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

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4
votes
1answer
79 views

Use of stochastic calculus outside finance?

I have noticed most of the books about stochastic calculus are targeted fo finance and derivatives. Are there any other areas outside finance where stochastic calculus is applicable?
6
votes
1answer
443 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
2
votes
1answer
26 views

What is a good book for learning Stochastic Calculus?

I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the ...
1
vote
1answer
57 views
+50

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
2
votes
2answers
161 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
1
vote
0answers
18 views

How to calculate the differential of the following stochastic integral?

Let $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ I want to compute $\mathsf dY_t$. This suggests me to consider how to find $\mathsf dY_t$ for $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ or $$Y_t=\int_t^T g(t,s)\ ...
1
vote
1answer
40 views

Understanding Quadratic Variation

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much ...
0
votes
0answers
13 views

Stopping of quadratic covariaton

I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...
2
votes
0answers
42 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
0
votes
0answers
38 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...
1
vote
1answer
25 views

Covariance of Ornstein-Uhlenbeck process

$U(t)=e^{-\mu t}W(\frac{\sigma^2e^{2\mu t}}{2\mu})$. The problem is to find $Cov[U(t),U(t+s)]$. I used the identity, $W(\frac{\sigma^2e^{2\mu t}}{2\mu})=W(\frac{\sigma^2e^{2\mu t}e^{2\mu s}}{2\mu ...
2
votes
0answers
32 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
0
votes
0answers
17 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
3
votes
0answers
19 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
2
votes
1answer
134 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
0
votes
0answers
12 views

Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
0
votes
0answers
8 views

ODE where the derivative is a function of a stochastic process

Suppose we have a linear ODE, $\frac{dv}{dt} = 1 + xv $, where the coefficient $x$ is an Ornstein-Uhlenbeck process, $dx = (x_0 - x)dt + \sigma d\omega$. Is there a way to express this ODE as an ...
0
votes
0answers
13 views

Finding a function to use for Ito's Lemma

The original problem was to show the following stochastic process has a global solution: $$dx_i = x_i\left(b_i-\sum_{j=1}^4 a_{ij}x_j \right)dt+\sigma_ix_idW_t$$ To do so, they considered the ...
2
votes
0answers
14 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
0
votes
1answer
35 views

How to find exact solution of this volterra equation?

I was working on numerical solution of this equation (by block pulse). $$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ...
2
votes
1answer
51 views

Gradient descent method with random perturbation

Suppose there is a function $f:\mathbb R^n \to \mathbb R$. One way to find a stationary value is to solve the ODE $\dot x = - \nabla f(x)$, and look at $\lim_{t\to\infty} x(t)$. However I want to ...
2
votes
0answers
27 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing about the term" normalization ". According to the Hammersley–Clifford theorem, an random $x$ can equivalently be characterized by a ...
1
vote
2answers
39 views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
0
votes
1answer
23 views

interchanging spatial integral and time integral in the Brownian context

The problem is the following My attempt is inspired in the following: Consider $$F_n(x) = \int_{-\infty}^\infty h(a) u_n(x - a)\,da $$ By Itô's formula: \begin{align} &F_n(W_t) = F_n(W_0) + ...
4
votes
1answer
75 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 ...
1
vote
1answer
51 views

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

I am working on an exercise and I am not sure how to deal with these 3 cases... For example, is $ds \, dt=0$? I know $(dt)^2=0$, but I am not sure when it is 2 different variables. And what about ...
2
votes
0answers
21 views

Stochastic exponentials

Let $X$ be a good integrator with $X_0=0$, then the process \begin{equation*} Z_t=\exp(X_t-\frac{1}{2}[X,X]_t)\prod_{0\leq s \leq t}(1+\Delta X_s) \exp(-\Delta X_s + \frac{1}{2}(\Delta X_s)^2) ...
0
votes
1answer
20 views

Doob Meyer decomposition for Super-martingales

Let $Z$ be a super-martingale with usual Doob-Meyer decomposition: $Z=M-A$. Is it true that : $A\leq M$ and therefore: $\mathbb{E}[A^2]\leq \mathbb{E}[M^2]$ ?
1
vote
2answers
30 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
1
vote
0answers
26 views

Show $Y(t)=X^{(1)}(t)-X^{(2)}(t)$ and $\lim_{t\to\infty} \mathbb{E}Y^2(t)=0$ , for $dX^{(i)}=\mu X^{(i)}dt+\sigma X^{(i)}dW$

I am trying to solve this exercise which my professor has "solved" (he says what the result but not how he gets it). This is in a problem sheet which is about the Euler-Maruyama scheme. What I get ...
0
votes
0answers
48 views

Ito's Lemma / Expected Value / Variance - Mathematical Finance

Assume an asset price $S_t$ follows the geometric Brownian motion $$\Bbb dS_t = \mu S_t\Bbb dt + \sigma S_t\Bbb dWt,$$ where $\mu$ and $\sigma$ are constants and $r$ is the risk-free rate. ...
1
vote
1answer
21 views

Burkholder's inequality for elementary stochastic integral

An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail ...
0
votes
1answer
16 views

Show that $X_n\in\mathcal{H}$, where $\mathcal{H}:=\{h(t):h(t)\text{ is an adapted process, }\mathbb{E}[\int_0^{\infty}h^2(t)dt]<\infty\}$

I am not sure if I got this exercise right... I have 2 questions: Have I obtained the final result correctly? If so, I used Wolfram Alpha to obtain the value of the series, but how else can I obtain ...
1
vote
0answers
28 views

Differentiating Stochastic Integral

I was wondering how to write the following integral in differential form: $$\int^t_0 f(s,t)dW_s$$ where $W_s$ is a standard Brownian Motion. In my understanding, if $f(s,t)$ can be written as ...
2
votes
0answers
25 views

Generator of a stochastic process

I have a question about the generator of a stochastic process. $T>0$: fix Let $b: \mathbb{R} \to \mathbb{R}$ be a bounded measurable function. Let $\left( (X_{t})_{t \in [0,T]}, \left(P_{x} ...
1
vote
0answers
17 views

Novikov condition, martingale

I have a question about Novikov condition and martingale. $T>0$: fix. Let $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P)$ be a filtered probability space and $(B_{t})_{t ...
0
votes
1answer
16 views

Application of Ito's isometry in deduction of Wiener Ito Chaos expansion

I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ...
1
vote
1answer
29 views

Calculate $\mathbb{E}[M_{\alpha}^{p}(t)]$ for all $p>0$ and $t>0$, where $M_{\alpha}(t):=e^{\alpha W_t-\frac{\alpha^2}{2}t}$, $t\ge 0$

I am going through this solved problem but I don't understand some steps. My professor is notorious for making errors very often so don't hold back if you think he's wrong... Or if I am wrong. I am ...
3
votes
1answer
27 views

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: ...
2
votes
0answers
22 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 ...
-1
votes
0answers
39 views

polar coordinate transformation

If we have an equation $\mathcal{L_I}=\prod \mathrm{exp}\bigg(-\lambda_j \displaystyle\sum\limits_{m=1}^{\Psi_{j}}\binom{\Psi_j}{m} ...
1
vote
0answers
31 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 ...
1
vote
1answer
1k views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
2
votes
1answer
28 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 ...
0
votes
0answers
18 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 ...
1
vote
0answers
29 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 $W$ ...
3
votes
5answers
144 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 ...
0
votes
1answer
52 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 ...
1
vote
1answer
56 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
4
votes
1answer
45 views

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...