# Tagged Questions

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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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 ...
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### What does it mean for a pdf to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ I have tried a lot to ...
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### Reference needed for properties of Convergence of Random Variables

Does anybody know a good reference for properties of convergence of random variables? For example, if $X_n$ converges almost surely (a.s) to $X$ and if $Y_n$ converges a.s to $Y$, then $X_n Y_n$ ...
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### How to deduce the expectation of a stochastic equation [closed]

I am having a difficult time deducing the expectation, $\mathbb{E}[R_t]$, of the following stochastic equation: $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ $R_0 = r$, with $r > 0$. Please help me ...
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### Finding Stochastic processes

I have the following differential equation dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$, X$_0$ = x, with x > 0. Here, r>0. I am having trouble figuring out how to find the ...
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### Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
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### stochastic differential equation solution

I find it difficult to solve this differential equation: $dX(t)=[aX(t)+b]dt+σX(t)dW(t)$ $X(0)=x$ where $W(t)$ is a Brownian motion and $a, b, σ, x$ are real constants The thing which confuses is ...
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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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### 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 ...