# Tagged Questions

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

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### Expected value of stochastic process

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad$$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
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### How to solve this SDE

I have been learning basic stochastic analysis, and we have only been taught about Ito formula. The professor told us how can we solve this question below using it, but I miss it. Can anyone help me? ...
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### For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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### $M_t=N_t - \lambda t$. Find $Var[\int_{a}^{b} M_t dM_t]$.

$M_t$ is the compensated poisson process. $N_t$ is a poisson process. $M_t=N_t - \lambda t$. Find $Var[\int_{a}^{b} M_t dM_t]$. I have a doubt. I read the book and it is dealing with the left ...
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### Definition of self-financing strategy

Consider a portfolio of two assets with prices $S_t$, $B_t$ and holdings $\Delta_t$ and $E_t$ respectively. So the portfolio value is $$\Pi_t = \Delta_t S_t + E_t B_t$$ The portfolio is defined to ...
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### Radon-Nikodym with respect to Stochastic Measure?

Question This question is now concerning stochastic processes. Let $(X_t)_{t\geq0}$ be defined on the probability space $(\Omega,\mathcal{F},P)$ with $\mathcal{F}_t=\sigma(X_s:s\leq t)$. Assume that ...
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### Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds}$$ where the ...
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The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
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### Distribution of a exponetial Random Variable

i have a stopping time $T$ of an Poisson Process $N$ with rate $\lambda$. Somehow this stopping time is exponential distributed. So we have $T \sim exp(\lambda)$. I want to know the distribution of ...
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### If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
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### Calculation probability of dynamic process model of capacity

I found this place really helpfull and now I got my first own question I cant solve. I want to unterstand the calculation of an Article im reading. Therefore we define a capacity process $C$ in a ...
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### Correlation between stochastic processes

Question: if $W(t)$ is a standard Brownian motion with $W(0)=0$, what is the linear coefficient between the stochastic processes $W(t)$ and $I(t)=\int_0^t W(s)ds$? I argued as follows: what we want ...
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### Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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### System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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### Let $Z_t=\int{W_s }ds$. Show that $Z_t=\int (t-s) dW_s$

Let $Z_t=\int_{0}^{t} W_s ds$. Use integration by parts to show that $Z_t=\int_{0}^{t} (t-s) dW_s$. I have tried and i can't get the answer.
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### Itô-isometry in the extended case?

It is shown when constructing the Itô-integral that if: $E[\int_0^T X_t^2dt]< \infty$. Then we have that Itô-isomtry: $E[\int_0^T X_t^2dt]=E[(\int_o^TX_tdB_t)^2]$. In the extended Itô integral, ...
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### unbounded variation of $\sin(x)/x$

How can I show that the variation of $sin(x)/x$ is unbounded? Could you please help me. I know that I have to use but how can I rough estimate that this is bigger than infinity?
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### What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
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### Preservation of ui condition [closed]

I have a stopping time $\tau_n$ with $\mathbb{P}(\tau_n=\infty)\rightarrow 1$ for $n \to \infty$. With this stoppingtime $M^{\tau_n}$ is a uniformly integrable martingale. I deduced that $M$ is a ...
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### How is the martingale property of Brownian motion used in the construction of the Ito integral?

I am trying to learn about Fractional Brownian Motion (and eventually integration with respect to FBM) and keep running into references to the fact that the construction of the Ito integral with ...
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### Conditions for limiting distribution to equal stationary distribution of SDE

I have SDE of the form $$dX_t=a\mathopen{}\left(X_t\right)dt+b\mathopen{}\left(X_t\right)dW_t,$$ where $W$ is Brownian motion. If the stationary distribution of $X$ exist is it equal to the limiting ...
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### Show that $\hat{Y}$ is an optimal linear estimator of Y

Relevant Information. Let $X(t)$, $t \in T$ be a second order process. Let $M_0$ be the set of random variables of the form $a + b_1X(s_1)+ \cdots + b_nX(s_n)$ for a positive integer $n$ and constants ...
I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let $X_t$ solve the Ornstein-Uhlenbeck equation $$dX_t = X_t\, ... 0answers 31 views ### Changing the order of integration for Brownian motion (outer integration over the range of inner integration) X_t is bounded Brownian motion and it can be even standard Brownian motion if you wish. I want to express E[\int_{0}^{T}\int_{0}^{t}X^{n}dsdt] as a function of E[\int_{0}^{T}X^{n}dt] For ... 2answers 53 views ### Quadratic covariation of two Itô processes If dX(t)=\Delta_x (t) + \ominus_x(t) dt and dY(t)=\Delta_Y(t) dW(t)+ \ominus_Y(t) dt, where X(t), Y(t) are two Ito processes. I need show that d[X,Y](t)=\Delta_x(t)\Delta_Y(t)dt, where \... 1answer 36 views ### Decision theory references for advanced undergrad/early grad students? I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ... 0answers 39 views ### Strong existence of solutions to SDE and continuity in the time variable I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE$$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$where a, ... 1answer 94 views ### When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0) Assume you have a Lévy process X. Let N(t,A) be defined as the number of jumps in the interval (0,t], such that the jumps size \Delta X_s \in A. It can be shown that if 0 \ne \bar{A}, then ... 0answers 55 views ### Convergence of a process this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ... 1answer 43 views ### Stochastic differential equation substitution reasoning? I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ... 0answers 45 views ### Time scaling birth process in Poisson process Given a birth process \{B_t:t\geqslant0\} with \lambda >0, define$$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i) if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
Here is my question statement. I cannot understand the last equality. Let $U=[-1,1]$. \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\...