Abstract
Stochastic differential equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. Strong, or pathwise, simulation of these models is required in various settings and long time stability is desirable to control error growth. Here, we examine strong convergence and mean-square stability of a class of implicit numerical methods, proving both positive and negative results. The analysis is backed up with numerical experiments.
Highlights
Stochastic differential equations (SDEs) arise in many disciplines. They are used in mathematical finance in order to simulate asset prices, interest rates and volatilities
In this work we extend that analysis to the case where jump magnitudes are random—a situation that is common in financial models [9, 10, 17, 25, 26]
We have extended previous results on strong convergence and mean-square stability [14] to the case of jump-SDEs where the jump magnitude is a random variable
Summary
Stochastic differential equations (SDEs) arise in many disciplines. In particular, they are used in mathematical finance in order to simulate asset prices, interest rates and volatilities. We believe that our treatment is of independent interest as it is based on the style of analysis in [16] and has potential to be extended to the case of nonlinear coefficients that are not globally Lipschitz In this context the use of implicit methods that can be guaranteed to produce moment-bounded solutions appears to be an essential ingredient in the error analysis. A key difficulty that we address in this work is that relative to previous analysis [14] a new source of error arises from the approximation of γ(t) by γ(t) This occurs because the exponentially distributed jump times do not coincide with the grid points.
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