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

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1answer
66 views

Payoff function

$$ \mathrm{d}S_t=μS_t\,\mathrm{d}t+σS_t\,\mathrm{d}B_t $$ The payoff function for a european call is: $$ f(S,T)=(S(T)-K)⁺. $$ When I graph this it is obvious it is continuous. How would I ...
0
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1answer
139 views

Confusion regarding Stochastic integral

I've a stupid doubt in the construction of stochastic integral of real scalar valued maps. Many times I've seen in books after the stochastic integral is defined in [$0,T$] for the integrand in $L^2$ ...
0
votes
1answer
188 views

stochastic exponential

For a semimartingale $X$, we want to solve the SDE $dZ_t=Z_{t-}dX_t$. I was able to prove that $$ Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t^c)}\prod_{0<s\le t}(1+\Delta X_s)\exp{(-\Delta X_s)} ...
3
votes
0answers
119 views

Stochastic calculus integral

How can I evaluate, or at least find an upper bound for, the following integral without the Hölder inequality, is there an alternate way anyone knows of: $$\mathbb{E}\left[\sup\left|\int_0^t\mu ...
4
votes
0answers
146 views

Integrating the inverse of a squared bessel process - integrability

Let $X_t$ be a 4-dimension Squared Bessel Process (BESQ-4). Let $M_t$ be a continuous true martingale. Question: Does $\int_0^t \frac{1}{X_s}dH_s$ exist? If so, is it only a local or a true ...
3
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0answers
85 views

For $X_{t}=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right\}$, do we have $\mathbb{E}[\int_{0}^{\tau_{b}}X_{s}dW_{s}]=0$?

Let $X_{t}$ denote the solution to the SDE: $$dX_{t}=(\mu -r)X_t dt+\sigma X_t d W_{t}, \ X_{0}=1$$ i.e. $X_{t}$ is the process: $$X_{t}:=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma ...
2
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0answers
28 views

For what $p$ do we have $Z_{T} \in \mathcal{L}^{p}$, where $Z_{T}$ is an exponential martingale?

Let me define $Z_{t}$ to be the stochastic exponential $\mathcal{E}(-\dfrac{B_{s}}{\sigma},B_{s})$, where $B_{s}$ is a standard 1-d Brownian motion and $\sigma$ is a positive constant, i.e. $Z_{t}$ is ...
0
votes
1answer
52 views

How to resolve this?

I've the following problem to model and program it: suppose that we have a central server that provides 3 different services($S_1,S_2,S_3$), there are $N$ machines connected to this server: each ...
1
vote
2answers
172 views

Generating function of the stopping time

Let $X_t$ be a generalized Wiener process with drift rate $\mu$ and variance $\sigma^2$, and let $\tau$ be the stopping time $$\tau:=\inf \left\{ t\geq0: X_t= b\right\}, \quad b\geq0 $$ Can anyone ...
0
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0answers
166 views

Ito's formula for irregular functions

Let's say we have \begin{align} Y_t=h(t,X_t) \end{align} and for simplicity \begin{align} dX_t=e\,dt+f\,dW_t \end{align} then by Ito's formula we have \begin{align} dY_t=\left(\frac{\partial ...
2
votes
1answer
234 views

SDE - removal of the diffusion coefficients

I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_\mathrm{loc}$. If I have \begin{align} dX_t=b(X_t) \, dt+\sigma \, dW_t, \end{align} where $b\in ...
0
votes
1answer
62 views

maximal inequality

Let $(W_{t})_{t\geq 0}$ be a standard Brownian Motion. Then it is easy to get the following inequality, using the Burkholder inequality: ...
1
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1answer
142 views

proof of a lemma on stopping times

Hi I got some problem with the proof of this lemma which is left as excercise Let $(F_t)$ be a filtration and S,T be stopping times, then show $ \{S=T\},\{S≤T\},\{S<T\} \in F_S\bigcap F_T$. Could ...
1
vote
2answers
142 views

Follow up to “Is this local martingale a true martingale?”

This question pertains to the topic titled: Is this local martingale a true martingale? (Find it here!) "Using the Ito's formula I have shown that $X_t$ is a local martingale, because $\mathrm{d}X_t ...
1
vote
1answer
101 views

Is independence preserved under the Girsanov transformation?

Let $(B_{t},\mathcal{F}_{t})$ be a standard 1-d Brownian motion on some $(\Omega, \mathcal{F}, \mathbb{P}) \ $, and let's assume $\mathcal{F}_{t}$ is the augmentation of $\mathcal{F}_{t}^{W}$. Let ...
2
votes
1answer
187 views

Prove that Brownian Motion absorbed at the origin is Markov

I'm trying to prove that Brownian motion absorbed at the origin is a Markov process with respect to the original filtration $\{\mathcal{F}_{t}\}$. To be more specific, let $(B_{t},\mathcal{F}_{t})_{t ...
0
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0answers
74 views

Simplification of an expression

I am trying to simplify the following expression: $X(t)-Y(t)=\int\mu(X\{s\}-Y\{s\})ds+\int\sigma(X\{s\}-Y\{s\})dB\{s\}$ ...
7
votes
2answers
657 views

Ito integral of a Brownian Motion w.r.t. an independent Brownian Motion.

Let $B$ and $W$ be independent Brownian motions, let $\tau$ be a stopping time adapted to $\mathcal{F}^{W}$, do we always have $E[\int_{0}^{\tau}B_{s}dW_{s}]=0$? I know that $\int_{0}^{t}B_{s}dW_{s}$ ...
0
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0answers
74 views

Expectation of $\int_0^t X(r) \, dW(r)$ where $dX=\mu \, dt+\sigma \, dW$

I have a questionlike: if $dX=\mu \, dt+\sigma \, dW$, where $W$ is a standard B.m. Then, is this expectation still o,$\int_0^t X(r) \, dW(r)$ ? Thank you all.
4
votes
1answer
177 views

Existence of solutions to stochastic differential equations by the Banach contraction principle?

I've read a proof for existence of solutions to stochastic differential equation from a book of Ikeda and Watanabe and have a question. Is it possible to prove existence (and uniquness) by means of ...
0
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0answers
99 views

How to transform a stochastic jump diffusion equation to a Levy stochastic differential equation?

If I have this type of stochastic differential equation : $$ dX(t) = A(X(t),t)\ dt +B(X(t),t)\ dW(t) + C(X(t),t)\ dP(t) $$ With $$ \begin{align} dW(t)& : \text{A wiener process}\\ dP(t)& : ...
2
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0answers
106 views

How to check if a process is a semimartingale?

Consider the process $X_t = \sum_{i=1}^{N_t} Y_i$. This is a Lévy process, hence Markov and so on ($N_t$ is a Poisson counting process). Now add some diffusion $D$ for each jump $Y_i$ that starts at ...
2
votes
0answers
183 views

Property of the sharp bracket process

I know that for a continuous local martinagle $M$ we have $\langle M\rangle^\tau = \langle M^\tau\rangle$ for any stopping times. Now if $M,N$ are two local martingale I know that there exists again ...
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0answers
148 views

problems in exponential martingale

Let $B\left(s\right)$ be a brownian motion and $\sigma\left(s\right)$ be the nondeterministic function. The following equation then holds $$ ...
0
votes
1answer
93 views

How to solve the expected value of such integral

$$ V_t = E^\mathbb{Q} \left[\int_t^{+\infty} e^{-r(u-t)}X_u \, du|X_t\right] $$ with a given process $ X_t $ satisfied: $dX_t = (\mu-\sigma^2 \gamma) X_t \, dt + \sigma X_t \, dW_t^\mathbb{Q}$
0
votes
1answer
124 views

A solution's variance and mean

I have a stochastic differential equation of this type : $$ \ dX(t) = a dB(t)+k c t^{c-1} \cos(\theta(t))dP(t). $$ with $$ \begin{align} a,k &:\text{ constants }\\ B(t) &:\text{ Brownian ...
2
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1answer
209 views

A question related to Novikov's condition

The well-known 'Novikov condition' says: Let $ L = (L_t)_{t \geq 0} $ be a continuous local martingale null at 0 and $ Z = \exp(L - \frac{1}{2} \langle L \rangle) $ its stochastic exponential. If ...
2
votes
1answer
72 views

Covariation Paradox??

we can see that $\left\langle \int_0^t \! W_s \, \mathrm{d} s ,W_t \right\rangle_t = 0$ However if I am to use the expression $$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
2
votes
0answers
126 views

Finding an SDE which satisfies $X(t)$

I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time! The question is as follows: "Let ...
2
votes
0answers
148 views

Solving a SDE and finding its related moments

I am attempting to answer this multi-part question, and hope you can provide any feedback on any of my workings. My apologies for the length and thank you in advance for any help! i) Let $g$ be a ...
1
vote
2answers
256 views

Why the set of stochastic process Ito Integrable has to be square integrable w.r.t time as well?

Ito Integral Consider a set of stochastic process $f(t)$ mainly such that a) $$ E\left(\int_0^{+\infty}f(t)^2 \,dt\right) < \infty. $$ Denote this set of stochastic process as $M^2$. ...
1
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0answers
105 views

Looking for an example of non-unique weak solution to a SDE but unique strong solution

Everything is in the title, I'm looking for an example of an SDE with a unique strong solution but with multiple weak solutions. Best regards.
4
votes
1answer
862 views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
4
votes
1answer
98 views

Bessel differential equation with random parameter

I know that the following differential equation: $$x^2\frac{d^2y(x)}{dx^2}+x\frac{dy(x)}{dx}+(x^2-\alpha^2)y(x)$$ has the solution: $$y(x)=C_1\cdot J_\alpha(x)+C_2\cdot Y_\alpha(x)$$ In my case, the ...
2
votes
0answers
68 views

Are affine SDEs invertible?

If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves ...
1
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0answers
196 views

existence of a strong solution for a sde

Suppose we want to study a SDE of the form $$ dX_t = a(t,X_t)dt + b(t,X_t)dW_t$$ and $X_0=Y$, on a filtered probability space $(\Omega,\mathcal{F}, \mathbb{F},P)$ and where $W$ is a $(P,\mathbb{F})$ ...
1
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1answer
1k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
0
votes
1answer
168 views

a question on Stochastic Calculus

I encounterred a question on Stochastic Calculus as following, but I don't understand the meaning of $\mathcal{N}$ here, can any expert explain me a little bit? Thank you very much in advance! ...
4
votes
2answers
510 views

Physical meaning of Ito integrals

I'm having trouble getting my head around the meaning of the stochastic Ito integral. Specifically: the intuitive meaning of "Stochastic Integral" to me is a function that takes a time $t$ and ...
2
votes
1answer
136 views

The variance of bilateral filtered random variables

I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized ...
3
votes
1answer
262 views

Karhunen-Loève expansion of Poisson process

Let $X_t,t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the Karhunen-Loève expansion of $X$ in interval $[0, T]$. How about the KL expansion of the centered process $X_t−\lambda ...
-1
votes
1answer
187 views

PDF for the integral of a Stochastic Process

My continuous-time, continuous step Stochastic Process P runs from time $t=0$ to $t=t_f$ and generates a path. I am able to observe its starting and ending position (so $P(0)=a$ and $P(t_f)=b$), but ...
3
votes
0answers
464 views

Variance of a Wiener process

Problem statement: a continuous wiener process $w(t)$ with unit incremental variance and $w(0)=0$ is given, and then we check the wiener process at every $h$ seconds, $h>0$ is a positive number. If ...
1
vote
1answer
111 views

Why is $ N^\tau ( M - M^\tau ) $ a continuous local martingale if $ M $ and $ N $ are?

Working through my stochastic calculus script, I encountered the following identity, for which no proof is given: $ \langle M, N^\tau \rangle = \langle M, N \rangle^\tau $, if $ M, N $ are continuous ...
4
votes
1answer
380 views

How to make this heuristic extension of Itô-Tanaka's formula valid

Here is my story, I have the following function : $$ g(x)=(1+x)\cdot\exp\left(-\frac{(\log(x+a)+c)^2}{2\sigma^2}\right)1[x\ge y]=f(x)\cdot1[x\ge y] $$ with $a,c,\sigma$ being "good" reals so that ...
3
votes
1answer
139 views

Solving SDE's on subsets of $R^n$.

It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\sigma(\cdot,\cdot):[0,T] \times \mathbb{R}^n ...
2
votes
1answer
265 views

Stochastic Heat Equation

Given the heat equation: $$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$ with the boundary conditions: $$\Phi(x,0)=\Phi_0$$ and a Neumann boundary condition of the kind: ...
3
votes
3answers
510 views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...