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

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1answer
26 views

Why is $\mathbb{P}(f \geq g) = \int_{\mathbb{R}} \mathbb{P}(f \geq t | g=t) \, D_g(t) \text{d}t$?

For random variables $f,g$ I have problems with understanding the equation $$\mathbb{P}(f \geq g) = \int_{\mathbb{R}} \mathbb{P}(f \geq t | g=t) \, D_g(t) \text{d}t, $$ where $D_g$ is the density ...
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0answers
10 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let W be a one-dimensional standard Brownian motion de fined on a filtered probability space (Omega; F; P). Using It^o's formula, find the ...
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0answers
12 views

conditional Variance and expectation of the stochastic differential equation (Q)

Let $W_t$ be a Brownian motion with corresponding filtration $\lbrace F_{t} \rbrace_{t\succeq 0} $.Use Itô's Lemma to obtain appropriate stochastic differential equations for the following processes, ...
-5
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1answer
29 views

Variance and expectation of the stochastic intergal [on hold]

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} ...
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0answers
54 views
+500

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
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2answers
31 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
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1answer
60 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 ...
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0answers
24 views

Calculus vs Analysis; When applying for a PhD involving stochastics, what do I say is my proposed field of research? [on hold]

When applying for a PhD involving stochastics, what do I say is my intended field of research? Do I say Stochastic Calculus? Stochastic Analysis? (Stochastic Processes?) Is there a difference? If so, ...
1
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1answer
24 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 ...
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2answers
44 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!
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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
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1answer
59 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) = {} ...
1
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1answer
68 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 ...
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0answers
16 views

The differential of the trace of two random matrices

I have two random matrices evolving in time, $X_{t}$ and $Y_{t}$. I know that $dX_{t} = X_{t}Adt + X_{t}dB_{t}$ and $dY_{t} = AY_{t}dt + Y_{t}dB_{t}$, where $A$ is a constant matrix and $dB_{t}$ is ...
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0answers
9 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 ...
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0answers
25 views

Stochastic integration by parts [on hold]

How does one go about computing the following stochastic integral: $$ \int f(t,w(t))\,dw(t) $$ where $w(t)$ is a Wiener process.
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0answers
23 views

Evaluating the stochastic integral $ \int \cos(t) \sin w(t)~dw(t) ~.$ [on hold]

Let $w(t)$ be a Wiener process. I would like help in computing the following stochastic integral: $$ \int \cos(t) \sin w(t)~dw(t) ~.$$ Hints are very much welcome.
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0answers
23 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} ...
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0answers
11 views

Stochastic CDF,PDF,DC,AC Calculation

I am having trouble with this problem as I do not understand on how to proceed and how to calculate the required parameters CDF,PDF,AC,DC for a given sawtooth waveform I have proceed with respect to ...
0
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0answers
16 views

Integral with stochastic Brownian motion integrand

let $B_{t}$ is standard brownian motion ,then how to compute this integral $\int_0^t B_{s} ds=?$ I was working on SDE and find some integral in form of $\int_0^t f(B_{s}) ds=?$ $ f(B_{s})$ can ...
2
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0answers
28 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
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2answers
43 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
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2answers
59 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 = ...
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1answer
32 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 ...
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0answers
21 views

Is quadratic variation dominates the sequare integrable continuous martingale almost surly?

Suppose $M(t)$ is a sequare integrable continuous martingale and denote $\left< M \right>_t$ as its quadratic variation process. Is the inequality $M(t)\le \left< M \right>_t, \forall ...
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0answers
34 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. $dX_t^1 = X_t^2dt + \alpha dB_t^1$ $dX_t^2 = -X_t^1dt + \beta dB_t^2$ $\alpha \ and \ \beta$ ...
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0answers
28 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 ...
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0answers
27 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 ...
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0answers
27 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 $V_t$ ...
3
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1answer
50 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 ...
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0answers
9 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
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1answer
17 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
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1answer
50 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
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2answers
75 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!
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0answers
13 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
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0answers
38 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 ...
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0answers
32 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 ...
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0answers
20 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
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1answer
43 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 ...
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2answers
74 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 ...
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0answers
22 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 ...
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0answers
17 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
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1answer
33 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, ...
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0answers
15 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
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1answer
39 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 ...
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0answers
41 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
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1answer
41 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 ...
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1answer
38 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
86 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
134 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 ...