that $\lim_{|P|\to 0}\sum_{k=0}^{n-1}\frac{W(t_{k+1})+W(t_k)}{2}\left[W(t_{k+1})-W(t_k)\right]=\frac{W^2(T)}{2}$

I have this problem which I am stuck in because it seems very obvious to me that the result is correct, but I don't know how $|P|\to 0$ can be used in the proof. Thanks a lot! QUESTION: Show that $$\...
2
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1answer
52 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
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Show $\mathbb{E}Xf(X)=m\mathbb{E}f(X)+\sigma^2\mathbb{E}f'(X)$, for any function $f$, where $X$ is a Gaussian random variable$

I have the following problem which I am struggling to solve. I have the solution, but I think I am using the formula wrong. Any help would be really appreciated, thanks a lot in advance! QUESTION: ...
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1answer
102 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets $$\...
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1answer
64 views

Expectation of a product involving Brownian motion

I would need to verify if this solution is fine. Let $W_t$ be a Brownian motion and $\lambda > 0, \text{ } \lambda \in \mathbb{R}$. Calculate $\mathbb{E} \left[W_t e^{(\lambda W_t)}\right]$. ...
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0answers
28 views

Considering the Black and Scholes model, check that $\ln(S_T)=2W_T$ in a particular case

I have the following problem with its solution, but I keep on getting it wrong. I would be really grateful if someone could please explain to me what I am doing wrong. Thanks! It is part C that I don'...
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1answer
46 views

Harmonic functions and Brownian motion

How can I prove that harmonic functions have the mean-value property using Brownian motion ${B_t}$? I know that I need to use the fact that $B_{t\wedge\tau}$ is a martingale where $\tau$ is a ...
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36 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

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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limit of quadartic variation [duplicate]

I am trying to understand why : $[\int_{0}^{t}a_s dB_s]=\int_{0}^{t}a_s^2 ds$ [] is the 2-variation process, $B$ is brownian motion in the proof I have seen they used Riemman-sums to get an ...
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44 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ $...
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88 views

Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
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66 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets $$\...
3
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83 views

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 ...
3
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1answer
95 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\...
3
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1answer
31 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation $E^...
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1answer
65 views

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 ...
4
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0answers
68 views

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}...
4
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1answer
35 views

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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1answer
49 views

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 ...
4
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0answers
127 views

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 ...