On a class of stochastic differential equations in a financial - PowerPoint PPT Presentation

On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University of California, Santa Barbara Part of research is joint work with Nils Detering & Jean-Pierre Fouque Thira May 2017 Thera Stochastics 1

Motivation: On ✭✡ ❀ ❋ ❀ ✭ ❋ t ✮ ❀ P ✮ let us consider R N -valued diffusion process ✭ X 1 ✭ t ✮ ❀ ✿ ✿ ✿ ❀ X N ✭ t ✮✮ , 0 ✔ t ❁ ✶ induced by the following random graph structure. Suppose that at time 0 we have a random graph of N vertices ❢ 1 ❀ ✿ ✿ ✿ ❀ N ❣ and define the strength of connections between vertices i and j by ❋ 0 -measurable random variable a i ❀ j (whose distribution may depend on N ) for every 1 ✔ i ✻ ❂ j ✔ N and fix a i ❀ i ❂ 0 for 1 ✔ i ✔ N . We shall consider N ❳ d X i ✭ t ✮ ❂ � 1 a i ❀ j ✭ X i ✭ t ✮ � X j ✭ t ✮✮ d t ✰ d B i ✭ t ✮ ❀ N j ❂ 1 for i ❂ 1 ❀ ✿ ✿ ✿ ❀ N , t ✕ 0 , 2

where ✭ B 1 ✭ t ✮ ❀ ✿ ✿ ✿ ❀ B N ✭ t ✮✮ , t ✕ 0 is the standard N -dimensional BM, independent of ✭ X 1 ✭ 0 ✮ ❀ ✿ ✿ ✿ ❀ X N ✭ 0 ✮✮ and of the random variables ✭ a i ❀ j ✮ 1 ✔ i ❀ j ✔ N . The randomness determined at time 0 affects the diffusion process ✭ X 1 ✭ ✁ ✮ ❀ ✿ ✿ ✿ ❀ X N ✭ ✁ ✮✮ . Deterministic A in the context of financial network : Carmona, Fouque, Sun (’13), Fouque & Ichiba (’13), ... The system is solvable as a linear stochastic system for X ✭ ✁ ✮ ✿❂ ✭ X 1 ✭ ✁ ✮ ❀ ✿ ✿ ✿ ❀ X N ✭ ✁ ✮✮ ✵ . Let A ✭ N ✮ ✿❂ ✭ a i ❀ j ✮ 1 ✔ i ❀ j ✔ N be the ✭ N ✂ N ✮ random matrix and B ✭ ✁ ✮ be the ✭ N ✂ 1 ✮ -vector valued standard Brownian motion. 3

Then the system can be rewritten as d X ✭ t ✮ ❂ � A ✭ N ✮ X ✭ t ✮ d t ✰ d B ✭ t ✮ ❀ N Diag ✭ A ✭ N ✮ 1 N ✮ � 1 1 A ✭ N ✮ ✿❂ N A ✭ N ✮ ❀ where 1 N is the ✭ N ✂ 1 ✮ vector of ones, and Diag ✭ c ✮ is the diagonal matrix whose diagonal elements are those elements in the vector c . Note that each row sum of elements in the matrix A ✭ N ✮ is zero by definition, i.e., ❳ a ✭ N ✮ a ✭ N ✮ ❂ � i ❀ i i ❀ j j ✻ ❂ i for each i ❂ 1 ❀ ✿ ✿ ✿ ❀ N , where a ✭ N ✮ is the ✭ i ❀ j ✮ element of the i ❀ j random matrix A ✭ N ✮ . 4

The solution to this linear equation is given by ❩ t ✭ N ✮ ✏ ✑ ✭ N ✮ X ✭ t ✮ ❂ e � tA e sA X ✭ 0 ✮ ✰ d B ✭ s ✮ ❀ t ✕ 0 ✿ 0 ✭ N ✮ Here we understand e tA is the ✭ N ✂ N ✮ matrix exponential. Given the initial value X ✭ 0 ✮ and A ✭ N ✮ , the law of X ✭ ✁ ✮ is conditionally an N -dimensional Gaussian law with mean ✭ N ✮ e � tA X ✭ 0 ✮ and variance covariance matrix Var ✭ X ✭ t ✮ ❥ A ✭ N ✮ ✮ . Q. How to understand the case N ✦ ✶ of large network, i.e., what happens if N ✦ ✶ ? N ✦✶ A ✭ ✶ ✮ and For example, if A ✭ N ✮ a ✿ s ✿ � � � � ✦ 5

if a ✭ ✶ ✮ ❂ � 1 for j ❂ i ✰ 1 , a ✭ ✶ ✮ ❂ 1 , a ✭ ✶ ✮ ✑ 0 , o.w., i.e., i ❀ j i ❀ i i ❀ j ✵ ✶ � 1 1 0 ✁ ✁ ✁ � A ✭ ✶ ✮ ✿❂ ❇ ❈ 0 � 1 1 0 ✁ ✁ ✁ ❆ ❀ ❅ ... ... ... d X 1 ✭ t ✮ ❂ ✭ X 2 ✭ t ✮ � X 1 ✭ t ✮✮ d t ✰ d W 1 ✭ t ✮ ❀ d X 2 ✭ t ✮ ❂ ✭ X 3 ✭ t ✮ � X 2 ✭ t ✮✮ d t ✰ d W 2 ✭ t ✮ ❀ . . . then how can we solve? ✎ Finite N case: Fernholz & Karatzas (’08-’09) studied flow, filtering and pseudo-Brownian motion process in equity markets. 6

Example as N ✦ ✶ For simplicity let us set X i ✭ 0 ✮ ❂ 0 . Given X 2 ✭ ✁ ✮ , we have ❩ t ❩ t e � ✭ t � s ✮ X 2 ✭ s ✮ d s ✰ e � ✭ t � s ✮ d B 1 ✭ s ✮ ❀ X 1 ✭ t ✮ ❂ 0 0 and also, given X 3 ✭ ✁ ✮ , we have ❩ s ❩ s e � ✭ s � u ✮ X 3 ✭ u ✮ d u ✰ e � ✭ s � u ✮ d B 2 ✭ u ✮ X 2 ✭ s ✮ ❂ 0 0 for t ✕ 0 , and hence substituting X 2 ✭ ✁ ✮ into the first one, ❩ t ❩ t ❩ s e � ✭ t � s ✮ d B 1 ✭ s ✮ ✰ e � ✭ t � u ✮ d B 2 ✭ u ✮ d s X 1 ✭ t ✮ ❂ 0 0 0 ❩ t ❩ s e � ✭ t � s ✮ e � ✭ s � u ✮ X 3 ✭ u ✮ d u ✰ 0 0 for t ✕ 0 . 7

By the product rule for semimartingales, we observe ❩ t ❩ s ❩ t e u ✭ t � u ✮ k e u ✭ s � u ✮ k � 1 d B ✭ u ✮ d s ❂ d B ✭ u ✮ ❀ k 0 0 0 for k ✷ N , t ✕ 0 , and hence ❩ t ❩ s ❩ t e u d B ✭ u ✮ d s ❂ e u ✭ t � u ✮ d B ✭ u ✮ ❀ 0 0 0 ❩ t ❩ s k ❩ s 1 ❩ t e u ✭ t � u ✮ k e u d B ✭ u ✮ d s 1 ✁ ✁ ✁ d s k ❂ ✁ ✁ ✁ d B ✭ u ✮ k ✦ 0 0 0 0 for k ✷ N , t ✕ 0 . Thus for the above example we have ❩ t ✶ ❳ e � ✭ t � u ✮ ✁ ✭ t � u ✮ k X 1 ✭ t ✮ ❂ d B k ✰ 1 ✭ u ✮ k ✦ 0 k ❂ 0 for t ✕ 0 . 8

❩ t ✶ ❳ e � ✭ t � u ✮ ✁ ✭ t � u ✮ k X 1 ✭ t ✮ ❂ d B k ✰ 1 ✭ u ✮ k ✦ 0 k ❂ 0 is a centered, Gaussian process with covariances ❩ s ✶ ❳ e 2 u E ❬ X 1 ✭ s ✮ X 1 ✭ t ✮❪ ❂ e � ✭ s ✰ t ✮ ✭ k ✦✮ 2 ✭ s � u ✮ k ✭ t � u ✮ k d u 0 k ❂ 0 ❩ s q ❂ e � ✭ t � s ✮ e � 2 v I 0 ✭ 2 ✭ t � s ✰ v ✮ v ✮ d v 0 for 0 ✔ s ✔ t , where I 0 ✭ ✁ ✮ is the modified Bessel function of the first kind with parameter 0 , i.e., ✶ ✏ x ✑ 2 k ✰ ✗ ❳ 1 I ✗ ✭ x ✮ ✿❂ 2 �✭ k ✰ 1 ✮�✭ ✗ ✰ k ✰ 1 ✮ k ❂ 0 for x ❃ 0 , ✗ ✕ � 1 . 9

In particular, ❩ t e � 2 v I 0 ✭ 2 v ✮ d v ❂ te � 2 t ✭ I 0 ✭ 2 t ✮ ✰ I 1 ✭ 2 t ✮✮ ❁ ✶ Var ✭ X 1 ✭ t ✮✮ ❂ 0 (it grows as O ✭ t 1 ❂ 2 ✮ for large t , also, E ❬ X 1 ✭ s ✮ X 1 ✭ s ✰ t ✮❪ ❂ O ✭ e � ✭ t � 2 ♣ ✭ t ✰ s ✮ s ✮ t � 1 ❂ 4 ✮ ✿ ✮ Thus X 1 ✭ ✁ ✮ is not stationary . The (marginal) distribution of X k ✭ ✁ ✮ , k ✷ N is the same as X 1 ✭ ✁ ✮ , and hence, we may compute (at least numerically) ❩ t ✂ X 2 ✭ s ✮ X 2 ✭ u ✮ ✄ d s e ✭ t � s ✮ E E ❬ X 1 ✭ t ✮ X 2 ✭ u ✮❪ ❂ 0 ❩ t ✂ X 1 ✭ s ✮ X 1 ✭ u ✮ ✄ d s e ✭ t � s ✮ E ❂ 0 and recursively, E ❬ X 1 ✭ t ✮ X k ✭ u ✮❪ , k ✷ N for 0 ✔ t ❀ u ❁ ✶ . 10

Sample path of ✭ X 1 ✭ ✁ ✮ ❀ X 2 ✭ ✁ ✮✮ generated from the covariance structure. X 1 ✭ ✁ ✮ X 2 ✭ ✁ ✮ ❩ t ❩ t e � ✭ t � s ✮ X 2 ✭ s ✮ d s ✰ e � ✭ t � s ✮ d B 1 ✭ s ✮ ❀ X 1 ✭ t ✮ ❂ 0 0 for t ✕ 0 and Law ✭ X 1 ✭ ✁ ✮✮ ❂ Law ✭ X 2 ✭ ✁ ✮✮ . 11

Weighted by Poisson probabilities An interpretation of ❩ t ✶ ❳ e � ✭ t � u ✮ ✁ ✭ t � u ✮ k X 1 ✭ t ✮ ❂ d B k ✰ 1 ✭ u ✮ k ✦ 0 k ❂ 0 ❩ t ✶ ❳ ❂✿ p k ✭ t � u ✮ d B k ✰ 1 ✭ u ✮ 0 k ❂ 0 for t ✕ 0 : Suppose N ✭ s ✮ ❀ 0 ✔ s ✔ t is a Poisson process with rate 1 , independent of ✭ B k ✭ ✁ ✮ ❀ k ✷ N ✮ . Then ❩ t ❤ ✶ ☞ ✐ ❳ ☞ X 1 ✭ t ✮ ❂ E 1 ❢ N ✭ t � u ✮ ❂ k ❣ d B k ✰ 1 ✭ u ✮ ☞ ❋ ✭ t ✮ ❀ 0 k ❂ 0 where ❋ ✭ t ✮ ✿❂ ✛ ✭ B k ✭ s ✮ ❀ 0 ✔ s ✔ t ❀ k ✷ N ✮ , t ✕ 0 . 12

If we replace the Poisson probability by compound Poisson probability, i.e., N ✭ t ✮ ❳ ❢ N ✭ t ✮ ✿❂ ✘ k ❀ k ❂ 1 where ✭ ✘ k ❀ k ✷ N ✮ are I.I.D. integer-valued R.V.’s with P ✭ ✘ 1 ❂ i ✮ ❂ p i , 1 ✔ i ✔ q , P q i ❂ 1 p i ❂ 1 for some q ✷ N , independent of N ✭ ✁ ✮ and ✭ B k ✭ ✁ ✮ ❀ k ✷ N ✮ , then ❩ t ☞ ❤ ✶ ✐ ❳ ☞ ❢ X 1 ✭ t ✮ ✿❂ E N ✭ t � u ✮ ❂ k ❣ d B k ✰ 1 ✭ u ✮ ☞ ❋ ✭ t ✮ 1 ❢ ❡ 0 k ❂ 0 ❩ t ✶ ❳ ❂ ❡ p k ✭ t � u ✮ d B k ✰ 1 ✭ u ✮ ❀ 0 k ❂ 0 where ✑✐☞ ❤ ✏ q ❳ ❅ k ☞ p i t ✭ z i � 1 ✮ p k ✭ t ✮ ✿❂ ❡ exp ☞ ❅ z k z ❂ 0 i ❂ 1 for k ✷ N , t ✕ 0 13

corresponds to the modified matrix ✵ ✶ � 1 ✁ ✁ ✁ 0 ✁ ✁ ✁ p 1 p 2 p q � � ❇ ❈ ... A ✭ ✶ ✮ ✿❂ ❇ ❈ 0 � 1 ✁ ✁ ✁ ❆ ❀ p 1 p 2 p q ❅ ... ... ... ... ... ... and q � � X k ✭ t ✮ ✰ ❳ ✁ d t ✰ d B k ✭ t ✮ d X k ✭ t ✮ ❂ p i X i ✰ k ✭ t ✮ i ❂ 1 with X k ✭ 0 ✮ ❂ 0 for k ✷ N , t ✕ 0 . ✎ In particular, if q ❂ 2 , p 1 ❂ p 2 ❂ 1 ❂ 2 , then ❜ k ❂ 2 ❝ ❳ e � t t k � j t ✕ 0 ❀ k ✷ N ✿ ❡ p k ✭ t ✮ ❂ 2 k � j ✭ k � 2 j ✮✦ j ✦ ❀ j ❂ 0 14

Another modification: d X 1 ✭ t ✮ ❂ ✭ X 1 ✭ t ✮ � X 2 ✭ t ✮✮ d t ✰ d B 1 ✭ t ✮ ❀ d X 2 ✭ t ✮ ❂ ✭ X 2 ✭ t ✮ � X 3 ✭ t ✮✮ d t ✰ d B 2 ✭ t ✮ ❀ ✁ ✁ ✁ We may use the same reasoning in this case to obtain ❩ t ✶ ❳ e t � s ✁ ✭ � 1 ✮ k ✭ t � s ✮ k X 1 ✭ t ✮ ❂ d B k ✰ 1 ✭ s ✮ k ✦ 0 k ❂ 0 with exponentially growing variance Var ✭ X 1 ✭ t ✮✮ ❂ te 2 t ✭ I 0 ✭ 2 t ✮ � I 1 ✭ 2 t ✮✮ ❀ t ✕ 0 ✿ 15