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5
votes
1answer
200 views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in ...
6
votes
0answers
97 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
2
votes
1answer
155 views

Derivation of HJB equation

I am trying to derive the HJB equation in a stochastic setting. Let me exemplify my problem with the simplest case where there is no control, just one state variable. Assume the payoff is given by $$ ...
2
votes
2answers
131 views

How can I make this portfolio self-financing?

$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$ $b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate So the value of the portfolio: $$V_t = ...
2
votes
1answer
88 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
1
vote
2answers
106 views

Why does the short rate in the Hull White model follow a normal distribution?

Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
3
votes
3answers
152 views

For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?

Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
1
vote
0answers
66 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + ...
3
votes
1answer
100 views

stochastic calculus - brownian motion

I don't know how to prove this : let be $X_t = \int_{0}^{t}\sigma_{u}dW_{u}$ where $\sigma_{t}$ is a predictable process. If $|\sigma_{t}| = c$ a.s. how can I prove that $X_{t}=c*\beta_{t}$ ...
2
votes
1answer
57 views

equality in distribution

I encounter the following problem : I have the equality in distribution: for all $\lambda >0, ((1/\lambda)*\int_{0}^{\lambda t}\sigma_{u}^{2}du,t\geq0)=(\int_{0}^{t}\sigma_{u}^{2}du,t\geq0)$ ...
2
votes
1answer
55 views

forward option, stochastic calculus

I encounter a problem to understand this: The price of a forward option is : $C(K,t,T)=\mathbb{E}[((S_{T}/S_{t})-K)+]$ OK The option should only depend on $T-t$ because the yield randomness (for a ...
3
votes
2answers
137 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
1
vote
0answers
26 views

Stochastic Optimal Control for ratios

Do you know any good papers on methods of Stochastic Optimal Control and Hamilton-Jacobi-Bellman(HJB) for optimization of different ratios(Sharpe, M2, Sortino, Sterling, etc.)? Meaning that using ...
2
votes
1answer
66 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
4
votes
6answers
600 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
2
votes
2answers
201 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 ...
1
vote
1answer
27 views

Complete Multiperiod Binomial model

I have the following deifnition of a Complete multiperiod binomial model: A multi period binomial model can be called complete if every derivative security can be replicated by trading in the ...
2
votes
2answers
117 views

Using Black-Scholes to price a geometric average price call

Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway. I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
6
votes
2answers
283 views

Ito, Stochastic Exponential and Girsanov

This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential. Whilst reading notes relating to Girsanov it is stated that the ...
2
votes
2answers
144 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: ...
5
votes
0answers
83 views

Stochastic control (HJB) for wealth process involving stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream. ...
2
votes
1answer
133 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
2
votes
1answer
69 views

What is the filtration described?

What is the filtration $(\mathfrak{F}_t)$ encircled below? Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$? Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in ...
3
votes
2answers
67 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...