SC. Ito Lemma (Vector Valued)

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28 Dec 201903:47

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TLDRThis video delves into the vector-valued Ito Lemma, expanding on the scalar version. It introduces the concept for n stochastic processes, each with its own differential equation involving potentially stochastic coefficients. The video focuses on a deterministic function 'f' applied to these processes and time, under certain smoothness conditions. The Ito formula is then used to derive the stochastic differential equation for the function's output, highlighting the role of instantaneous covariation between processes and its simplification to the product of volatilities and the correlation coefficient.

Takeaways

📊 This video introduces the vector-valued formula for Ito's lemma.
📘 Assumes prior knowledge of the scalar version of Ito's lemma from a previous video.
🔢 Considers n stochastic processes X_i with i ranging from 1 to n.
🔍 Each X_i follows an Ito stochastic differential equation with potentially stochastic processes alpha_i and theta_i.
🕰️ Defines Y_t as the output of a smooth deterministic function f at time T.
🧮 The function f takes the stochastic processes X_i and time T as inputs, making Y_t equal to f(T, X_1(T), ..., X_n(T)).
📝 Smoothness conditions: f must be continuously differentiable in T and twice continuously differentiable in X_i.
📏 If these assumptions hold, Ito's formula states that the output of f solves the specified Ito SDE.
⚖️ Instantaneous covariation term (bracket dX_i, dX_j bracket) is explained as the product of volatilities and correlation coefficient.
🔗 Correlation between Brownian motions i and j (dW_i and dW_j) impacts the instantaneous covariation term.
📐 The covariation term simplifies to theta_i * theta_j * Rho * dt, representing the instantaneous covariance between X_i and X_j.

Q & A

What is the vector-valued formula for Ito's lemma?
-The vector-valued formula for Ito's lemma is an extension of the scalar version, which is used to find the differential of a function of multiple stochastic processes. It involves the application of the lemma to a deterministic function f that takes time and n stochastic processes as inputs.
What are the assumptions made for the stochastic processes X_i in the script?
-The script assumes that each stochastic process X_i, where i is an element of 1 to n, follows an Ito stochastic differential equation with potentially stochastic processes alpha_i and theta_i.
What is the role of the function f in the context of Ito's lemma?
-Function f is a smooth deterministic function that takes time and the stochastic processes X_i as inputs and is assumed to be continuously differentiable in time and twice continuously differentiable in the stochastic processes.
What does the notation F(T, X_1(T), ..., X_n(T)) represent?
-This notation represents the output of the function f at time T with the stochastic processes X_1, ..., X_n at time T as its arguments, which is equivalent to Y_T.
What are the smoothness conditions on the function f according to the script?
-The smoothness conditions on f are that it must be continuously differentiable in time T and twice continuously differentiable in each of the stochastic processes X_i.

Outlines

00:00

📚 Introduction to Vector-Valued Ito's Lemma

This paragraph introduces the concept of the vector-valued Ito's Lemma, building upon the scalar version previously discussed. It assumes the presence of 'n' stochastic processes, \( X_i \), each following an Ito stochastic differential equation with potentially stochastic coefficients, \( \alpha_i \) and \( \theta_i \). The focus is on a deterministic function 'f' that is smooth and takes both time and the stochastic processes as inputs. The Ito formula is then applied to this function, leading to a new stochastic differential equation. The paragraph emphasizes the importance of understanding the instantaneous covariation term, which is a key component in the formula.

Mindmap

Keywords

💡Ito Lemma

Ito Lemma is a fundamental result in stochastic calculus, particularly in the theory of stochastic differential equations (SDEs). It provides a way to find the differential of a function of a stochastic process. In the video, the lemma is discussed in the context of both scalar and vector-valued functions, emphasizing its importance in analyzing stochastic processes.

💡Stochastic Processes

Stochastic processes are random processes that evolve over time. In the video, the processes X_i are assumed to follow Ito stochastic differential equations, which are crucial for modeling various phenomena in finance and other fields where randomness plays a significant role.

💡Smooth Deterministic Function

A smooth deterministic function is a function that is continuously differentiable. In the video, the function f is applied to the stochastic processes X_i and time t, generating an output Y_t. The smoothness of f ensures that it meets the required differentiability conditions for applying Ito Lemma.

💡Instantaneous Covariation

Instantaneous covariation refers to the immediate correlation between the changes in two stochastic processes. In the video, it is represented as [dX_i, dX_j] and is related to the product of their volatilities and the correlation coefficient of their driving Brownian motions.

💡Brownian Motion

Brownian motion, denoted as dW_i in the video, is a fundamental stochastic process that models random movement, often used to represent the underlying noise in financial models. The video discusses the covariation of processes driven by Brownian motions and their correlation.

💡Volatility

Volatility is a measure of the dispersion of returns for a given security or market index. In the video, volatilities θ_i are part of the stochastic differential equations and play a role in the calculation of instantaneous covariation between processes.

💡Correlation Coefficient

The correlation coefficient, denoted as ρ in the video, quantifies the degree to which two stochastic processes move in relation to each other. It appears in the calculation of instantaneous covariation, affecting the interaction between different stochastic processes.

💡Differentiability Conditions

Differentiability conditions refer to the requirements that a function must be continuously differentiable in its arguments. In the video, these conditions are necessary for applying Ito Lemma to the function f, ensuring it is sufficiently smooth.

💡Ito Stochastic Differential Equation

An Ito stochastic differential equation (SDE) is a type of differential equation used to model stochastic processes. In the video, each X_i follows such an equation with terms α_i and θ_i, which may themselves be stochastic processes.

💡Deterministic Function

A deterministic function is one that produces the same output for a given input every time, without any randomness. In the video, the function f is deterministic and smooth, taking stochastic processes X_i and time t as inputs to produce Y_t.

Highlights

Introduction to the vector-valued formula for Ito's Lemma, assuming prior knowledge of the scalar version.

Assumption of n stochastic processes X_i, each serving an Ito stochastic differential equation.

Potential for alpha_i and theta_i to be stochastic Ito processes themselves.

Y_T as the time T output of a smooth deterministic function f, taking stochastic processes and time as inputs.

Smoothness conditions on function f: continuous differentiability in T and twice continuous differentiability in xi.

Ito's formula states the output of function F solves a specific Ito's stochastic differential equation.

Analysis of the last term in the equation, the instantaneous covariation between processes X_i and X_j.

Instantaneous covariation written as [DX_i, dX_j], coinciding with the product of theta_i and theta_j.

Thetas' role in the equation, leading to the correlation between Brownian motions.

Correlation between Brownian motions I and J denoted by rho_dt.

Expectation of DW_i times DW_j equals rho_dt, indicating the correlation.

Instantaneous covariation simplifies to theta_i times theta_j times rho_dt.

Transcripts

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