Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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2answers
228 views

A question on Ito

If we know the dynamics of $S$, then we can estimate the value of $S$ at a time point, $t$. Here, I have a question concerning how to solve for $S_t$ by Itô because I obtained different results by ...
7
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1answer
579 views

How to show that this process is “normally distributed”?

Say we have following SDE (Vasicek): $$dr(t) =(b-ar_t) dt + \sigma dW_t$$ I am able to reach an integral form of this SDE : $$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
7
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1answer
164 views

Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution

I have to show that, if $W_t$ is a 1-d Brownian motion then $\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on ...
7
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1answer
489 views

Baxter & Rennie HJM: differentiating Ito integral

From Baxter and Rennie, page 138: $$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$ $$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$ $$dZ_t=...
7
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1answer
772 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
7
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1answer
3k 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 ...
7
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1answer
163 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: $P(\tau < t) \geq (|W_t|>2)\\$ ...
7
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2answers
2k 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 ...
7
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1answer
765 views

Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ ...
7
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1answer
236 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. ...
7
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1answer
361 views

What is a good Computer Algebra System for financial engineering?

I would like to know if there exists some computer algebra systems adapted to calculate pricing based on particular models, i.e. pricing YoY Inflation Swap under Jarrow Yildirim Model. I know that ...
6
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2answers
452 views

close form for stochastic integral

I am new to stochastic calculus. Can I know how to compute the close-form solution for $$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$ and $$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$ I encounter ...
6
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2answers
2k views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price $Z(t)...
6
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2answers
382 views

Variance of $\int_{t=o}^{T}\sqrt{|B(t)|}$ $dB(t)%$

I'm new to stochastic calculus. Could someone please explain how I would calculate the variance of $\int_{t=o}^{T}\sqrt{|B(t)|}$ $dB(t)%$ I'm aware that I would first have to calculate the ...
6
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2answers
428 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
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1answer
3k views

Extended Hull White Interest Rate Model for Zero Coupon Bond

Let's take the following three SDEs: $$dr=u(r,t)dt + w(r,t)dX$$ $$u(r,t)=a(t)-br$$ $$w(r,t)=c$$ where $b$ and $c$ are constants and $a(t)$ an arbitrary function of time $t$. If Zero Coupon Bond $Z(...
6
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1answer
185 views

Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
6
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1answer
2k views

Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
6
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1answer
732 views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
6
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1answer
207 views

Derivation of the Stochastic Vol PDE

I'm trying to follow the derivation of the stochastic vol pde for an option price - as given in Gatheral (The vol surface), Wilmott on Quant Finance and many other places. As usual one starts off with ...
6
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1answer
131 views

Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ...
6
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2answers
2k views

How to compute the variance of this stochastic integral?

I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true. I am trying to compute the variance of the ...
6
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2answers
330 views

Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
6
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1answer
297 views

Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
6
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1answer
428 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
6
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1answer
147 views

Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for the first time

Let say I have a zero-mean OU process as follows: $dX_t = -\alpha X_t + dW_t$ The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for ...
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0answers
114 views

Random variable minus Integral of Ito Generator is a Martingale under what conditions?

I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
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152 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends $n-...
5
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1answer
1k views

CIR Process from Ornstein–Uhlenbeck process

The wikipedia entry on the CIR Model states that "this process can be defined as a sum of squared Ornstein–Uhlenbeck process" but provides no derivation or reference. Can any one do that? I could only ...
5
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2answers
217 views

Why is $Y(t)V^h(t)$ a martingale?

Let $\lambda$ be the market price of risk: $\frac{a - r}{\sigma}$, and define $Y(t) = e^{-\lambda W(t) - (r + \frac{\lambda^2}{2})t}$. Let $V^h(t)$ be the value process of any self-financing portfolio....
5
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1answer
444 views

Martingale representation theorem

Let $r_t, \theta_t$ denote some stochastic processes driven by a $N$ dimensional Brownian motion $W_t$ (they are of course assumed adapted to the natural filtration $\mathcal{F}_t$ of that Brownian ...
5
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1answer
380 views

Square of arithmetic brownian motion process

We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation ...
5
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1answer
316 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
5
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1answer
115 views

Expected value of exponential of hitting time of GBM

We have a stopping time $$ \tau=\inf\{t\geq 0: S_0e^{\sigma B_t+(r-\sigma^2/2)t}=S^* \} $$ where $S_0,\sigma,r,S^*$ are constants and $S^*<S_0$, and $B_t$ is a brownian motion. I wish to compute ...
5
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1answer
120 views

Why $W_{t}^3$ is not a martigale?(by Definition)

If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
5
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1answer
227 views

What is the probability that a OU process hits an upper barrier U before a lower barrier L?

What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
5
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1answer
239 views

Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?

Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$? If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
5
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2answers
482 views

Ito vs. Stratonovich: Why is it the exact midpoint that renders Ito-correction zero?

Perhaps I am approaching this from the wrong direction but I was just thinking about the relationship between Ito and Stratonovich integrals: It is a well known result that to convert one into the ...
5
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2answers
1k views

Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
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2answers
2k 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 ...
5
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2answers
375 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 $ ...
5
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1answer
178 views

Explicit solution SDE

I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$ where $W_{t}^{...
5
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1answer
223 views

What is the stochastic differential of a general semimartingale?

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3: $$H = B + H^c + h(x) \...
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0answers
100 views

Complete Financial Market: Integrability condition for Contingent Claims

Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
4
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3answers
909 views

Variance of Brownian Motion

Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$ I had tried using the following property with no luck: $E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
4
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3answers
421 views

Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
4
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3answers
348 views

Determine $E[W_p W_q W_r]$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let 0 < p < q < r. Determine $E[W_p W_q W_r]$. ...
4
votes
2answers
2k views

Differential equation for log-returns

I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian ...
4
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1answer
3k views

Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
4
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1answer
873 views

What's the variance of this Ito integral?

I am reading stochastic calculus and I have understood that the process $$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$ has normal distribution with mean zero. How can I find the variance of $X$?