Mathematical modelling of financial market is based on construction of stochastic processes representing particular securities. The problems arising require finding prices of some contingent claims based on the securities modelled. To this end the theory of partial differential equations becomes a valuable tool. This is related to the basic fact linking stochastic analysis and PDEs, namely the Feynman-Kac formula. As a result we may solve the pricing problem by solving the corresponding PDE. This approach is particularly useful if the underlying securities are given by means of a nonlinear stochastic differential equation and when the noise includes the Poisson process resulting in diffusions with jumps. These cases are relevant when addressing the credit risk issues, like pricing defaultable bonds and credit derivative securities.

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