The stochastic-integrals tag has no wiki summary.
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2answers
46 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
votes
1answer
322 views
Stochastic Calc
(a) Consider the process
$$
d\sqrt{v} = = (\alpha - \beta\sqrt{v})dt + \delta dW
$$
Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that
$$
dv = (\delta^2 + 2\alpha\sqrt{v} - ...
0
votes
1answer
38 views
Rewriting SDEs - “Multiplication on both sides”
I have a question concerning a calculus "trick" sometimes used in stochastic calculus (e.g. in the Book on Arbitrage Theory in Cont. Time of Bjoerk). There they do the following in the proof of Prop. ...
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1answer
48 views
Product rule of stochastic exponents
we know that for standard exponents, $(e^x)(e^y)=e^{(x+y)}$. What is the product rule for stochastic exponents?
$E_n(U)E_n(V)=E_n(U+V+[U,V])$ where $U$ and $V$ are stocchastic sequences, $E_n$ is the ...
3
votes
1answer
128 views
Multidimensional infinitesimal generator of a jump-diffusion
Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE
$$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$
where $\mu, \sigma$ and $\beta$ are ...
0
votes
1answer
25 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
1
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0answers
57 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
2
votes
0answers
35 views
Product of predictable process and a characteristic function is integrable
Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that
$$\int_0^T\theta_u dS_u\ge -a$$
for a $a>0$. Furthermore ...
3
votes
1answer
441 views
Covariance of Brownian Bridge?
I am confused by this question. We all know that Brownian Bridge can also be expressed as:
$$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$
Where the Brownian motion will end at b at $t ...
0
votes
1answer
50 views
Itō's Lemma neglecting terms
In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$
I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small.
...
2
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1answer
73 views
Integrating a Brownian Bridge conditioned above a linear boundary
The Setup:
A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t ...
0
votes
1answer
36 views
Integral: Is there a closed form?
I wonder whether there is a closed form or way to compute explicitly:
$$\int_0^t e^{\alpha s} dB_s$$
where $\alpha$ is just a real number and the integral is in the Itô sense.
Thank you very much!
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0answers
18 views
Solve a special non-linear Backward SDE
It is straigtforward to solve a linear Backward SDE. i.e.
$dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.)
How can I solve $dY_t=Z_tdW_t+ ...
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1answer
177 views
Stochastic integral inequality
Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that
...
1
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1answer
64 views
Approximation of stochastic integral
Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
3
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1answer
221 views
Itô Integral has expectation zero
I have a question about the following property, which I didn't know so far:
Why does the Itô integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
5
votes
1answer
46 views
Is this stochastic integral well defined?
Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
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2answers
85 views
Conditional expectation of a functional of an Itô's semimartingale under its equivalent martingale measure
Consider a probability filtered space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfying the habitual conditions and is generated by $1 d $- ...
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0answers
20 views
Supermartingale Lemma + related problems
Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
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0answers
16 views
a probleme about transformation of homogeneous SDE
I have a question about the characteristics of homogeneouse SDE:
$$dX_t=\beta(X_t)+\alpha(X_t)dW_t\ \ (1)$$
where $W$ implies a standard Brownian motion.
To be more specific, given a general SDE:
...
1
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1answer
66 views
Martingale inequality
Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define
$$
Y^r_t := \int_0^t f(r,s) dW_s
$$
For each fixed $r$, ...
5
votes
1answer
126 views
Ito's Lemma and Brownian Motion
Show by using Ito's Lemma, for $k \geq 2$ the following result hold.
$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$
where $W(t) = N(0,t)$ is standard Brownian motion.
I think ...
0
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0answers
35 views
Intuition: integration of function with respect to stochastic process
Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function.
With the above conditions, the following equality ...
6
votes
1answer
111 views
Very basic doubt about Itô's lemma
While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following
$$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$
I had some doubt concerning the application of ...
4
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1answer
65 views
Computation of basic stochastic integral.
I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage,
$$\text{Cov }(x_s,x_t)=\sigma^2 ...
2
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1answer
59 views
Laplace functional of a Poisson random measure with stochastic intensity
This is one of the problems from Cinlar's 2011 book - "Probability and Stochastics" (Chapter VI, page 262, exercise 2.36) : Let $N$ be a Poisson random measure on $R^{+}$, defined by
$N(\omega, B) = ...
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2answers
204 views
Area enclosed by 2-dimensional random curve
Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
2
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1answer
90 views
Covariances of $\int_0^t h(s)\;dB_s$ process
Let $h:[0,\infty) \to \mathbb{R}$ be a measurable, square integrable function on $[0,t]$, for all $t \geq 0$.
I want to show that if $H_t = \int_0^t h(s)\;dB_s$, where $(B_t)_{t\geq0}$ is a standard ...
1
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1answer
39 views
Rephrasing a Stochastic Process as a Stochastic Differential Equation
I have a continuous-time stochastic process $X$, described as follows:
(1) If the process is at $x_0$ at time $t_0$, then the function $f(t_f, x_f \, | \, t_0, x_0)$ is a PDF in the parameter $x_f$ ...
2
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1answer
138 views
Ito integral almost sure and $L^2$ limit
why does one define the Ito integral as the $L^2$ limit, although it can be shown by Doob's martingale inequality and Borel-Cantelli lemma that there exists a t continuous version, which is ...
14
votes
2answers
445 views
Brownian bridge expression for a Brownian motion
Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as
$$
Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s}
$$
for $0\leq t<1$. Here ...
0
votes
1answer
105 views
Stochastic process as an Ito integral with time-dependent integrand
Will the following process $$r(t)=\int_0^ta(s,t)dW(s)$$ be adapted to the Brownian motion $W(s)$? Will $r(t)$ be an Ito process?
Edit: Maybe I should rephrase it a bit. The question is: does ...
1
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0answers
24 views
Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$
I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
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votes
1answer
409 views
Stochastic Diff Eq SDE
enter link description here
Consider the following SDE
$$d\sigma = a(\sigma,t)dt + b(\sigma,t)dW $$
The Forward Equation (FKE) is given by
$$\frac{\partial p}{\partial t} = ...
1
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1answer
80 views
properties about stochastic integral
I have a question about stochastic / Lebesgue Stieltjes integrals. I'm following Revuz / Yor. The space $H^2$ is the space of all $L^2$ bounded continuous martingales. If $M\in H^2$ then they call ...
0
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1answer
60 views
How can I prove it is a martingale when there is a jump process
Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale.
Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $
$\tau_i$ ...
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2answers
214 views
Ito Isometry and quadratic variation
Here is a confusion regarding stochastic integrals. Let
$Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
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1answer
47 views
One stochastic integrability problem
On a lecture notes, there is a following arguement:
To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
1
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1answer
114 views
Expectation and variance of this stochastic process
I am trying to compute the expectation and variance of the following stochastic process:
$$
Z_t = \exp \left( \frac{1}{2} \int_0^t W_s \, dW_s \right)
$$
where $W_t$ is a standard Brownian motion. I ...
8
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1answer
159 views
Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?
I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
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3answers
283 views
Limit of a Wiener integral
How to show that
$$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$
where $\left (B_s ...
2
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1answer
78 views
About stochastic differential equations
Consider, for all $x \in \mathbb R $, the process $\left( X_t^x\right)_{t\geq 0} $ unique solution of the following SDE:
$$ X_t ^x =x + \int _0 ^t \sigma\left( X_s^x\right) ~dB_s + \int _0 ^t ...
1
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1answer
182 views
Is continuous L2 bounded local martingale a true martingale?
I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...)
I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
6
votes
1answer
271 views
Expectation of an integral w.r.t. Brownian Motion
I know the following statement:
if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
3
votes
1answer
99 views
convergence ito integral
It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$
That means I showed that $\int_0^T S_n \, ...
4
votes
1answer
29 views
How to deal with differential in Itô
Suppose I have two Brownian Motion $W$ and $B$ which are connected through Girsanov, i.e. $W_t=B_t-\int_0^t v(u,T)du$. Furthermore I have the following expression
$$\exp{(\int_0^tv(u,T)-v(u,S) ...
2
votes
0answers
81 views
How do I derive the Gaussian Mixture distribution of an Ito Integral?
I have a question about the distribution of an Ito Integral. Consider the integral
$$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$
where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
2
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2answers
97 views
Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$
Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
2
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2answers
118 views
Basic stochastic integral
I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process?
Thanks!
0
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2answers
136 views
Explicit solution of a SDE
I'd like an explicit formula as a function of $W_t$ (standard brownien motion) and $\lambda >0$ for the solution of the following SDE:
$$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$
...
