# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

669 questions
Filter by
Sorted by
Tagged with
2answers
807 views

### ARMA-GARCH model, bset model selection and confidence levels calculations

I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch. So, 2 main questions. 1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it ...
2answers
3k views

### Intergral of Brownian motion w.r.t. Brownian motion

I don't understand why $S$ (highlight on picture), I learned $$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t$$ everyone please explain for me. Thank you
1answer
310 views

### Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
2answers
896 views

1answer
4k views

### Implications of shifting the lognormal model for forward rates from a probability perspective

I have a question regarding the application of a shift to the Black-Scholes formula for negative forward rates. I am reading in the Brigo book that "increasing the shift $\alpha$ shifts the ...
0answers
49 views

### Regularity requirement for convergence of Euler scheme for stochastic integral?

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process $\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$ which is like a naive hedge of a European put, which does not work in ...
1answer
142 views

### Second Moment of Stock Process

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then ...
3answers
309 views

### How to understand nonrandom/random process in Shreve book? [closed]

I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given ...
2answers
418 views

### Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
2answers
3k views

### Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*): $$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$ (with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$) and the SDE of ...
2answers
945 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 ...
2answers
313 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 ...
2answers
198 views

### Problem with deriving the dynamics of a process

I'm trying to solve the following problem. Given a process $X_t$ and a process $Z_t$, with the dynamics of $X_t$ as $$dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t$$ and $Z_t$ defined ...
0answers
165 views

1answer
81 views

### Please help me with this problem of double exponential distribution

please help me with this problem of double exponential distribution
1answer
375 views

### Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
2answers
83 views

1answer
226 views

0answers
528 views

### How to compute the stochastic integral of log-normal process?

How do you compute the following integral: $$\int_0^t e^{\mu s + \sigma W_s} ds$$ or $$\int_0^t e^{\mu s + \sigma W_s} dW_s$$ ? Are those integrals stochastic processes of some well-know type (...
1answer
501 views

### Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
3answers
269 views

### Is a bond expiring at $T$ clean or dirty price a martingale under the $T$-Forward measure?

When we say Bond prices are martingale under T-Forward measure, do we mean their Clean Price is a martingale or is it their dirty price. I guess it should be dirty price, as clean price is just a ...
1answer
205 views

### Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $P(\tau < t) \geq (|W_t|>2)\\$ ...
1answer
262 views

2answers
818 views

### how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ‎...
1answer
352 views

### Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
2answers
4k views

### Distribution of stochastic integral

Suppose that $f(t)$ is a deterministic square integrable function. I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$. I want to know if the following approach is ...
3answers
492 views

### Stochastic Calculus Rescale Exercise

I have the following system of SDE's $dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t$ If $\sigma_B > \sigma_A$ I ...