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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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148 views

Problem with derivating integral

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
251 views

HJM framework problem - showing that HJM drift condition implies that $b(z)=b+βz$ and $(ρ)^2=α$

Hi I am looking for some general clarification to Heath–Jarrow–Morton framework. I am analyzing a problem where the forward rate is modeled as $$f(t,T)=e^{\beta(T-t)} Z_t+h(T-t) \tag{1}$$ for some ...
414 views

Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
295 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1$ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
214 views

54 views

$\beta = 1$: Simulation of SABR and whether a solution is *exact*

Quick question regarding the conditional distributions (SABR is just an example here) Consider $$dS_t = \sigma_tS_tdW_t$$ $$d\sigma_t = \alpha\sigma_tdV$$ $$dW_tdV_t=\rho dt$$ Hence a SABR process ...
74 views

737 views

Discounted Stock Price

I have the following Question : Prove that under the risk-neutral probability p the stock and the banjaccount have the same average rate of growth. In other words, if $S_0 , S_N$ are the initial ...
208 views

Transformation into Martingale

If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes: \begin{...
446 views

Bachelier model: number of stocks in replicating strategy

Given: Consider a two-asset, continuous time model (B,S) where \begin{equation} dB_t = B_t r dt, \quad dS_t = \mu dt + \sigma dW_t. \end{equation} The question is: Show that there exists a trading ...
46 views

Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
37 views

Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as $$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\... 0answers 55 views The Ho-Lee Model (1986)'s Bond Call Option Pricing [closed] (My Question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, W_t is a S.B.M. (the details in this ... 0answers 39 views How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables? The integral is shown below: And how to use python to calculate pi (better if we don't need to code for each pi)? 0answers 50 views The Ho-Lee Model (1986) (My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, W_t is a S.B.M. (Thank you for your ... 0answers 100 views Term structure equation in the Vasicek model Consider the SDE$$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$Let$$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r). (F can be interpreted as price of a zero coupon bond with ...
Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...