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

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4
votes
1answer
274 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
2
votes
1answer
61 views

Determine for which values of some parameters a stochastic integral is a Brownian motion

Let $W_t$ be a Brownian motion on $(\Omega, F, (F_t)_t, P)$. Find all values of $a$ and $b$ such that the stochastic integral $$X_t=\int_0^t a+\frac{bu}{t} \;dW_u$$ is a Brownian motion. 1)So I need ...
0
votes
0answers
93 views

Transition density of a Geometric Brownian-motion

The solution to SDE $$dS(t)=\sigma S(t)dW_t$$ is $$S(t)=S(0)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ the transition density for this martingale is $$p(S(t),t;S(0),0)=\frac{1}{S(t)\sigma \sqrt{2\pi ...
1
vote
0answers
31 views

Convergence in finite-dimensional distributions of some integral

Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ ...
1
vote
1answer
54 views

If $M_t$ is a martingale, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$

If $M_t$ is a martingale, for $0<t<T$, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$. I can think of $LHS=M_T \Bbb E \left[ \int_t^T h_s ...
2
votes
1answer
66 views

Proving the identity $P( X + Y = a)= \int_{-\infty}^{\infty} P( X + y = a)f_Y(y) \, \text{d}y $

Suppose $\lambda_1, \lambda_2, a \in \mathbb{R}$ and $X,Y$ are random variables. If it is needed, I can assume that $X$ and $Y$ are independent. I want to show, that the identity ...
2
votes
0answers
38 views

Differentiate probability max function

I have function as following $d(a,b):=pr(x-a>max{(y-b,0)})$ where a and b are constant and x and y are random variable. As this is a max function, it will have kink point hence, will not be ...
3
votes
1answer
58 views

Why is $\mathbb{P}(F\geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g$?

For random variables $F,G$ I have problems with understanding the equation $$\mathbb{P}(F \geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g, $$ where $D_G$ is the ...
0
votes
1answer
64 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
-3
votes
1answer
55 views

Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
8
votes
2answers
349 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
1
vote
2answers
48 views

Verifying $S(t)=S(0)e^{rt} + \sigma e^{rt} \int_0^t e^{-rs} dW(s) $ satisfies $dS(t) = rS(t)dt + \sigma dW(t)$

Consider the SDE $$ dS(t) = rS(t)dt + \sigma dW(t). $$ To solve this, I let $f(t,x) = xe^{-rt}$, so $\frac{\partial f}{\partial t} = -rxe^{-rt}$, $\frac{\partial f}{\partial x} = e^{-rt}$ and ...
2
votes
1answer
134 views

Quadratic Variation of a square-integrable Lévy process

I am having a problem with the following question. I have tried using the definition of square integrable martingales and quadratic variation, but just can't seem to get anywhere. Can anybody offer me ...
1
vote
1answer
49 views

Stochastic Differential Equation Question

So I'm again working on doing something similar to this paper and could use some help. In the paper they worked with the equation $N(t)[(a(t)-b(t)N(t))dt + \alpha(t)dB(t)]$. It's a normal logistic ...
0
votes
2answers
60 views

Limit of time integral of brownian motion

Can someone help explain the following, $$ \lim \limits_{t \to 0} \frac{1}{t} \int_0^t W_u\, du=\lim \limits_{t \to 0} \frac{W_0t}{t}=W_0=0\,? $$ Thanks!
1
vote
0answers
17 views

Simulation of Brownian Motion on Borel Spaces

I am studying stochastic calculus on my own, and currently stuck to the following issue. Say my probability space is $(\Omega, \mathcal F, \mathbb P)$. Now when my $\Omega$ has sequences of finite ...
7
votes
1answer
73 views

Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} ...
2
votes
2answers
225 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
0
votes
0answers
22 views

Brownian Bridging Time Series Variance

Suppose I have a time series of daily levels $(X_t)_{t\geq 0}$. I want to create Brownian Bridges between these levels, such that variance is preserved. I assume that $X_t$ diffuses as, $dX_t=\mu ...
1
vote
0answers
35 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...
2
votes
0answers
49 views

Stcochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I wjust want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
2
votes
2answers
57 views

Brownian Motion $dW_t \, dt=0$ proof!

I am facing a bit weird issue here. I am going through Shreeve book on stochastic calculus and faced the following theorem, while proving $dWdt=0$. $\sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))(t_{j+1}-t_j)$ ...
4
votes
2answers
82 views

Integral of time with respect to Brownian motion

I am trying to compute $\int_0^T t\ dB_t$ where $B$ is the standard Brownian motion. To this end I define the sequence of simple predictable functions $$ f_n = ...
0
votes
1answer
62 views

What to read after Shreve's “Stochastic calculus for finance 2”?

I am finishing the last pages of Shreve's Stochastic calculus for finance 2, and I was wondering what would be the best book to follow. I would like to go on with a book introducing more technical ...
1
vote
0answers
35 views

What is the solution to these SDP?

I am in trouble with my homework, the quesetion is to solve a pair of stochastic differential equation. $dX_t^1 = X_t^2dt + \alpha dB_t^1$ $dX_t^2 = -X_t^1dt + \beta dB_t^2$ $\alpha \ and \ \beta$ ...
0
votes
0answers
35 views

Is the variance of an Ito process strictly increasing?

Consider the Ito equation: $dX_t = f(t, X_t) dt + G(t, X_t) dW_t$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in ...
0
votes
0answers
47 views

Regularity of a parabolic equation

Consider the following parabolic equation on $\mathbb{R}^d$: \begin{equation} \partial_t\mu=\mathrm{div}(b\mu) + \mathrm{div}(D\nabla\mu), \end{equation} where the drift ...
0
votes
0answers
37 views

Is this simply assuming an Ito semimartingale.

I am reading a paper where they start by assuming some process follows $$ \frac{dX_t} {X_{t-}} = \alpha_t dt + \sqrt{V_t} dW_t + \int_{x > -1} x \tilde{\mu}(dt, dx) $$ with $\alpha_t$ and $V_t$ ...
4
votes
1answer
83 views

Solve a PDE with Feynman-Kac Formula

So there is the following PDE given: $\frac{\partial}{\partial t}f(t,x) + rx\frac{\partial}{\partial x}f(t,x)+\frac{\sigma^2 x^2}{2}\frac{{\partial}^2}{\partial x^2}f(t,x) = rf(t,x)$ With boundary ...
1
vote
0answers
16 views

Which filtration do we use with the bracket process of two local martingales.

I've read the following result without proof from my lecture notes: Let $X$ and $Y$ be two continuous local martingales (on the same probability space) with reducing sequences and filtrations ...
0
votes
1answer
29 views

Calculation of quadratic covariation of stopped processes

I am stuck in computing the quadratic covariation of the following two processes: Let $0< y <r$ and let $(B_t)$ be a Brownian motion started at $y$. Let $T_0 = \inf \{ t \geq 0 : B_t = 0 \}$ ...
0
votes
1answer
85 views

Derivation of Kolmogorov Forward Equation

By Ito's formula we have that for any suitable function $v(t,x)$, $$ v(T, X_T) = v(t,X_t) + \int_t^T\left( v_s(s, X_s)+ b(s, X_s)v_x(s,X_s)+\frac{1}{2}\sigma^2(s, X_s)v_{xx}(s, X_s) ...
3
votes
2answers
183 views

Good introductory book for stochastic calculus / Itō calculus?

I am looking for recommendations of a good first book to read on stochastic calculus / Itō calculus, say at the advanced undergraduate level. Does anyone have a favorite? Thanks so much!
0
votes
0answers
27 views

Quantile of the product of two random variables

Suppose two independent random variables for whom I have enough historical data to get statistical significance, but do not fit into a normal distribution. I want to get the 0,95 quantile of the ...
2
votes
0answers
42 views

Calculating a stochastic differential

Let $f$ be a real-valued function with bounded continuous second derivative $f''$, and $w(t)$ be a Wiener process. Let $$ V(t,w(t)) = f(w(t)) - \frac{1}{2} \int_a^t f''(w(s))ds. $$ I want to apply I ...
1
vote
0answers
48 views

An application of Ito-Doob differential formula

I want to apply the following formula $$dV(t,w(t)) = \left(\frac{\partial}{\partial t}V(t,w(t)) + \frac{1}{2} \frac{\partial^2}{\partial x^2}V(t,w(t))\right)dt + \frac{\partial}{\partial ...
0
votes
0answers
44 views

Stratonovich SDE and generator in divergence form

Let $a:\mathbb{R}^d\rightarrow\mathcal{S}_{d\times d}$ be a smooth map that takes its values in the space of $d\times d$ symmetric matrices and suppose there exists a $d\times d$ matrix valued ...
1
vote
1answer
71 views

Prove that the quadratic covariation is a bilinear form

If we take $X,Y,Z$ to be square integrable martingales starting at zero, we want to show that for any $\alpha\in\mathbb{R}$ we have $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z ...
1
vote
2answers
91 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
1
vote
0answers
28 views

Preservation of the Markov Property for SDEs

Let $X$ be a continuous Markov process on $\mathbb{R}^d$ that is also a semimartingale. Let $V=(V_1,...,V_d)$ be a collection of suitably nice vector fields on $\mathbb{R}^d$ such that there exists a ...
0
votes
0answers
32 views

Expectation of Compound Poisson Process

$\mathbb{E}[e^{(\sigma-\lambda)X_t } \mathbb{1}\{X_t \geq X^*\}] $ I am not too sure how to compute the expectation of a compound Poisson process multiplied with a indicator function. The Question ...
5
votes
1answer
49 views

Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
1
vote
0answers
19 views

Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale?

Question: Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale with respect to the filtration generated by $B_t$? In order to determine whether the above expression is a ...
2
votes
1answer
47 views

What is the stochastic integral of $\frac{dW_t}{W_t}$

Does anyone know the solution to the Ito integral with the scaling factor on $dW_t$ being $\frac{1}{w_t}$? In other words what is: $\int \frac{dW_t}{W_t}$ ? It looks dangerously close to what ...
0
votes
0answers
53 views

Solving infinitesimal operator in stochastic process

I am trying to understand a notion in a paper (p. 4) about identities in stochastic processes. The author uses the following infinitesimal generator of a diffusion $Y_{t}$, $t \geq 0$: $$ ...
1
vote
1answer
63 views

Integration by parts formula for Wiener integral

Hi I need an help understanding "integration by parts" in Wiener integral. I've defined this integral as in the following: let $T=[0,t]\subset \mathbb R$ we want to define $\int_T f(s) dB_s$ where ...
-1
votes
1answer
49 views

what is the answer of this stochastic integral? [closed]

as we know "ito integral "$$\int_{0}^{t}B_sdB_s=\frac{1}{2}B_s^2-\frac{1}{2}t$$now, I am searching for the solution for this one :$$\int_{0}^{t}B_s^2dB_s$$or$$\int_{0}^{t}B_s^4dB_s$$ $B_t$ is standard ...
2
votes
1answer
92 views

Question related to Kolmogorov equations

Let $d X_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$ be an Ito diffusion. If we choose a continuously twice twice differentiable function $f$ with compact support and define $u(t,x) = E( f(X_t) | X_0 = x)$ ...
4
votes
2answers
155 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
1
vote
1answer
35 views

Time scaled polynomial Brownian Motion

I want to choose constants $a$ and $b$ such that the process $$X_t = t^aP\left(\frac{B_t}{t^b}\right)$$ is a martingale, where $B_t$ is a Brownian Motion and $P(y)$ is a polynomial of degree n. Thus ...