BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets
- PDF / 2,286,904 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 140 Views
BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets Yushi Hamaguchi1 Received: 20 March 2020 / Revised: 20 August 2020 / Accepted: 31 August 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution. We also consider the hedging problem in bond markets and prove that, for an approximately attainable contingent claim, the sequence of locally risk-minimizing strategies based on small markets converges to the generalized hedging strategy. Keywords Backward stochastic differential equation · Cylindrical martingale · Bond market · Locally risk-minimizing strategies Mathematics Subject Classification 60H05 · 91G15 · 91G30
1 Introduction In mathematical finance, backward stochastic differential equations (BSDEs) have been studied and applied to the theory of option hedging and portfolio optimization problems in stock markets where a finite-number of assets are traded. El Karoui et al. [13] studied hedging problems, recursive utilities, and control problems in terms of finite-dimensional BSDEs. Besides, infinite-dimensional (forward) SDEs have also been extensively studied and applied to mathematical finance; see Da Prato and Zabczyk [5] for a summary of infinite-dimensional SDEs and Carmona and Tehranchi [4] for applications to bond markets. In this paper, motivated by hedging problems
This work was supported by JSPS KAKENHI Grant Number JP18J20973. * Yushi Hamaguchi [email protected]‑u.ac.jp 1
Department of Mathematics, Kyoto University, Kyoto 606‑8502, Japan
13
Vol.:(0123456789)
Y. Hamaguchi
in bond markets, we study the following infinite-dimensional BSDE driven by a cylindrical martingale 𝕄:
Yt = 𝜉 +
∫t
T
f (s, Ys , ℍs ) dAs −
∫t
T
ℍs d𝕄s −
∫t
T
dNs , t ∈ [0, T],
where ∫ ℍs d𝕄s is the so-called “generalized stochastic integral” with respect to the cylindrical martingale 𝕄. As mentioned by Björk et al. [1], in the continuous-time bond market, unlike in the stock market, there exists a continuum of tradable assets (zero-coupon bonds parametrized by their maturities), and the time evolution of the price curve is described by an infinite-dimensional stochastic process. To describe the portfolio theory in this model, we have to consider trading strategies in which possibly a continuum of zero-coupon bonds of each maturity can contribute. Hence, in the bond market, stochastic integrals with respect to infinite-dimensional (semi) martingales naturally arise and the term “trading strategy” has to be generalized in the infinite-dimensional setting. A theory of stochastic integration with respect to cylindrical martingales that is suitable for this purpose has been
Data Loading...
