Questions on the calculus of stochastic processes, or processes that have a random component.

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30 views

Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
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21 views

Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
3
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0answers
17 views

Calculate expectation of stochastic integrals

I am trying to calculate $$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$ where $(B_t)_{t\geq 0}$ is a brownian motion, $h>0$ and $\lambda > 0$ is some ...
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0answers
34 views

Distribution of “range” of a process

Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value $X_t$ and a past value $X_{t-100}$ is $$M_t ...
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1answer
34 views

What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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1answer
61 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
2
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1answer
42 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that $\...
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31 views

Parallel Shifts of Forward Curve and Arbitrage Opportunities

I came accross a phrase in the Paul Glasserman, Monte Carlo Methods In Financial Engineering, page 153 : "a model in which the forward curve makes only parallel shifts admits arbitrage opportunities :...
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17 views

How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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0answers
16 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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1answer
34 views

Martingale Poisson [closed]

Can somebody help me with working out: $$E[(N_{t}-\lambda t)^2\mid F_{s}]$$ where $N_{t}$ is a Poisson process and $F_{s}$ the $\sigma$-algebra generated by $N_{s}$, $0 \leq s < t$.
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1answer
40 views

Showing martingale, submartingale or supermartingale with log

Can somebody help me with determining whether $Z_{n}=\log(2n+S_{n})$ is a martingale, supermartingale or submartingale with $S_{n}=\sum_{i=1}^{n}X_{i}$ and the are i.i.d. random variables with $P(X_i ...
5
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1answer
57 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
3
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0answers
31 views

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
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0answers
58 views

Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where $\...
2
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1answer
56 views

HJM model - Differentiation Problem

starting from the folowing SDE (HJM model): $$df(t,T)=\left(\sigma(t,T)'\int_t^T{\sigma(t,u)du}\right)dt+\sigma(t,T)'dW_t$$ And having $r(t)=f(t,t)$, I have two questions : 1) how do we obtain the ...
2
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0answers
38 views

Counterintuitive result on quadratic variation

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then $...
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0answers
74 views

Gaussian Random Walk

Suppose $\{Z_{i}\}_{i=1,2,\ldots}$ are iid normally distributed with mean zero and positive variance random variables ($\sim N(0,\sigma^{2})$). Write the discrete time stochastic process as: $$N_{0}(...
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18 views

Can a Brownian motion be defined for negative time?

I was just looking at fractional brownian motions on this page. The definition of $B_H(t)$ requires integrating on a negative time domain on $dB(t)$ where $B(t)$ is a Brownian motion! Could you please ...
2
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0answers
48 views

Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 \int_0^...
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1answer
40 views

self financing strategy

how could one prove the following proposition from stochastic calculus applied to finance? Proposition : Let $\Phi$ a trading strategy. Then, $\Phi$ is self financing if and only if $D(0,t)V_t(\Phi)=...
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1answer
11 views

Tower property of conditional expectations - Application Question

How could I prove the folowing using the tower property of conditional expectations? $$E\left(E\left[\frac{D(t,T)D(T,S)H}{P(T,S)}|F_T\right]|F_t\right)=E\left(\frac{D(t,T)H}{P(T,S)}E[D(T,S)|F_T]|F_t\...
2
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0answers
100 views

How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(x_0)\...
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2answers
42 views

Reference for stochastic calculus with jumps

All the standard books I know on stochastic calculus work almost exclusively with continuous martingales. What are the standard references for the general theory (with jumps)?
4
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58 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
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0answers
32 views

Why $d\langle X \rangle_t = d X_t dX_t$ if $X_t$ is a semimartingale?

Following this question, proving the equivalence between equation $(1)$ and $(2)$, I deduced that $$d\langle X \rangle_t = d X_t dX_t$$ (where $X_t$ was an Ito's process, hence a semimartingale). I ...
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41 views

How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, $\...
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1answer
78 views

$dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
2
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1answer
31 views

An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
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0answers
15 views

Manipulating a log normal variable

I am wondering given: and is it possible to state: $$\text{Jdq}_t s_t-\text{dq}_t s_t=\text{dq}_t \log (J) s_t$$ And if it is the case can we show how this argument is done?
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37 views

Brownian Motion Hitting Time?

So my problem is the following. Take a 2D Brownian motion $(W_{1t}, W_{2t})$ such that it starts at $(1,1)$. With probability 1 it will hit the x-axis. What is the probability that it will hit the ...
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24 views

Does Ito isometry hold pointwise?

It is known that the stochastic integral satisfies the following property $$ \mathbb{E}\left[\left\langle \int_0^{\cdot}X(s)\,dM(s) \right\rangle_t\right]= \mathbb{E}\left[ \int_0^t X^2(s) \, d\left\...
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69 views

Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
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27 views

How to evaluate the expectation of the exponential of reflected brownian motion

How do you compute this expectation $\mathbb{E} \left [ e^{\varepsilon|W_t|} \right] $ where $W_t$ is a Brownian Motion Do I need to expand the absolute value? Can I use the standard Taylor series ...
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1answer
16 views

Independent stochastic processes

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks
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1answer
45 views

Exponential martingale and change of measure

$\newcommand{\qq}{\mathbb{Q}}\newcommand{\ee}{\mathbb{E}}$ Denote $Z_t= \exp( \theta B_t - \frac{1}{2}\theta^2t )$ Given the probability measure $\qq(A) := \ee[ Z_t \mathbb{1}_A ]$ I must ...
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1answer
89 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. Given ...
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0answers
22 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
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2answers
75 views

Linear non-homogenous SDE

I'm struggling to understand how to resolve the following SDE: $$dX(t)=(\sin(t)-2X(t)) dt + (1+X(t))dB(t)$$ I understand that I should use the Ito formula but I have no idea how the $F(X(t),t)$ should ...
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1answer
45 views

A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and $E^*...
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33 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space $(X_t)_{...
2
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1answer
64 views

Rigorous meaning of conditional expectation in Feynman-Kac formula/in general

In Wikipedia https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula and plenty of other books/sources, Feynman-Kac formula is expressed in a form of the type $$f(t,x)=E(f(T,X_T)\mid X_t=x)$$ What ...
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22 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
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8 views

Testing numerical solvers for multidimensional SDEs with non-commutative noise

I am implementing the multidimensional Milstein scheme to solve SDEs. I am trying to test the solver on benchmark equations but I cannot find an analytic solution for the case with a 2-d state vector ...
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1answer
54 views

How to solve a SDE defined via a Markov Process?

I have to solve the following SDE. $$ \mathrm dY_t= f(X_t) \mathrm dt, \tag{1} $$ where $X_t$ is a two-state Markov Process possesses states $a$ and $b$. Moreover, I would like to solve $$ \mathrm ...
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0answers
25 views

Proving bounded expectation and switching order of stochastic integrals

I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-...
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0answers
66 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
4
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1answer
33 views

Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$ \int_0^T W(t) dt = \int_0^T (T-t) dW(t) $$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ (...
2
votes
0answers
19 views

Distribution MSR

We have $Y_i = \beta_0 +\beta_1(X_i -\bar X )+\epsilon_i$ for i=1,...,n $$\epsilon_i \sim N(0,\sigma^2)$$ We know that $SSR= Y^T P_xY - n\bar Y^2=Y^T (P_x -n^{-1} J_nJ_n^T)Y$ $$J_n=\big[\begin{...
2
votes
1answer
34 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...